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8.1.3. Newtonian reflector diagonal flat   ▐    8.2.1. General aberrations
 

8.2. all-reflecting Two-mirror telescopes

AberrationsClassical and aplanatic - Dall-Kirkham - Loveday - Miscollimation and close focusing 

As the title implies, this section as about mirror-only telescope systems. Mirror/lens systems are presented in section 10. Catadioptric telescopes.

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. 121), 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).

FIGURE 121: TOP: Left: Gregorian reflector consists from a pair of concave mirrors. The secondary, placed axially outside of the primary's focus F1, forms properly oriented final focus F by re-focusing diverging light cone coming from the primary. In the classical arrangement, primary is paraboloid, hence for the secondary to maintain zero spherical aberration it has to be an ellipsoid with its near focus coinciding with the primary's focus.
Right:
Cassegrain reflector also consists from a pair of mirrors, but the secondary is convex, and placed inside the primary's focus. In the classical arrangement with paraboloidal primary, secondary has to be hyperboloidal, with its near (virtual) focus coinciding with the primary's. The image is reversed, left-to-right and up-side-down. Effective focal length is determined by extending marginal ray converging from the secondary to the final focus backwards to the point of intersection with the incident marginal ray, or
ƒ=s/k, k being the relative (in units of aperture radius) height of marginal ray on the secondary. Since the primary is paraboloid in the classical arrangement, its focus in either Gregorian or Cassegrain has to coincide with the near (conic) focus of the secondary (not to confuse with secondary's Gaussian focus F2, for object at infinity), in order for the latter to form an aberration-free axial image. BOTTOM: Projected image of the primary is, effectively, the object for the secondary. It is located at a distance from secondary's surface, which re-images it to the final focus. The final image is constructed by a reverse projection of the reflections from the concave side of the secondary of two imaginary rays, one parallel with the axis (blue) and the other extended to secondary's vertex (red). The final top point image forms at the point of their intersection. The primary is the exit pupil for the secondary (i.e. chief, of central ray CR for the off-axis image point is coming from the primary's center), and it is also the aperture stop. The stop-to-secondary separation σ, in units of the secondary radius of curvature R2 (secondary-to-primary separation s=σR2) is, according to the sign convention, negative; it is a factor in calculating off-axis aberrations at the secondary (due to the chief ray - and with it the entire converging wavefront - being shifted off the center of the secondary, as a result of displaced stop). Optical image of the primary formed by the secondary is the system exit pupil (ExP2); it is a factor - aperture stop - in calculating aberrations of the tertiary mirror in three-mirror systems, due to the chief ray of off-axis points appearing as if coming (onto the tertiary) from its center, and the marginal rays (1 and 2) appearing as if coming from its boundary.

The final system focal length,

is determined by the 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 the 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 Renι Descartes (La Gιometriι, 1637) - 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 the 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 the 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, F1 the primary's focal ratio, α the field angle, h the height in the focal plane and ƒ1 the primary's focal length.

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 the 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 in units of its radius of curvature (Ω=R2/, see FIG. 121, 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 the 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.

The next step in two-mirror system evolution was modifying mirror conics for the correction of coma, in order to obtain wider well corrected photographic field. Coma-free two-mirror systems are termed aplanatic Cassegrain (Ritchey-Chrιtien) and Gregorian. A two-mirror system interesting from the practical point of view is Dall-Kirkham, consisting of ellipsoidal primary and spherical secondary.

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).
 

Front baffle tube

Although formally not a part of their optical system, axially configured two-mirror telescopes regularly use baffle tube in front of the primary, as well as baffle tube around secondary, to protect the final image from stray light.

Front baffle tube should be extended forward from the primary as much as possible, not to intrude into the light converging from the primary. As illustration at left shows, the limit is set by the plane where diameter of the cone converging from the secondary equals diameter of the hollow core (caused by obscuration by the secondary) in the cone converging from the primary. The common diameter, in units of the aperture diameter P, is given as:

             T=(1+η)/[m+(1/o)]

where η is the back focal length B in units of primary's f.l. and o is the relative size of central obstruction in units of the aperture (o=S/P).

