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11.5.1. SCT close focusing

Close object observing 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 shape 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 near-collimated bundles of 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 formed by it 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 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. 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 in systems that use conventional focusing via focuser tube.

If the 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. It slightly increases the marginal ray height on the primary, as determined by its maximum refraction angle δ (Eq. 107), with this ray being then directed toward the new common focus, somewhat inside primary's paraxial focus.

With, say, a 200mm f/2 spherical primary, combined with a 0.707 neutral zone corrector at 0.4R in front of it, marginal ray on the primary will be only ~0.23mm higher (for ~0.72 corrector power in an all-spherical arrangement), focusing ~1.55mm inside the mirror's paraxial focus (mid point of its D(mm)/32F longitudinal spherical aberration). In effect, the marginal ray height at the secondary, placed at 3/4 primary's f.l. from the primary, is ~0.22mm smaller. In other words, the primary effectively becomes an f/1.988 mirror with the secondary-to-primary radius of curvature ratio ρ reduced from 0.3125 (in an f/10 mirrors-only system) to 0.3106, the relative ray height at the secondary in units of the aperture radius k reduced from 0.25 to 0.247, and the resulting secondary magnification m reduced to 4.88. If the two mirrors without corrector would form an f/10 system, optical effect of the corrector changes it into ~f/9.7.

However, starting with this configuration, influence of the corrector on the parameter change with varying object distance is negligible. So, simply assuming a two-mirror system with the stop at the corrector location (but without corrector) gives very close approximation of the change in system parameters with object distance.

Back focus distance η in units of the primary's f.l. in a two-mirror system is given by η=(m+1)k-1. The secondary magnification m=ρ/(ρ-k) changes in both parameters as object distance diminishes. The R2/R1 mirror radii ratio ρ effectively decreases with the primary cone extension, in proportion to (1-ψ), with ψ=ƒ1/o being the the reciprocal of the object distance in units of the primary's focal length. On the other hand, the ray height on both, primary and the secondary increases, the former by a factor 1/(1-2σψ), and the latter by 1-(1-1/k)ψ. Assigning to new parameters a prime mark, they are given as D'=D/(1-2σψ), ρ'=(1-ψ)ρ, k'=k+(1-k)ψ and m'=ρ'/(ρ'-k').

Since the back focal distance (primary to final focus separation) in a two mirror system is given by (m+1)k-1, the back focal distance for a close object is closely approximated by (m'+1)k'-1. Using typical 8" commercial SCT values, k~0.25, r~0.3125, m~5 and η=0.5, observing object at 100 times the primary focal length (ψ=0.01) will increase the cone width at the secondary to 0.2575D, secondary magnification to m'=5.96 and back focal distance to 0.79f1'. That places new focus at (0.79-0.5)=0.29f1' farther out. In this example, it amounts to nearly 120mm, likely already out of the focuser range. Note that the new effective focal length of the primary f1' is larger than its original focal length f1 by a factor of 1/(1-ψ).

At this point, the amount of spherical aberration induced is likely to be moderate. In order to find out more specifically how much it is, we need to examine the change of aberration contribution from each of the three SCT's optical elements. Contributions from the corrector and primary mirror do not depend on the focusing mode, while that of the secondary is different in conventional vs. mirror focusing.

At the corrector, the amount of spherical aberration will change for close objects due to their spherical wavefronts being modified differently by the Schmidt surface than flat wavefronts it is optimized for. As illustrated on FIG. 90, right, the in-glass path increases more for wavefront point 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 depth profile is changed from z(ρ) in Eq.101 to z(ρ'), with ρ' being approximated by the height at which the object ray intersects the Schmidt surface. Taking the 0.707 neutral zone Schmidt surface profile, the greatest change in pre-corrected wavefront is for the marginal ray. Even with an object as close as 25 primary's focal lengths, a 200mm corrector with an f/2 primary and corresponding zonal ray incident angle β=0.57 degrees will have the maximum increase in its wavefront pre-correction P-V error of  ~1/20 wave (for n=1.5).

The following is not a rigorous ray-tracing procedure, but does include all main factors, and gives useable results. For the reciprocal of the object distance in units of the primary's focal length ψ=ƒ1/o, the secondary mirror parameters ρ and k change into ρ'=(1-ψ)ρ and, for fixed focus location, k'=(1-ψ)k. This changes the effective secondary magnification to m'=ρ'/(ρ'-k'). Since the secondary aberration coefficient is (2ρ-k)2k2/ρ3, it now becomes (1-ψ)(2ρ-k)2k2/ρ3. In other words, secondary contribution changes with (1-ψ), as a result of the change in ρ and k. In addition, it also changes due to the change in the effective aperture diameter of the primary.

