9.2.1. Semi-apo and apo lens objective examples

PAGE HIGHLIGHTS
• LA graphs, all-spherical    • Spherical vs. aspherized    •Spherochromatism and lens spacing
•Spherochromatism and lens shape   • Common glass apos   • Dialyte apochromats

FIGURE 149: Secondary spectrum and spherochromatism in the objectives from Table 12. While formally defined as the longitudinal separation between the common red blue (F/C) focus and green (e-line), all paraxial (here, on the horizontal scale; note that the size of scale varies), actual location of these foci is shifted due to the presence of spherochromatism. In general, the paraxial blue focus is shifted closer, and the red focus about as much farther away from where that common focus would  have occurred. Obviously, if  they would be nearly coinciding, the wavefront error would have been significantly greater in the shorter wavelength, more so due to it being overcorrected (when reading these plots, it is important to keep in mind that all the zones at the aperture radius - represented as the vertical scale - focus at the horizontal scale; the foci are merely projected up to the zonal height, forming the
longitudinal aberration plot).
The magnitude of F/C paraxial separation for given level of spherical aberration is somewhat arbitrary; it is often such that F and C-line ray spots are nearly equal in size. Since in that case the wavefront error is greater in the F - not the best mode considering that eye sensitivity is higher in the blue line, even in the photopic mode, and more so toward mesopic mode - refracting objective here are generally brought to a near equal wavefront error in F and C.
Plots for objectives #1-5 show F and C lines clearly separated from the e-line, indicating the presence of secondary spectrum (these five objectives only show these three lines, since r and , particularly, g, are far enough to squeeze F, C and e too close together on this image scale). Secondary spectrum is significantly reduced in #6, a fast triplet using somewhat abnormal short flint for its central element (typically for short flint combinations, the violet end runs away). It is still more reduced in #7, a triplet combo of the early NASA Christen triplet combo; #8, also an Astrophysics' triplet combo from the 1980s, uses another short flint, originally made for NASA (reportedly, a bit better than Schott's KZFSN4 available today only as a special melt); it does form a typical apo C/F/e crossing  at about 76% zone, but some residual secondary spectrum is still evident, since F and C paraxial foci are at unequal separation from paraxial e focus (by slightly strengthening the negative power, F could be brought to coincide with C but, since C is farther away from e than F, this common focus would be be falling somewhat behind e). In #9, an early Zeiss triplet, secondary spectrum is nearly eliminated, but higher order spherical creeps in due to the strong surface radii, even at f/15. A more recent doublet, the older Meade "apo" at #10, with Hikari's FK01 ED glass (near-equivalent to Ohara FPL51), still has some relatively significant secondary spectrum (at 127mm f/9 its is still quite close to 0.95 visual polychromatic Strehl). At #11, very similar Schott FK51 ED glass, in combination with a common crown, has secondary spectrum all but eliminated, but clearly shows spherochromatism at f/7 (both combinations, with the ED first, and following the crown, are shown). Using ED with higher Abbe number (i.e. lower dispersion), like FPL53 (#12), or calcium fluoride (#14), reduces spherochromatism, and even more does using the triplet form (#13). Spherochromatism with these same glasses in a doublet form is, as expected, significantly reduced at slower f-ratios (#15, 16). Larger fast objectives require triplet form for good chromatic correction: #17 shows plots for TEC 140mm f/7 triplet combination optimized for visual mode (the actual instrument is said to have 0.92 polychromatic Strehl, which implies that the violet is being pulled closer, at the expense of the optimum F/C correction, to improve its photo/CCD performance). If such triplet was made with ZKN7 for the 1st and 3rd element (at a small increase in the glass cost), secondary spectrum is practically eliminated, but the overall gain is relatively small, because it is spherochromatism, i.e. spherical aberration in non-optimized wavelengths that causes most of the chromatic error (#18).
For the same reason, replacing FPL53 with fluorite and ZKN7 with one of fluorite's matching crowns (K7), produces nearly identical correction (#19). With somewhat wider air gap between the front and second lens element, it is possible to practically eliminate spherochromatism (#20, Takahashi TOA type triplet). Using the same glasses as in #18 in a 2-doublet Petzval arrangement reduces spherochromatism by reducing the lower-order residuals due to index variations and eliminating higher-order spherical, but it is still present (#21). However, the added bonus of the Petzval is the flattened image surface (which is curved to about 35-40% of the focal length in doublets and triplets); residual astigmatism left in for that purpose is still less than half the astigmatism of the triplets. 

