8” f/12 Stevick-Paul — Exact Raytrace & Layout

Deliverables generated by sp_model.py (raytracer + geometry) and make_outputs.py (drawings). All values derived by exact 3-D raytracing (general conic surfaces, tilted/decentered, reflection law at each surface).

Outputs

File	Contents
`sp_layout.png` / `.svg` / `.dxf`	CAD-style meridional layout with all dimensions, tilts and radii
`sp_spotdiagrams.png`	Spot diagrams at field 0.00°, 0.25°, 0.50° with the Airy disk circle
`sp_airy.png`	Airy disk diameter vs wavelength and vs f-number, with the spot RMS marked
`sp_model.py`	Reproducible model: conic raytracer + `build_sp()`
`make_outputs.py`	Regenerates all figures

Regenerate with: python3 make_outputs.py

1. How the geometry was reconstructed

The original ATMJ article and Stevick’s own raytracing program are not available, so the geometry was recovered from three primary sources and then refined by exact raytracing:

The design equations and prescription in the archived SEDS/AmSky design manual (DESIGN.md).
The construction rule: “draw the three mirror surfaces at right angles to the bisectors of the principal-ray angles” — i.e. each mirror’s surface normal is the bisector of the incident and reflected principal ray, and every mirror axis points toward its centre of curvature.
Bob Novack’s labelled Folded-Stevick-Paul diagram, used to fix the fold topology.

The topology (a Cassegrain-like fold) is:

incoming (-z)  ->  PRIMARY  ->  SECONDARY  ->  TERTIARY  ->  focus (needs diagonal)
                    (+z)          (-z)           (+z)

The key numbers fall out of the design equations exactly:

R1 = 2·N·D = 192", R2 = R3 = 0.4·R1 = 76.8" (mating spheres)
e1 = F1+F2 = 57.6", e2 = R3 = 76.8", e3 = F3 = 38.4"
T1 = C/N = 2.9725° (C=35.67 for 8”), T2 = 1.75·T1 = 5.2019°, T3 = T2−T1 = 2.2294°
lateral offsets a1 = 6.0", a2 = 12.0", a3 = 3.0"

The principal-ray angles at each gap are then fixed by the offsets and spacings, and one further angle — the direction out of the tertiary — was solved for by scanning for the aberration null. It is −9.000°, which is exactly the focal-plane tilt (Stevick’s approximation 4·T3 = 8.92°, raytraced exact ≈ 9.0°). This closed the system: the tertiary beam leaves at the focal-plane-tilt angle, and the focal plane is perpendicular to the incoming light, so the chief ray strikes it 9° off normal.

Result: all third-order aberrations vanish, giving a diffraction-limited spot (RMS 0.77 µm, versus the 8.05 µm Airy radius). This confirms the design’s claim.

2. Exact prescription (8” f/12, A = 0.40)

Coordinate system: z = optical axis (incoming light travels in −z), y = the “walk”/meridional direction, x = sagittal (out of plane). All mirror axes lie in the y–z plane.

Surface	Diameter	Vertex (y, z)	Radius	Conic	Axis tilt	Spacing to next
Primary — paraboloid	8.0”	(0.0000, 0.0000)	+192.000”	−1	+2.9725°	e1 = 57.600”
Secondary — convex sphere	4.25”	(5.9658, 57.2902)	+76.800”*	0	+0.7431°	e2 = 76.800”
Tertiary — concave sphere	5.0”	(11.9364, −19.2774)	+76.800”*	0	−6.7294°	e3 = 38.400”
Flat diagonal	1⅝” minor	in converging beam	∞	—	42.77° (T4)	—
Focal plane	(clear field 0.62”)	(5.9293, 18.6499)	—	—	tilt 9.00°	—

* the signed radius convention: the mirror axis points toward the centre of curvature; the secondary is convex as seen from the incoming beam, the tertiary concave.

Principal-ray beam angles

primary → secondary: 2·T1 = 5.9450° from +z
secondary → tertiary: 2·T3 = 4.4587° from −z
tertiary → focus: 9.0000° from +z (= focal-plane tilt)

Design parameters (Stevick): T1=2.9725°, T2=5.2019°, T3=2.2294°; a1=5.966”, a2=11.936”, a3=2.983”; central beam dia A·D = 3.2”.

