10.2.1.3. Busack-Honders-Riccardi cameras and telescopes

EXAMPLE: A sort of  re-construction of an actual Honders design - this particular one created by Massimo Riccardi, thus could be referred to as Honders-Riccardi - a 300mm f/3.7 catadioptric dialyte with a positive front lens, catadioptric rear element, and positive field lens, all three elements of BK7 crown (n=1.52, V=64.4). Based on the final system already known - which makes it easier - it illustrates basic design principles and its optical architecture.

Starting with the front refractive surface of the catadioptric element (CE), its radius R3 is to be somewhat longer than the desired final focal length, considering the need for a positive field lens to correct for lateral color. The separation S between the front lens and CE should be S~900mm, to make image accessible using a diagonal flat mounted onto the front lens. If ~1100mm is desired for the system, R3~-1200mm is as good a starting value as any. Standard center thickness of glass element with reflective surface is 1/6 (so called "full thickness") to 1/10, so the CE's thickness t2 should be in that range. These values are the basis for approximating needed radius of curvature for the reflective surface R4 as somewhat smaller than 2R3, which would be needed for the R4 value if the light falling at it would be collimated. Since we can expect the light falling at it to be somewhat diverging, in order for the final secondary spectrum to be balanced at the positive field lens, the first approximation of R4 should be somewhat smaller, say, 10-20%, for R4~1.7R3~-2000mm. Note that both radii are, according to the sign convention, numerically negative, and the thickness is positive for light traveling from left to right.

With the power of R3 approximately known, we can also approximate needed power of the small field lens to cancel its lateral color. Approximately, lateral color is proportional to hd/fV, where  h is the height of the chief ray on the surface, d is the effective distance past the refracting element in the direction of light travel, f is its focal length and V its Abbe number.

For the field lens, the chief ray height hF is determined by the position of its exit pupil ExF which, with the initial assumption that light has little or no interaction with R3 on its way to the field lens, is the image of the exit pupil Ex3 for R3 formed by the reflecting surface. The chief ray appears as if coming from the center of the exit pupil, thus determining the chief ray angle at its origin on the reflecting surface. With the exit pupil Ex3 for R3 being at the front lens, its image of it, whose location is determined from Eq. 1, is the exit pupil Ex4 for the reflecting surface R4, which re-mages it into the field lens' exit pupil ExF.

Using the approximate values of S~900mm, R3~-1200 and R4~2000 gives Ex3=-984mm and Ex4=72,400mm (for the pupil distance for the reflecting surface -984-t2). The positive Ex3 value implies that the chief ray refracts toward axis after passing R3. The Ex4 value implies that it reflects from R4 nearly parallel with the optical axis, slightly diverging. For the R3 surface on its return, it will appear as if coming from Ex4. For the field lens, it will appear as if coming from the exit pupil ExF, formed by R3 imaging of Ex4 (unlike the axial rays, R3 surface does affect off-axis rays of the light reflected from R4). With -1.52 for n, and -1 for n' (due to light traveling from right to left), Eq. 1 gives the exit pupil distance for the field lens ExF=2240mm, to the right of the CE. Since the chief ray appears as if coming from the pupil center, it is diverging toward the field lens, and will hit its surface approximately at the height h~[1-1000/2240]h3~1.45h3.

Also, by diverging light, this second refraction at R3 is opposite in effect to the first one, and stronger. Since the height of chief rays is nearly identical for both, the combined effect with identical dispersion power is determined by the respective surfaces' optical power, i.e. focal lengths. For air-to-glass surface (first refraction at R3) the focal length is fA=nR/(n-1)=-3510mm, and for glass-to-air surface (second refraction) it is given by fG=R/(n-1)=-2310, with the combined (diverging) power of fC=-6760mm.

As a result of the negative refractive power after second refraction at R3, the chief rays of opposing (vs. optimized green) wavelengths will effectively reverse their divergence, turning toward each other, red (long wavelengths) moving lower and blue (short wavelengths) upward. Since the new negative divergence is nearly half as strong as the positive divergence of the first refraction at R3, the red and blue will cross after they travel about double the CE thickness (t2). The point of this divergence is now the relevant distance d past refractive surface (or element) for R3. Since the combined power of both refraction is nearly three times weaker than that of the first R3 refraction, this point of convergence will lie nearly 3 times the double center thickness of the CE, or ~200mm from R3, toward the field lens. Thus the effective distance d3 past the imaginary combined refractive surface can be approximated as d3~S-200~700mm (note that the reflection on R4 does change the geometry of color divergence, but since it results from its magnification factor and doesn't actually influence the chromatism, it can be neglected, for simplicity).

