Ray Tracing on a Real Lens Bench

Detailed abstract

Purpose

A ray-diagram trainer built on a realistic optics bench, for introductory physics (conceptual or algebra-based). It connects the textbook three-ray construction to what a student actually sees on a bench: - where the image forms and why; - real versus virtual images; - magnification and inversion; - why real thick lenses differ slightly from the thin-lens equation.

What it can do

• Bench hardware:
• a 1200 mm Thorlabs-style dovetail rail on a breadboard;
• an LED-backlit arrow (with a side flag, so the up/down and left/right flips show) as the object;
• one or two N-BK7 lenses, or one spherical mirror, in any of six sizes: Ø1", Ø2", Ø3" (default), Ø4", Ø8" or Ø10". Ø1"–Ø3" sit in Thorlabs LMR-series mounts; Ø4"–Ø10" in large lens cells (verify). Each size has its own catalogue: the same lens types and focal lengths, thicker for the bigger diameters, and only lenses whose surfaces are not near-hemispheres. The Ø3" and larger lenses are bigger than the candle, so the parallel ray from the flame tip gets through;
• a white screen on a post.
• Objects: the object has a fixed size (arrow 15 mm, candle 28 mm, apple 28 mm); its up/down slider moves it above or below the axis (the arrow within its Ø2" mask, the candle and apple on their post). An LED-backlit arrow, a candle (self-luminous flame lighting its own wax) or an apple (lit by room light). The candle and apple are 3D: their front surfaces are traced point by point, so depth and colour show up in the image.
• Firefly: a generic lightning bug (about 14 mm long) clinging to a grass stem with its underside toward the lens. Only its 3 × 4 mm lantern glows; the body is dark. The lantern can flash (a stylised 0.3 s flash every 2 s; real species differ, e.g. the common eastern firefly Photinus pyralis flashes for about half a second every 5–6 s), stay on for alignment, or go dark. When it is dark, no rays are drawn and the screen goes black.
• Lens choices:
• plano-convex lenses f = 75–300 mm (Thorlabs LA-series), bi-convex f = 100 and 200 mm, plano-concave f = −100 and −200 mm, and bi-concave f = −100 mm;
• cylindrical lenses (plano-convex f = 100 and 200 mm, plano-concave f = −100 mm) and a plano-toric (astigmatic) lens with f = 100 mm in one plane and 150 mm in the other.

Each lens is drawn with its true thickness and surface shape. Any lens can be flipped or turned about the optical axis; gold ticks mark a cylinder's axis. - Idealized thin lens: a checkbox per lens swaps the real glass for the textbook lens: a zero-thickness plane with the same focal length that changes every ray's slope by exactly −h/f (no aberration, no colour). The ray through F then leaves exactly parallel and every ray from a point meets at one image point, so you can compare the textbook picture with the real thick lens. It has no surfaces, so no C is drawn for it. - Mirror mode: a concave (f = 100, 150, 250, 500 mm) or convex (f = −100, −250 mm) spherical mirror on the rail, facing the object. Rays reflect exactly off the true sphere (law of reflection at the local normal); C, F, the parallel, vertex, focal and through-C rays, real images caught on the screen and virtual images behind the mirror on the probe plane. The screen catches only the returning light (in the lab: a small card off to the side, or the mirror tilted slightly). Presets: object beyond C, object inside F (makeup mirror), convex mirror. - Ray toggles: a row of switches over the 3D view turns each diagram ray (and the full bundle) on or off, with "all off" and "all on". - Two ray planes: the diagram rays and the 2D view can be shown in the vertical (y) plane or the horizontal (x) plane. That is how you see astigmatism: each plane has its own focus, and the screen shows a line focus at each. - Moving things: drag the object, the lenses, the screen and the probe plane along the rail in 3D (or use the sliders). Carriers cannot pass through each other; the probe plane can, because it is not physical. - Turning a lens on its post: drag the blue ring on its carrier (or Shift-drag the lens, or use its slider), up to ±45°. The traced rays and the screen image include the tilt (you see coma and astigmatism appear); the paraxial readouts do not. - Ray-diagram rays, traced exactly through the glass: - parallel → through F′; - through the centre (nodal point) → undeviated direction; - through F → parallel; - to the lens edge; - toward the centre of curvature C of the first curved surface: it meets that surface at normal incidence, so it is not bent there, only at the flat back face. - Both faces get a C: C₁ for the front face, C₂ for the back face, marked where they are. A flat face has its C at infinity, and the diagram says so on that side. - Rays clipped by a mount stop there (dot). - Dashed grey backward extensions show where a virtual image is. - Optional full bundle of 15 rays from the tip. - White screen: shows what really lands on it, computed by tracing thousands of rays from every point of the arrow: - sharp at the real image; - blurred elsewhere; - coloured fringes with the white LED (chromatic aberration). - Probe plane: a see-through, non-physical plane that plots where the outgoing rays, or their backward extensions, cross. It is the only place a virtual image appears. - Real image in space: shown as a glowing arrow where light actually converges (and the intermediate image in two-lens setups). - 2D ray diagram with true lens profiles, mount apertures, F, F′, 2F, 2F′ and principal planes H, H′. An optional thin-lens textbook construction can be overlaid for comparison. - Readouts: - focal length at the chosen wavelength; - centre and edge thickness; - principal-plane offsets; - s and s′ from the principal planes, with a check of 1/s + 1/s′ = 1/f; - the thin-lens guess measured from the lens centre; - image position, height and magnification; - traced best focus and blur; - blue versus red image position. - Presets: - object beyond 2F, at 2F, between F and 2F, at F; - inside F (magnifier); - diverging lens; - 4f relay; - two-lens microscope; - converging + diverging pair. - Generated parts table and CSV.

