10.2.2.4.2. SCT focusing errors
Focusing error in a Schmidt-Cassegrain telescope results from deviations from the optimum mirror separation in order to: (1) shift focus location to accommodate use of accessories, or even different eyepieces or (2) keep focus location nearly fixed while focusing on close objects. Both are an issue with SCT systems that use mirror focusing - typical for a commercial SCT. For comparison, error induced while observing close object will be included for an SCT with conventional focusing (fixed mirrors) as well.
Strictly talking, commercial SCT using primary mirror focusing are misaligned every time the two mirrors are not at a single optimum separation. It is interesting to determine how it affects wavefront quality. For any change ∆ in mirror separation, the final focus shifts axially by -∆m2/[1-∆(m-1)/g], m being the secondary magnification at the optimum separation, and g the secondary mirror to primary focus separation, negative in sign for the Cassegrain (note that the mirror shift ∆ is negative when reducing the primary-to-secondary separation which is, according to the sign convention, positive in sign). For ∆ up to ~1mm, good approximation of the resulting focus shift is given by -∆m2. In general, reduction in mirror separation results in a positive focus shift (away from the primary), while the increase in mirror separation shifts the focus closer to primary mirror.
Depending on the accessories used, required shift from the optimum focus can be more or less than that. Since moving the primary only changes the aberration contribution of the secondary, it is the error differential at the secondary determining the resulting system error. The resulting system error is, in effect, the despace error for two mirror system. For spherical secondary (K2=0) it is from Eq. 91.3 given as:
as the P-V wavefront error at best focus, where m is the secondary magnification and F1 the primary focal ratio ƒ1/D.
The relation implies that, for nominal primary mirror shift ∆, error induced by extending back focus is independent of aperture. It only depends on secondary magnification and primary's focal ratio. For F1=2, m=5, and K2=0, every mm of reduction in mirror separation (∆=1), or nearly for every inch of focus extension, induces ~1/23 wave P-V wavefront error of over-correction, and as much of under-correction for widening the separation. Aspherizing the secondary would result in the error of similar magnitude, only the sign of aberration would reverse: focus extension (i.e. reducing mirror separation) would induce under-correction.
This level of error remains nearly unchanged with the change in aperture. This is due to any given mirror shift being producing identical nominal change of the height of marginal ray at the secondary. In other words, the relative ray height change is proportionally smaller for larger aperture. Since the aberration contribution of the secondary is in proportion to the fourth power of its effective aperture (Eq. 78.1), the relative change in its aberration contribution is approximated by 1+(4∆'/D), ∆' being the change in marginal ray height, ∆'=∆/2F1. Hence, the relative increase in secondary mirror contribution is proportionally smaller for larger aperture, but since its contribution is also proportionally greater, the final nominal change in its aberration contribution remains at the nearly same level.
Spacing error also induces coma. For reduced mirror separation, it adds to the system coma, but not significantly (less than 10% for 5 inch focus extension).
However, this only applies to the change in mirror separation to accommodate the use of various accessories, while the object of observing remains at a large distance. In addition to the change in mirror separation, focusing on relatively close objects has to deal with optical consequences of the change in shape of the incident wavefront (which is now convex spherical) as well. Follows more detailed consideration of how it affects performance level of a typical commercial SCT.
Observing close terrestrial objects is not what telescopes are made for. Any optical system can give optimum performance only within a limited object distance range or, in terms of geometric properties of incident light, for a certain degree of curvature of the incident wavefront, determining geometry of rays projected from it. Telescopes are optimized for distant objects observing, hence for a near-flat incident wavefront, or bundles of near-collimated (parallel) rays. The smaller object distance, the more convex-spherical shape of the wavefront, and the more diverging the rays. This change in light geometry, without an adequate change in properties of the optical surfaces of telescope objective, results in the wavefront it forms being different from spherical - the closer object, the more so.
In an SCT optimized for infinity, near-zero level of spherical aberration is achieved by balancing aberration contributions of its three optical elements, Schmidt corrector, primary and secondary mirror, for near-flat incoming wavefront. As the shape of the incident wavefront changes, so do the individual aberration contributions of these three elements. The problem is that they do not change in proportion that would preserve the near-zero balance. Instead, certain amount of aberration is generated, increasing exponentially with the reduction in object distance.
Typically, the most significant aberration generated while observing close object with telescopes is spherical. The same applies to the SCT. Change in its off-axis aberrations is insignificant.
Spherical incoming wavefront affects the three SCT optical components differently. Due to its low power, changes to the wavefront at the corrector are small enough to be neglected. Not so with the primary mirror. New wavefront/ray geometry result in larger effective mirror diameter and extension of its converging cone. These, in turn, increase the width of converging cone at the secondary, extending its converging cone significantly more. Most importantly, the aberration contributions to the wavefront at the primary and secondary change in different proportions, resulting in the aberrated final wavefront (FIG. 177).
In order to establish more specifically the amount of spherical aberration generated in an SCT used for close object observing, we need to track down changes in the aberration contribution for each of its three optical elements. While the change in contribution of the corrector and primary mirror is independent of the type of focusing, the aberration at the secondary mirror will be different in SCT systems that use mirror focusing (the common commercial type) from that generated by systems that use conventional focusing via focuser tube.