Lastly, will take a distance dF from the field lens to the final image to be dF~100mm~. With these results, the condition for near-zero lateral color, (hFdF/fFVF)~-(h3d3/fCV3), with VF=V3 and hF~1.45h3 reduced to (1.45dF/fF)~(700/fC), gives fF~1.45dFfC/700~fC/4.8. With fC=~-6760mm, needed power of the field lens to neutralize lateral color resulting from two refractions at R3 is fF~1400mm. Since neutralizing divergence still doesn't correct lateral color, unless the colors nearly cross at the neutralizing element, the field lens needs to be twice as strong, thus with the focal length f/font>F~700mm (analogy to the optical power: a lens of identical opposite power will collimate diverging beam; for reversing it into conversion, it needs to be twice as strong). With BK7 glass and plano-convex lens, this translates into RF=(n-1)fF/n~240mm (the sign depending on lens orientation).

Taking one more hint from the actual design to save approximating best lens shape for balancing monochromatic aberration, will approximate the radii as R1/R2~-0.5 for the front lens, and near plano-convex field lens. With the front lens' focal length put at ~6 times its separation from the CE, or f1~5400mm, by substituting R2~-2R1 into Eq. 1.2, this gives R1~4200mm  and R2~-8400mm. Similarly, RF~-(n-1)fF/n~-240mm (convex to the right). The respective lens thicknesses are t1~30mm and tF~10mm. With the parameters already approximated, S~900mm, R3~-1200mm, R4~-2000mm, t2~30mm, this completes the first contour of the final system. Since the lens power, so far, has been adjusted for minimizing lateral color, additional adjustments for minimizing secondary spectrum, through separations/radii adjustments, are likely to be needed.

Plot to the left shows the result after the above parameters ("first approximation") were input in OSLO . Mainly due to relying on an established main frame, the resulting 300mm f/3.4 (somewhat faster than planned) system is ready for optimization. Spherical aberration is only 1/4 wave P-V, lateral color roughly minimized, coma, and to a smaller degree astigmatism significant farther out in the field, with the secondary spectrum being by far the worst of the aberrations. Nearly optimized 300mm f/3.7 Honders system to the left designed by Massimo Riccardi (SPECS)  has all of the aberrations minimized to negligible, or near-negligible by relatively small adjustments in radii and separations. Secondary spectrum is at the level comparable to 4" f/70 standard achromat, and can be further reduced by a factor of 3 with a few minor tweaks (framed inset on the right).

Expectedly, correction of aberrations gets even better with slower systems of this type. As the plot shows, there is no aberrations to speak of with a 300mm f/7.4 Honders-Riccardi variant (SPECS). The field is limited to that usable with 2" eyepiece barrel, but even at 2f (about 3 inches) off-axis, the combined aberrations are still well below "diffraction-limited" level. This particular system is not optimized for the minimum central obstruction size, which is here about 0.25D. As small as 0.15D central obstruction is possible with re-arranged systems, but at a price of sacrificing sufficient illumination in the outer field. In that respect, longer-focus Honders variants such as this one do not differ appreciably from other catadioptric systems with full aperture lens corrector (Schmidt-Newtonian, Maksutov- and Houghton-Newtonian). With mildly curved spherical surfaces, they are relatively easy to fabricate, and somewhat more compact than other comparable systems.

What makes Honders telescope really stand out is its unsurpassed correction level; Houghton-Newtonian is closer to it than the other two types, but still suffers from appreciably stronger astigmatism and field curvature (not a factor in visual observing, but a minus in astrophotography). Visual chromatism defined by F/C (blue and red line, respectively) is entirely negligible; Toward the ends of visual spectrum, correction is still excellent, although unbalanced: at the best focus, h-line (405nm) and r-line (707nm) RMS wavefront errors are 0.1 and 0.012, respectively (it can be balanced by using another ordinary glass, PK2, for the field lens, to 0.04 and 0.026 wave RMS, respectively).

This Honders-Riccardi system is also fairly tolerant to miscollimation: the most sensitive element in this respect, the CE, doesn't induce appreciable aberrations if despaced or decentered up to a couple of mm. A 0.1 degree tilt only induces lateral color with F/C line separation of about the Airy disc diameter, due to effectively misaligned field lens (in comparison, a 300mm f/5 paraboloid with the same amount of tilt would have 2/3 wave P-V of coma in the field center). All these qualities make this type of catadioptric dialyte a very viable alternative for high-quality instruments.