Mirror Systems

It should be apparent that the equations in Chap. 1 and the systems in the preceding sections apply to and can be executed using mirrors.

We have previously noted that (1) the focal length of a mirror is one-half of its radius, (2) a concave mirror has positive power, and (3) the principal points are at the mirror surface. But when raytracing mir-rors, there is a bit of complexity. For surface-to-surface raytracing [Sec. 1.5; Eqs. (1.14) and (1.15)] the spacing is negative if a following surface is to the left, and the index is negative when the ray travels from right to left (i.e., as after a reflection). However, for component-by-component raytracing [Sec. 1.7; Eqs. (1.19) and (1.20)], we use the

“air-equivalent distance,” which is simply the distance divided by the

d = 0.0 f = 5.0

d = 0.2 f = 2.78

d = 0.4 f = 1.92

d = 0.6 f = 1.47

d = 0.8 f = 1.19

d = 1.0 f = 1.0

(a)

Figure 2.21 Schematic of two two-component zoom systems, showing the motions

d = 0.0 f = 5.0

d = 2.2 f = 0.51 d = 0.2 f = 2.78

d = 0.4 f = 1.92

d = 0.6 f = 1.47

d = 0.8 f = 1.19

d = 1.0 f = 1.0

d = 1.2 f = 0.86

d = 1.4 f = 0.76

d = 1.6 f = 0.68

d = 1.8 f = 0.61

d = 2.0 f = 0.56

(b)

Figure 2.21 (Continued) (B) The reversed arrangement allows both a greater zoom range and a good back focus clearance

dis-(a)

(b)

Figure 2.22 Three-component zoom systems. (A) A negative component moving between two positive components to change the focal length is a very common arrangement. Either of the positive components may be moved to compensate for the focus shift introduced by moving the inner lens. (B) A moving positive lens can also produce a zoom.

index. Thus, when both are negative, we take the spacing as positive.

This is the case for most two-mirror systems. This rule applies not only to component-by-component raytracing but also to the equations for combinations of two components [Sec. 1.10, Eqs. (1.27) through

With that formality out of the way, we can discuss a number of popu-lar two-mirror systems. Remembering to use a positive sign for d (per the paragraph above), and using the same notation as in Eqs. (1.27) to (1.37), we have the following equations for two-mirror systems:

f_ab
(2.21)

B
(2.22)

where r_aand r_b are the mirror radii, d is the spacing, B is the back focus, and f_abis the focal length of the combination. Note that we use the usual sign convention for the mirror radii; i.e., r is positive if the center of curvature is to the right of the surface.

To determine the mirror radii, given the combined focal length, the back focus, and the spacing, we can use

c_a

(2.23)

c_b

(2.24)

where c is the surface curvature, equal to 1/r.

Several of the commonly encountered two-mirror arrangements are diagramed in Fig. 2.23. Figure 2.23A is called a Cassegrain and is the mirror equivalent of the telephoto arrangement, where a long focal length is produced in a compact package. The Cassegrain is perhaps the most widely used of all the two-mirror systems. Figure 2.23B shows what is called a Schwarzschild; it is the mirror analog of the

B d  fab

Figure 2.22 (Continued) (C) This is an afocal zoom which can be used as an attachment to make a zoom projection lens from an ordinary fixed-focus lens. It goes out of focus, but in use it can be refocused after the picture size is adjust-ed.

fab

B d

(a)

(b)

(a)

(a)

(b)

B

fab

d

(b)

Figure 2.23 Three common arrangements for two-mirror systems: (A) The Cassegrain objective with a concave primary mirror and a convex secondary is a compact system with a long focal length, in a sort of telephoto configuration. The most commonly used arrangement. (B) The Schwarzschild system has a convex primary and a concave sec-ondary, producing a long working distance and a short focal length (at the expense of a secondary mirror diameter which is several times the beam diameter). Often found in short-focal-length microscope objectives for ultraviolet and infrared work.

retrofocus, since it has a long working distance compared to its focal length. It is often used as an infrared or ultraviolet microscope objec-tive. The gregorian arrangement (Fig. 2.23C) is the mirror equivalent of a positive component forming an image which is then relayed and magnified by a relay lens. The focal length f_abof the gregorian is nega-tive, since P₂is to the right of F₂. Mirror systems have the advantage that a mirror has no chromatic aberration and that they do not require high-quality optical materials to transmit the wavelengths of interest.

Note that in all of these systems, the central part of the beam is obscured by one of the mirrors, and the other mirror has a central hole for the beam to pass through. The Cassegrain and the gregorian are usually executed with aspheric surfaces on both mirrors (to cor-rect aberrations). For the Schwarzschild, the aspherics are not neces-sary; it can be made from two spheres. As indicated in Chap. 4, the central obscuration of the beam has the diffraction effect of signifi-cantly reducing the image contrast.

Afocal mirror systems are shown in Fig. 2.24. A common arrange-ment is to use “confocal” paraboloid mirrors as shown in Figs. 2.24A and B. These are of course the mirror equivalents of the galilean and keplerian telescopes, respectively. Some mirror systems use an off-axis aperture stop in order to avoid obscuring the center of the beam.

Figure 2.24C and D shows this arrangement for confocal paraboloids, (b)

fab

B d (b)

(a)

(c)

Figure 2.23 (Continued) (C) The gregorian arrangement uses a concave secondary mir-ror to relay the image through a hole in the primary and erect the image. Rarely used.

Axis

Axis (a)

(b)

Axis

(c)

Axis

(d)

Figure 2.24 (A), (B), (C), and (D) are “confocal” systems and also afocal systems. (A) and (B) are the mirror analogs of the galilean and Kepler telescopes, respectively.

(C) and (D) are the same except that the apertures are decentered in order to avoid an obscuration in the center of the beam. These four are often made with parabolic surfaces to achieve a system without spherical, coma, or astigmatism.

parabola” (the aperture is off-axis, not the paraboloid) used as a colli-mator.