The Stevick-Paul Telescope — Design Notes

Collected from David Stevick’s original pages (archived at https://spider.seds.org/scopes/dstevick/stevpaul.html) and the associated design manual and Folded Stevick-Paul writeup.

Overview

The Stevick-Paul telescope (SPT) is an unobstructed, all-reflecting, tilted-component telescope (TCT). It has aspects of the Mersenne telescope (two confocal paraboloids) and of the Schmidt camera. Despite a superficial resemblance to a TCT, all mirrors are coaxial; the unobstructed view comes from using the optics off-axis.

Its defining property: in axial form it is a paraboloid freed of all third-order aberrations except a gentle field curvature (Petzval curvature). Used off-axis it therefore yields paraboloidal-quality images across the field.

4 optical surfaces minimum: paraboloidal primary, spherical convex secondary, spherical concave tertiary, plus a flat diagonal to extract the focus.
All higher-order aberrations, distortion, image (focal-plane) tilt and field curvature are minor.

History

1935 — Maurice Paul (France) first described the system as a zero-power corrector for large astronomical telescopes: two spherical mirrors of mating curve with the concave tertiary nominally in contact with the paraboloidal primary. Because of the large central obstruction of that axial arrangement, Paul regarded it as of theoretical value only.
1960s — Dr. J.G. Baker independently rediscovered it. He moved the two spherical mirrors farther from the primary (reducing their diameter and obstruction, a condition allowed by Paul’s equations) and added an approximate fifth-order correction by turning the edge of the convex secondary. Such designs became known as Baker-Paul three-mirror systems.
1991 — David Stevick, while raytracing TCTs to improve the Tri-Schiefspiegler, rediscovered the arrangement. He found that when the secondary–tertiary spacing equals the tertiary radius of curvature, all aberrations vanish (spot diagrams collapse to points). Dr. Richard Buchroeder confirmed the work and identified it as an off-axis form of the Paul system. Dr. Baker allowed the name Stevick-Paul telescope since Stevick’s design does not use a turned edge.
Also: German amateur Michael Brunn independently discovered it (unpublished) while exploring 4-mirror Tetra-Schiefs (US Patent 5,142,417); he set it aside for lack of back focal length for his binocular attachments.
1993 — A.L. Woods demonstrated a pioneering 6” f/12 SPT at the Chicago Astrofest.
Prototype: 10” f/6.2 by Stevick (used existing paraboloidal primary; only two spherical mirrors had to be made).

Optical configuration and layout rules

Mirror roles

Primary — paraboloidal. Same f-ratio as the final instrument.
Secondary — convex, spherical, positioned just to the side of the entering beam (leaving room for a light baffle). Positioned afocally: it sends parallel light to the tertiary.
Tertiary — concave, spherical, of mating curve to the secondary. Positioned exactly twice as far to the side of the entering beam as the secondary, and at its own radius of curvature distant from the secondary. Receiving parallel light, it forms an image at its focus just as it would with starlight.
Diagonal (flat) — throws the focus out of the telescope at 90° to the secondary–tertiary beam. Introduces an even number of reflections (correcting the reverted image from the odd count of the three curved mirrors). Does not obstruct light.

Key principle

The radius of the spherical mirrors is 0.4 × the radius of curvature of the primary (the “radius ratio” A = 0.4 is optimum). This ratio is found to be optimum but other ratios are possible.

Design equations

Let:

D = primary diameter
d = diameter of the clear field
A = radius ratio = (r.o.c. of spherical mirrors) / (r.o.c. of primary), normally 0.4
F1, F2, F3 = focal lengths of primary, secondary, tertiary (F2 negative)
R3 = tertiary radius of curvature

Mirror diameters

Eqn 1 — Secondary diameter: A·D + (1 − A)·d
Eqn 2 — Tertiary diameter: A·D + (3 − A)·d

Central beam diameter (same at secondary and tertiary)

Eqn 13 — A·D

Mirror tilts (primary tilt determines all others)

Eqn 3 — Primary tilt: T1 = C / N, where N = primary f-ratio and C is taken from the table below (values are empirically derived to assure freedom from stray light, valid for radius ratio 0.4 only).

Primary dia.	C
4.25”	37.08
6”	36.24
8”	35.67
10”	35.29
12.5”	34.97
16”	34.67

Eqn 4 — Secondary tilt: T2 = k·T1, with k = (1 + A) / (2A). For A = 0.4, k = 1.75.
Eqn 5 — Tertiary tilt: T3 = T2 − T1
Eqn 6 — Focal-plane tilt ≈ 4·T3
Eqn 14 — Diagonal tilt: T4 = 45° − T3 (measured against the principal ray through the diagonal center).

