9.1 REFRACTING TELESCOPE OBJECTIVE: DOUBLET 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 R1~0.61f, R2~-0.35f, with the focal length of about 0.44f, and R3~-0.36f and R4~-1.48f, with the focal length of about 0.78f, R1-4 being the lens surface radii from the front to the rear, and f being the final system focal length (alternatively, the curvatures can be expressed in the inverse form, f/Ri, as c1~1.64, c2~-2.86, c3~-2.78 and c4~-0.68). This also approximates the lens element focal lengths as f1~0.43f for the crown and f2~-0.76f for the flint.
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, 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 (R1~0.43f, R2~-0.224f, R3~-0.223f and R4~-f 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 0.5° 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 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).
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, but before that a quick look on the effect of stopping down a fast achromat.
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.