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◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ►
 

9.1 REFRACTING TELESCOPE OBJECTIVE: ACHROMATS

PAGE HIGHLIGHTS
• Doublet achromat types   • Petzval-type achromat

As mentioned on previous page, the inventor of a doublet achromat was Chester David Hall in 1730s, but it was John Dolland who had it patented in 1758. 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.

A long time standard for doublet achromats is the Fraunhofer doublet. It is relatively easy to make, free from coma and, as any other doublet achromat, about as well corrected for secondary spectrum as a doublet made of ordinary glasses can be. The doublet consists from the positive front crown element, and negative rear flint element. The radii of lens curvature vary somewhat with the particulars of a doublet; for the standard crown and flint combination (BK7/F2) they are approximately R1~0.61ƒ, R2~-0.35ƒ, with the focal length of about 0.44ƒ, and R3~-0.36ƒ and R4~-1.48ƒ, with the focal length of about 0.78ƒ, R1-4 being the lens surface radii from the front to the rear, and ƒ being the final system focal length (alternatively, the curvatures can be expressed in the inverse form, ƒ/Ri, as c1~1.64, c2~-2.86, c3~-2.78 and c4~-0.68). This also approximates the lens element focal lengths as ƒ1~0.43ƒ for the crown and ƒ2~-0.76ƒ for the flint.

The inner two radii of a Fraunhofer can be equalized, without significant change in the correction level, on or off axis (in order to minimize ensuing spherical aberration, R4 is slightly weakened, and lens spacing slightly widened). Such modification is known as Baker doublet.

Another coma-free doublet with reversed order (flint in front) is the Steinheil, which requires significantly more strongly curved surfaces (R1~0.43ƒ, R2~-0.224ƒ, R3~-0.223ƒ and R4~-ƒ for F2/BK7 glasses).

Two other doublet achromat types of mostly historical significance are the Littrow, requiring even more strongly curved surfaces than the Steinheil, with more coma than comparable paraboloid, and the Clark, with somewhat less coma than the Littrow, but more lateral chromatism. Another older doublet type is the Cooke, which consists of the biconvex front and biconcave rear element; it has more than double the coma of Littrow, while no advantage of easier fabrication.

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 0.5ฐ off-axis (e-line) and best image curvature radius.


FIGURE 146: Doublet achromat objective types for the common crown/flint combination, 100mm f/10 examples. The standard choice is Fraunhofer doublet, an aplanat consisting from biconvex front lens followed by a negative meniscus, with the secondary spectrum, defined as the longitudinal separation between the red/blue (C/F) focus and green (e-line) focus, of about 0.00055ƒ (about 0.0005ƒ measured from d-line). The only remaining monochromatic aberration is negligible astigmatism; best image curvature is about -0.36ƒ (concave toward the objective). Its inverse form, with the flint element in front, known as the Steinheil, is nearly identical in performance, but requires more strongly curved lenses for given focal length. Another variation, with the inner radii equalized, and somewhat wider spacing to compensate for the spherical aberration it induced, is known as the Baker doublet and does not differ in its performance level from the other two. Unlike them, it is not strictly a contact doublet, but the air gap is relatively small. The original Littrow objective consists from the equconvex front lens followed by the negative lens of the same inner radius and flat last surface. With three identical radii and a flat surface, it is the easiest to fabricate, but at a price of some residual coma and spherical aberration. Since the astigmatism is nearly identical in all doublets and near negligible, nearly all of the 0.133 wave RMS at 0.5ฐ off axis is coma, which makes it more than twice greater than in a comparable paraboloid (since the size of linear quality field with respect to coma changes with the third power of mirror's F-number, it makes this Littrow's coma comparable to that in a paraboloid with 30% larger ƒ-ratio (ƒ/7.7 for ƒ/10 lens). Not a concern in visual observing, and neither is the residual spherical, which is here little over 1/12 wave P-V. The air gap is reduced to zero, in order to keep this residual spherical at its minimum. With larger lenses, the latter may become significant, in which case one of the two inner radii may need to be slightly changed in order to have it minimized or cancelled. Note that the last surface in a common crown/flint combination cannot be flat; it is mildly convex, since the -ƒ1/ƒ2=V2/V1 achromatic condition (Eq. 42) for the Littrow implies V2=(n2-1)V1/2(n1-1), where n, V and 1, 2 refer to the refractive index, Abbe number, front and rear lens, respectively, hence such lens configuration requires specific glass match (the only nearly matching flint for the flat-rear Littrow using BK7 is F15). A Littrow modification by Clark&Sons, known as Clark doublet, has the two lenses more widely separated (~15% of the focal length, according to Sidgwick), in order to make the inner surfaces accessible for cleaning without taking lenses out. Since widening the separation effectively weakens the rear relative to front lens, with the former in the Littrow already slightly too weak for cancelling spherical aberration, the Clark requires somewhat stronger (relative to R1 and R2) third surface (like the Littrow, the last surface can be flat only with a specific glass match; for the common crown/flint combinations it is mildly convex). Clark's coma is somewhat lower than Littrow's, but its lateral color is, due to the wider lens separation, significantly larger, although still acceptable (less than 1/4 of the Airy disc diameter between F and C lines at 0.5ฐ off axis for the above system). Also, best image surface is somewhat more strongly curved, indicating a bit stronger astigmatism: the respective Zernike coefficients for primary coma and astigmatism at 0.5ฐof 0.252 and 0.107 indicate 0.252/√8=0.088 and 0.107/√6= 0.044 wave RMS error, thus astigmatism about half as strong as coma, which is about a third lower than in the Littrow. Clark configuration can be coma-free with the first radius made stronger than the second one, and also stronger fourth radius. Finally, the alternative aplanatic doublet solution, consisting of a positive and negative meniscus, is the Gauss doublet. Since a meniscus requires significantly more strongly curved surfaces to achieve given power, this objective type is, in addition to be more difficult for fabrication, unsuitable for all but long focal ratio instruments. As raytrace shows, higher order spherical originating at the strongly curved surfaces - particularly R4 - makes it unacceptable already at 100mm ƒ/10, with 0.057 wave RMS design limit in the e-line, and more than a third larger chromatic error due to spherochromatism (residual coma noticeable at the off-axis wavefront map cannot be further minimized without strengthening the radii even more, which would double the spherical error). Vice versa, accepting significantly more coma would allow for the weaker radii and reducing higher-order spherical to insignificant, but it would be still inferior to the other doublets, both, for more difficult fabrication and lower field quality. The positive is somewhat lower astigmatism - about 1/3 of the Clark objective, and nearly half of the other doublets' - and correspondingly less strongly curved best image surface.
 

