13.2 The Design of an Air-Spaced Dialyte Lens

We could, of course, determine the bendings of the two thin elements in the rear half-system for spherical and astigmatic correction, but it is best to insert thicknesses first and assemble the two components before doing this. To assign thickness we must decide on the relative aperture of the finished lens, and f/6 is a good value to adopt. This makes the aperture of the rear component about f/12, and a diameter of 1.0 is suitable. The thicknesses of the two lenses will then be 0.06 for the flint and 0.20 for the crown.

For our starting bendings we may assign 40% of the total flint curvature to the front face of the rear negative element, and 25% of the crown curvature to the front face of the rear positive element. However, because of the finite thicknesses, we must scale each thick element to restore its ideal thin-lens power, and the air space must be adjusted to maintain the ideal separation between adjacent principal points. With the stop at a distance of 0.12 from the flint element, the whole lens becomes

The lens in its present state is drawn to scale as was shown in Figure 13.3. The focal length is 5.6496 and the Petzval sum (for f 0 ¼ 10) is 0.04039. Tracing f/8.5 rays in F and C light, with Y1 ¼ 0.3323, gives the zonal chromatic aberration as 0.03312. The increase in Petzval sum is due to the finite thicknesses of the lenses.

We must now restore the desired values of Petzval sum and chromatic aberration by changing the power of the two crown elements and the two outer air spaces, maintaining symmetry about the stop and letting the focal length go. Of course we could equally as well vary the power of the flint elements, but we must adopt a fixed procedure or we shall never reach a satisfactory solution. A double graph is a great convenience here, plotting the zonal chromatic aberration as ordinate and the Petzval sum for f 0 ¼ 10 as abscissa. The starting point will be (0.0404, 0.0331) and the aim point will be (0.034, 0). A trial change of the outer air spaces by 0.05 gives zonal chromatic aberration ¼ –0.0016 and

Figure 13.4 Double graph for chromatic aberration and Petzval sum.

Ptz ¼ 0.0368, and then weakening both surfaces of the crown elements by 2% gives that the zonal chromatic aberration ¼ 0.0133 and Ptz ¼ 0.0331. The graph shown in Figure 13.4 tells us that we should have increased the original air spaces by 0.061 and weakened the crown lens surfaces by 1.24%. These changes give zero chromatic aberration and Petzval sum ¼ 0.0339.

At this stage we find

We desire to have the marginal and zonal spherical aberrations equal and opposite, or LA0 þ LZA ¼ 0, and we would also like to have X t0 ¼ 0 for a flat tangential field. We proceed to accomplish this by bending both crowns and both flints in such a way as to maintain symmetry.

In the double graph of Figure 13.5, we see that at the start X t0 ¼ 0.4216 and LA0 þ LZA ¼ 0.1764. The aim point is (0, 0). Bending the crowns by Dc1 ¼ – 0.02 toward a more nearly equiconvex form gives X t0 ¼ 0.1344 and LA0 þ LZA ¼ 0.1742. Then bending the flints by Dc3 ¼ þ0.02 toward a more nearly equiconcave form gives X t0 ¼ 0.0254 and LA0 þ LZA ¼ 0.0494. The graph indicates that we should have used Dc1 ¼ –0.0190 and Dc3 ¼ þ0.0282. These changes gave X t0 ¼ –0.0043 and LA0 þ LZA ¼ 0.0022, both of which are acceptable. The Petzval sum for f 0 ¼ 10 is now 0.0341 and the zonal chromatic aberration is 0.0010; hence both are virtually unaffected by the small bendings that we have applied to the lenses.

Petzval sum ¼ 0:0341; for f 0 ¼ 10

LA0 þ LZA ¼ 0:0158

Xs0 ¼ 0; X 0t ¼ 0:0304

Since the last change has slightly altered the spherical aberration and the field curvature, we return to the graph in Figure 13.5 and apply Dc1 ¼ –0.0034 and Dc3 ¼ –0.0027 to restore these. The system is now as follows:

0.6350

0.21.6109

0.2411

0.1366

0.4638

0.061.6053

0.3517

0.12

0.12

0.3517

0.061.6053

0.4638

0.1366

0.2508

0.21.6109

0.6610

with focal length for half lens ¼ 5.4212, zonal chromatic aberration ¼ 0.00011, Petzval sum ( f 0 ¼ 10) ¼ 0.0342, LA0 þ LZA ¼ –0.0022; for 22 : X s0 ¼ –0.0088, X t0 ¼ –0.0005, distortion ¼ 0.189%, lateral color ¼ –0.00017; stop diameter for f/6 ¼ 0.7896. Everything is thus known except coma, which we must now investigate.

