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Non-Normal-Incidence Reflection and Transmission Polarizers

This is a continuation from the previous tutorial - dichroic and diffraction-type polarizers.


By far the largest class of polarizers used in the infrared and ultraviolet spectral regions (where dichroic sheet polarizers and calcite polarizing prisms cannot be used) is the so-called pile-of-plates polarizers from which light is reflected (or transmitted) at non-normal incidence.

Since most of these polarizers operate at angles near the Brewster or polarizing angle, they are frequently called Brewster angle polarizers.

The plane-parallel plates which are used for Brewster angle transmission polarizers are generally thick enough to ensure that although multiple reflections occur within each plate, the coherence of the light beam is lost and there are no interference effects.

However, another class of non-normal-incidence transmission polarizers makes use of interference effects to enhance their polarizing properties. These include interference polarizers and polarizing beam splitters. A relation which is frequently used in connection with non-normal-incidence reflectance measurements is the Abeles condition.


Brewster Angle Reflection Polarizers

Most reflection-type polarizers are made of plates which are either nonabsorbing or only slightly absorbing. The angle of incidence most often used is the Brewster angle at which the reflection of the \(p\) component, light polarized parallel to the plane of incidence, goes to 0. Thus the reflected light is completely plane polarized with the electric vector vibrating perpendicular to the plane of incidence (\(s\) component).

The polarizing efficiency of reflection-type polarizers can be experimentally determined.

Brewster angle reflection polarizers for the infrared are made from the semiconductors silicon, germanium, and selenium which are transparent beyond their absorption edges and have high refractive indexes.

Table 3 lists various infrared polarizers which have been described in the literature. All involve external reflections except the Ge-Hg polarizer described by Harrick, in which light undergoes two or four reflections within a bar of germanium.

While Harrick’s polarizer has attractive features, it depends on maintaining polarization in the germanium, so that great care must be taken to obtain material with a minimum of strain birefringence.


Table 3.  Infrared Brewster Angle Reflection Polarizers


In the ultraviolet, materials such as LiF, MgF2, CaF2, and Al2O3, can be used as polarizers. Biotite, a form of mica, has also been found to perform very well in the 1000- to 6000- Å region. In the extreme ultraviolet , metallic films, particularly Au, Ag, and Al, have been used as polarizers. Table 4 lists various non-normal-incidence ultraviolet reflection polarizers as well as authors who have made calculations and measurements on various materials for ultraviolet polarizers.


Table 4.  Ultraviolet Reflection Polarizers and Polarization Measurements


The most versatile non-normal-incidence reflection polarizer would be one which does not deviate or displace the beam from its axial position. One convenient arrangement would be a symmetric three-reflection system in which the light is incident on one side of a triangle, reflected to a plane mirror opposite the apex, and back to the other side of the triangle, as was done by Horton et al., and Barchewitz and Henry.

If the polarizer must have a good extinction ratio and the light beam is highly convergent, two of the reflections could be from the polarizing material and the third from a silvered or aluminized mirror. If the beam is highly collimated or more throughput is required, only one reflection may be from the polarizing material.

The throughput can also be increased by using a plane-parallel plate for the polarizing reflection. The major drawback to a reflection polarizer is the extreme length of the device required to accommodate a beam of large cross-sectional area.

For example, if a germanium polarizer were used at the Brewster angle (\(76^\circ\)) and the beam width were about 25 mm, each Ge plate would have to be about 25 by 100 mm and the overall length of the polarizer would be greater than 200 mm if a three-reflection axial arrangement such as that described above were used.

The Abeles condition, which applies to the amplitude reflectance at \(45^\circ\) angle of incidence is useful for testing the quality of reflection polarizers. Schulz and Tangherlini apparently rediscovered the Abeles condition and used the ratio \(R_s^2/R_p=1\) as a test to evaluate their reflecting surfaces. They found that surface roughness made the ratio too small but annealing the metal films at temperatures higher than \(150^\circ\)C made the ratio larger than unity.