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 ƒ/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.

TABLE 11:  ACHROMAT DOUBLETS: GENERAL SPECIFICATIONS FOR BK7/F2 GLASS (in units of focal length)
Type aplanat R1 t1 M1 R2 AIR (M2) R3 t3 M3 R4
FRAUNHOFER YES 0.6 0.011 BK7 -0.36 0.001 -0.363 0.007 F2 -1.51
STEINHEIL YES 0.442 0.007 F2 0.229 0.001 0.2285 0.009 BK7 -15
BAKER YES 0.582 0.011 BK7 -0.363 0.003 -0.363 0.007 F2 -1.59
LITTROW NO 0.45 0.01 BK7 -0.45 0.00005 -0.45 0.006 F2 -7.7
CLARK NO 0.428 0.01 BK7 -0.428 0.015 -0.4 0.0065 F2 -8
GAUSS (~ƒ/14 and slower) YES 0.129 0.00665 BK7 0.3345 0.0003 0.14206 0.0042 F2 0.10357
COOKE NO 0.373 0.009 BK7 -0.563 0.001 -0.527 0.006 F2 4.7

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 dialyte objectives, defined as those employing widely separated elements.

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.

Petzval-type achromat is a design that uses two groups of lenses, with the rear group at 1/3 to 2/3 of the focal length of the front group behind (approximately; in the original Petzval configuration rear doublet is at half the focal length of the front doublet apart, its focal length half that of the front doublet, and the combined f.l. also half that of the front doublet). If using common crown and flint glasses, such arrangement can reduce secondary spectrum by approximately 15%; apparently, generally not considered worth the extra expense. With the second lens group closer to the focus of the front group, which is not Petzval configuration, rather one with sub-aperture corrector, secondary spectrum can be reduced somewhat more, up to about 30%.

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 (FIG. 147C).


FIGURE 147: Standard Fraunhofer doublet achromat can be combined with an auxiliary widely separated second doublet for enhanced correction (A) 100mm ƒ/10 objective alone, for comparison (B) Petzval-type achromat with a cemented rear doublet. The front lens is an ƒ/10 doublet. The only significant change is that the front doublet is bent to generate coma cancelling the coma of rear doublet, which in turn cancels out astigmatism. As a result field curvature equals the focal length, and there is no other monochromatic aberrations to speak of. Since the astigmatism is cancelled, best image surface coincides with the Petzval surface. (C) Another ƒ/10 doublet in front, with a doublet sub-aperture (field) air spaced corrector. Again, the main modification is bending the front doublet to induce some coma, offsetting that of the corrector which, in turn, flattens the field w/o introducing significant astigmatism. In fact, astigmatism is lower than in a comparable achromat (D). Chromatism is also lower in both double achromats (somewhat more in the one with field corrector), without appreciably larger lateral color (note that the Airy disc for
e-line spot at 0.5ฐ off-axis is enlarged for clarity).

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 semi-apo and apochromatic (apo). Following page gives several examples, including some that were, or still are  marketed.
 

◄ 9. REFRACTING TELESCOPES   ▐    9.2. Refracting telescope objectives: Apo and semi-apo ►

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