The easiest way to evaluate the coma is to trace several oblique rays and draw the meridional ray plots at two or three obliquities and look for a parabolic trend, although in general this will be mixed with a cubic tendency due to oblique spherical aberration, and a general slope caused by inward or backward tangential field curvature. If the parabolic trend is not particularly noticeable, the amount of coma is probably negligible in view of the other aberration residuals that are unavoidably present. However, if coma is the dominant aberration it is necessary to reduce it by bending the two crown elements in the same direction, and not symmetrically about the stop as was done previously to correct the spherical aberration and field curvature.

Figure 13.6 Meridional ray plots for dialyte ( f 0 ¼ 5.42).

In the present example, meridional ray fans were traced at the three obliquities: 10 , 16 , and 22 , as shown in Figure 13.6. The abscissa is the A values of the rays, that is, the height of the incidence of each of the rays at the tangent plane to the front surface.5 To locate the endpoints of the curves we must decide which clear aperture we should allow at the front and the rear of the complete lens.

It is customary in a short lens such as this to give all eight lens surfaces the initial aperture of the marginal beam, which in our case is f/6 or 0.904. The limiting rays at each obliquity are then found by trial such that the lower rim rays meet r1 at a height of –0.452, and the upper rim rays meet r8 at a height of þ0.452. These limiting rays are marked V in Figure 13.6, their paths being shown in Figure 13.7. If there were no vignetting, the upper and lower rays would be limited only by the diaphragm, these rays being marked S in Figure 13.6. It is clear that the vignetting has proved to be very beneficial, especially for the lower rays at 22 ; these would cause a bad one-sided haze due to higher order coma if they were not vignetted out in this way.

A glance at this graph reveals that there is a small residual of negative coma, requiring a small negative bending of the front and rear crowns to remove it. A Dc of –0.005 was found to be sufficient to make all three curves quite straight. To gild the lily, trifling bendings were applied to remove small residuals of

Figure 13.7 An f/6 dialyte objective.

LA0 and X t0, namely, Dc1 ¼ –0.0005, and Dc3 ¼ –0.0025. The final lens was then scaled to a focal length of 10, with the following specification:

with f 0 ¼ 10.0, l0 ¼ 9.1734, Petzval sum (10) ¼ 0.0342, LA0 ( f/6) ¼ 0.0143, LZA ( f/8.5) ¼ –0.0193; for 22 : Xs0 ¼ –0.0155, X t0 ¼ 0, distortion ¼ 0.218 %, lateral color ¼ –0.0004; zonal chromatic aberration ¼ 0.0001.

The aberrations of this final system were shown in Figure 13.7. One interesting point here is that after all the changes in powers and bendings that have been made, radii r2 and r7 are almost identical. It might be a significant manufacturing economy to make them identical by a further trifling bending to one or both of the positive elements.

Lenses of this dialyte type perform admirably and can be designed with apertures up to about f/3.5; the field, however, is limited to about 22 to 24 from the axis.

13.3 A DOUBLE-GAUSS–TYPE LENS

The mathematician Gauss once suggested that a telescope objective could be made with two meniscus-shaped elements, the advantage being that such a system would be free from spherochromatism. However, this arrangement has other serious disadvantages and it has not been used in any large telescope. Alvan G. Clark tried to use it with no success, but with considerable insight he recognized that two such objectives mounted symmetrically about a central stop might make a good photographic lens. He patented6 the idea in 1888, and a lens of this type called the Alvan G. Clark lens was offered for sale by Bausch and Lomb from 1890 to 1898. The same type was also used in the Ross Homocentric, the Busch Omnar, and the Meyer Aristostigmat. An unsymmetrical version was later used by Kodak in their Wide Field Ektar lenses.