Aspherizing for reduction of higher-order spherical aberration

All above systems utilize only spherical surfaces. While it is a plus with respect to ease of fabrication, it is also a limitation with respect to correction of higher (6th) order spherical aberration (secondary astigmatism is generally negligible, and secondary coma low in comparison, easy to reduce to negligible by balancing it with the primary coma). Aspherizing a surface does not appreciably change the ray height on it, but it does change surface profile, bringing it either closer or farther away (depending on the sign of the aspheric) from the 6th order surface approximation (FIG. 37), as well as changing the incident angle. When significant, higher-order spherical affects not only the level of correction in the optimized wavelength, but also adds to spherochromatism (since it changes surface profile, it also affects longitudinal chromatism, and may require slight adjustment in the radii power balance). Removing 6th order spherical by way of aspherizing also tends to lessen the violet 4th order error, producing better color balance. The effect is generally small, and may be marginally significant in oiled objectives, due to the stronger aspherization required to remove 6th order spherical aberration on an oiled surface.
The manner in which aspherization affects spherochromatism is somewhat different in triplets vs. doublets.
In doublets, the typical form of secondary spherical is undercorrection. With it uncorrected, and with primary spherical corrected in the optimized wavelength, a doublet is more undercorrected in the red (which is undercorrected in primary spherical for zero primary spherical in optimized wavelength, thus the two aberrations add up), while to some extent balanced in the blue (which is overcorrected for zero primary spherical in optimized wavelength, hence opposite in sign to that of secondary spherical). When this residual secondary spherical is balanced with primary spherical of opposite sign (overcorrected) by a slight adjustment in one of the inner radii, it can be minimized only in one, optimized wavelengths, while added primary overcorrection reduces error in the red (tending to produce predominantly higher order spherical, since added primary overcorrection partly offsets its primary undercorrection component) and increases it in the blue/violet, by enlarging its primary overcorrection component. Taking as an example the FPL53/ZKN7 doublet (#12), those two stages are shown as first and second from left, respectively. Since most of the lens' secondary spherical is generated by the slight excess on the second vs. third surface, it can be eliminated by slightly aspherizing either of the