Rabinovitch et al. made use of the Abeles condition to determine the polarization of their Seya-Namioka vacuum-ultraviolet monochromator. They measured the reflectance at \(45^\circ\) of a sample whose plane of incidence was perpendicular or parallel to the exit slit.

From these measurements they deduced the instrumental polarization by assuming the Abeles condition. Values of instrumental polarization obtained using carefully prepared gold and fused-silica samples were in excellent agreement, showing that neither of these materials had surface films which invalidated the Abeles condition.

Surface films usually have relatively little effect on the Abeles condition in the visible region but become important in the vacuum ultraviolet. Hamm et al. eliminated the effect of instrumental polarization from their measurements of the reflectance of a sample in unpolarized light at \(45^\circ\) angle of incidence by making use of the Abeles condition.

Although McIlrath did not refer to the Abeles condition as such, he used it to determine the instrumental polarization of his vacuum-ultraviolet apparatus so he could measure the absolute reflectance of a sample at \(45^\circ\) angle of incidence.

Thonn and Azzam have calculated the polarizing properties of dielectric-coated metal mirrors at 2.8 μm in the infrared. Reflections from 2, 3, or 4 such mirrors at the Brewster angle should give excellent performance, although the polarizer would be quite long.


Brewster Angle Transmission Polarizers

To help overcome the beam-deviation problem and the extreme length of reflection-type polarizers, Brewster angle polarizers are often used in transmission, particularly in the infrared, where transparent materials are available.

At the Brewster angle, all of the \(p\) component and an appreciable fraction of the \(s\) component are transmitted. Thus, several plates must be used to achieve a reasonable degree of polarization. The higher the refractive index of the plates, the fewer are required.

Conn and Eaton have shown that the formulas which assume incoherent multiple reflections within each plate and none between plates give the correct degree of polarization for a series of Zapon lacquer films (\(n=1.54\)) and also for a series of eight selenium films, whereas the formula of Provostaye and Desains predicted values which were much too low.

These authors also point out that the number of multiply reflected beams between plates that enter the optical system depends on the spacing between plates and the diaphragm used to limit the number of beams. One can use a fanned arrangement, as suggested by Bird and Shurcliff, to eliminate these multiply reflected beams. Internal reflections within each plate can be removed by wedging the plates.

Most of the infrared Brewster angle transmission polarizers described in the literature have been made of selenium, silver chloride, or polyethylene sheet; they are listed in Table 5.


Table 5.  Infrared Brewster Angle Transmission Polarizers


For wavelengths longer than 3 μm, where calcite polarizing prisms become highly absorbing, to about 10 μm, beyond which wire-grid polarizers have good extinction ratios, Brewster angle transmission polarizers are the most useful, since the better-extinction, on-axis reflection-type polarizers are impossibly long.

Some of the interference polarizers are superior if the beam-convergence angle is small. Ultraviolet Brewster angle transmission polarizers are not nearly as common; LiF and CaF2 have mainly been used from about 1500 to 2500 Å (see Table 6).


Table 6.  Ultraviolet Brewster Angle Transmission Polarizers


In the wavelength region where calcite polarizing prisms are usable (\(\gt\) 2140 Å), Brewster angle polarizers have the advantage of a larger linear aperture and less absorption.

Low-absorption glass pile-of-plates polarizers have been used in the visible spectral region by Weiser, in preference to more absorbing Glan-Thompson prism polarizers, to increase the power output of giant-pulse ruby lasers. Weinberg calculated the degree of polarization of glass and silver chloride plates, but he did not calculate the transmittance of his polarizers.


Interference Polarizers

When the sheets or films constituting a non-normal-incidence transmission polarizer are thin and have very smooth surfaces, the internally reflected beams can interfere constructively or destructively. In this case, the transmittance of the \(p\) component remains unity at the Brewster angle (where \(R_p=0\)) and only oscillates slightly (with respect to wavelength) for angles close to the Brewster angle.

However, the \(s\) transmittance varies from a maximum of unity to a minimum of \((1-R_s)^2/(1+R_s)^2\) whenever \(\lambda\) changes by an amount that will make the quantity \((nd\cos\theta_1)/\lambda\) change by \(\frac{1}{2}\).

