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9. REFRACTING TELESCOPES
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9.2. Refracting telescope objectives: Apo
and semi-apo
► ## 9.1 REFRACTING TELESCOPE OBJECTIVE: DOUBLET ACHROMAT
PAGE HIGHLIGHTS As mentioned on the previous
page, the idea of a doublet achromat probably comes from
Chester Moore Hall in the
1730s, but it was John Dollond
who
had it patented in 1758 and succeeded in producing working doublet
achromats. The original Dollond's doublet - possibly as the one claimed
by Hall (order not specified) - had flint (negative element)
in front, followed by crown, with all surfaces spherical. His son
Peter modified it by putting crown in front, and correcting residual
spherical aberration by aspherizing the 4th surface, polishing off its
peripheral area (
From what is avalable, it seems fair to conclude
that it was John Dollond who made the crucial step from the idea
of combining two different glasses to actually producing lens objective
with chromatism reduced to the level of an achromat (in court proceedings
from 1765, Hall's doublet was described as consisting of a plano-convex
crown and plano-concave flint element, with the latter having focal length
three times longer; such doublet would not just have only marginally lower
chromatism than a singlet lens, but would also suffer from unacceptably
high spherical aberration due to a large differential between two radii).
Experimenting with prisms, John Dollond determined approximate value of
the focal lenghts ratio for the two lenses as 3:2 (flint-to-crown), and
minimized spherical aberration by making the flint element a
negative meniscus, which allowed for the stronger concave radius,
nearly equal to the crown element's inner radius. Below is raytrace of
one of his achromats (data from
It is a small 32mm f/24 lens with low residual spherical aberration.
As most of the objectives from that time, it is somewhat overcorrected
(vs. singlet, which has shorter wavelengths focusing closer to the
objective than the longer ones). The likely reason is that it was
compensating for chromatism of the 5-element terestrial eyepiece
used at that time ( After that, a number of doublet types have been developed. An achromat, by definition, uses two common glasses - crown and flint - to reduce primary chromatism (chromatism of a single lens). Since chromatic correction of a doublet depends mainly on the glass combination, and common glasses span relatively narrow range of properties (i.e. refractive index and dispersion), doublets achromats have similar level of chromatism and differ mostly in their correction of monochromatic aberrations. Typical astronomical doublet is air-spaced. Air space makes it possible to optimally correct for monochromatic aberrations with any glass combination. Cemented doublets, on the other side, require specific glass combinations that allow optimal correction with equal inner radii (for instance, BK7 and SF5; such doublets perform just as well with a small air gap). Another obstacle for cemented doublets fabrication is that in larger sizes - generally around 100mm in diameter, and larger - thermal expantion and contraction of the glass becomes detrimental for the bond between lenses.
A long time standard
for doublet achromats is the
The inner two radii of the Fraunhofer can be equalized,
without significant change in the correction level, on or off axis (in
order to minimize ensuing spherical aberration,
Another coma-free doublet
with reversed order (flint in front) is the Two other doublet achromat
types of mostly historical significance are the Diagrams below illustrate basic properties of the main achromatic doublet types: longitudinal aberration plot for five spectral lines spanning most of the visual spectrum (g-436nm, F-486nm, e-546nm, C-656nm and r-707nm), axial F-e-C ray spots, P-V wavefront error at 0.5° off-axis (e-line) and best image curvature radius.
Following table gives design specifics for doublets achromats consisting of BK7 crown and F2 flint, with all measures in units of the focal length. The specs are based on 100mm f/10 objectives, but are scalable within the normal range of refractor apertures, up to about 50% faster or any slower objective (by applying the desired focal number ratio to 10 directly to the radii), with only minor raytrace adjustments. Substituting similar glasses should also require only minor adjustments; glass thickness is generally not a significant factor, the exception being the Gauss objective.
A doublet with air gap wide as Clark's can be made aplanatic, in which case it is jus another variant of the Baker, with the third radius somewhat more strongly curved. Relatively unusual achromat designs are triplet
achromats, as well as those with more than one group of lenses. The
latter include Petzval-type achromats and those with the second lens
group closer to the focal plane. By their basic form, they belong to Triplet achromats require more glass and work, but offer no significantly better correction of aberrations - either monochromatic or chromatic - than doublet achromat. The only possibly beneficial use of the triplet would be for very fast, large achromats with significant level of higher-order spherical aberration.