Equations 4–6 are valid for all radius ratios; the Eqn 3 C table is valid for A = 0.4.

Spacings (note F2 is negative)

Eqn 7 — Primary–secondary: e1 = F1 + F2
Eqn 8 — Secondary–tertiary: e2 = R3
Eqn 9 — Tertiary–focal plane: e3 = F3

Axis distances (offsets between mirror centers)

Eqn 10 — First axis distance: a1 = sin(2·T1)·e1
Eqn 11 — Second axis distance: a2 = 2·a1
Eqn 12 — Third axis distance: a3 = a1/2

Field / image quality constraints

f-ratio: Diffraction-limited performance possible down to about f/5.6, but focal-plane tilt increases as f-ratio decreases. Stevick restricts designs to f/10 or longer, keeping image anamorphism under 2% and focal-plane tilt under 10°.
The prototype 10” f/6.2 works only if the eyepiece is square (fat f/6 cone won’t tolerate a tilted eyepiece); top/bottom of field go out of focus. Shallower cones allow the eyepiece to be tilted to match the focal plane.
Radius ratio tradeoffs:
- A < 0.4 → more compact (down to ~0.33, below which increasing primary–secondary spacing lengthens the instrument again), smaller primary tilt, smaller spherical mirrors, but a more steeply tilted focal plane.
- A > 0.4 → longer, larger primary tilt, larger spherical mirrors, less steeply tilted focal plane.
With the preferred A = 0.4, the eyepiece falls close to the balance point, in a convenient viewing position with a short arc of travel.
Spherical aberration tolerance is the same as a Newtonian of the same primary f-ratio (the primary f-ratio is preserved in the final system). Therefore any long- focus primary that could be left spherical in a Newtonian may also be left spherical here (e.g. an f/15 design can use a spherical primary).

Design field / clear field table

The clear field is the diameter over which the design is fully illuminated; the design field has a ½-magnitude loss at its edge. Smaller telescopes use a downsized design field so they retain enough free cone.

Primary size	Design field diam.	Clear field diam.
4.25”	1/2”	0.28”
6”	3/4”	0.45”
8”	1”	0.62”
10”	1”	0.50”
12.5”	1”	0.36”
16”	1”	0.18”

Clear-field values should be considered minimums; instruments above 8” or with long f-ratios may support up to the design field while retaining enough free cone.

The diagonal mirror / baffling

The focal plane lies inside the system of mirrors; the flat diagonal throws it clear.
If the diagonal is placed to catch all light for a 1” field while not infringing the field rays between secondary and tertiary (upper position), insufficient residual cone remains to get the focus free. Increasing all tilts would lengthen the cone but worsen focal-plane tilt.
Solution: nestle the diagonal against the central beam between secondary and tertiary (middle position) — it intrudes into the lower edge of the field but is clear at center and upper field — then slide it along the central beam toward the tertiary (lowest position) until it catches only enough of the top-edge rays to produce a ½-magnitude vignetting loss.
Residual cone = length of cone the diagonal can field.
Free cone = the part of the residual cone on the observer’s side of the parallel secondary–tertiary beam. Stevick looks for at least 4 inches of free cone (less requires a low-profile focuser).
Diagonal minor axis = major axis / 1.4.
An alternative is to send light out the side panel (focuser near the altitude axis, nearly stationary eyepiece); the diagonal then sits on its side against the parallel beam.
Exact raytraced placement (8” f/12): the flat’s upper edge rides tangent to the collimated secondary→tertiary beam and it captures the whole converging cone — centre at 26.5” from the tertiary, surface 38.3° from the axis, half-length 0.70” (raytrace-verified: 100% cone capture, 0 rays blocked in any through-beam). This supersedes the approximate T4 = 45−T3 = 42.77° for construction; see TRACE.md §5.
Light baffle: surrounds the converging cone from the primary, fitting into the notches where light paths cross. Sized to just pass the central beam intact, so vignetting technically begins immediately off-center but blocks <10% of light at the field edge. The diagonal’s vignetting dominates.
Primary tilt is chosen ~1” larger than the obscuration-free minimum to allow an effective baffle; by raytracing the light just clearing the baffle, Stevick ensures it approaches no closer than 1” to field center, giving a 2” diameter stray-light-free field (protects even 2” O.D. eyepieces).

Worked designs

Example A — Stevick’s own 8” f/11.75 (from the Design Procedures)

Target was 8” f/12 (f.l. 96”); ground primary came out at 94” f.l. → f/11.75.

Primary r.o.c. = 188” → spherical-mirror curve = 0.4 × 188 = 75.2” radius.