These transmittance oscillations are only \(\pm0.225\) for a single film of refractive index 1.5 but can become as large as \(\pm0.492\) when \(n=4.0\).

Since the \(p\) transmittance remains essentially constant, the extinction ratio will vary cyclically with the \(s\) transmittance, as can be seen in the upper curve of Fig. 17 for a 2.016-μm-thick selenium film.


Figure 17.  Calculated extinction ratios for a series of selenium films (\(n=2.46\)) as a function of wavelength from 2.5 to 30 μm. Light is incident at the Brewster angle, \(67.9^\circ\), and multiply reflected beams interfere within the film. The upper curve is for a single film 2.016 μm thick; arrows indicate positions of maxima for three thinner films: \(t_1\) = 1.080 μm, \(t_2\) = 1.440 μm, and \(t_3\) = 1.800 μm. The lower curve is the extinction ratio for the four films in series assuming no reflections between films. The calculated \(p\) transmittance for each film (and for four films in series) is unity at the Brewster angle.


If a transmission polarizer with a good extinction ratio is needed for use over a limited wavelength range, it can be made of several uniform films of a thickness that yields a minimum extinction ratio in the given wavelength region.

The extinction ratio for a series of \(m\) films is \((T_s/T_p)^m\) when there are no multiple reflections between them. In this way only half as many films would be needed to achieve a given extinction ratio as would be necessary if interference effects were not present.

This rather surprising result can be seen from the expressions for \((T_s)_\text{sample}\) for \(m\) plates with and without interference effects.

Assuming no multiple reflections between plates, the expressions are \([2n^2/(n^4+1)]^{2m}\) and \([2n^2/(n^4+1)]^m\), respectively. Hertz achieved a degree of polarization of 99.5 percent in the 6- to 17-μm region using three unbacked selenium films 0.95 μm thick. Conn and Eaton obtained only a slightly better performance with eight thicker nonuniform selenium films.

As can be seen in Fig. 17, the calculated extinction ratio for the 2.016-μm-thick film goes to unity at 3.0, 4.6, and 9.2 μm, indicating that the \(s\) as well as the \(p\) transmittance at these wavelengths is unity.

This ratio will remain unity at the above wavelengths if there are several nonabsorbing films of the same thickness. Even if the films have slightly different thicknesses, or if their surfaces are somewhat rough, interference effects may still persist, adversely affecting polarizer performance. Such effects have been observed by Elliott et al., Barchewitz and Henry, Duverney, Mitsuishi et al., and Walton et al.

By choosing films of appropriate thicknesses, interference effects can be used to advantage. The lower curve in Fig. 17 shows the extinction ratio obtained if four selenium films of thicknesses 1.08, 1.44, 1.80, and 2.02 μm are used at the Brewster angle as a transmission polarizer. (The wavelengths at which maxima occur for the three thinner films are indicated by arrows in the upper portion of the figure.)

In this example the extinction ratio for the four films in series is better than \(2\times10^{-2}\) from 2.5 to 30 μm and at most wavelengths is better than \(10^{-2}\) (corresponding to a degree of polarization in excess of 98 percent).

In the 11- to 27-μm wavelength region the extinction ratio is better than \(10^{-3}\). Four thick or nonuniform selenium films without interference effects have a calculated extinction ratio of about \(10^{-2}\), and six films are required to change this ratio to \(10^{-3}\).

Thus, in the 11- to 27-μm wavelength region, four selenium films of appropriate thicknesses with interference have a superior extinction ratio to six selenium films without interference. If one wishes to optimize the extinction ratio over a more limited wavelength range, the film thicknesses can be adjusted accordingly and the extinction ratio improved.

Unfortunately, the gain in extinction ratio is offset by a more sensitive angular function, so that the incident beam must be very well collimated.

Interference effects can also be used to advantage in other types of non-normal-incidence polarizers. Bennett et al. made a transmission polarizer from a series of four germanium films (ranging in thickness from 0.164 to 0.593 μm), evaporated onto strain-free plates of sodium chloride.

