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8.2. Twomirror
telescopes
▐
8.2.2/3. Classical and aplanatic
► 8.2.1. Twomirror telescope aberrations
PAGE HIGHLIGHTS The basic concept used for more detailed presentation of twomirror aberrations is that given in "Astronomical Optics" from Daniel J. Schroeder. It requires only two parameters for computing system aberrations: one is secondary magnification m and the other, alternatively, either relative back focal distance η, or the height of marginal ray at the secondary (i.e. minimum relative secondary size) k (FIG. 122). The former is defined as the primarytofinal focus separation in units of the primary focal length, or η=B/f1=(m+1)k1, where B, f1 and m are the back focal length in units of the primary mirror's focal length, primary's focal length and secondary's magnification, respectively, and the latter by k=1s/f1., with s being the primarytosecondary separation (both f1 and s numerically negative according to the sign convention). Use of parameters in their relative ("dimensionless") form generally simplifies the relations, also making calculation quicker and more convenient by leaving out large numbers (focal length, radii, etc.).
According to the sign convention, secondary magnification m is positive for the Cassegrain, and negative for the Gregorian, while the relative back focal distance η is positive for the final focus to the right from the primary mirror, and negative for the final focus to the left.
Following table details twomirror system
parameters determining its geometric and optical properties.
Note that k and ρ are also positive for the Cassegrain and negative for the Gregorian. With the Cassegrain, the secondary will form real image with ρ>k; for ρ=k, the secondary will produce collimated beams, and for ρ<k reflected beams will be diverging (i.e. magnification is numerically negative). Likewise, with the Gregorian, the condition for the secondary to form real image is k>ρ, with positive magnification values indicating secondary too weak to form a real image.
Twomirror system parameters , with respect to their sign, are
summarized in the following table.
TABLE 9: Numerical sign for main parameters of the Cassegrain and Gregorian twomirror systems. General aberration coefficients for the twomirror telescope are obtained as a sum of aberration coefficients for the primary (with the stop at the surface) and secondary mirror (with the stop at the primary). For cancelled lowerorder spherical aberration in a twomirror system, for object at infinity, primary and secondary mirror conics  K1 and K2, respectively  have to relate as:
This relation is derived by setting a sum of the spherical aberration coefficients for the primary (Eq. 9.2) and secondary mirror (Eq. 9), equal to zero. The sum itself, from Eq. 7, determines expression for the system PV wavefront error of spherical aberration at the best focus for twomirror systems in general as:
With values for K1 and K2 satisfying Eq. 80, the result is zero value for Ws. In Eq. 81, the (K1+1) factor inside the main brackets is the aberration contribution of the primary, while the complex righthand factor is the aberration contribution of the secondary. Deviations from values needed for a zerosum in the four contributing elements shown in bottom relation  mirror conics, secondary location and radius of curvature  can either add up or partly offset one another. For instance, it is evident that change in the minimum relative secondary size k (i.e. change in its separation from the primary) will be mainly offset by a similar relative change in ρ (i.e. the secondary radius of curvature). But that will change secondary magnification. Offaxis aberrations in twomirror systems are coma, astigmatism and field curvature (distortion is usually negligible). General twomirror system aberration coefficient for lowerorder coma is a sum of the aberration contributions of the primary (Eq. 15.1) and secondary mirror (Eq. 15.2). After substitutions for the secondary's conic, stop (i.e. the primary) separation σ (FIG. 76, bottom) and the object (i.e. primary's image) distance factor Ω, in terms of the selected dimensionless parameters it can be written as:
Similarly, the twomirror system aberration coefficient for lowerorder astigmatism is:
with f being the system focal length. The PV wavefront error at the best focus is, from Eq. 12 and Eq. 18, given by Wc=csαD3/12 and Wa=as(αD)2/4, respectively, with α being the field angle in radians. Good approximation for the level of coma in a twomirror system from Eq. 82 gives system coma approximately changing in proportion to [2+(K1+1)m3/(1+η)]/2F12, F1 being the primary mirror focal number F=f1/D. For K1=1 (paraboloidal primary), the coma changes as 1/F12, inversely to the square of the primary mirror Fnumber. For an aplanatic (comafree) twomirror system, needed primary mirror conic is approximated by K1~12(1+η)/m3 (of course, coma of the primary is conicindependent as long as the stop is at its surface; the indication is indirect, due to primary's conic actually compensating for spherical aberration induced by aspherizing the secondary as needed to offset primary's coma). Twomirror system Petzval and best (median) image field curvature are:
respectively, with the system focal length f being, as mentioned in the beginning, numerically positive for the Cassegrain and negative for Gregorian. For K1~1, good approximation of the median field curvature, for η set to zero in the denominator, is given by:
The sign of Rm is negative for the Cassegrain, and positive for the Gregorian, indicating that median field curvature is concave toward secondary for the former, and convex for the latter. Graph below shows Petzval and median field curvature for classical Cassegrain and Gregorian (K1=1), for back focal length η=0.25 (note that the system focal length is positive for the Cassegrain and negative for the Gregorian, thus the sign of field curvature is as appears for the former, but opposite to it for the latter).
