DESIGNER NOTE

When using a computer-based lens design program to design such a lens as a Bravais, an easy method to create a “virtual” object, that is, a converging beam, is to use an ideal lens focused at the location of the virtual object. Most lens design programs provide such an ideal lens and may be called an ideal lens, perfect lens, paraxial lens, and so on. Should your program not include such a feature, you can use either of the Section 6.1.8 aspheric planoconvex lenses since they are free from spherical aberration or perhaps a parabolic mirror or elliptical (source at one focus and image at other) mirror. Of course if the lens designer has the objective lens available, that can be included instead. It should be noted that, in general, the objective optics and Bravais lens are designed separately since the Bravais lens most often is moved in and out of the system as needed. Consequently, image quality of the objective optics must be acceptable without the Bravais lens; the Bravais lens must not degrade the image quality.

14.4 THE PROTAR LENS

In 1890 Paul Rudolph of Zeiss13 had the idea of correcting the spherical aberration of a new-achromat landscape lens by adding a front component resembling the front component of a Rapid Rectilinear but with very little power. The thought was that the strong cemented interface in the front component could be used to correct the spherical aberration, and that it would have little effect on field curvature because the principal rays would be almost perpendicular to it. The cemented interface in the rear component would be used to flatten the field as in the new-achromat landscape lens. It is noted that in the Protar patent, one form of the rear component was a cemented triplet although the primary component form was a cemented doublet.

This leaves us with four other radii to be determined. The fourth and sixth radii can be used for Petzval sum and focal length, as in the design of a new achromat, leaving the first and third radii for coma and distortion correction. The two chromatic aberrations are controlled by the final selection of glass dispersions.

As an example, we will first select suitable refractive indices. The two doublets comprise four elements that we denote as (a) to (d). For the outer elements (a) and (d) we may assume that ne ¼ 1.6135. In the Schott catalog we find many glasses having ne lying close to this figure, with values of Ve ¼ (ne – 1)/(nF – nC) lying between 37.2 and 59.1. For the inner elements (b) and (c) we choose similarly ne ¼ 1.5146 for which Ve values are available between 51.2 and 63.6. Suitable thicknesses are, respectively, 0.25 and 0.4 for the front component

and 0.1 and 0.4 for the rear component; the center space is set at 0.4 with the diaphragm midway. We do not use diaphragm position as a degree of freedom since we have enough degrees of freedom already; it is, however, advisable to keep the center space small to reduce vignetting.

For a first trial we may choose c1 ¼ 0.5, c2 ¼ 1.2, and c5 ¼ 0.5. We solve c3 to make the front component afocal, and we determine c4 and c6 by trial and error to make the focal length equal to 10 and the Petzval sum 0.025. The Setup A is as follows:

0.5

0.25 1.6135

1.2

0.4 1.5146

0.417646

Stop

0.2

0.2

0.626156

0.1 1.5146

0.5

0.4 1.6135

0.572960

with f 0 ¼ 10, l0 ¼ 9.8120, Ptz ¼ 0.025, trim diameter ¼ 1.5. A scale drawing of this lens is shown in Figure 14.16. Tracing an f/8 marginal ray from infinity and a principal ray passing through the center of the stop at a slope of –20 gives these starting aberrations:

Since the main function of radius r2 is to control the spherical aberration and the main function of r5 is to flatten the field, we next proceed to vary c2 and c5 in turn by 0.05 and plot a double graph by means of which the spherical aberration and the tangential field curvature can be corrected. We assume that the desired values of these aberrations are LA0 ¼ þ0.15 and Xt0 ¼ 0. The double graph in Figure 14.17 indicates that we should make the following changes from the original Setup A:

1.Dc2 ¼ 0.034. But c2 was 1.2, so therefore try new c2 ¼ 1.234.

2.Dc5 ¼ 0.009. But c5 was 0.5, so therefore try new c5 ¼ 0.509.

These changes give Setup B, the thicknesses and refractive indices remaining as before:

c d

with f 0 ¼ 10, Ptz ¼ 0.025, LA0 ( f/8) ¼ 0.1361, LZA ( f/11) ¼ –0.0724; for 17.91 :

Xs0 ¼ –0.0668, Xt0 ¼ –0.0023.

Before making any further changes in LA0 and Xt0, we must decide whether the aim point that we have chosen is the best. Certainly the zonal spherical aberration is about right, and so we will maintain our aim for spherical aberration at þ0.15. However, to study the field requirements, it is necessary to trace several more principal rays at higher obliquities and plot the astigmatism curves. These rays give the tabulation shown in Table 14.7. A plot of these field curves (Figure 14.16) indicates at once that a much better aim point for the sag of the tangential field Xt0 would be at –0.03, and this value will be used from now on (second aim point in Figure 14.17).

We next proceed to correct the coma and distortion. The OSC of Setup B is found to be –0.00399 at f/8, and both the coma and distortion are clearly excessive. Our free variables are now c1 and the power of the front component. By using y ¼ 1 for the paraxial ray, the power of the front component is given directly by u30 , and any desired value of this angle can be obtained by solving for c3. Assuming for a start that both the OSC and the distortion should be

Table 14.7

Astigmatism and Distortion for Setup B

Distortion (percent)

Figure 14.18 Double graph for coma and distortion.

zero, we plot a second double graph (Figure 14.18), changing c1 and u30 . This graph indicates that we should make the following changes in the system:

1.Dc1 ¼ 0.1227. But c1 is 0.5, so therefore we should try c1 ¼ 0.6227.