Limitations

• Geometric optics only: no diffraction, so the "sharp" image is limited only by aberrations and the screen's pixel size.
• Rays and surfaces:
• surfaces are perfect spheres;
• no reflections from lens surfaces (ghosts), no coating losses, no scatter from the ground-glass edge.
• Object illumination: the light is assumed to fill the first lens (in the real setup a diffuser does this). Light that misses the lens is not drawn on the screen.
• Probe plane: it accepts the full lens aperture. Your eye's pupil (2–7 mm) would pick a small part of it, so a virtual image looks sharper to the eye than the probe's full-aperture picture.
• Cylindrical and toric lenses: Thorlabs cylindrical lenses are mostly rectangular; here they are cut round to fit the same Ø2" mount (verify part numbers). Off-axis rotations use the paraxial cos²/sin² power split for the readouts; the traced rays and images are exact.
• Candle and apple: sizes are typical, not measured. The apple is treated as a lit surface with fixed colours (no shading changes when it moves).
• Firefly: the bug and lantern sizes are typical, not measured; the lantern is a uniform yellow-green patch with a fixed spectrum. The screen uses automatic exposure, so it shows the firefly image as bright as the lamp image; in the lab the firefly is far fainter (a dark room and a dark-adapted eye or a long camera exposure are needed).
• Glass data: N-BK7 index from the Schott Sellmeier formula. Lens geometry is computed from the catalogue focal length with a 3 mm edge thickness (positive lenses) or 5 mm centre thickness (negative lenses). This reproduces LA1050 (R = 51.5 mm, tc = 9.7 mm) but is ASSUMED for the others.
• Part numbers: bi-convex and concave Ø2" part numbers are given as series with "verify".
• Hardware dimensions: rail, carrier and mount dimensions are close to Thorlabs drawings but simplified.
• Screen exposure: the screen and probe use automatic exposure (like a phone camera), so brightness differences between setups are not to scale.

How realistic is it?

• Best case: image positions, sizes, orientations and the real/virtual distinction are exact for the modelled lenses. Spherical and chromatic aberration come out of exact Snell's-law tracing, not from formulas. A student can predict on paper and check on the bench to within a few millimetres.
• Likely case: a real bench differs by the lens tolerance (±1 % in f), how well the lens is centred, and where the student judges "best focus" by eye (several mm for slow lenses).
• Hidden burdens:
• Everything must be centred on one height. With Ø2" optics on posts that means setting every post to the same height first.
• A bright, even object (LED + diffuser) is what makes the screen image visible in a lit room.
• Virtual images have to be seen by eye, looking back through the lens. The probe plane stands in for that.

The modeling: core physics and how it is handled

Core physics and how it is handled

1. Refraction at each surface (Snell's law). At every spherical glass surface a ray bends by n₁ sin θ₁ = n₂ sin θ₂, measured from the surface normal.

Handled: exact 3D vector tracing: - intersect the ray with the sphere (or plane); - compute the normal; - refract with t = η d + (η cos θᵢ − √(1 − η² sin² θᵢ)) N.

Rays outside the clear aperture of the retaining ring hit the mount and stop (45.7 mm for Ø2" in LMR2/M + SM2RR; 69 mm for the default Ø3" in LMR3/M + SM3RR, ASSUMED). Rays below the mount hit the post. Rays above the mount pass by.

1b. Surface shapes. Every surface is a biconic with curvatures c_y and c_z:

s(y, z) = (c_y y² + c_z z²) / (1 + √(1 − c_y² y² − c_z² z²))

This is exactly a sphere when c_y = c_z and exactly a cylinder when one curvature is zero; for the toric lens it is the standard biconic approximation. A turned lens is handled by rotating (y, z) into the lens frame.

