telescopeѲptics.net .......................................................................................... CONTENTS


◄ 8.1.1. Newtonian reflector diagonal flat   ▐    8.2.1. Classical and aplanatic ►
 

8.2. Two-mirror telescopes

8.2.1. Classical and aplanatic - 8.2.2. Dall-Kirkham and Loveday - 8.2.3. Miscollimation and close focusing 

While the Newtonian does use a pair of mirrors, one of them is flat and, therefore, a passive imaging element. Systems with both mirrors active in shaping up the wavefront - in other words, with both mirrors curved - are termed two-mirror telescopes. Classical two-mirror arrangements are Gregorian - the very first reflective telescope design, conceived by James Gregory in 1663 - and Cassegrain (FIG. 59), probably introduced in 1672 by Laurent Cassegrain. In the former, concave secondary mirror is placed outside the focal point of the primary mirror, and in the latter the secondary is convex, placed inside the primary mirror focus. In either case, the final focus is made accessible either by focusing through an opening on the primary, or by inserting small diagonal flat in front of it, reflecting converging cone out to the side (Nasmyth arrangement).


FIGURE 59: (a) Gregorian reflector consists from a pair of concave mirrors. Secondary mirror, placed outside the primary mirror focus, forms the final focus by re-focusing diverging light cone coming from the primary. The image is properly oriented. (b) Cassegrain reflector also consists from a pair of mirrors, but the secondary is convex, and placed inside the primary mirror focus. The image is reversed, left-to-right and up-side-down. Although formally not a part of their optical system, axially configured two-mirror telescopes regularly use baffle tube in front of the primary, to protect image plane from stray light. It is extended as much as possible, without causing obstruction in the light cone converging from the primary. The maximum baffle front extension (measured from the primary) in units of the primary mirror f.l. is given by [(1+η)/(1+1/mc)]-η, with c being the relative size of central obstruction in units of the aperture, m the secondary magnification and η the back focus distance (from the primary) in units of the primary's focal length. Width of the cone converging from the secondary at this point equals the hollow core diameter in the axial cone converging from the primary, both given by (1+η)/(m+1/c), in units of the primary mirror diameter. At this baffle position, field of full illumination is limited to the axial spot. Pulling front baffle opening back toward the primary by x longitudinally allows it to increase by x/2F1 in radius, with the resulting field of full illumination radius of ~x/2(1-L/S)F1, with F1 being the primary mirror focal ratio number, L the secondary-to-final-image separation and S the front baffle opening to final image separation.

The final system focal length ƒ is determined by mirror focal lengths and their separation, which determine secondary magnification as m=-ƒ2/(ƒ1-ƒ2-s), where ƒ1 and ƒ2 are the focal length of the primary and secondary mirror, respectively, and s is the mirror separation. Note that if the final system f.l. is set positive when forming a real image (converging), then that of the primary mirror (ƒ1) is negative, while the secondary f.l. ƒ2 is positive in Gregorian arrangement, and negative in the Cassegrain; the separation s is negative for both, Cassegrain and Gregorian. If, for convenience, the final system focal length is is set to be positive when converging, then it is given by ƒ=-mƒ1 for the Cassegrain, and by ƒ=mƒ1 for the Gregorian; if it is negative, the final cone is diverging (too weak secondary).

Back focal distance, from the vertex of the primary (thus positive), is B=[ƒ(ƒ1-s)/ƒ1]-s.

It took about half a century for mirror making to advance from the spherical mirror shape to paraboloidal, which was accomplished by John Hadley and James Short, both from England. Due to the difficulty of making convex aspheric surface, the Cassegrain was the last to come into efficient use, but soon became the most popular two-mirror arrangement.

Image quality level produced by either of these systems is determined by the particulars of optical configuration of the two mirrors. In the very beginning, both mirrors were spherical, suffering from all of the three primary "image quality" aberrations - spherical, coma and astigmatism. Such telescope still could produce useable image, but only when using mildly curved surfaces, that is, with very small relative apertures. Advancing to paraboloidal primary and the secondary adequately aspherized for zero system spherical aberration was significant improvement, also resulting in reduced off-axis aberrations. Two-mirror systems using paraboloidal primary are termed classical Cassegrain, or Gregorian.

The next step was modifying mirror conics for added correction of coma. Coma-free two-mirror systems are termed aplanatic Cassegrain (Ritchey-Chrιtien) and Gregorian. Variation of the basic two-mirror arrangement worth mentioning is the Gregorian with three reflections, which can be made anastigmatic aplanat (corrected for primary spherical aberration, coma and astigmatism).

The concept used for two-mirror aberration relations is that given in "Astronomical Optics" from Daniel J. Schroeder. It reduces the expressions to only two parameters: secondary magnification m and back focal distance h, defined as the primary-to-final focus separation in units of the primary focal length (FIG. 60).


