# Nondispersive Prisms

This is a continuation from the previous tutorial - half-shade devices and miniature polarization devices.

### 1. Glossary

Here are the glossaries we are going to use in this tutorial.

• $$\delta$$ - angular deviation
• $$\phi$$ - phase
• $$\omega$$ - radian frequency of rotation

subscripts:

• A, B, C, D, d - prism dimensions
• $$t$$ - time
• $$x$$, $$y$$ - rectangular components
• $$\alpha$$ - angle
• 1, 2 - prism number

### 2. Introduction

Prisms of various shapes and sizes are used for folding, inverting, reverting, displacing, and deviating a beam of light, whether it be collimated, converging, or diverging.

Prisms, rather than mirrors, are often used for the applications discussed here, since they make use of reflecting coatings at what amounts to an interior surface. The coatings can be protected on their backs by other means, and do not tarnish with age and exposure. Even better, some prisms do not need such coatings if the (internal) angle of incidence exceeds the critical angle.

In these applications, chromatism is to be avoided. Thus, the arrangements either make use of perpendicular incidence or compensating angles of incidence.

Almost all of these prisms are meant to be used with collimated beams. Most of the operations are somewhat equivalent to the use of plane parallel plates, which displace but do not deviate a collimated beam. However, such plates have both chromatism and spherical aberration in a convergent beam.

### 3. Inversion, Reversion

A reverted image shifts the image left for right. An inverted image is upside down. A reinverted image or inverted-reverted image does both.

### 4. Deviation, Displacement

The beam, in addition to being inverted and/or reverted, can also be displaced and/or deviated.

Displacement means that the beam has been translated in $$x$$ or $$y$$, but it has not changed the direction in which it was traveling.

Deviation indicates that the beam has been caused to change its direction. Deviation is measured in angular measure; displacement in linear measure.

If a beam has been deviated, displacement is not important. If a beam has been displaced, it usually has not been deviated. Although the two can occur simultaneously, it seldom happens in optical prisms.

### 5. Summary of Prism Properties

Table 1 is a listing of the prisms that are described in this section. The first column has the name of the prism. The second column indicates whether or not the image has been reverted; the second column, whether it has been inverted. The third column indicates the extent to which the prism displaces the beam. This value is given in terms of the dimension $$A$$ that is indicated for each of the prisms. It is a characteristic, normalizing dimension.

The next column indicates the angular deviation of the beam in degrees. Some entries have two angles. This indicates that the beam is deviated in two directions. The first is the horizontal deviation; the second is the vertical deviation, $$h$$ and $$v$$.

These are the conventional planes. Of course, if the prism is rotated, the directions are reversed. The general deviation prisms have a range of deviation angles that can be obtained by reasonable changes in the prism angles. Thus, each of them is a representative design for a range of angles. The final column is for comments.

A prism that has neither deviation nor displacement is a direct-vision prism.

### 6. Prism Descriptions

Each diagram shows at least one view of the prism and a set of dimensions. The A dimension is a reference dimension. It is always 1.00, and the rest of the dimensions are related to it.

The refractive index is almost always taken as 1.5170, a representative value for glass in the visible. Prism dimensions can change somewhat if the refractive index is modestly different from the chosen value. If, for instance, germanium is used, however, the prism might be drastically different.

#### Right-angle Prism

Perhaps the simplest of the deviating prisms is the right-angle prism. Light enters one of the two perpendicular faces, as shown in Fig. 1a, reflects off the diagonal face, and emerges at 90° from the other perpendicular face. The beam has been rotated by 90°, and the image has been inverted.

If the prism is used in the other orientation, shown in Fig. 1b, then the image is reverted. The internal angle of incidence is 45°, which is sufficient for total internal reflection (as long as the refractive index is greater than 1.42).

#### Porro Prism

A Porro prism, shown in Fig. 2, has a double reflection and may be considered to be two right-angle prisms together. They are often identical. Often, two Porro prisms are used together to invert and revert the image.

The incidence angles are the same as with the right-angle prism, so that total internal reflection takes place with refractive indices larger than 1.42. It is a direct-vision prism.

Abbe's version of the Porro prism is shown in Fig. 3. The resultant beam is inverted and reverted and is directed parallel and in the same direction as the incident beam.

#### Abbe's Prisms

Two versions of prisms invented by Abbe are shown. they both are direct-vision prisms that revert and invert the image. One version is symmetrical; the other uses three different prism segments. They are shown in Figs. 4 and 5.

#### Pechan Prism

The Pechan prism (shown in Fig. 8) performs the same function as the Dove, but it can do it in converging or diverging beams. The surfaces marked $$B$$ are silvered and protected. The surfaces bordering the air space are unsilvered.