On the other hand, Gregorian can't have flat Petzval (obviously, since employing two concave mirrors), but can have flat best astigmatic surface, for m~1.6. However, such system is not practical, since the corresponding secondary is, from k=(1+η)/(m+1)=2.083, more than twice the diameter of the primary. Practical Gregorian systems have m<3.5, i.e. secondary's minimum size not larger than about 50% of the primary. For the same criterion, the lower magnification limit for the Cassegrain is m~1.5. Thus, while for given system fratio the Gregorian will have flatter best image surface (and larger secondary)  its median curvature is nearly identical to the Petzval of the Cassegrain, and its Petzval is fairly similar to the Cassegrain's median curvature  for given primary's fratio it is the Cassegrain that can reach faster system fratio and have a flatter best image surface not only in terms of the system focal length, but in absolute terms as well. For instance, with an f/3 primary the fastest that the Gregorian can go with the minimum 50% linear obstruction is ~f/11, vs. ~f/4.7 for the Cassegrain; the former has the best image surface radius ~0.15f, and the latter ~0.45f, or nearly 30% weaker. Variations in the back focal length from 0.25 are relatively small for practical systems (even in the Nasmyth arrangement and η~0 there is no significant change in the median field curvature), which means that these plots are representative of the two systems in general. Incident ray angle and the final image Incident ray angle at any offaxis point in the final image of a twomirror telescope is significantly larger than the corresponding incident angle at the aperture (primary mirror). This is a consequence of the ray reflected from the secondary appearing as if coming from the center of the exit pupil  image of the primary  projected by the secondary. Larger image forming angles are not an aberration factor in the final image itself, but are an aberration factor for the eyepiece, or other optical element following the image. Given system focal ratio, eyepiece field aberrations in a twomirror system are larger than in a Newtonian, or a refractor. Secondary mirror in a twomirror system forms the system exit pupil (i.e. virtual image of the primary mirror) in front of the primary, at a distance E=(1k)ρ/(1k+ρ) from the secondary, in units of the primary's focal length, with k being the height of marginal ray at the secondary (i.e. minimum secondary size) in units of the aperture radius, and ρ the secondary radius of curvature in units of the primary's (the negative value of E inicating it is at left from the secondary, for the primary oriented to the left). Since the chief ray appears as if coming from the center of the exit pupil, the incident angle at the offaxis image point is given by h/(Es), h being its linear height in the image plane and s the secondarytofinalfocus separation (positive in sign). At the same time, incident angle at the aperture is defined by h/f, with f=mf1 being the system focal length and f1 the primary f.l. Consequently, magnification factor of the incident angle at the final image is given by f/(Es), with the minus sign to make it positive. If expressing all in units of primary's focal length, it is m/(E+s), m being the secondary magnification (since the primary f.l. is negative, E turns positive, and since s is positive too, sign in the brackets is a plus).
Thus, for average values of k~0.25 and m~4, the effective incident ray
angle at the final image is over 3 times larger than the actual incident
angle at the primary. It is greater for the Gregorian, since both,
secondary magnification m and relative size k are
numerically negative. ◄ 8.2. Twomirror telescopes ▐ 8.2.2/3. Classical and aplanatic ►