Obviously, the distance from this plane to the final focus is a sum of its separation from the primary L and back focal length B. Since L+B=FT, with F being the final system focal ratio, L=FT-B. Substituting in this relation T=(1+η)P/(m+1/o) and dividing with ƒ1 - the primary focal length - gives the limit to the front baffle extension in dimensionless parameters, measured from the primary, in units of the primary mirror f.l. as:

where ε=L/ƒ1. As mentioned, the width of the cone converging from the secondary at this point equals the hollow core diameter in the axial cone converging from the primary, caused by obstruction, given by

in units of the primary mirror diameter. At this baffle position, field of full illumination is limited to the axial spot. Moreover, baffle tube won't protrude into the light converging from primary only if with zero wall thickness (assuming its inner diameter equaling T, in order to allow all axial light from the secondary to pass through). Thus, the front opening of an actual front baffle tube will have to be pulled somewhat toward primary, to avoid light intrusion. Since the inner hollow cone converging from the primary has the effective focal ratio Fc=F1/o, with F1 being the primary's focal ratio, every mm of shortening the front baffle tube clears 1/2Fc mm for the baffle wall thickness. So, if baffle wall at the opening is 1mm thick, baffle tube extension should be a bit more than 2Fc mm shorter than what Eq. 79 implies.

    This will also create some "breathing room" between baffle opening and cone converging from the secondary. For instance, a system with ƒ/3 primary (F1=3) and P/3 central obstruction diameter (o=0.33), will have Fc=9 and need some 10mm shorter front baffle tube. If the secondary magnification is 4 (so F=12), that will create 10/2F=0.4mm gap between the converging cone and inner edge of the baffle opening. As SCT systems show, such tight front baffle opening causes negligible vignetting (of more concern is the rear baffle tube opening).

     This baffle gap g will also create a small 100% illuminated image field radius i100, which is well approximated with i100~(ƒ1+B)g/(ƒ1-L) or, in dimensionless units, i100~(1+η)g/(1-ε). Pulling the front opening closer to the primary would create wider gap, and increase 100% illuminated field, but would likely require increase in the secondary baffle diameter, i.e. central obstruction. The two baffles, around secondary and in front of primary have to work in concert, i.e. have to be tightened so that no ray passing next to the secondary baffle pass the baffle tube and fall directly onto the image. For that to happen, a ray projected from the bottom of baffle tube rear opening, that touches top of its front opening, is not allowed to clear the secondary baffle. For that to occur, opening of the secondary baffle has to be at a distance d from the rear baffle tube opening not exceeding:

 d=H/[2F1b+1],

preferably a bit smaller, where H=ƒ1+E, with E being the length of baffle tube behind the primary, and b is the ratio of baffle tube opening vs. length.

    Since the position of this plane determines the width of cone converging from primary, it also determines the minimum needed central obstruction. If the secondary is already larger, the two baffles will be well matched. If, however, the secondary is smaller than the width of this cone cross section, secondary baffle and corresponding central obstruction need to be larger.

    If the secondary - i.e. its cell - is larger than this cross section, the baffle should extend toward primary by
2(s-s
m)F1, with s being the actual secondary's radius (or size of its cell), and sm the radius of its minimum size.  If it is smaller, its cell needs to be made larger, to fit the cross section of the primary's cone at distance d from the rear end of the baffle tube. Similarly to the front baffle tube opening, that needs to be corrected for the actual baffle wall thickness, with the actual baffle extension being shorter by a bit more than 2F1t, t being the wall thickness (slight vignetting induced to off-axis image points as a consequence of the tight secondary baffles negligible even in photographic applications).
 

8.1.3. Newtonian reflector diagonal flat   ▐    8.2.1. General aberrations
 

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