2.Du30 ¼ 0.0117. But u30 is zero, so therefore we should try u30 ¼ 0.0117.

With these changes our lens becomes Setup C:

0.6227

0.25 1.6135

1.234

0.4 1.5146

0.570553

Stop

0.2

0.2

0.473548

0.1 1.5146

0.509

0.4 1.6135

0.453187

with f 0 ¼ 10, Ptz ¼ 0.025, power of front ¼ þ0.0117, OSC ( f/8) ¼ –0.000402, distortion (18 ) ¼ –0.017%.

We will assume for the present that zero is a good aim point for the distortion, but we must investigate the coma further. To do this we trace a family of rays entering the lens at –17.23 , and plot a graph connecting the height of incidence of each ray at the stop against the height H0 of the ray at the paraxial focal plane. This graph, Figure 14.19, indicates the presence of some negative primary coma with an upturn at both ends of the curve due to positive higher-order coma. Since the ends of the curve will probably be cut off by vignetting, it might be better to aim at, say, þ0.002 of OSC at f/8 instead of zero. This will represent the aim point for OSC on future double graphs.

The spherical aberration of Setup C is þ0.0908 and the field curvature is given by Xt0 ¼ þ0.1531. Reference to the first double graph, using the new aim point, indicates these changes:

1.Dc2 ¼ þ0.026. But c2 ¼ 1.234, so therefore we should try c2 ¼ 1.260.

2.Dc5 ¼ þ0.0916. But c5 ¼ 0.509, so therefore we should try c5 ¼ 0.6006.

These changes give us Setup D:

c

0.6227

1.260

0.564736

0.466835 0.60060.435009

with f 0 ¼ 10, Ptz ¼ 0.025, front power ¼ 0.0117, LA0 ( f/8) ¼ 0.1422; for 17.24 : Xs0 ¼ –0.0704, Xt0 ¼ –0.0331.

Figure 14.19 Meridional ray plot for Setup C (17 ).

The spherical aberration and field curvature of this system are acceptable. However, we find that the OSC at f/8 has now become –0.00313 and the 17 distortion –0.009%. Reference to the second double graph enables us to remove these residuals, and we then return to the first graph for spherical aberration and field curvature, and so on back and forth several times until all four aberrations are acceptable. The final Setup is E:

0.6445

0.25 1.6135

1.2466

0.4 1.5146

0.628369

Stop

0.2

0.2

0.383337

0.1 1.5146

0.5856

0.4 1.6135

0.395628

with f 0 ¼ 10, Ptz ¼ 0.025, power of front ¼ 0, LA0 ( f/8) ¼ 0.1529, LZA ( f/11) ¼ –0.0487, OSC ( f/8) ¼ 0.00204. The astigmatism and distortion values are as shown in Table 14.8.

These aberrations are plotted in Figure 14.20, as well as the 17 meridional ray plot for the study of coma. It is clear from these graphs that we should have aimed at about þ0.2% of distortion at 17 and about þ0.001 of OSC at f/8. These values should be adopted in any future changes. The field and spherical aberrations are just about right.

Table 14.8

Astigmatism and Distortion for Setup E

Height Y in stop

Figure 14.20 Aberrations of final Protar design.

Our next task is to investigate the correction of the two chromatic aberrations by choice of glass. We first attempt to correct the (D – d) Dn sum of each component separately, since this is the proper thin-lens solution to the problem. For this we use the true Dn ¼ nF – nC of each likely glass, ignoring the fact that the catalog refractive indices are not quite equal to those we have assumed so far. This gives what is shown in Table 14.9.

No suitable glasses were found by which we could have reduced the negative (D – d) Dn sum in the rear component.

To calculate the lateral color at 17.18 , we apply the same arithmetic error to the individual F and C indices as we have assumed for the e indices. This gives the numbers shown in Table 14.10.

From these data we find that HF0 – HC0 ¼ þ0.001086. Now the lateral color in any lens takes the same sign as the longitudinal color of the rear component and the opposite sign to the longitudinal color of the front component. Hence to improve both the longitudinal and lateral color aberrations simultaneously we must make the (D – d) Dn sum of the front component more positive. To do this we need a glass in lens (1) having a lower V number or a glass in lens (2) having a higher V number. Inspection of the chart enclosed with the glass catalog indicates that the only possible choice is to use BK–1 in place of K–1 in lens (2), because all other possible glasses have a refractive index differing too much from the e indices we assumed for the aberration calculations. This glass has Dn ¼ 0.00805 giving a value of P (D – d) Dn equal to þ0.00005953 in the front component, or –0.00006696 for the whole system, and a lateral color of HF0 – HC0 ¼ þ0.000790. We must accept these residuals in the absence of other more extreme glass types.

Of course, the final stage is to repeat the design using the true ne refractive indices, and then to adjust the clear apertures to give the desired degree of vignetting.

Table 14.10

Lateral Chromatic Error for Setup E