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telescopeѲptics.net
.......................................................................................... CONTENTS
8.2.3. Two-mirror telescopes: miscollimation, close focusingMiscollimation sensitivity in two-mirror systems Whenever the primary and secondary mirror in a two-mirror system are not optimally positioned, it induces certain amount of axial and off-axis aberrations. In general, induced aberration is proportional to the linear misalignment. While either of the two, or both mirrors can be misaligned, the system sensitivity can be simply shown as misalignment of the secondary relative to the primary. It can be expressed separately for tilt, decenter and despace. By far the dominant aberration resulting from the first two forms of misalignment is axial (independent of the field height) coma. Depending on the sign, it can add up with the "regular" off-axis system coma, or subtract from it, but the most troublesome part is its presence in the field center. As the P-V wavefront error, it can be expressed separately for the tilt as: Wt= -(m-1)(1+h)Dτ/48F2 (91.1) and decenter: Wdc= ∆[(K2-(m+1)/(m-1)](m-1)3/96F3 (91.2) with τ being the tilt angle in radians, ∆ the linear decenter and F the system focal ratio number. The RMS wavefront error is smaller by a factor 1/321/2. Tilt and decenter are usually both present, so the two errors combine, with the final error given by their sum. Whether they will add or subtract depends on the sign of τ and ∆. Misaligned secondary doesn't induce coma when τ=-∆[1-(m-1)K2/(m+1)]/R2, R2 being the secondary radius of curvature. Despace (separation) error s in two-mirror systems - positive for larger separation, and vice versa - results in change in the relative height of the marginal ray at the secondary k into k*=k+(s/ƒ1), and secondary magnification m into m*=ρ/(ρ-k*). Resulting wavefront error is obtained by substituting k* and m* for k and m in Eq. 81 (top), or simply k* for k in either of the bottom two expressions. The bottom equation gives the P-V wavefront error at best focus as: Wds=-[K2k*2+(k*-2r)2]k*2D/2048(rF1)3 (91.3) with F1 being the primary's focal ratio number. Alternately, it can be expressed in terms of secondary magnification alone as: Wds=[(m2-1)+(m-1)3K2]s/512F1F3 (91.4) It shows that Dall-Kirkham with K2=0 has significantly lower sensitivity to despace compared to both, classical and aplanatic Cassegrain (Ritchey-Chrétien), with the latter being the most sensitive. However, despace sensitivity is lowest in the Gregorian arrangement, for which the sum in the brackets is smaller than in a comparable Dall-Kirkham, due to numerically low negative secondary conic of the former.
Note that the separation change
s is
negative when the mirror separation decreases, resulting in larger
minimum relative secondary size k* (conventionally, primary's
f.l. is positive for convenience, against the strict sign convention
rule). From Eq. 91.4 it is easy to see that larger than optimum
mirror separation (i.e. positive s) induces under-correction
(positive in sign wavefront error at best focus), and that smaller
mirror separation (negative s) induces over-correction. Close objects error For relatively close objects, when magnification of the primary - given by m1=-ƒ1/(o-ƒ1), ƒ1 being the primary focal length and o the object distance - is appreciably greater than zero, Eq. 9 applies to both mirrors. The aberration contribution of the primary changes from (K1+1) into [K1+(1-2ψ)2], with ψ=f1/o being the primary focal length in units of the object distance. Also, due to the extended converging cone of the primary, both, relative height of the marginal ray at the secondary k and secondary magnification m increase. The height k becomes k'=(1-ψ)k+ψ, and secondary magnification m becomes m'=1/(1-k'/ρ'), with ρ'=(1-ψ)ρ. For the secondary, the effective primary focal ratio number is now F1/(1-ψ). With these changes, after substituting k', r' and m' for k, r and m in Eq. 81 (bottom) the system P-V wavefront error of spherical aberration at best focus becomes: W's= W1'+W2'=-{K1+(1-2y)2 -[K2k'2+(k'-2r')2]k'2/r'3}D(1-y)3/2048F13 (92) Since the secondary conic K2 is a factor for the aberration contribution of the secondary, sensitivity of two-mirror telescopes to reduction in object distance varies with the system type.
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