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10.1.2. Sub-aperture
corrector examples
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10.2.1.2. Schupmann
medial telescope
► 10.2. CATADIOPTRIC TELESCOPES WITH FULL-APERTURE CORRECTORSUnlike sub-aperture catadioptric telescopes, combinations with spherical primary mirror are the most frequent form in arrangements with full-aperture corrector. Such combinations are attractive not only for the ease of mirror fabrication, but also the possibility of influencing coma of the spherical mirror by varying its stop position - something that sub-aperture corrector catadioptrics can't take advantage of. The corrector can be used with either a single-mirror (Newtonian or camera) or with two-mirror (usually Cassegrain) arrangements. While there is a number of possible forms of the full-aperture corrector, the three types most often used for amateur telescopes are Schmidt corrector, Maksutov meniscus corrector and Houghton two-element corrector. The most common configuration are catadioptric Newtonian - Schmidt-Newtonian (SN), Maksutov-Newtonian (MN) and (quite rare) Houghton-Newtonian - as well as catadioptric Cassegrain systems: Schmidt-Cassegrain (SCT), Maksutov-Cassegrain (MCT) and (never saw one) Houghton-Cassegrain telescope (HCT). Performance level achievable, as well as ease of manufacture, differ somewhat from one to another.
Before taking a closer look of these three full-aperture corrector types, it will be well worth the time to revisit the very first form of catadioptric telescope, technically in between sub-aperture and full-aperture catadioptrics. This is catadioptric dialyte (generally, a lens objective consisting of two widely separated elements), in which the rear lens has been replaced by catadioptric element - a negative meniscus concave toward front lens, with its rear surface reflecting converging beam back toward front lens, to form the final focus. The earliest of these designs date back to the 19th century. In their simplest form, they consist of only two elements, yet can approach very high level of optical correction. The catadioptric element in these dialyte designs is often referred to as Mangin mirror, although they are not quite the same thing. Mangin mirror was invented in 1876 by a French officer whose name it carries, as an alternative to paraboloid (overcorrection from the concave front lens surface can be calibrated to cancel under-correction of the spherical reflecting concave surface). Not only that it is originally intended to be a single image-forming element, it also historically follows, not precedes, the use of technically similar optical element in catadioptric dialytes. For those reasons, in the absence of information on who did actually introduce the lens/mirror element in early dialytes, it is better to refer to it simply as catadioptric element. Among the early catadioptric dialytes, those that are still viable systems - and both original and exceptional with respect to their correction level - are Hamiltonian, Schupmann and - much more recent, but belonging to the same family - Honders telescope and astrograph. 10.2.1. 1. Hamiltonian telescopeW. F. Hamilton conceived his idea of a dialyte catadioptric at the very beginning of the 19th century, and have it patented in 1814. In its simplest form, it consists from only two single glass elements: a single lens at the front end, followed by a widely separated negative meniscus with silvered convex rear surface. Hamilton was the first to utilize the advantage of this type of arrangement, which allows for weaker lens surfaces than standard achromats, due to its power being mostly produced by the reflecting surface. In fact, the first three refraction in a Hamiltonian produce a collimated, or near-collimated beam of light, which is then focused by the reflecting surface, and actually slightly weakened by the fourth refraction. Resulting secondary spectrum is typically 2 to 4 times lower (possibly more) than in a comparable standard doublet achromat. Both, chromatic and monochromatic aberrations of this arrangement are easy to compute, being a sum of the aberrations of a single lens for object at infinity, a pair of single refracting surfaces for which the object is the image formed by the preceding element (this is the same front surface of the catadioptric element, for the entering and exiting light), and a single reflecting surface, for which the object is the image formed by the concave refracting surface (third refraction ) in front of it - typically at infinity (collimated or near-collimated beam). While various configurations are possible, the usual one is with the rear flint element at half the focal length of the front crown lens. As a result, the flint element is about half the diameter of the front lens, and its front concave radius - if producing collimated beam - has negative focal length of about half the front lens' focal length (this places its object - the image projected by the front lens - at its focus). The final system focal length is mainly determined by the radius of curvature of the rear reflecting surface. It may be - and usually is - somewhat affected by the second refraction at the glass surface preceding the reflector. If the final focus distance from the rear element is φ, in units of the element separation, the system focal length in this arrangement is approximately ƒ~φƒ1, ƒ1 being the front lens' focal length, and the system's physical length is φƒ1/2.