two radii (prolate ellipsoid for the second, and oblate for the third radius). To compensate for the induced primary spherical imbalance, one of the other two surfaces, with the relatively low secondary-to-primary aberration ratio, needs to be aspherized. The result is shown on the third plot. Removing secondary spherical by aspherizing had no appreciable effect on spherochromatism. The only significant change is in the correction level in the optimized wavelength, but since it is sufficiently low with all surfaces spherical, aspherization is not worth the extra cost. Even if the 8TH order residual (about 0.05 wave RMS, minimized to 0.012 wave by balancing it with the opposite in sign primary spherical) is removed by putting a Schmidt-type aspheric on one of the surfaces, the overall correction is not significantly better.
In the typical triplet, the residual secondary spherical is most often overcorrected, while for cancelled primary spherical in the optimized wavelength, the red and blue/violet are, like with the doublet, undercorrected and overcorrected in the primary spherical, respectively. Hence balancing this residual overcorrected secondary spherical by adding primary spherical of opposite sign now lessens the error in the red, producing a partially balanced 6th/4th order aberration, while increasing it in the blue/violet. Since the latter typically already has significantly larger error, this creates more of a red vs. blue/violet imbalance than in the doublet. Thus lessening this imbalance through removing the lens' secondary spherical residual by aspherizing one or two surfaces will generally produce more of a beneficial effect on overall correction than in a doublet. The difference may and may not be significant.
It can be illustrated on the ZKN7/FPL53/ZKN7 triplet (#18). While triplets with equal pairs of inner radii are usually intended for oil-type lens, they were given above as a cemented objective, without spacing and oil specs, because small spacing of this type (usually a small fraction of a mm), with or without oil, does not appreciably change lens' output (in this particular case it only added about 1/20 wave P-V of primary spherical, which can be corrected by a slight bending of the inner two radii pairs, to 220/496mm). That's because the aberrations at the inner four radii, which are dominant, tend to change evenly, in proportion. However, the conics that works with air-spaced triplet (which are nearly identical for the cemented as well) will not work with the oil-spaced one. For instance, this particular triplet if air-spaced will have its higher-order spherical aberration excess nearly eliminated by putting -0.03 conic on the second surface, and the primary spherical induced by that offset by putting -0.225 conic on the front surface. And if lens' interspaces is filled with oil, the needed conics are -0.5 and -.538 for the third and first surface, respectively (third surface requires significantly less aspherizing in this case). The output is also very different for the air-spaced and oiled objective, as shown below (since OSLO Edu doesn't list lens oils, prescription uses Schott KF6 crown which has similar refractive index to some commonly used oils).



Compared to the non-aspherized air-spaced lens with all other specs identical, removing higher-order spherical with aspherization did somewhat reduce the error in the blue vs. red, but only by about 10%. Thus removing 6th order spherical aberration residual with aspherization wouldn't be justified with this particular lens, less so considering that it would require double aspherization. The results are, however, very different with the oiled lens. Due to the refraction being very much suppressed by the higher refractive index in the interspace (1.52 for KF6 vs. 1 for air), the inner surfaces act as of significantly weaker radii, requiring a higher nominal conic for canceling the 6th order spherical imbalance. But the required change of surface profile to remove 6th order spherical here also affects spherochromatism in the 4th order spherical, simultaneously removing both, 6th and 4th order spherochromatism. The result is a nearly zero-spherochromatism lens, which is at f/7 better corrected than TOA-type lens at f/7.7 (note that a slight change in radii was required to offset the effect of aspherization on longitudinal spherochromatism; it increased the focal length about 1%). Unfortunately, putting -0.5 conic on such a strongly curved surface (f/0.8 if mirror) - more so considering it is a hard to work with glass - is highly unappealing in terms of related cost. The alternative of aspherizing non-FPL53 surfaces (0.4 conic on the first and -39 last surface) would eliminate the 6th order residual, but with no reduction in spherochromatism.
However, the manner in which the 6th order excess is generated varies with the glass combinations and lens configuration. The above example is not a "good candidate" for aspherization, with double aspherization required, and little to gain from it. The reason for double aspherization here is that one surface needs to be aspherized in order to minimize 6th order spherical, but this cannot be compensated for by bending the mid element, due to the 6th order spherical aberration total for the two inner surface pairs remaining significant. Thus compensatory aspherization of another surface is necessary. This may change with different glass combinations, different lens arrangement, or both. For instance, this same glass combination but in the positive-negative-positive (PNP) element arrangement - hence with the matching glass sandwiched between two ED elements - only requires a single aspherization, as shown below, can have its 6th order spherical aberration residual eliminated by aspherizing a single non-FPL53 surface, with the primary



spherical induced by aspherizing neutralized by the aberration of opposite sign induced by reshaping the mid element (first and last radius are changed in order to minimize coma which was to a smaller extent also induced by reshaping the mid element). As the LA graphs and RMS numbers show, spherochromatism is significantly reduced with aspherizing (note that F/C correction can be made still better, by slightly weakening the positive vs. negative power, i.e. shifting red to the left, and blue/violet to the right, at a price of g/r error increase). The negatives are the relatively strong aspheric required, and two ED glass element needed in the PNP triplet arrangement.
In order to be viable for aspherization, an NPN triplet has to use different matching glasses. Here's an example where the matching crowns are also Ohara, as well as the glass used to simulate the oil (nearly identical to the Schott KF6 in the index and dispersion).
 