The plates were inclined at the Brewster angle for germanium and arranged in the form of an X so that the polarizer would have a large square aperture and would not deviate the beam. An extinction ratio better than \(3\times10^{-3}\) was measured at 2.5 μm, and the plates transmitted from 2 to 13 μm. (Calculated extinction ratios in this wavelength range vary from \(1\times10^{-3}\) to \(2\times10^{-4}\) for radiation incident at the Brewster angle.)

Polarizers consisting of a high refractive index transparent film on a lower refractive index transparent substrate have been suggested for use in the visible wavelength region by Schroder and Abeles. These still have a Brewster angle where \(R_p=0\), and furthermore \(R_s\) at this angle is greatly increased over its value for an uncoated low refractive index substrate. Thus, a large-aperture, high-efficiency polarizer with no absorption losses is possible, which should find numerous applications in laser systems.

One polarizer of this type, suggested independently by Schroder and by Abeles, would consist of high refractive index titanium dioxide films (\(n\approx2.5\)) evaporated onto both sides of a glass substrate (\(n=1.51\)). At the Brewster angle, \(74.4^\circ\), \(R_s\approx0.8\), making this polarizer equivalent to one made from a material of refractive index 4.

Two glass plates coated on both sides with TiO2 films should have an extinction ratio of about \(1.6\times10^{-3}\) at 5500 Å and about twice that value at the extreme ends of the visible region, according to Abels.

Schroder measured the degree of polarization as a function of angle of incidence for one such TiO2-coated glass plate and found values comparable to the calculated ones.

Kubo calculated the degree of polarization, reflectance, and transmittance (as a function of angle of incidence and wavelength) of a glass plate (\(n=1.50\)) covered with a thin transparent film of index, 2.20. His results are similar to those of Abeles and Schroder.

Schopper, Ruiz-Urbieta and Sparrow, and Abeles have also investigated making non-normal-incidence reflection polarizers from a thin transparent or absorbing film deposited onto an absorbing substrate. Zaghloul and Azzam proposed using silicon films on fused silica substrates as reflection polarizers for different mercury spectral lines in the visible and ultraviolet regions.

Abeles designed some specialized reflection polarizers for use in the vacuum ultraviolet. Unfortunately the wavelength range covered by such a polarizer is very narrow; for one polarizer it was 25 Å at a wavelength of 1500 Å. However, the spectral range could possibly be increased by using several thin films instead of one.

Multilayer film stacks have also been used to produce non-normal-incidence reflection or transmission polarizers by Buchman et al.. Buchman later improved the design performance of his polarizers by adding antireflection layers between the repeating groups of layers.

Although this type of polarizer has a relatively narrow operating bandwidth, a small angular acceptance, tight wavelength centering, and layer thickness uniformity requirements, it can be used successfully in high power laser systems as shown by Refermat and Eastman.

Songer described how to design and fabricate a Brewster angle multilayer interference polarizer out of a titanium dioxide, silicon dioxide multilayer on BK 7 glass for use in a 1.06-μm laser beam.

Blanc, Lissberger, and Roy designed, built, and tested multilayer zinc sulfide– cryolite-coated glass and quartz polarizers for use with a pulsed 1.06-μm laser.

Recently, Maehara et al. have reported excellent performance for a pair of polarizers coated with 21 ruthenium and silicon films on a silicon wafer over a wide wavelength range in the soft x-ray region. In several designs of multilayer film stacks, both the reflected and transmitted beams are used.


Polarizing Beam Splitters

Polarizing beam splitters are a special form of non-normal-incidence interference polarizer in which the beam is incident on a multilayer dielectric stack at \(45^\circ\).

The transmitted beam is almost entirely plane-polarized in the \(p\) direction, while the reflected beam is nearly all plane-polarized in the \(s\) direction.

Generally the alternating high and low refractive index dielectric layers are deposited onto the hypotenuses of two right-angle prisms, which are then cemented together to form a cube.

The beam enters a cube face normally and strikes the multilayers on the hypotenuse (the high refractive index layer is next to the glass), and the reflected and transmitted beams emerge normal to cube faces, being separated by \(90^\circ\).