General optical scheme of the Hamiltonian is a positive crown lens at the front end, followed by flint catadioptric element at the rear end. Its optical prescription begins with the lens shape factor q needed to generate spherical aberration and coma that will nearly offset those generated by the rear element. The q value is approximately in the range 1.2<q<1.4. It determines the lens' surface radii ratio as R2/R1=(1+q)/(q-1), where R1 and R2 are the front and rear lens surface radius of curvature, respectively. Substituting it in Eq. 1.2, gives the radii to focal length relation as R1=(n1-1)(1-R1/R2)ƒ1, n1 being the front lens' refractive index. The front convex surface of the catadioptric element placed at half the front lens' focal length from it produces collimated beam when its focal length is -ƒ1/2. Thus, radius of curvature for this surface that will produce collimated beam is, from Eq. 1, R3=-(n2-1)ƒ1/2, n2 being the glass refractive of the rear element. According to Eq. 100.1, dialyte doublet is achromatized (i.e. two widely separated wavelengths brought to the same focus) when Abbe numbers of the front and rear element relate as
where S is the element separation, numerically positive, 1/ƒf=(n1-1)[(1/R1)-(1/R2)] is the front lens' focal length and ƒr is either focal length of the third surface alone, ƒr=ƒ3=R3/(n2-1), when light reflected from R4 passes through R3 without refraction (i.e. has its focus coinciding with the center of curvature of R3), or 1/ƒr=[1/(n2-1)R3]-1/ƒ', when refraction is taking place, with ƒ' being the additional negative refracting power at R3, obtained from 1/ƒ'=(1/ƒ3)+[(2/R4)-(1/I)], where 2/R4 is the focal length of the reflecting surface and I is the final image separation from the rear element (all three numerically negative). For ƒr=ƒ3=ƒ1/2=S, the needed Abbe number for the flint V2=V1/2, and for somewhat stronger ƒr with the additional final refraction at R3, the needed Abbe number V2 for the rear element is nominally somewhat larger (i.e. of lower dispersive power) than V1/2. This is what makes crown/flint combination suitable in a typical Hamiltonian arrangement with the element separation - and the front (refractive) surface of the rear element focal length ƒ3 - of about 1/2 the front lens' focal length. If the reflective surface R4 focuses at the center of curvature of R3, its radius of curvature is given by R4=2(R3+t), t being the rear element center thickness. If the final focus is to be at ƒ1/2 from the reflective surface which, with marginal ray height at the catadioptric element being 1/2 the aperture radius, results in the final system focal length nearly equal to that of the front lens, needed radius of curvature of the reflecting surface R4=[2nR3/(n'-1)(n'-n)]+2t, with n=-n2 and n'=-1 being the incident and index of refraction at the glass surface (it is also obtained from Eq. 1, with the mirror focus distance from R3 as object distance, and the final image distance from R3 as its image distance). For any given final image separation I from R3, needed reflecting surface radius is
EXAMPLE: A 150 ƒ/12 Hamiltonian with the rear lens at 1/2 the front lens' focal length separation, and final focus at the front lens. The system focal length is nearly identical to that of the front lens, thus we can start with ƒ1=1800mm. For BK7 crown (n1=1.5187, V1=64.4) front lens with the shape factor q=1.3, the lens surface curvature radii relate as R2=7.67R1, with R1=(n1-1)[(1-(R1/R2)]ƒ1=812mm and R2=6228mm.
Needed radius of curvature of the F2 flint (n2=1.624,
V2=34.4)
rear element's front surface R3=-(n2-1)ƒ1/2=-562mm,
and needed radius of curvature of the fourth, reflecting surface for the
image separation I=-900 and element thickness t=10mm is R4=-1482.