The aspherized LA plot is very similar to that of the published plot for the AP 140mm f/7.5 EDF triplet apo (in the published lines, which are somewhat different, with the 460nm instead of 436nm, no 706nm and unspecified line in between e and F). There is no way to know how close are these specs to the actual ones - except that we know it is a NPN arrangement with FPL53 as the mid element - but this fast and large triplet had to be aspherized in order not to show the 6th order residual. While the spherochromatism is somewhat reduced in the aspherized version, the gain is rather small. The red/blue-violet imbalance is also slightly smaller with aspherizing (which should result in a slightly whither white-star image), but even both combined make for a rather small difference. It is important to note, though, that the g-line LA is 25% smaller in the aspherized version, hence optimizing for the violet would give the aspherized version a slight edge in this respect. Still, the only rationale for aspherizing here would be to bring the lens as close to perfection as possible (note that the mid element in the AP lens is not equi-convex - the first surface is more strongly curved, which may make it possible to aspherize only the 5th surface, perhaps less than in this example - but the point here is to examine the gain from aspherizing, which should remain similar, considering that the aspherized version's LA graph is nearly identical to the published data for the AP lens).
However, this is only so in the context of the error magnitude expressed as RMS wavefront error, or corresponding Strehl. Neither accounts for the differences in the pattern of the disturbed energy and its possible effects. Looking from the standpoint of distribution of the energy lost from central maxima, the benefit from aspherization can be significant even in this case (as well as in the case of the doublet example). Even if the RMS error decrease is relatively small, the effect on ensquared energy just from replacing balanced secondary spherical - and to a smaller extend its partially balanced forms - with primary spherical at best focus can result in appreciable benefit in imaging applications. It could also be beneficial in visual use, since balanced 6th/4th order (i.e. secondary) spherical aberration throws energy lost to the Airy disc significantly farther away than primary spherical of similar magnitude (FIG. 98). This generally makes the wavelengths with the largest error - commonly the blue/violet - more readily visible as a separate fringe coloration, particularly on bright objects. While contrast-wise this shouldn't have significant effect if the magnitude of aberrations remains similar, it could still produce a noticeable cosmetic, i.e. color purity improvement in such instances. It could also enhance imaging performance in these wavelengths. Combined with the possible reduction in the 4th order spherical due to aspherization, it could produce tangible overall benefit in the performance level.

Spherochromatism and lens spacing

General rule for secondary spectrum is that it doesn't change appreciably for any given combination of glasses, regardless of how many elements we put together, or how we change the configuration. With spherochromatism, however, things are different: change in configuration can cause change in the level of different orders of spherical aberration. Certain configurations can have these different orders have compensatory effect, significantly reducing spherochromatism. It is more pronounced with the change of lens spacing. An example are Takahashi TOA objectives, but since they use unortodox configuration with two ED glass elements, the effect will be illustrated on a stundard NPN triplet with a single ED glass.

Picture below shows a triplet modified to reduce spherochromatism (top) vs. standard triplet form (middle). The former was posted on Cloudy Nights forum, by user RayG, and uses very wide second spacing to constructively change the proportions of spherical aberration orders (all graphs are with the same scale, so they can be directly compared; the ray spot plots are twice smaller for the mid and bottom objective). As a result, the LA graph is significantly tighter than for the standard triplet, as well as the OPD (optical path difference, i.e. wavefront error) curves. Reduction in spherochromatism is reflected in the respective polychromatic Strehl values, 0.951 vs. 0.934 photopic and 0.849 vs. 0.788 average CCD. Chromatic focal shift graphs - which show how paraxial foci of different wavelengths deviate from the paraxial focus of optimized wavelength - are noticeably different, with the modified triplet having, expectedly, smaller deviations, except in the infrared (chromatic focal shift is not the primary indicator when spherochromatism dominates, but becomes more meaningful as spherochromatism diminishes).


