STEM Column Ray Trace

How to read it. Solid rays are the real illumination, traced by the Ray Optics engine from a point source through the VOA. Teal dashed lines are sample images, orange dashed lines are diffraction planes. Both are read off the post-sample transfer matrix [[A,B],[C,D]]: an image plane is where B = 0 and a diffraction plane is where A = 0. Neither depends on the condenser, so both stay correct in TEM and STEM alike. The illumination crossovers listed in the readout are a different thing — images of the source. With a focused probe they land on the sample images; with parallel illumination they land on the diffraction planes instead, which is why they cannot be used to classify planes.

The number under each label is the primary beam radius at that plane. The VOA is fixed at 0.40 mm radius, which together with C2 and C3 sets the convergence semi-angle, the probe size, and which rays are physically blocked.

Lens sliders are logarithmic in focal length, spanning 10000 mm down to the 2 mm strength limit at about 0.86% per step, with off at the bottom of travel and stronger upwards. That limit is what bounds the EELS solver: reach goes roughly as M_max ≈ 800/f_min, so 2 mm buys magnification to about 400× and camera length to about 8000 mm. It is the lens strength that binds, not the spacing between the projector lenses.

Radial exaggeration K is cosmetic. It (currently K×) is pure vertical zoom — the beam is a fraction of a mm across in a 1500 mm column, so it has to be stretched to be seen. That is exact rather than a cheat: an ideal thin lens is perfectly linear in (y, dy/dz), so scaling every radial quantity while leaving z and every focal length alone reproduces the identical diagram, and no reported number moves.

Sample defocus dz shifts the specimen off the OL1/OL2 midpoint, positive toward OL2, up to ±1 mm — enough to run a confocal depth series. Note that the diffraction planes do not move when the sample does: a ray leaving the sample parallel to the axis stays parallel until OL2, so they are set by OL2 and downstream only. The image planes do move.

Reference plane vs specimen. The reference plane at z = 640, midway between OL1 and OL2, is the objective's nominal object plane and never moves — every conjugate, camera length, magnification and solve is referenced to it, just as a real column's projectors are aligned to a fixed height. The specimen sits at 640 + dz and affects only the Bragg cone origin, the beam radius on the specimen, the probe-defocus readout and its own marker. Sweeping dz therefore moves nothing optical; it just takes the probe off the specimen.

The two illumination modes are both C2 + C3, because the condenser is what sets illumination. Those two lenses together fix the mode and the convergence angle: α = |marginal ray height at OL2| / f_OL2, the slope OL2 imparts to the marginal ray. Focused, that is the usual convergence semi-angle; parallel, it is the angle OL2 focuses the illumination to, and the illuminated radius is exactly r = α f_OL2. The α slider spans 0.55–50 mrad and applies live, keeping whichever mode you last chose; the VOA stays fixed at 0.40 mm. Below about 0.14 mrad focused the solve refuses. The slider bottoms out at 0.55 mrad, comfortably above that floor, so every position solves. Shrinking the aperture is what buys smaller angles on a real column, which is why the 5 mrad default is set that way rather than by straining the condensers.

The two EELS coupling modes are exact mirrors of one another, and mutually exclusive: diffraction-coupled puts the diffraction pattern on the entrance aperture and the sample image on the EELS focal plane; image-coupled swaps them. Either pairing uses up two degrees of freedom, so a two-lens solve would leave the third quantity as whatever fell out — PL1 is the lens that buys it back. In diffraction coupling the knob is camera length L and Mag = 200/L follows; in image coupling the knob is magnification M and L = 200/M at the focal plane follows (200 mm being the gap between the two EELS planes). Because both conditions live entirely in the post-sample matrix, matching works identically under focused and parallel illumination — the same PL1/PL4/EL solve either way.

Where A and B come from. Write the ray-transfer matrix from the reference plane to whichever plane you care about as [[A, B], [C, D]]. A ray leaving the reference plane with height y and slope θ arrives at height Ay + Bθ. At a diffraction plane A = 0, so it arrives at Bθ — position depends only on angle — and that B is the camera length L in r = L θ. At an image plane B = 0, so it arrives at Ay, and that A is the magnification. Both are shown at both EELS planes and greyed out wherever the plane is not of the matching kind, because that is the only place each means what it is called.

Camera length defined this way is the same number as the textbook L = f_OL2 M — OL2's back focal plane at z = 720 is conjugate to the aperture at M = −1, so 40 × 1 = 40 mm. It is perfectly meaningful in STEM: it is what decides which scattering angles get inside the aperture.

Bragg cones are a construction: the illumination re-emitted at ±θ_B, amber for +1 and violet for −1. Each cone ray leaves from its own primary ray's position on the sample, so a focused probe gives cones from a point and parallel illumination gives cones across the whole illuminated width. They merge at every image plane and spread into separated discs at every diffraction plane, which is what a CBED pattern shows — but the exact plane positions come from the paraxial solver, not from the cones.

Ideal thin lenses only; no spherical or chromatic aberration, no space charge, no lens bore limits. Positive focal lengths only. Element positions are fixed.

Sean Kung · sean.kung@ubc.ca · bundles Ray Optics Simulation 5.4 (Apache-2.0, see NOTICE)