Note the physical mirror-axis tilts (right column) are NOT equal to Stevick’s T1,T2,T3: because the fold is near-retro-reflective at the secondary and tertiary, the surface normal (bisector) sits near ±90°, so Stevick’s “tilts” are the half-angle of each principal-ray kink, not the geometric axis angle. Both are listed for that reason.

3. Raytrace results

Full 8” aperture, λ = 550 nm, diffraction reference r = 1.22·λ·F = 8.05 µm (diameter 16.1 µm):

Field	Image shift (y)	Geometric spot RMS	vs Airy radius
0.00°	5.929”	0.77 µm	0.10 × Airy
0.25°	6.354”	0.90 µm	0.11 × Airy
0.50°	6.781”	0.99 µm	0.12 × Airy
1.00°	7.639”	1.06 µm	0.13 × Airy

The spot is a fraction of the Airy disk at all field points → diffraction limited over the design field, exactly as the design promises. sp_spotdiagrams.png shows the spots (tiny) inside the red Airy circle; sp_airy.png shows the Airy diameter (16.1 µm at 550 nm / f12) dwarfing the 0.77 µm geometric RMS.

4. Notes, assumptions and limitations

Reconstruction, not the original file. The original ATMJ/QBasic–Zemax prescription was reconstructed from the archival prose plus Novack’s diagram, then verified to be diffraction-limited. It reproduces Stevick’s published values (R1=192, R2=R3=76.8, e1=57.6, e2=76.8, e3=38.4, a1=6, a2=12, T1–T3) and his claimed aberration-free image.
Third-order vs higher-order. The residual 0.77 µm is fifth-and-higher order (spherical); it is well within the Airy disk and is the reason the design eventually needs slower f-ratios — see BUILD_NOTES.md.
Diagonal / exit. The layout shows the flat at T4 = 42.77° throwing the beam out of the meridional plane. The diagonal is a fold only and does not affect image quality.
Vignetting / baffling. Not modelled here — the layout is the on-axis design; the ~½-magnitude edge vignetting and the light baffle are design/mechanical issues covered in DESIGN.md and BUILD_NOTES.md.
Units: inches unless stated; wavelength in nm/µm for the diffraction terms.

5. Diagonal placement & obstruction check

The flat diagonal sits inside the converging tertiary→focus cone and must thread between that cone and the collimated secondary→tertiary beam, which run within a fraction of an inch of each other.

Original cartesian placement was wrong. Placing the diagonal at 0.72·e3 along the chief ray with a ±1.4″ extent put its upper end at y = 8.64”, while the lower edge of the collimated beam is at y ≈ 8.13” there. Raytrace: 63 of 613 aperture rays (~10 %) of the collimated secondary→tertiary beam were blocked — a real aperture obstruction.

Corrected placement (now used in sp_layout.png / .svg / .dxf), chosen so the diagonal’s upper edge rides tangent to the collimated beam and it still catches the whole cone:

centre on the chief ray at 26.5” from the tertiary (z ≈ 6.9”),
surface 38.3° from the axis, half-length 0.70”,
verified: 100 % of the converging cone captured, 0 rays blocked in the incoming, primary→secondary, or secondary→tertiary beams.

Why this is delicate. Because the collimated beam and the converging cone leave the tertiary nearly together (they are tangent around z ≈ 9” and overlap by up to ~0.9” closer to the tertiary), the diagonal has only a narrow window in which to sit. This is exactly the situation Stevick describes — “nestle the diagonal up against the central beam … slid along the central beam toward the tertiary … until it catches only enough of the rays forming the top edge of the field to produce a one-half magnitude loss” — and is the reason he insists on the full-scale layout before cutting metal. The margin is small; any builder should re-derive the diagonal position from their own full-scale drawing and check the collimated-beam clearance.

Caveat: this reconstruction’s on-axis image is diffraction-limited, but the collimated and converging beams pass within a fraction of an inch and are not separated by a wide margin as in a fully-detailed mechanical model. Treat the diagonal position as indicative and verify against the finished layout.