In addition to somewhat stronger astigmatism/field curvature, a price to pay for the spherochromatism reduction is the increased lateral color, about three times larger than in the standard triplet. Diffraction simulations show its affect at 0.5 degrees off axis, for the five wavelengths and even sensitivity. While maybe tolerable visually, it is not acceptable for CCD. Since lateral color here is caused by the rear gap, the only way to lower it is a reduction in the lens separation. By nearly halving the gap, and strengthening the gap radii in order to maintain reduction in spherochromatism, lateral color is reduced by nearly 40%, with even somewhat higher Strehl values vs. wide gap (despite more than twice wider LA graph span in the former; bear in mind that for any given longitudinal error, the wavefront error for spherical aberration at the best focus location is eight times smaller than for defocus). Note that the Takahashi TOA, due to its more sophisticated design, doesn't suffer from a significant lateral color error.

Spherochromatism and lens shape

Change in the lens shape also affects spherocromatism, primarily with triplets, since doublets have their lens shape determined by constraints related to their single gap. As an example here is taken BSM81/FPL53/BSM81 air-spaced triplet, using Ohara super-low-dispersion glass in combination with the closest to it on the RPD diagram lanthanoid. This is probably a combination used for the Astro-Physics 110mm f/6 StarFire GTX apo refractor. Starting point is configuration with all four inner radii equal, with their nominal value determined by such needed to bring all colors close together. In this case it is aproximatelly R_2,3,4,5~210mm (absolute value; weaker radii would bring blue-violet closer while pushing the red end farther away, and vice versa). However, with all four radii equal spherical aberration is grossly uncorrected. In order to minimize it, at least one inner radius has to be either changed, or aspherized (#1,2,3). Also, more often than not, it is possible to minimize spherical aberration by reshaping the middle element, while keeping the paired radii equal. In this case, keeping them equal would require significantly widening the gaps (#4). Keeping the lenses in near contact requires one radius changed to correct for spherical aberration (with every change, the first and last radius also need to be adjusted in order to minimize coma). Changing either R2 (#5) or R5 produces similar output, with the former being a tad better. Changing both makes it a bit less well corrected than either (#6). Aspherizing is, of course, also an option, producing similar rezult (#7). Finally, arrangement similar to #5, but having a bit better correction, best of all variations shown (it is very similar to the design posted on Cloudy Nights forum by RayG, only with R4 and R5 made equal, and some minor changes to compensate for the effect).

To make objectives comparable, they are nearly balanced in F and C line, and have the central line error at near-minimum. The first two variants, having only a single radius altered, are less well corrected than the rest, particularly #2, mainly because of its relatively significant higher-order spherical residual. Aspherizing R5, with all four inner radii equal, produces the best correction (#3), although not significantly better than #1. Yet better correction is achieved by reshaping middle element (#4) but the gaps required, although still moderate, are not "near contact" anymore. Practically the same level of correction offers its near-contact option with a single radius changed (#5). Adjusting two radii produces slightly lower correction level (#6), as well as aspherizing R5 (#7). The best option (#8) has the F/C Strehl average at 0.95+, and polychromatic photopic Strehl at 0.98. That is numerically equal to the arrangement with somewhat wider gaps (#4), and only slightly better than #5. No eye would be able to tell the difference (making the rear two radii another 5mm weaker in #5 doesn't change the poly-Strehl value, implying it is the design limit).

In all, differences in the level of correction do exist, but are near negligible, except when there is a significant increase in the higher-order spherical residual.