Example

Suppose we wish to lay out an optically compensated zoom having R ¼ 3, with X ¼ 2.2. Then Eqs. (3-24) and (3-25) give

S ¼ 0:3; fb2 ¼ 3:84; fc2 ¼ 11:25

Since R is greater than 1, the two moving lenses will be positive and the fixed lens will be negative. Taking square roots gives

fb¼ 1:95959 and fc¼ 3:35410

Assuming that the initial separation between lenses a and b is to be 3.0, we find that the focal length of the front lens must be 7.15959, and the rear air space d 0b will be initially 1.69451. Using the (y – u) method, we calculate the data shown in Table 3.8.

Table 3.8

Performance of Example Zoom Lens

It is clear that the image plane passes through the three designated positions corresponding to D ¼ 0, 1, and 2, but it departs from that plane for all other values of D. These departures, commonly called loops, will be very noticeable if the system is made in a large size, but they can be rendered negligible if the zoom system is made fairly large and is used in front of a small fixed lens of considerable power, as on an 8-mm movie camera. It will be noticed, too, that the law connecting image distance with zoom movement is a cubic (Figure 3.27).

3.5.4 A Four-Lens Optically Compensated Zoom System

We can drastically reduce the sizes of the loops between the in-focus image positions by the use of a four-lens arrangement, as shown in Figure 3.28. Here we have a fixed front lens, followed by a pair of moving lenses coupled together

Shift of image

Figure 3.27 Image motion with a three-lens optically compensated zoom system.

H

fh

Figure 3.28 Layout of a four-lens optically compensated zoom system.

with a fixed lens between them. The algebraic solution for the powers and spaces of the four lenses is similar to that already discussed, but it is vastly more complicated.

We now designate the separations between adjacent focal points in the three airspaces by H, X, and S, the X and S serving the same functions as before. The initial lens separations are

d 0h ¼ fh þ fa H d 0a ¼ fa þ fb þ X d 0b ¼ fb þ fc þ S

The initial focal length and the initial back focus are

Shift of image

Figure 3.29 Image motion with a four-lens optically compensated zoom system.

These can be used to set up a system, taking the positive root for fa ¼ fc and the negative root fb. The H is the initial distance from the fixed object to the anterior focus of the front lens, the initial object distance being therefore (H – fa). The initial lens separations are respectively ( fa þ fb þ X) and ( fa þ fb þ X – 3).

Example

As an example, we will design such a zoom system with H ¼ –8 and X ¼ 2. The preceding formulas give fa ¼ fc ¼ 6.157183 and fb ¼ –2.667455. The separations are, respectively, 4.667455 and 1.667455 at the start; they will, of course, be increased or decreased as the zoom elements are moved to change the magnification. The overall distance from object to image is equal to 2(14.157183 þ 4.667455) ¼ 37.6493. Tracing rays by the (y – u) method gives the data shown in Table 3.10 on the next page.

The image shift is shown graphically in Figure 3.30. It will be noticed that as we are now moving a pair of positive components, the quadratic curve is in the opposite direction to that for the previous example, in which we moved a pair of negative lenses.

98 Paraxial Rays and First-Order Optics

Table 3.10

Performance of Example Optically-Compensated Enlarger or Printer Zoom Lens

Figure 3.30 Image motion for a zoom enlarger system.

If it is desired to cover a wider range of magnifications, the value of H should be reduced, and if the lenses come too close together, then X can be somewhat increased. Obviously there is no magic about the size, and if a different object-to-image distance is required, the entire system can be scaled up or down as needed. The fixed negative component is very strong and in practice it is often divided into a close pair of negative achromats, but we leave this up to the designer.