Since the total refracting power at the rear element
ƒ3=-660,
best Abbe number for it is, according to Eq. (k), V2=0.58V1=37.15
(assuming its refractive index unchanged; stronger index will require
yet higher Abbe number, and vice versa), which is somewhat higher,
numerically, than
F2 flint's 36.6. It indicates that F3 flint, or even F9, would probably
be a better match with respect to minimizing secondary spectrum, but in either case minor adjustment
will be necessary to compensate for less than perfect glass match and
tighten up the red and blue foci.
Other alternatives for minimizing secondary spectrum are better glass match, varying the element thickness or front lens' power, etc. Secondary spectrum of the adjusted system (SPECS) is 0.33mm (ƒ/5400). It is 2.7 times lower than in a comparable Fraunhofer doublet, respectively, which puts this 150mm ƒ/12 Hamiltonian at the level of ƒ/32 Fraunhofer, with respect to the size of secondary spectrum.
Unfortunately, as common to most dialytes, the field remains crippled by
lateral chromatism. Its cause is in the
significant element separation, resulting in the chief ray from the
front lens arriving at the rear lens well off its center. Since the two
surfaces at this height have different powers, different wavelengths do
not exit the lens nearly parallel, as they do when passing through a
lens near its center, but at an angle, and consequently are being
focused at different heights in the image plane. This lateral color is
created by the same dispersive power needed to correct the secondary
spectrum, thus cannot be avoided (unlike secondary spectrum, it is only
mildly affected by relatively small changes in element separation, or
power). Solution to this problem suggested by Hamilton is to compensate - or significantly reduce - lateral
color error by the use of matching eyepieces (a simple singlet's doublet at the
bottom of FIG. 145,
right, would be appropriate for this purpose). It
can also be eliminated by achromatizing either catadioptric element
alone, or both front and rear elements. Various arrangements of the two Hamilton elements are possible. The original configuration suffers from excessive lateral color, but it can be remedied by adding a single-lens corrector before the focal plane (FIG. 164).
The good news is - that is quite possible. A single positive field lens can accomplish just that. Shown to the right on FIG. 164 is a system with such simple corrector added. No calculations were made, no serious attempts to optimize the arrangement; merely a positive (plano-convex) F2 lens was placed in the cone converging toward final focus, knowing that it should suppress lateral color ( SPECS). So it did, turning a system unusable without matching eyepieces into one with lateral color practically disappearing from this same 0.25° diameter field. In order to compensate for corrector's secondary spectrum, the first two elements need to have it appropriately imbalanced (in this case, red and green focusing together, blue farther out). It can be accomplished by a small adjustment of the separation between the two main elements. It is almost certain that the arrangement with field corrector for lateral color can be further optimized. Secondary spectrum is already exceptionally low - less than ƒ/9000, placing this 150mm ƒ/7.4 system at a level of an ƒ/32 standard achromat. Both, remaining lateral color and coma visible in the outer field will likely be nearly cancelled in design optimization.More examples of the Hamiltonian can be found on Roger Ceragioli's "medial" telescopes' page. With four glass elements, as Ceragioli shows, correction is essentially perfect, including lateral color.
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Plugging
this data in OSLO gives nearly expected system, with focal length
ƒ=1821mm, 1/17.5 wave P-V of 3rd order spherical aberration, residual
astigmatism which can be somewhat minimized, but not eliminated, and
traces of coma. Coma can be reduced to zero, but it wouldn't result in a
visible improvement, with astigmatism being main determinant
of the outer field quality. Diffraction field diameter set by
astigmatism and coma exceeds 0.8 degrees, which mean that no off-axis
aberrations would
be noticeable in actual observing. As expected, secondary spectrum needs
optimization. The easiest way to tighten up the red and blue foci is to
adjust the separation between elements; in this case, it is needed to
increase it to 921mm (no change in the system's focal length). A small
change in spherical aberration caused by this change in separation had it practically reduced to zero;
aberrations in general are not appreciably affected by the adjustment.
FIGURE
164: LEFT: Unusually fast
150mm ƒ/7.6 Hamiltonian (