 FIGURE 60:Two-mirror system in the right-hand Cartesian coordinate system. Main parameters are the relative height of marginal ray at the secondary k, secondary-to-primary's focus separation s, and secondary-to-final-focus separation s' (s1, s1' for Cassegrain, and s2, s2' for the Gregorian arrangement). With s and s' expressed in units of the primary's focal length, k=1+s and s'=ms, m being the secondary magnification. Back focus fb in units of the primary's focal length is η=k(m+1)-1. The sign of parameters is determined by their direction: positive for left-to-right and upward from the axis, negative for the opposite directions. The negative sign in the relation for secondary magnification is a consequence of magnification being defined as the ratio of image vs. object height. Since the object for the secondary is the image formed by the primary mirror - reversed bellow the axis, thus of negative sign - the relation is defined with a minus sign. Consequently, secondary located inside the focus of primary (Cassegrain arrangement), and forming reversed image bellow the axis, has positive magnification, while secondary located outside the focus (Gregorian), forms erect image above the axis, whose magnification is, by definition, negative.

 According to the sign convention, m is positive for the Cassegrain, and negative for the Gregorian, while h is positive for the final focus to the right from the primary mirror, and negative for the final focus to the left. The two parameters define the rest of system parameters:

- height of marginal ray at the secondary in units of the aperture radius, k=(1+h)/(m+1)

- ratio of the secondary to primary radius of curvature R2/R1=r=mk/(m-1)

- secondary magnification m=r/(r-k) = 1/(1-k/r)

Note that k and r are also positive for the Cassegrain and negative for the Gregorian.

General aberration relations for the two-mirror telescope are obtained from aberration coefficients for the primary (with the stop at the surface) and secondary mirror (with the stop at the primary) given with each aberration. For corrected lower-order spherical aberration in a two-mirror system, for object at infinity, primary and secondary mirror conics - K1 and K2, respectively - have to be related as:

                                         K2=-[(m+1)/(m-1)]2 +(K1+1)m3/k(m-1)3                        (80)

This relation is derived from the spherical aberration coefficients for the primary (Eq. 9.1) and secondary mirror (Eq. 9), setting their sum equal to zero. The sum itself, from Eq. 7, determines expression for the system P-V wavefront error of spherical aberration at best focus for two-mirror systems in general as:

                              Ws={K1+1-k[K2+(m+1)2/(m-1)2](1-1/m)3}D/2048F13               (81)

                             ={K1+1-[K2+(1-2r/k)2]k4/r3}D/2048F13

                                          ={K1+1-[K2k2+(2r-k)2]k2/r3}D/2048F13

results in zero value for Ws. In Eq. 82, the (K1+1) factor inside the main brackets is the aberration contribution of the primary, while the complex right-hand factor is the aberration contribution of the secondary.

General two-mirror system aberration coefficient for lower-order coma is

                                                  cs=[1+(K1+1)Pm2/2]α/4ƒ2                                   (82)

and for lower-order astigmatism

                            as={[(m2+η)/(m(1+η)] - mP2(K1+1)/4}α2/2ƒ                         (82.1)

with P=(m-η)/(1+η), ƒ being the system focal length, and α the field angle in radians. The P-V wavefront error at best focus is, from Eq.12 and Eq. 18, given by Wc=csαD3/12 and Wa=as(αD)2/4, respectively.

Good approximation for the level of coma in a two-mirror system can be obtained by neglecting h in the numerator of P, and using reduced form of Eq. 82. This gives coma approximately changing in proportion to [2+(K1+1)m3/(1+η)]/2F12, F1 being the primary mirror focal number F=ƒ1/D. For K1=-1 (paraboloidal primary), the coma changes as 1/F12, inversely to the square of the primary mirror F-number. For an aplanatic (coma-free) system, needed primary mirror conic is approximated by K1~-1-2(1+η)/m3 (of course, coma of the primary is conic-independent as long as the stop is at its surface; primary's conic actually compensates for spherical aberration induced by aspherizing the secondary as needed to offset primary's coma).

Two-mirror system Petzval and best (median) image field curvature are Rp=ƒ/[P-(m+1)/(1+h)]  and Rm=ƒ/{[(m2-2)/m] + [(m+1)/(1+h)] - [m(K1+1)P2/2]}, respectively.
 

Incoming ray angle at the final image

Incoming ray angle at any off-axis point in the final image of a two-mirror telescope is significantly larger than the corresponding incoming 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 - formed by the secondary. Larger image forming angles are not an aberration factor in the final image itself, but are for the eyepiece, or other optical element placed after the image. Given system focal ratio, eyepiece field aberrations in a two-mirror system are larger than in a Newtonian, or a refractor.

Secondary mirror in a two-mirror system forms the system exit pupil (i.e. image of the primary mirror) in front of the primary, at a distance p=(km2)/(m+k-1) from the final image, 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 m the secondary magnification. Since the chief ray appears as if coming from the center of the exit pupil, the incoming angle at the off-axis image point is given by h/p, h being the height in the image plane. At the same time, incoming angle at the aperture is defined by h/ƒ, with ƒ=mƒ1 being the system focal length  and ƒ1 the primary f.l. Consequently, magnification factor of the incoming angle at the final image is given by f/p or, after substituting for ƒ and p, by (m+k-1)/mk.

Thus, for average values of k~0.25 and m~4, the effective incoming ray angle at the final image is over 3 times larger than the actual incoming angle at the primary.
 

◄ 8.1.1. Newtonian reflector diagonal flat   ▐    8.2.1. Classical and aplanatic ►
 

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