Handled: the ray–surface intersection is found by Newton's method from the vertex plane; the normal comes from the gradient (1, −∂s/∂y, −∂s/∂z). In each plane the paraxial power uses c_plane = c_y cos²φ + c_z sin²φ (φ = lens rotation).

2. Glass index and dispersion. n(λ) for N-BK7 from the Sellmeier formula: n_d = 1.5168, about 1.524 at 470 nm and 1.515 at 625 nm.

Handled: each wavelength is traced separately. With the white LED, three colours (470, 550, 625 nm) are traced and added.

3. Thick-lens cardinal points (paraxial). Each surface has power P = (n₂ − n₁)/R. Glass of thickness t is a translation t/n. Multiplying the 2 × 2 ray matrices gives [[A, B], [C, D]], and from it: - f = −1/C; - F′ = x_b − A/C; - F = x_a + D/C; - H′ = x_b + (1 − A)/C; - H = x_a + (D − 1)/C.

The image of an object at distance s₀ before the first vertex is at s_i = −(A s₀ + B)/(C s₀ + D) after the last vertex, with magnification m = A + C s_i.

Handled: exact (paraxial). This is what the readouts and the "move to image" buttons use. For real images, the traced best focus (minimum RMS blur of the full-aperture tip bundle) is used instead.

4. The diagram rays from the arrow tip: - parallel to the axis; - aimed at the front principal (= nodal, in air) point H; - along the line through the front focal point F; - to 92 % of the clear radius.

Handled: launched from the paraxial cardinal points, then traced exactly. Because they are real rays, they miss the ideal image slightly (aberration) and can be clipped by the mount, exactly as on a bench.

5. Real versus virtual image. After the last lens, the rays from one object point either converge (real image: light is there, and a screen shows it) or diverge (virtual: only their backward extensions meet).

Handled: - Screen: stops the light; it records only the points where rays actually arrive. - Probe: takes each ray's final straight line and evaluates it at the probe position, forward or backward. It also ignores the screen, because it is not a physical object.

6. Image formation. For each of about 80 points on the arrow, about 400 rays fill the first lens aperture (stratified random samples, rotated per point), per wavelength. Each landing point is spread bilinearly onto a 300 × 300 grid (100 mm screen, 120 mm probe), lightly smoothed and auto-exposed.

Firefly light. Fireflies make light by bioluminescence: the enzyme luciferase oxidizes luciferin, and the excited product emits yellow-green light peaking near 560 nm. The process is very efficient, with little heat (it is often called "cold light"). The lantern is an extended, incoherent source like the candle flame: every point on it sends light in all directions, so the lens images it exactly like any other object (a small, inverted glowing patch at the image plane). Handled: the lantern is a grid of emitting points across a 3 × 4 mm patch, with a yellow-green colour (mostly the green band, some red, almost no blue); the body emits nothing, and when the lantern is dark the object has no points at all.

How the simulation is built

• src/rt_model.js:
• N-BK7 Sellmeier formula;
• lens catalogue (geometry from the lensmaker equation);
• surfaces, ABCD and cardinal points, image location;
• exact 3D tracing;
• diagram rays, image splatting, best focus.

Pure functions; runs in Node. - src/rt_app.js: - three.js bench (breadboard, rail, carriers, post holders, posts, LMR2/M mounts with retaining rings, lathe-profiled lenses, LED/diffuser/mask object, screen, probe plane); - drag-on-rail; - 3D rays and images, 2D diagram, readouts, parts table. - src/rt_shell.html + src/build.py: page; python build.py inlines everything into ray_tracing_bench.html.

How to recreate it

1. Write the four surfaces (two per lens) as {vertex x, R, n₁, n₂}.
2. Multiply the ray matrices to get f, F, F′, H, H′ and the paraxial image.
3. Trace rays exactly: - intersect each sphere: |p + t d − c|² = R², taking the root nearer the vertex; - refract with the vector form of Snell's law; - stop rays outside the clear aperture.
4. Launch the four diagram rays from the arrow tip using F, F′ and H. Draw backward extensions of the final segments when the image is virtual.
5. For the screen, trace bundles from many object points and histogram where they land at the screen. For the probe, histogram the final lines at the probe position.
6. Checks: - LA1050 (computed R = 51.7 mm vs the catalogue's 51.5 mm, tc = 9.7 mm) gives f = 100.0 mm at 587.6 nm, so the catalogue lens is reproduced to about 0.4 %; - an object at 2F images at 2F with m = −1 (measured from the principal planes); - inside F gives a virtual, upright, magnified image that the screen never shows.

Key parameters