◄ 8.1.3. Newtonian reflector diagonal flat   ▐    8.2.1. General aberrations ►
 

8.2. Two-mirror telescopes

Aberrations -  Classical and aplanatic - Dall-Kirkham - Loveday - 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. 76), 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, to reflect converging cone out to the side (Nasmyth arrangement).

The final system focal length,

is determined by focal lengths and separation of the two mirrors. It determines secondary magnification as

where ƒ1 and ƒ2 are the focal length of the primary and secondary mirror, respectively, and s is the mirror separation. According to the above relation, for light moving from left to right the final system focal length is numerically positive in the Cassegrain, and negative in the Gregorian. Primary mirror focal length (ƒ1) is always negative, while the secondary mirror focal length ƒ2 is positive in Gregorian arrangement, and negative in the Cassegrain; mirror separation s is negative for both.

In order to correct for spherical aberration, the two mirror must have specific conic forms. Although Gregory established - based on conic properties established by René Descartes - that in his arrangement primary and secondary need to be paraboloid and ellipsoid, neither him, nor anyone else at the time knew how to make aspheric surfaces. For some time, it was difficult to produce even spherical mirrors accurately. As a result, early instruments were suffering from all of the three primary "image quality" aberrations - spherical, coma and astigmatism. It took about half a century for mirror making to advance to that next level, when John Hadley, in 1720, built first two-mirror system close to its theoretical specifications. It eliminated most of spherical aberration. Not long after, James Short was the first to make a true Gregorian system, with paraboloidal primary and ellipsoidal secondary. He went on to produce well over a thousand Gregorian telescopes.

Due to the added difficulty of making strongly aspheric convex surface (hyperboloid) for the secondary, the Cassegrain was the last to come into efficient use, but soon became the most popular two-mirror arrangement. First significant telescope of this type was probably James Nasmyth's 20-inch in 1845.

Two-mirror systems using paraboloidal primary are termed classical Cassegrain, or Gregorian. Nowadays,  unlike the telescope pioneers, we can determine lower-order aberrations of such a system at best focus location with a few simple relations. The system aberration is a sum of aberrations at the primary and secondary mirror. For the former, the P-V wavefront error at best focus for object at infinity is, from Eq. 7, 12 and 18, given by:

for spherical aberration, coma and astigmatism, respectively, with K1 being the primary conic, D1 the aperture diameter, F the focal ratio, α the field angle, and h the height in the focal plane.

For the secondary, the effect of close object distance and displaced aperture stop make the aberration relations more complex. Change in stop position doesn't affect the coefficient for lower-order spherical, but the object distance does; thus Eq. 9.1. applies (note that 2ψ=Ω). With coma, both stop separation and object distance affect the aberration coefficient, thus the appropriate expression is Eq. 15.3.  And with astigmatism, stop separation affects the coefficient, but object distance doesn't, as given with Eq. 22.

Substituting these coefficients in their respective wavefront aberration functions which are, same as for the primary, given by Eq. 7, 12 and 18, results in the following P-V wavefront error at best focus for spherical aberration, coma and astigmatism, respectively:

with K2 being the secondary conic, D2 the secondary minimum aperture (the cone width at the secondary), R2 the secondary radius of curvature, Ω the inverse of the secondary's surface-to-object distance ℓ in units of its radius of curvature (Ω=R2/ℓ, see FIG. 76, bottom), and σ the secondary to the aperture stop (i.e. primary) separation in units of secondary's radius of curvature (according to the sign convention, σ is numerically negative in Cassegrain, while positive in the Gregorian, and Ω is positive in both).

Some useful identities are (Ω-1)=(m+1)/(m-1)=(2ρ/k)-1, with k being the relative height of marginal ray at the secondary in units of the aperture radius, and ρ=R2/R1, the secondary radius of curvature in units of the primary's (both parameters extensively used on the next page).

The system P-V wavefront error at best focus for each aberration is given by a sum W=W1+W2.

It is obvious from Eq. 78-78.1 that with paraboloidal primary, secondary conic for corrected spherical aberration is K2=-(1-Ω)2. Spherical secondary has coma of opposite sign to that of the primary in both, Cassegrain and Gregorian, but significantly smaller; aspherizing it to correct for spherical aberration adds additional offset and lowers the system coma to the level of a comparable paraboloid.

Astigmatism of spherical secondary is, on the other hand, of opposite sign to that of the primary in the Cassegrain, and of the same sign in Gregorian; aspherizing it for corrected spherical adds astigmatism of the opposite sign in both, resulting in the two ending at a similar system astigmatism levels. As the astigmatism relation in Eq. 78.1 shows, it is independent of object (i.e. primary's image) distance.

According to Eq. 78-78.1, needed secondary mirror conic for zero coma is σK2=-(R22D13/R12D23)-(1-Ω)(1-σ), and the corresponding conic for the primary (to compensate for spherical aberration induced by changing the secondary conic), K1=-1-[K2+(1-Ω)2]R13D24/R23D14.

Alternately, needed mirror conics in terms of the primary and system focal lengths, and mirror separation are given as  K1=-1, K2=-[(ƒ-ƒ1)/(ƒ+ƒ1)]2, and K1=-1-2(ƒ1-s)ƒ12/sƒ2, K2=-[(ƒ-ƒ1)/(ƒ+ƒ1)]2-2ƒƒ13/s(ƒ+ƒ1)3 for the classical and aplanatic arrangement, respectively. Note that the primary focal length ƒ1 and mirror separation s are always numerically negative, while the system focal length ƒ is numerically positive for the Cassegrain, and negative for the Gregorian.