The right-angle prism possesses the simple geometry of a 45-degree right triangle (see Figure 1), and is one of the most commonly used prisms for redirecting light and rotating images. This interactive tutorial explores light reflection and image rotation, inversion, and reversion by a right-angle prism as a function of the prism orientation with respect to incident light.

The tutorial initializes with the right-angle prism positioned to act as a Porro prism with light emitted from an orientation mark entering the hypotenuse face. The orientation mark contains a red sphere at the top, a blue block on the left arm, a stub at the bottom, and a yellow cone on the right arm. Light emitted by the orientation mark and reflected by the prism is represented by a single red line in the tutorial. Inversion, rotation, and reversion of the orientation mark can occur as a result of light passing through a right-angle prism, and is dependent upon the prism orientation. To operate the tutorial, use the Prism Orientation slider to change the orientation between a Right-Angle Prism, Porro Prism, or Dove Prism. The Orientation Mark slider can be utilized to rotate the orientation mark by any angle between zero and 360 degrees around the central axis.

In a right-angle prism, a parallel bundle of light waves entering one of the smaller prism faces (or legs) at a perpendicular angle is reflected from the hypotenuse (longest) face and exits through the other leg. Provided the prism is constructed from a material having a refractive index greater than the square root of 2 (approximately 1.414), the light will undergo total internal reflection at the glass/air boundary while inside the prism.

This feature renders prisms an excellent substitute for mirrors, because there is no requirement for metallic or dielectric coatings on the reflecting surface, which serves as a nearly perfect reflector. The only light scattering and loss that occurs (usually only a few percent) is due to minute surface imperfections, absorption by the prism material, and reflections at the entrance and exit legs of the prism. Careful polishing of the surfaces and the application of a suitable anti-reflection coating to the legs will minimize even these small light losses. In this orientation, the right-angle prism acts as an image inverting system with the top face performing the duties of a plane mirror by producing left-handed images from right-handed images, and vice versa. Note in Figure 1(a) that the pointed end of the alignment marker has been flipped, but the left and right sides remain in the same position.

Reorienting the right-angle prism, so that light now enters and exits through the hypotenuse face, produces a non-reversing mirror, as illustrated in Figure 1(b). Often termed a Porro prism, the light beam in this configuration undergoes two internal reflections after it enters the prism and is deviated by 180 degrees upon exiting. As a result, images are inverted top to bottom, but are not reversed right to left. When a prism is utilized in this manner, it is often referred to as a constant deviation prism because the incident and emerging light rays will be parallel, regardless of the angle at which light enters the prism. Porro prisms are often employed in traditional binocular configurations, where they are doubled together orthogonally to first invert and then reverse light beams to produce erect or upright images. The twin prisms fold the light path of an optical system and also displace the image both horizontally and vertically by half the length of the hypotenuse in each direction. Binocular prisms are usually manufactured with rounded corners to reduce weight and size, and have a small slot cut into the hypotenuse face to obstruct light rays that are internally reflected at glancing angles.

A third orientation of the right-angle prism with respect to the incident light beam (Figure 1(c)) is commonly referred to as a dove prism, which is useful as an image rotator. Dove prisms often have the unnecessary triangular apex section removed, both to save weight and to reduce stray internal reflections. A bundle of light rays enters the dove prism parallel to the hypotenuse face, and is refracted downward at the first leg toward the longer internal surface. Upon being totally reflected by the hypotenuse face, the light is then refracted again as it exits the prism through the other leg and proceeds in the same direction that it was traveling before entering the prism. Because the dove prism introduces a substantial amount of astigmatism when convergent light is passed through, it is used almost exclusively with collimated light. The dove prism does not deviate or displace an image, but it can be utilized to either invert or reverse an image.

Although at first glance the dove prism appears to be a good candidate for dispersion (due to the angular entrance of the light beam), transmission of light through the prism is actually equivalent to passage through a slab of glass with the side benefit of image rotation. An interesting effect of the dove-style geometry results as the prism is rotated along the longitudinal axis. In the orientation presented in Figure 1(c), light passing through the dove prism forms an image that is inverted from top to bottom and reversed from right to left. However, if the prism is rotated 45 degrees, the resulting image is rotated through 90 degrees, and when the prism is rotated another 45 degrees (for a total of 90 degrees, in effect, being placed on its "side"), the image is now rotated by 180 degrees. Thus, the image is rotated twice as fast as the prism. In practice, two dove prisms are often cemented together at the hypotenuse (after placing a mirrored surface on these faces) to produce a bi-prism with the ability to change the direction of sight for telescopes, periscopes, and other optical instruments.

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