9. LENS OBJECTIVE: ACHROMAT and APOCHROMAT

In addition to the effect of secondary spectrum, this section will give more details on the effects of astigmatism and spherochromatism on the refracting telescope performance. There are many types and variations of refractors, but the consideration here is limited to a contact doublet objective, either achromat or apochromat (FIG. 95).

FIGURE 95: Refractor doublet objective, contact type, in its standard form with positive front lens and negative rear lens. The final image is formed by the negative rear lens, which is in effect re-imaging the image formed by the front lens to the final focus. The fictitious image formed by the front lens at F1 can be considered to be the object for the rear lens, and the final image is real as long as the front lens has more power; the top point of the (reversed) final image at F is formed at the intersection of the chief ray, which can be approximated as passing through the doublet's center point, and a ray extended from the focus of the negative rear lens F2 (when the two lenses are of identical opposite powers, the two are nearly parallel, and the image forms at infinity). Depending on glass types used for the objective, chromatic correction of a lens doublet can be anywhere from an ordinary achromat level to that of an apochromat. Astigmatism and spherochromatism are negligible in long-focus systems, but may become significant at larger relative apertures.

The idea that two lenses of different glass types can help break through severe limitations imposed by chromatism came from Chester David Hall in 1730s, but it wasn't until John Dolland had it patented in 1758, that doublet achromat came to life. A number of doublet types have been developed, with various levels of correction of chromatism and monochromatic aberrations.

A long time standard for doublet achromats is the Fraunhofer doublet. It is relatively easy to make, free from coma and 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, R3~-0.36ƒ and R4~-1.48ƒ, 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.

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 four time as much coma as comparable paraboloid, and the Clark, with half as much coma as the Littrow, but more lateral chromatism.

Details of the secondary spectrum and field curvature of a doublet are given earlier. What is left to address are its astigmatism and spherochromatism. According to Eq. 23, the P-V wavefront error of astigmatism for a contact doublet, in general, is given by Wa= [(1/ƒ1)+(1/ƒ2)](αD)2/8, with α being the field angle, ƒ1 and ƒ2 the front and rear lens focal length, respectively and D the aperture diameter. Since (1/ƒ1)+(1/ƒ2)=1/ƒ, ƒ being the combined doublet focal length, the P-V wavefront error of lower-order astigmatism simplifies to:

Wa=Dα2/8F or Wa=h2/8fF2 (95)

with h=αƒ being the linear height in the image plane, and α the field angle in radians. For a common 100mm ƒ/12 doublet 1-degree off axis, it gives Wa=0.00032, or 0.58 wave for λ=0.00055mm. With the RMS error for best focus astigmatism being smaller than the P-V error by a factor of √24, that gives the RMS wavefront error of 0.118, in units of the wavelength. This amount of astigmatism is acceptable that far off-axis, however, since the error increases in proportion to the aperture and ƒ-ratio, it may become significant with large fast systems.

The P-V wavefront error of spherochromatism of a contact doublet can be obtained from Eq. 48-50. It can be simplified by substituting typical values for the position factor p and the shape factor q for the Fraunhofer doublet: for the front and rear lens, respectively, p1~-1, q1~0.26 and p2~2.5, q2~1.65 (p values are for the object at infinity). With this in place, the P-V wavefront error of primary spherochromatism at best focus for a system optimized for the spectral e-line, using BK7 crown and F2 flint glass, is given as:

Ws=(s1+s2)D4/2048 (96)

with s1 and s2 being aberration coefficients for the front and rear lens, respectively, given by:

with n1, n2 and ƒ1, ƒ2 being the refractive indici and focal lengths of the front and rear lens, respectively. For direct calculation, the individual focal lengths can be replaced by their system focal length ƒ relative equivalent, as ƒ1=0.432f and ƒ2=-0.761ƒ. For accurate results, the radii need to be exact within a tiny fraction of mm. In practice, this level of surface radius accuracy is not necessarily met.

