In addition to the effect of secondary spectrum, this section will give more details on the effects of monochromatic aberrations and spherochromatism on the refracting telescope performance. There are many types and variations of refractors, but the consideration here is limited to the general case of a contact doublet (FIG. 141), or triplet objective, either achromat or apochromat.
FIGURE 141: 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.
For over a century since the first recorded refracting telescope with a singlet lens was invented, the only significant advancement to it was that by Johannes Kepler, who just a few years later, in 1611, explained its working principles, and proposed significantly better configuration, replacing the negative eye lens of the Galilean refractor with a positive one. In order to minimize chromatism of a single-lens objective, refractors of this type grew in length well beyond practical limits, but their aperture remained quite limited: the 57mm 11-foot lens with which Christian Huygens discovered Titan in 1655 was comparable to a 100mm ƒ/3.5 achromat chromatism-wise. His largest, 8-inch 210-foot lens had chromatism comparable to that in today's 100 ƒ/6 standard achromat (these are two different forms of longitudinal chromatism, primary vs. secondary spectrum, respectively, but the comparison is possible via polychromatic Strehl). Similar instruments used by Cassini, Hevelius, and others have the same limitations, and were very difficult to use.
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.
Details of the secondary spectrum and field curvature of a doublet are given elsewhere. Doublet's monochromatic aberrations are a sum of the aberrations of each lens. They are commonly expressed in terms of lens shape factor q and position factor p. Since the two lens elements of a doublet can be considered a thin lens (i.e. one whose thickness is negligible with respect to object and image distance), at least in first approximation, aberration coefficients for primary spherical aberration and coma for either doublet lens are given by Eq. 8 and Eq. 14.1, respectively; once the coefficients are known, the corresponding wavefront error is found from Eq. 7 and Eq. 12, in the same order.
The same principle applies to the triplet objective.
Graph below (FIG. 142) illustrates the relation between these two aberrations, lens shape and object distance, for the lens' refractive index n=1.5 (common variations in the index value do not affect form of the graph, which remains parabolic, only cause relatively small changes in its vertical position). Minimum spherical aberration for any given lens shape q varies with the position factor p (relative object distance); for n=1.5, spherical aberration is at the minimum when qmin=-(5/7)p, with the corresponding aberration coefficient s=[9-(3p2/7)]/32ƒ3 dropping to zero for |p|=√21. Thin lens is coma free when q=-0.8p, which means that a coma-free thin lens corrected for infinity (p=-1) will also have near minimum spherical aberration.
That would apply to the front lens of a doublet; for the rear lens, the object is the image formed by the front lens, so its p value - and the coma-free shape factor q - varies depending on whether the lens is positive (p~0) or negative (p>1). Correspondingly, q~0 for cancelled coma and minimum spherical aberration for the positive rear lens, and q<-0.8 or q<-(5/7) for the negative. For the usual negative rear lens, with the front lens' image somewhat inside its focus (thus, according to FIG. 69, with the position factor p somewhat larger than 1), the corresponding zero coma (and near minimum spherical aberration) shape factor q is roughly somewhat over -2.
A q~0.8 shape factor makes the front lens biconvex, with the strong first radius and weak second. The q<-2 makes the rear lens a negative meniscus, convex toward the front lens. Coma of such a doublet is negligible, but spherical aberration is still very large. Most of it comes from the front lens which, despite being near its possible minimum, is still extraordinarily high (for instance, in a 100mm ƒ/10 doublet like this, it would be near 18 waves P-V). For that reason, the approach in designing a doublet is different; in principle, the two lenses are shaped up so that their spherical aberration and coma (for an aplanat), very significant at each element separately, nearly cancel out when combined.
Astigmatism of a thin lens, with the coefficient given by a=n/2ƒ, is only dependant on the lens' focal length. According to Eq. 23, the P-V wavefront error of astigmatism for a contact doublet 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.
Doublet also can be affected by a specific form of chromatism at a refractive surface, resulting from the increase in spherical aberration toward non-optimized wavelengths: spherochromatism. The P-V wavefront error of spherochromatism of a contact doublet can be obtained from Eq. 48-50 (note that it is the lower-order component of spherochromatism, or primary spherochromatism; it can be accompanied with the higher-order component if higher-order spherical is significant). 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 the best focus for a system optimized for the spectral e-line, using BK7 crown and F2 flint glass, is given as:
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.
Eq. 96 implies that the primary 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 for 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. FIG. 143 illustrates the magnitude of defocus in the achromat vs. defocus and spherochromatism in an apo doublet.
Spherochromatism in lens objective, sometimes called tertiary spectrum, can be more serious problem with apochromats. It becomes the dominant form of chromatic aberration, increasing exponentially with the 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. 144). 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. 144 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 aberration dominant in the shorter wavelengths; for the optimized wavelength, the lower and higher-order aberration are nearly optimally balanced).
roughly, defocus equaling 1/4 of the longitudinal aberration from best focus location for spherical aberration doubles the error. If it is, as mentioned, roughly 0.1 wave for the red and blue lines in the apo, and their focus is at about 1/4 of their entire respective longitudinal aberration from best focus, the F/C 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 wavelengths 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 the result of more strongly curved lens surfaces; it is much more of a problem with the doublet than with triplet objectives, and the primary source of higher-order spherical in the former are the two strongly curved inner surfaces. This simply means that the 4th order surface approximation (primary spherical) is not sufficiently accurate anymore, and the 6th order surface term (secondary spherical) has to be included. This can be done by tweaking surfaces to re-introduce an 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. When the two are optimally balanced, which requires similar amounts of the two aberration forms of opposite sign, the higher-order aberration, as the RMS wavefront error, is reduced nearly six fold.
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. 145). Note that the patterns are for moderately bright telescopic star (i.e. the first dark ring in aberration-free aperture is clearly visible, but noticeably fainter than central maxima); pattern appearance will change with significantly brighter or fainter stars.
The actual lens objectives may have their higher-order spherical aberration (HSA) balanced with lower-order spherical (LSA) - or balanced higher-order spherical (BHSA) - in the optimized wavelength well below the "diffraction-limited" level, but as little as λ/50 wave RMS, is still detectable in the star test. For given P-V wavefront error, the BHSA 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, noticeably different intra- and extra- focal patterns in an apochromatic objective do not necessarily indicate compromised performance level, and it generally indicates lower error than in the case when similar apparent difference is caused by 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.