Clapham et al. have a good discussion of polarizing beam splitters, which were invented by S . M . MacNeille and developed by Banning. Banning’s beam splitter was made with three zinc sulfide and two cryolite layers on each prism; the polarization for white light was greater than 98 percent over a \(5^\circ\)-angle on each side of the normal to the cube face for both the reflected and transmitted beams.

Variations on this design have since been proposed by Dobrowolski and Waldorf, Monga et al., and Mouchart et al., primarily to improve the laser damage resistance of the device and increase the angular field of view.

Dobrowolski and Waldorf designed and built a polarizing beam splitter consisting of a multilayer coating of HfO2 and SiO2 deposited onto fused silica and immersed in a water cell that acted like the MacNeille cube. Tests with a 0.308 μm excimer laser showed a high laser damage threshold.

The multi-wavelength polarizing beam splitters designed by Monga et al. could be made in large sizes and could withstand high laser power levels. The modified MacNeille cube polarizers designed by Mouchart et al. had angular fields of view that could be increased to about \(\pm10^\circ\) when the polarizers were used with monochromatic light sources.

Lees and Baumeister designed a frustrated total internal reflection beam splitter that had a multilayer dielectric stack deposited onto the hypotenuse of a prism. Their designs, for use in the infrared spectral region, consisted of multilayer stacks of PbF2 and Ge deposited onto a germanium prism and covered by a second germanium prism.

Azzam designed polarization independent beam splitters for 0.6328 μm and 10.6 μm using single-layer coated zinc sulfide and germanium prisms. The devices were found to be reasonably achromatic and their beam-splitting ratio could be varied over a wide range with little degradation in polarization properties.

Azzam also proposed coating a low-refractive-index dielectric slab on both sides with high-refractive-index dielectric films to make an infrared polarizing beam splitter.

Various high- and low-refractive-index materials have been successfully used in the multilayer stacks. In addition to zinc sulfide and cryolite on glass by Banning and Schroder and Schlafer, layers of a controlled mixture of silicon dioxide and titanium dioxide have been alternated with pure titanium dioxide on fused-silica prisms by Pridatko and Krylova, thorium dioxide and silicon dioxide have been used on fused-silica prisms by Sokolova and Krylova, chiolite (a mixture of sodium and aluminum fluorides) and lead fluoride have been used on fused-silica prisms by Turner and Baumeister, bismuth oxide and magnesium fluoride have been used on EDF glass prisms by Clapham et al., and zirconium oxide and magnesium fluoride have been used on dense flint-glass prisms by Clapham et al.

The calculations involved in optimizing these beam splitters for good polarizing characteristics, achromaticity, and relative insensitivity to angle of incidence are quite involved. Clapham et al. and Turner and Baumeister discuss various calculation techniques frequently used. Clapham also gives the measured characteristics of a high-performance achromatic polarizing beam splitter made with zirconium oxide and magnesium fluoride multilayers.

Although polarizing beam splitters are generally designed so that the \(s\) and \(p\) polarized beams emerge at right angles to each other, Schroder and Schlafer have an ingenious arrangement in which a half-wave plate and mirror are introduced into the path of the reflected beam to make it parallel to the transmitted beam and of the same polarization. Other optical schemes to accomplish the same purpose have been described in a later paper.

For some purposes it is desirable to have a beam splitter that is insensitive to the polarization of the incident beam. Baumeister has discussed the design of such beam splitters made from multilayer dielectric stacks of alternating low- and high-refractive-index materials.

One of his designs is composed of six dielectric layers for which the extinction ratio \(T_s/T_p\) varies from 0.93 to 0.99 in a bandwidth of about 800 Å, with a \(\pm1^\circ\) variation in the angle of incidence.

In principle, any multilayer filter which is nonreflecting at normal incidence will be nonpolarizing at all angles of incidence, according to Baumeister. Costich has described filter designs for use in the near infrared which are relatively independent of polarization at \(45^\circ\) angle of incidence.


The next tutorial discusses in detail about retardation plates


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