It is immediately apparent that the spherochromatism wavefront error changes in proportion to the fourth power of the aperture, and in inverse proportion to the third power of the system focal length. Or, in proportion to the aperture size, and inverse proportion to the cube of F-number. However, same as with secondary spectrum - or spectral defocus - the energy spread caused by chromatic spherical aberration is a subject to a final "validation" by the eye and its spectral response. Both aberrations are present in an achromat, and their combined effect is cumulative, although the effect of spherochromatism remains negligibly low in long-focus systems. In order to begin to significantly contribute to the chromatic defocus error, the spherochromatism nominal wavefront error needs to become a significant fraction of the nominal wavefront error of chromatic defocus - roughly, about half as large, or larger.

Taking longitudinal aberration of the red/blue foci in an achromat to be ƒ/2000, the P-V wavefront error of defocus is, from Eq. 51, given by D/16,000F. For a 4" ƒ/12 achromat, that gives about 1 wave P-V error of the F/C chromatic defocus; for a 4" ƒ/6 twice as much. At the same time, the nominal wavefront error of spherochromatism, given by Eq. 96-96.2, is ~1/20 wave P-V at ƒ/12 and comparatively (to the chromatic defocus) still relatively insignificant with ~0.4 wave P-V at ƒ/6.

Spherochromatism, sometimes called tertiary spectrum, can be more serious problem with apochromats. It becomes the dominant form of chromatic aberration, generally increasing with relative aperture. Apochromats, and particularly doublets, usually combine glasses with more closely matched dispersive powers than achromats; as a consequence, lens surfaces (typically the inner two radii) have to be significantly more curved in order to bring color foci together, which in turn increases higher-order spherical aberration (FIG. 96). Fast, larger apochromats can face the prospect of significant amounts of spherochromatism, both primary and of higher orders (L.A. curves of the achromat in FIG. 96 are nearly entirely primary spherochromatism, while those of the apochromat, flattened for mid-zones and more strongly curved toward the edge, reveal a dominant higher-order aberration in the longer wavelengths, with lower-order spherical dominant in the shorter wavelengths; for the optimized wavelength, the lower and higher-order aberration are nearly optimally balanced).