Mirror	Diam.	Radius	Figure	Tilt (deg)	Dist. to next surface
pri.	8”	188	para	3.04	56.4
sec.	4.25”	−75.2	sph	5.32	75.2
tert.	5”	75.2	sph	2.28	28.75
diag.	1.63”	flat	flat	42.72	8.85
f.p.	1”	—	—	9.1	—

k = 1.75 at A = 0.4 → T2 = 1.75 × 3.04 = 5.32°; T3 = 5.32 − 3.04 = 2.28°.
Focal-plane tilt (Eqn 6) = 9.12° (raytrace exact value 9.07°).
Axis distances: a1 = sin(2·3.04)·56.4 = 5.974” → rounded to 6”; a2 = 12”; a3 = 3”.
Central beam diameter = A·D = 3.2”.
Minimum mirror diameters (clear field 0.62”): secondary = 3.572”, tertiary = 4.812” → uses standard 4.25” and 5” blanks.
Diagonal minor axis ≈ 1.6” → specified next standard size 1-5/8”.
Diagonal–tertiary separation along principal ray = 28.75”; residual cone 8.85”; nearly 5” of free cone available.

Example B — A.L. Woods’ 6” f/12

Pioneering SPT; primary left spherical (f/12 long enough). Displayed publicly at Astrofest 1993.

Example C — Bob Novack’s Folded Stevick-Paul (FSP), 6” f/10

Radius ratio A = 0.45 (slightly higher than 0.4 → slightly less focal-plane tilt).
Uses the flat to fold the light path, making the design more compact and eliminating the separate diagonal mirror (still 4 mirrors: primary, secondary, tertiary, folding flat). The flat sits in the collimated beam between secondary and tertiary (primary → secondary → flat → tertiary → focus), and the primary tilt is larger than the A = 0.4 table gives (2·T1 = 8.77° at f/10). See FSP.md.
Functions essentially the same as the standard SPT. An f/15 version can use a spherical primary.
Aberration balance: on/off-axis astigmatism and coma cancelled by mirror locations and tilts; spherical aberration is removed by the primary, put back by the secondary, and finally removed by the tertiary. Remaining distortion, image tilt and field curvature are minor.
Building notes: primary parabolized (soft Foucault shadows — used a very narrow slit); secondary & tertiary interferometrically compared after nulling the concave; flat via Ritchey-Common against the spherical primary; all four mirrors given 96% enhanced coating. Baffled between entering beam and secondary with black-velvet-covered plate; plain black baffle in the triangular area between secondary and flat.
Bob’s summary: for purely visual use without an image-tilting lens, the fastest practical SP is about f/10; f/10–f/15 is still fast relative to most TCTs. A folded f/12–f/15 with three spheres may be the easiest off-axis scope to make at reasonable speed (avoids very long radii).

Mersenne telescope — two confocal paraboloids (SPT shares this aspect).
Paul system (1935) — zero-power corrector; concave tertiary nominally contacted to the paraboloidal primary; large obstruction.
Baker-Paul three-mirror system — moved spheres farther from primary; turned edge on the convex secondary for 5th-order correction.
Tetra-Schiefspiegler (Michael Brunn) — US Patent 5,142,417; where Brunn independently found the arrangement.

Key reference

D. Stevick, “The Stevick-Paul Off-Axis Reflecting Telescope,” Amateur Telescope Makers Journal (ATMJ), Issue 3, pp. 4–9 (1993).

Exact raytrace companion

TRACE.md + the figures and code in this directory contain an exact raytrace of the 8” f/12 version (A = 0.40): CAD layout (sp_layout.png/.svg/.dxf), spot diagrams at 0.00°/0.25°/0.50° field (sp_spotdiagrams.png), and Airy-disk plots (sp_airy.png). Key verified results:

Principal-ray angles: 2·T1 = 5.945°, 2·T3 = 4.459°, 9.000° out of the tertiary (= the focal-plane tilt). All spacings at design values (57.6/76.8/38.4”).
Diffraction limited: geometric spot RMS 0.77 µm on-axis, rising only to ~1.06 µm at 1° field, versus the 8.05 µm Airy radius (λ=550 nm, f/12).
Physical mirror-axis tilts: primary +2.97°, secondary +0.74°, tertiary −6.73° (near-retro fold — these differ from Stevick’s T1/T2/T3, which are kink half-angles).
Diagonal clearance checked by raytrace: the flat threads between the collimated secondary→tertiary beam and the converging cone without blocking either (see TRACE.md §5). A naive cartesian placement obstructed ~10% of the collimated beam.