At the primary, aberration contribution changes as the combined effect of object distance and stop separation. Reduction in object distance extends primary's focus separation by a factor 1/(1-ψ), lowering its under-correction by a factor (1-2ψ)2. This is partly offset by the effect of displaced stop, effectively increasing mirror diameter (this assumes the primary slightly larger than the corrector aperture, which is usually the case) and its effective relative aperture by a factor (1+2σψ), σ being, as before, the primary-to-stop separation in units of the primary's radius of curvature (FIG. 90). In effect, the displaced stop increases primary's aberration contribution by a factor of (1+2σψ)4, setting the final primary's aberration contribution in proportion to (1-2ψ)2(1+2σψ)4.

Changes in primary mirror properties also affect the aberration contribution of the secondary. Combined with the focus extension factor 1/(1-ψ), the effective primary's focal ratio number changes in inverse proportion to (1+2σψ)(1-ψ). Since the secondary aberration is affected by the inverse third power of the primary's focal ratio, and in proportion to the diameter, its contribution changes in proportion to (1+2σψ)4(1-ψ)3. Since it also changes with (1-ψ) due to the change in ρ and k, the final aberration contribution of the secondary is in proportion to (1+2σψ)4(1-ψ)4.

Changes in aberration contributions from the primary and secondary disturb the zero balance contribution of the three optical surfaces, given with Eq. 113. The result is a system wavefront error. For the typical f/2/10 SCT arrangement with spherical mirrors (k~0.25, ρ~0.31), σ~0.4, and the combined system error resulting from the object distance is approximated by:

                           W~[-0.7+(1-2ψ)2(1+0.8ψ)4-0.3(1+0.8ψ)4(1-ψ)4]D/2048F13

as the P-V wavefront error of primary (i.e. 3rd order) spherical aberration at best focus. The quantity within the main bracket is the system aberration coefficient, a sum of the contributions from the three active SCT optical surfaces, corrector, primary and secondary mirror, respectively. Aberration contribution of the corrector - the numerical factor 0.7 times D/2048F13 common factor - doesn't change appreciably with object distance.  Contribution of the spherical primary - (1-2ψ)2(1+0.8ψ)4 times the common factor - diminishes with object distance, and so does the aberration contribution of the spherical secondary - 0.3(1+0.8ψ)4(1-ψ)4 times the common factor. Note that the primary is under-corrected, and the secondary is over-corrected.

Consequently, reduction in the contribution of the primary effectively induces over-correction, while the reduction of the secondary mirror contribution induces under-correction. The two partly offset in the system error. In fact, both primary's and secondary mirror aberration contribution are reduced by a nearly identical ratio, 1/(1-2ψ)2(1+0.8ψ)4 vs. 1/(1+0.8ψ)4(1-ψ)4, respectively, leaving corrector's over-correction under-matched. Thus the combined error of spherical aberration resulting from close object distance with an SCT is over-correction.

Fitting the above approximation into a simple empirical expression for object distance induced spherical aberration in the typical commercial SCT gives:

W~-D/3600(o-5)F13

at best focus, for the aperture diameter D in mm and the object distance o in units of the primary focal length.

This means that for 1/20 wave P-V wavefront error of spherical over-correction (w=0.05) induced as a result of the object distance of, typical 8-inch ƒ/10 commercial SCT would need of~50, or object placed some 50 system focal lengths away (for star testing the optics, the object distance induced error shouldn't be greater than ~1/20 wave P-V; for collimation purposes, there is no strict rule but, in general, you don't want the correction error to be significantly greater than the collimation tolerance error).

With eyepiece focusing, there is no mirror movement, so k'= k+(1-k)ψ. In other words, the effective area of the secondary used by the expanding cone converging from the primary becomes greater as the object distance diminishes. As a consequence, while the corrector and primary mirror aberration contributions remain as for the mirror focusing above, the secondary mirror relative contribution actually increases as object distance diminishes. This means that reduced under-correction at the primary is met by increased over-correction at the secondary, resulting in larger system error. Simple empirical error approximation is

W~D/64oF13

for D in inches, as the P-V wavefront error of spherical aberration at best focus (o is the object distance, not 0).

While not practically important for relatively small values of ψ (the reciprocal of the object distance in units of the primary's f.l.), it puts conventional (fixed mirror) focusing with SCT at a definite disadvantage for close objects observing. More so considering the extent of axial image shift. At the 20 system focal lengths object distance, focus extension with an 8" SCT is already likely to be over 4", and the induced P-V error of spherical aberration at best focus ~0.3 wave.

Note that the above considerations are for the lower-order spherical aberration alone; typical SCT systems have certain amount of higher-order spherical, and the actual error varies depending on how the two combine. In general, higher-order spherical should be significant factor only at very low third-order aberration levels, which means it is not much of a factor (as long as the corrector is made properly, taking care of the higher order differential between the primary and secondary mirror).
 

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