The design is suitable for a low-aperture wide-angle objective, the design procedure following closely the design of the dialyte just described. However, the glasses must be much further apart on the V n chart than before, possible types being

(a)Dense flint: nD ¼ 1.6170, V ¼ 36.60, nF ¼ 1.62904, nC ¼ 1.61218

(b)Dense barium crown: nD ¼ 1.6109, V ¼ 57.20, nF ¼ 1.61843, nC ¼ 1.60775

Using these glasses, the formulas given in Eqs. (13-5) and (13-6) can be solved for the two lens powers in the rear component, assuming zero Lch0 and a smaller Petzval sum such as 0.028 for a focal length of 10. The powers are much smaller than before and the air space much larger:

in their turn upset the spherical and field corrections, so we have to make further small bendings for these, and so on back and forth until all four aberrations are corrected. The system then is as follows:

0.6733

0.31.6109

0.1183

0.145 air

0.4768

0.11.6170

0.9671

0.15

(symmetrical)

with focal length ¼ 5.9394, Ptz(10) ¼ 0.0278; for 32 : Xs0 ¼ 0.0674, Xs0; ¼ –0.0134; LA0 ¼ –0.0015, LZA ¼ 0.0000, zonal chromatic aberration ¼ –0.0008.

Finally, we come to the correction of distortion and lateral color:

These were reduced by shifting 3% of the power of the front crown element to the rear crown, giving 0.70% distortion and –0.0001 of lateral color. However, since this move upset everything, it was necessary to return to the previous graphs and repeat the whole design process once or twice more. After scaling up to a focal length of 10.0, the final system is as follows:

0.38600

0.5083 1.6109

0.06787

0.2355

0.28732

0.1694 1.6170

0.57670

0.2542

0.2542

0.57670

0.1694 1.6170

0.28732

0.2355

0.07201

0.5083 1.6109

0.40990

with f 0 ¼ 10.0, l0 ¼ 8.9971, Ptz(10) ¼ 0.0279, zonal chromatic aberration ¼ 0.00030, LA0 ( f/8) ¼ 0.00046, LZA ( f/11) ¼ 0.00225, stop diameter ( f/8) ¼ 0.5149. The results are shown in Table 13.1.

Table 13.1

Astigmatism, Distortion, and Lateral Color for Example Double-Gauss Lens

A scale drawing of this lens, together with its aberration graphs, is shown in Figure 13.10. It is evident that the zonal aberration is of the unusual overcorrected type and that the crown elements are quite unnecessarily thick.

To complete the work we must determine how much vignetting should be introduced, mainly to cut off the ends of the curves in Figure 13.11. Our procedure will be to decide to accept a maximum departure of the graphs from the principal ray by, say, 0.025, and cut off everything beyond that limit. The vignetted rays are shown in the lens diagram in Figure 13.10. The limiting rays are marked V on the ray plots (Figure 13.11), and the extreme unvignetted rays that just fill the diaphragm are marked S. It will be seen that the 15 beam is unvignetted. In view of this, the limiting surface apertures shown in Table 13.2 are recommended for this lens. This completes the design.

If a higher aperture than f/8 is desired, it is necessary to thicken the negative elements considerably, and introduce achromatizing surfaces into them. The process for the design of the f/2 “Opic” lens of this type has been described by H. W. Lee.9 Using the Buchdahl coefficients s3; s4; m10; and m11, Hopkins10 has shown that it is possible to calculate the image height, relative to the Gaussian image height, where the sagittal and tangential field curves intersect one another. This of course assumes that higher-order astigmatic terms are negligible.

Beyond this intersection height, the two field curves rapidly diverge from one another, as can be observed in Figures 13.2, 13.7, and 13.10, and is shown in Figure 13.14 in the next section (page 372). Attempting to use the lens beyond this image height will be unfruitful. This image height, Hn is derived by equating the linear terms in aperture of Eqs. (4-6) and (4-7) through fifth order, such that

½ð3s3 þ s4ÞHn2 þ m10Hn4&r ¼ ½ðs3 þ s4ÞHn2 þ m11Hn4&r