If the two mirrors are stationary, the amount of aberration induced by close object observing is practically limited by the available focuser travel. Since closer objects are imaged farther back, at a certain point the image simply runs out of a focuser reach. The limit to object distance set by given focuser travel length can be approximated by treating the SCT as a two mirror system with the aperture stop at the corrector location. The effect of the corrector plate on ray geometry is minor, as described in 11.5. Schmidt-Cassegrain telescope.
For given mirror separation s, back focus distance - the separation from mirror surface to the final focus - in a two-mirror system is given by B=s+(ƒ1-s)ƒ2/(ƒ1-s-ƒ2), with ƒ1 and ƒ2 being the primary and secondary mirror focal length, respectively (note that, according to the sign convention, all three parameters are negative). For the typical commercial SCT configuration, with s~0.75ƒ1 and (ƒ/ƒ1)~5, the back focal distance is well approximated by B~0.23(0.33+ψ)ƒ1/(0.06-ψ), with ψ=ƒ1/O being the the reciprocal of the object distance O in units of the primary's focal length. For an 8" ƒ/10 SCT with f1~400mm, the focus shift between ψ=0 (object at infinity) and ψ=0.01 (object at 100ƒ1, or 40m) is 0.3ƒ1, or 120mm - most likely already out of the focuser range.
For ψ~0.06, back focus distance approaches infinity (the secondary forms collimated beam of light), and for ψ>0.06 back focus distance has negative value (secondary forms diverging beam of light).
To determine the level of spherical aberration induced to a commercial SCT at this and other object distances, it is necessary to determine changes in aberration contributions of its three optical elements. For a typical commercial SCT configuration with spherical mirrors, the system wavefront error, as the P-V error of lower-order spherical aberration at best focus, can be approximated from Eq. 113.2, by
with P, 1 and S being the proportional contributions of the corrector, primary and secondary mirror, respectively, and aperture diameter D in mm. The sum of contributions - under-correction by the primary, over-correction by the corrector and secondary - is near zero in a well corrected system (the right side of the equation). Relatively small deviation from perfect figure on any of the three optical elements can be cancelled out by matching deviations on other two elements, but it is probably as hard to achieve high correction level doing this as it is to produce three near-perfect matching surfaces. For an ƒ/2/10 200mm system, a sum in the brackets of 0.011 would result in ~0.25 wave of spherical aberration, over-correction if positive, under-correction if negative.
Follows simplified analysis of the change in these contributions for close objects. Its primary goal is to illustrate the mechanism of change, but it should also give useful quantitative approximation of the effect.
As already mentioned, change at the corrector is small enough to be neglected. The amount of spherical aberration here changes due to the spherical wavefronts coming from close objects being modified by the Schmidt surface differently than nearly flat wavefronts from distant object that it is optimized for. As the result of diverging (convex) wavefront from close objects, the in-glass path for the 0.707 neutral zone Schmidt surface profile increases more for wavefront points toward corrector's edge. Consequently, these points are delayed more, increasing the P-V wavefront error of the pre-corrected wavefront. The extent of the change is, however, small. The effective corrector's depth profile is changed from z(ρ) in Eq. 101 to z(ρ'), with ρ' being closely approximated by ρ'~ρtan(β/n). That gives the maximum additional retardation - at the marginal ray - with the object distance of 25 primary's focal lengths (10m), 200mm corrector (n=1.5), ƒ/2 primary and corresponding zonal ray incident angle β=0.57 degrees, of about 1/20 wave. According to OSLO, corrector's best focus error is still smaller, probably the result of refocusing. In any instance, the error generated at the corrector can be neglected; the errors at the primary and secondary mirror are much more significant.
For the mirrors, we see from Eq. 9-9.2 that spherical aberration changes in proportion to (m+1)/(m-1) squared, and in proportion to the fourth power of the diameter. At the primary, for the object distance o in units of the primary's focal length ƒ1, o=O/ƒ1=1/ψ, with O being the object distance, object magnification changes as m1=ψ/(ψ-1), and the aberration coefficient changes with (1-2ψ)2. Since the effective aperture of the primary, due to displaced stop, changes in proportion to (1+2σψ), σ being the corrector-to-primary separation in units of the primary's radius of curvature, the combined change in its contribution due to these two factors is in proportion to (1-2ψ)2(1+2σψ)4.
For the secondary, magnification is given by m2=ƒ2/(ƒ2-ƒ1+s), with ƒ1 being smaller than the sum of ƒ2 and mirror separation s for the converging final cone (recall that all three parameters are negative values). Hence, the effective increase in ƒ1 with close objects results in increased secondary magnification. In a typical SCT, s:ƒ2:ƒ1~0.75:0.31:1, and the secondary magnification can be approximated by m2~0.3(1-ψ)/(0.06-ψ), giving (m2+1)/(m2-1)~(0.36-1.3ψ)/(0.24+0.7ψ). The square of relative change in (m2+1)/(m2-1) - and the aberration contribution due to the change in magnification - is then given by [(1-3.6ψ)/(1+2.9ψ)]2. With the ratio of change in the effective secondary diameter (determined by the width of the axial cone at its surface) given by 1+(2σ-1+1/k)ψ, with k being the relative height of marginal ray at the secondary for object at infinity, the secondary mirror aberration contribution to the lower-order spherical aberration of the system is now changed approximately in proportion to [(0.36-1.3ψ)/(0.24+0.7ψ)]2[1+(2σ-1+1/k)ψ]4.