#### Amici (Roof) Prism

This more complex arrangement of surfaces inverts the image, reverts it, and deviates it 90°. It is shown in Fig. 9. Since this prism makes use of the roof effect, it has the same angles as both the right-angle and Porro prisms, and exhibits total internal reflection for refractive indices larger than 1.42.

#### Schmidt Prism

The prism will invert and revert the image, and it will deviate it through 45°. It is shown in Fig. 10.

#### Leman Prism

This rather strange looking device, shown in Fig. 11, reverts, inverts, and displaces by 3A, an image.

#### Penta Prism

A penta prism has the remarkable property that it always deviates a beam by exactly 90° in the principal plane. This is akin to the operation of a cube corner. The two reflecting surfaces of the penta prism, shown in Fig. 12, must be reflectorized, as the angles are 22.5° and therefore require a refractive index of 2.62 or greater for total internal reflection. Some penta-prisms are equipped with a roof to revert the image.

#### Reversion Prism

This prism operates like an Abbe prism, type A, but does not require parallel light. It is shown in Fig. 13.

#### Wollaston Prism

This prism does not invert, revert, or displace. It does deviate a beam by 90°, allowing a tracing to be made. It is shown in Fig. 14.

#### Carl Zeiss Prism System

This arrangement of three prisms allows the image to be reverted, inverted, and displaced, but not deviated. The amount of deviation is adjustable. The system is shown in Fig. 15.

#### Goerz Prism System

This is an alternate to the Zeiss system. It does the same things. It is shown in Fig. 16.

#### Frankford Arsenal 1

This prism, shown in Fig. 17, reverts, inverts, and deviates through 115°.

#### Frankford Arsenal 2

This prism reverts, inverts, and deviates through 60°. It is shown in Fig. 18.

#### Frankford Arsenal 4

This prism reverts the image and deviates it 45° upward and 90° sidewards, like Frankford Arsenal 3. It is shown in Fig. 20.

#### Frankford Arsenal 5

This prism inverts the image while deviating it 90° sideways and 60° upwards. It is shown in Fig. 21.

#### Frankford Arsenal 6

This prism inverts, reverts, and deviates 90° horizontally and 60° vertically. It is shown in Fig. 22.

#### Frankford Arsenal 7

This prism neither reverts nor inverts, but deviates 90° horizontally and 45° vertically. It is shown in Fig. 23.

#### Brashear-Hastings Prism

This device, shown in Fig. 24, inverts an image without changing the direction of the beam. Since this is a relatively complicated optical element, it does not see much use.

#### Rhomboidal Prism

A rhomboidal prism, as shown in Fig. 25, displaces the beam without inverting, reverting, deviating, or otherwise changing things. The reflecting analog is a pair of mirrors at 45°.

#### Risley Prisms

Risley prisms are used in two ways. If they are slightly absorbing, they can be used as variable attenuators by translating one with respect to the other perpendicular to their apexes.

They can also be rotated to generate a variety of angular deviations. A single prism deviates the beam according to its wedge angle and refractive index. If rotated in a circle about an axis perpendicular to its face, it will rotate the beam in a similar circle.

A second, identical prism in series with it, as shown in Fig. 26, can double the angle of the beam rotation and generate a circle of twice the radius. If they rotate in opposite directions, one motion is canceled and a line is generated.

In fact, all sorts of Lissajous-type figures can be obtained; some are shown in Fig. 27. The equations that govern the patterns are

$\tag{1}\delta_x=\delta_1\cos\omega_1t+\delta_2\cos(\omega_2t+\phi)$

$\tag{2}\delta_y=\delta_1\sin\omega_1t+\delta_2\sin(\omega_2t+\phi)$

where $$\delta_x$$ and $$\delta_y$$ are the beam deviations, $$\delta_1$$ and $$\delta_2$$ are the individual prism deviations, $$\omega$$ is the rotation rate, $$t$$ is time, and $$\phi$$ is the phase of the prism position.

For relatively monochromatic applications, the prisms can be "Fresnelled," as shown in Fig. 28, and the mirror analogs, shown in Fig. 29, can also be used.

#### Retroreflectors

The familiar reflective cube corner (not corner cube), that sends a ray back in the direction from which it came, has its refractive analog, as shown in Fig. 30. The angles are so that total internal reflection occurs. The angular acceptance range can be large.

#### General Deviation Prisms

Figure 31 shows a 60°-deviation prism. Other angles are obtainable with appropriate changes in the prism manufacture, as shown for example in Figs. 32 and 33.

The next tutorial discusses in detail about polarization maintaining fibers.