FIGURE 96: Secondary spectrum and spherochromatism in a 4" ƒ/15 Fraunhofer doublet achromat (left) and a more or less typical 4" ƒ/9.7 apochromat refractor (right). TOP LEFT: Grossly exaggerated refraction of e (green), C (red) and F (blue) lines in the achromat and apochromat (the actual ray height at the last apo surface is only ~1.3mm smaller than at the front, and the blue-to-red separation at the 4th surface is little over 0.005mm). The wavelengths actually remain much tighter in the achromat's objective, but what matters is their height vs. angle when they exit the rear lens: it determines how close they focus in the image plane. RIGHT: Longitudinal (top) and transverse (bottom) aberration in the two systems. Best foci - marked by small squares - for the achromat are at 70% radius. For the blue F and red C line they are close to each other, but roughly 0.8mm - the nominal secondary spectrum - far from the optimum green focus. Longitudinal C/F spherochromatism in the achromat is approximately 0.14mm, which is ~1/50 wave P-V level. In comparison, the apochromat has its best green, red and blue foci placed much tighter in their orthogonal projection onto the axis: approximately 1/10 of the achromat's (the apo plot scale is larger by nearly four times). At the same time, its spherochromatism is greater than that of the achromat: roughly 1/10 wave P-V the red/blue average. The g violet line (436nm) in the achromat focuses more than three times farther from the e-line (to the right) than the red/blue lines. Note that best foci for lower-order spherical - characterized by the L.A. plot bending steadily to one direction toward the edge - coincides with the 70% zone focus. Higher-order spherical L.A. plot also bends in one direction, but is noticeably flatter centrally, and curving more strongly toward the edge. Best focus here coincides with ~75% zone focus. For mixed lower and higher order spherical of opposite signs, the LA plot at some point reverses and bends in the opposite direction. Best focus location for this form of aberration varies with the proportion of the two aberration forms; for a desirable, nearly balanced form (the green wavelength in the apo plot), best focus coincides with 87% and 50% zone focus. The F/C defocus in the apo is nearly 7 times smaller than in the achromat, but the error here is not defocus, rather defocused spherical aberration; roughly, defocus equaling 1/4 of the longitudinal aberration from best focus for spherical aberration doubles the best focus error. If it is, as mentioned, roughly 0.1 wave for the red and blue lines in the apo, their error at the optimized focus is nearly 0.2 wave P-V, or only about four times less than in the achromat. However, farther wavelengths remain nearly as close in the apo as red and blue, providing much better chromatic correction in the expanded range. BOTTOM: Strehl as a function of the wavelength for the two refractors. While the achromat has a slight edge over the wavelength of peak eye sensitivity, chromatic correction toward both ends of the visual spectrum is dramatically superior in the apochromat. This, however, has much smaller effect on visual observing than what it appears. The reason is that eye sensitivity also dramatically decreases toward those wavelengths that are far from the achromat's best focus, effectively offsetting better part of its defocus error. Since the sensitivity plots are also normalized to 1, the effective Strehl S is approximated by 1-(1-S)E, where E is the normalized eye sensitivity for given wavelength. For instance, Strehl at the 495nm wavelength is about 0.25 (S=0.25) for the achromat, nearly equal nominally to the corresponding photopic eye sensitivity (E=0.25). Therefore, the effective Strehl is approximately 0.81. It is also higher in the apo, but the difference here is much less significant: 0.92 nominal vs. 0.98 effective visual Strehl (i.e. 6.5% contrast improvement, vs. 324% improvement in the achromat). SPEC'S

The emergence of higher-order spherical aberration in apochromats is a result of strongly curved lens surfaces. This simply means that the 4th order surface approximation is not sufficiently accurate anymore, and the 6th order surface term has to be included. Formally, this can be done by tweaking the 4th order term to produce optimal amount of the 4th order aberration of opposite sign, in order to minimize the total aberration. Hence, both form of the aberration are present to some degree, reshaping the wavefront into some intermediate shape. This changed aberration form produces different star test pattern. In general, the difference between inside and outside focus patterns is noticeably more pronounced at a given error level, possibly indicating insufficient degree of correction even if the actual error is quite small (FIG. 97).

FIGURE 97: Star test patterns of lower-order spherical aberration (LSA) alone, and combined with higher-order spherical aberration (HSA), for ~4λ of defocus on both sides (focused patterns are magnified by ~2x). Actual instruments should have balanced LSA and HSA (LHSA) - in the optimized wavelength well below 0.1λ wave RMS, but often more than λ/50 wave RMS, which is still detectable. For given P-V wavefront error, the LHSA has nearly 40% smaller RMS wavefront error than LSA. The reason is that, as shown on the inset to the left, the deformation it causes affects larger, outer wavefront area less. But, due to more strongly curved edge area, it does comparably more damage to defocused patterns. Thus, with the level of LHSA in most apochromatic objectives being low, noticeably different intra- and extra- focal patterns do not indicate as much compromised performance level, as with LSA alone. However, the notion that these instruments don't need to have perfect star test in order to have a perfect correction level is, of course, unfounded. The key for the proper test interpretation is in determining how well balanced are the two aberration forms. HSA alone has both, wavefront form and pattern appearance similar to that of the LSA, so the more unbalanced it is, the more alike LSA the star pattern.

In other words, damage to the pattern appearance can be significantly greater than damage to the instrument's performance. Some other telescope types, notably Maksutov-Cassegrain, also need to have this factor taken into account when interpreting defocused patterns, due to relatively significant presence of higher-order spherical aberration.

◄ 8.4.3. Off-axis Newtonian ▐ 9.1. Designing doublet achromat ►

Home | Comments