The reason for this is that the blue and red exit the front lens group - which is assumed to be doublet achromat - only slightly separated, but at different angles: as FIG. 96 hints, the red just below, and blue just above the green ray, with the former two converging to a common focus (or nearly so) and emerging above the green ray at some distance toward the focus. Thus there is no appreciable effect on longitudinal chromatism by the second group of lens, until it is far back enough for the red/blue rays to raise above the green ray, and get refracted more strongly at the rear lens group, focusing slightly shorter relative to the green light. Additional advantage of the dialyte form in an achromat
is that astigmatism can be manipulated, either cancelled for less curved
image field, or added in the opposite sign, still very low, in order to
flatten the field, as illustrated below
As the examples above indicate, required change of the basic (front) doublet achromat in an optimized 2-doublet arrangement are relatively small. It mostly limits to bending the lenses to obtain either flat field or cancel astigmatism. Its chromatic correction can remain unchanged (infinity-corrected) if the rear achromat's is set for its object distance (which is, effectively, equal to the separation between rear doublet and virtual image formed by the front doublet). The former generally needs to be somewhat overcorrected in the blue (i.e. with blue and green having nearly a common focus, and the red focusing farther away), as well as somewhat overcorrected spherical-aberration-wise vs. doublet corrected for infinity (unless the front doublet induces offsetting aberrations). Use of special glasses for the rear corrector does not appreciably improves chromatic correction of such systems, because it is already being generated by the front doublet. For significantly improved chromatic correction, rear doublet has to be designed so that it offsets the chromatism induced by the front doublet, which requires complex lens systems. Alternately, both doublets have to be made with special glasses. A system with apochromatic correction can be made with common crown and flint, but such system would require three groups of lenses (two separated doublets and a positive lens closer to the focal plane for lateral color correction), and would be significantly longer than the effective focal length. It would also require strongly curved surfaces, generating significant higher order spherical aberration.
Use of special low-dispersion glasses in combination with common glass
types makes possible much higher level of chromatic correction in a
doublet, triplet or Petzval refractor. The
degree of improvement is determined by the respective properties of the
two glasses combined, and can vary significantly. Somewhat informally,
such refracting objectives are referred to as EFFECT OF THE CENTRAL LINE ERROR ON CHROMATIC CORRECTION
Presence of spherical aberration in the central line means that it is inevitably carries over to all others. Since spherical aberration in the red and blue are usually of opposite sign at the minimized central line error, any induced sphrical aberration (from miscollimation, or radius error) will add to one, and subtract from the other one. In general, spherochromatism in achromats is low, dominated by chromatic defocus, hence a small to moderate central line error will not significantly affect their chromatic correction. As raytrace below shows, that is the case with as fast as a 100mm f/5 unit (top).
1/5 wave P-V of spherical aberration only slightly increased P-V error in the red, and have it slightly reduced in the blue. However, in a correction mode with colors tightly bound together - i.e. apochromatic correction - the effect can be significant. As little as 1/7 wave of overcorrection in a triplet shown (Ascar 185mm f/7) caused the error in the central line to exceede error in the red C line, which has the error reduced, while also appreciably increasing the blue and violet error. EFFECT OF STOPPING DOWN ACHROMAT Stoping down achromat generally reduces all its aberrations, but it will likely cause chromatic disbalance which may not be significant, but could be noticeable vs. achromat unit with the same aperture and focal ratio that has the colors optimally balanced.
The longitudinal aberration plot shows that the color curves are
being literally trimmed off at the mask radius level. Despite no
change in the respective paraxial foci location, that effectively
pulled the blue closer to the green focus, while pushing the red
farther out. As the OPD plots show, the error is significantly
reduced in both, F and C line, due to the larger Airy disc and
decreased defocus sensitivity, but significantly more in the former.
Spherical aberration is also significantly smaller. It is of no
practical consequence in this case, with it being negligible at a
full aperture, but can be significant if the error at full aperture
is not negligible. For primary spherical aberration, the wavefront
error changes in proportion to the 4th power of the aperture, but
in fast achromats with a mix of lower and higher order SA,
reduction factor will vary somewhat; in general, it is smaller. ◄ 9. REFRACTING TELESCOPES ▐ 9.2. Refracting telescope objectives: Apo and semi-apo ► |