With the effect of changes in D and F being accounted for within the brackets, the P-V error of lower-order spherical aberration at best focus as a function of the object distance in an SCT with fixed mirrors (i.e. standard focuser) can be expressed as a lengthy but simple approximation:
with the aberration contributions from the corrector, primary and secondary mirrors given in green, blue and red, respectively. For ψ=0.01, or object at 100 primary mirror's focal lengths away, D=200mm, F1=2, σ=0.4R1 and k=0.25, it gives the error as W~(0.71-0.9915+0.2955)D/2048F13~0.000171mm, or 0.31 wave of over-correction in units of 550nm wavelength. This is nearly identical to what OSLO gives for such a system; however, there is a small discrepancy, since OSLO includes small higher-order aberration contribution, accounting for about 3% of the total aberration (the coefficient).
In the SCT with mirror focusing, mirror separation s is increased - usually by moving the primary - in order to lower secondary magnification back to its infinity level, and bring the focus for close objects back to its infinity location. Obviously, for this to occur, shift of the primary has to effectively place the secondary at nearly identical separation from the primary's image as with object at infinity. This indicates that the change in the effective magnification of the secondary resulting from the refocusing is small enough to be neglected in first approximation. The effective minimum secondary diameter in new configuration is slightly smaller, in proportion to the effective decrease in the focal ratio of the primary, given by (1+2σψ)/(1-ψ). Since it affects the aberration coefficient in the fourth power, it is significant enough to be included in calculation.
With this addition, the P-V error of lower-order
spherical aberration at best focus induced by reduced object distance
For the same object distance O of 100 primary's focal length (ψ=0.01), the error for the above system is now W~0.14 wave of over-correction. Again, practically identical to the 0.14 wave P-V error given by OSLO. Since the error is, as in the fixed mirror system, nearly entirely lower-order spherical, the corresponding RMS wavefront error is smaller by a factor of ~0.3.
Thus, correction error induced by close focusing is considerably smaller in SCT systems with mirror focusing (FIG. 178). Partial exceptions seem to be aplanatic systems, where the error with mirror focusing, while smaller for moderately close objects, grows at a faster rate with the reduction in object distance, exceeding the error with conventional focusing for very close objects.
Simple empirical formulas roughly approximating close focusing P-V wavefront error in units of 550nm wavelength for mirror focusing and fixed mirror SCT systems are:
W~D/1.8oF13 and W~D/oF13 (120.3)
respectively, with o being the object distance in units of the
primary mirror focal length (o=
Significantly larger error induced by close object observing puts conventional (fixed mirror) focusing SCT at a definite disadvantage for close objects observing. More so considering the extent of axial image shift, which could make this type of observing either difficult, or impossible.
SCT baffle system and extended focus
One of the conveniences of the SCT is the ability of focus extension, in order to accommodate use of various accessories. This, however, may have two negative effects: (1) increased vignetting of the outer field, and (2) stopping down effective aperture. Both result from tightly fit SCT baffles.
Extending SCT back focus by moving the primary closer to the secondary places somewhat wider converging cone at the secondary baffle; with the cone converging from the secondary being also extended, the relative widening of converging cone is even more pronounced at the front and rear opening of the baffle tube. At the front opening, it can result in effectively reduced aperture. Also, since angles of divergence toward off axis field points are larger for the cone converging from the secondary (due to secondary's magnification), vignetting at off axis field points is more pronounced at the baffle tube openings.
Scheme below is OSLO analysis of the consequences of focus extension in a Celestron C9.25 unit.
Specs for the baffle system are entered according to actual measurements, since they are not supplied by the manufacturer. At ƒ/10, focus location is nearly 230mm behind the primary (back focus), and about 100mm from the rear end of the baffle tube (A). For this focus location, converging cone tightly clears the front baffle tube opening, and the only significant vignetting within 0.5° field radius is for points close to field edge. Since the converging cone width at the front tube opening is nearly identical to the opening diameter, point illumination is given by I=(β/90)-c(sinβ)/π, with cosβ=c/2, where c is the relative overlap of the converging cone in units of the cone radius, and β is in degrees. At 0.5° off axis, the cone overlaps the opening by about 40% of its radius (c~0.4), giving illumination of about 0.75, i.e. 25% illumination drop, or 0.31 magnitudes loss (B).
With the focus extended by 108mm from its ƒ/10 location, the corresponding focal length is 2670mm. Converging cone at the front baffle tube opening is is wider than the opening, resulting in the aperture effectively reduced by about 7%, to ~220mm. Hence, the effective focal ratio is now ƒ/12.1.