The entrance pupil walks with the angle

The hole a scene actually sees located the centre of projection of a lens. It is not the aperture stop and not any piece of glass: it is the entrance pupil, the stop imaged by whatever glass sits in front of it. For a thin lens with its stop 18 mm behind, the chief rays from every object distance between 0.9 and 60 metres crossed the axis at one point, 28.1 mm behind the lens, to $3.6 \times 10^{-15}$ mm. A picture is a projection from there.

That essay ended with a list of what its model did not have, and one item on it is the subject here: its pupil is a perfect image of the stop rather than one that shifts and distorts with field angle. A thin lens images the stop without aberration, so every chief ray, from any direction, extended in front of the lens, passes through the same point. Real glass is thick and curved, and a curved surface bends a ray arriving from far off the axis differently from one arriving near it. The chief rays of a real wide-angle lens need not share a point at all.

The front of a wide-angle design #

A wide-angle lens gets its field by putting a strongly curved negative element at the front, which takes rays arriving from far off the axis and bends them toward it before they reach the stop. The entrance pupil depends only on the glass in front of the stop, so the smallest model that can show the effect is that front element and the stop behind it.

The model here is a negative meniscus, convex toward the scene: a first surface of radius 42 mm, a second of radius 14 mm, 3.5 mm apart, in glass of index 1.6204, with the aperture stop 22 mm behind the first surface. It is the front of a wide design, not a whole lens, and it is enough.

A chief ray is defined by passing through the centre of the stop, so it can be traced exactly by starting there and working outward: leave the stop’s centre at a stated angle, meet the second surface, bend by Snell’s law, meet the first, bend again, and leave into the scene. Its line in front of the glass, carried back toward the lens, crosses the axis somewhere. For a thin lens that crossing is the same place for every chief ray. The trace here uses the full vector form of the law at each sphere and makes no small-angle assumption anywhere, which is the same method what a ray does at a surface used to show a pool’s floor, seen at eighty degrees, appearing at a fifth of its depth.

They do not cross at one place. The section draws four chief rays, at 12°, 33°, 54° and 74° of field, and carries each back to the axis: they cross at 15.00 mm, 14.52 mm, 13.46 mm and 11.58 mm behind the front vertex. The further off the axis a chief ray arrives from, the nearer the front its crossing lies.

The paraxial pupil, and the walk from it #

The pupil the textbooks locate is the paraxial one: the crossing made by chief rays of vanishing angle, computed by the small-angle method that imaging formulas use. The chief rays traced at the very smallest angles converge on it, and the two routes — a paraxial calculation that never traces a real ray, and the limit of real traces — agree to about $10^{-5}$ mm. For this front group the paraxial entrance pupil sits 15.07 mm behind the front vertex.

That agreement is worth dwelling on because it is the check that matters. The walk measured below is a difference from the paraxial pupil, and a difference is only as trustworthy as its origin. An origin placed wrongly by the full distance to the front vertex would make every walk look like fifteen millimetres and every seam error look enormous, while the differences between rays still looked sensible; comparing the paraxial calculation with the limit of the traced rays is the test that would catch it.

Measured from there, the crossing walks. At 20° of field it is 0.20 mm nearer the front than the paraxial pupil; at 40°, 0.85 mm; at 60°, 2.11 mm; at 80°, 4.23 mm. The walk grows roughly as the square of the angle at first — doubling the angle from 20° to 40° multiplies it by a little over four — and faster further out, which is the shape a spherical aberration of the pupil takes. The entrance pupil is a segment on the axis, four millimetres long across eighty degrees of field.

The control is a front group of the same thickness and index whose two surfaces are both centred on the stop. Every chief ray from the stop’s centre meets each surface along its normal, so it is not bent at all, and every one crosses the axis at the stop itself, 30.00 mm behind the front vertex, at every angle traced. Its pupil does not walk, to the arithmetic floor. The walk above is therefore the curvature of the meniscus doing something to rays that a concentric element does not, and not an artefact of the tracing.

The same effect, on a ball of water #

The phenomenon has appeared before, in a different subject, and the parallel is exact enough to be worth drawing.

A ball of water has no eye either fitted a common point to the rays refracted through a sphere of water and found that they missed it by 1.3 mm on a ball of fifty-millimetre radius, with their axis crossings spread over 18.6 mm at seven tenths of its aperture — while at two per cent of the radius the same fit returned 24 nanometres. A ball does have a centre, that essay concluded, one at zero aperture and none by the time it is gathering any light. A picture through water has no viewpoint had found the same for a flat interface, and the centre a scroll does not have for a painter who walks, where the miss is the spread of the walk itself.

The entrance pupil is the same statement about a lens, sorted by field angle rather than by aperture. A lens does have a centre of projection — at zero field angle — and at any other angle its chief rays cross somewhere slightly different. The pinhole model of a camera is exact for the rays near the axis and approximate for the rest, and the approximation is worst exactly where a wide-angle lens earns its name.

It is also the same shape as a mirror without a focus. Where the focus went found that reflected rays which do not meet at a point meet each other in pairs, and the curve they are all tangent to is what a spherical mirror has instead of a focus. The chief rays in this section are a family of the same kind: each neighbouring pair meets somewhere, the meeting points trace an envelope rather than collapsing to one point, and by the symmetry of the section that envelope reaches the axis at the paraxial pupil. What the walk figure plots is the other thing a family of lines has, where each member crosses the axis, which is the quantity a rotation point is set by.

What the walk costs a panorama #

The practical consequence is for anyone who rotates a camera to stitch a panorama, and the eye is a place, not a point set out why the rotation point matters: turn about any point other than the centre of projection and near and far objects slide against each other in the overlap, by an amount that falls as one over their distance.

A walking pupil means there is no single centre to turn about. Consider two frames of a panorama yawed 60° apart about a pivot, and a point on the seam between them. Each frame sees that point 30° off its own axis if the point is level with the lens, and further off if it is above or below — at 40° of elevation along the seam, the field angle reaches 48°. Each frame therefore sees the point from wherever its chief ray for that field angle crosses its axis, and those crossings, in the two frames, are in different places in space whenever the pivot is not at them.

Pivoting exactly at the paraxial pupil, the two sight lines to a point one metre away disagree by 1.57 arcminutes where the seam is level with the lens and by 4.39 arcminutes at 40° of elevation — 4.39 px on a picture with 60 px per degree, a visible doubled edge in a stitched close-up. The paraxial pupil is the right pivot for rays near the axis, and the seam is not near the axis.

The best single pivot for this seam is the middle of the walk the seam’s field angles span. Those angles run from 30° to 48°, across which the pupil walks 0.82 mm, and a pivot 0.87 mm in front of the paraxial pupil sits in the middle of it. There the worst misregistration along the seam is 1.41 arcminutes — a third of the paraxial pivot’s, and still not zero. No pivot can reduce it to zero, because no point is the centre of projection for every direction on the seam at once.

This is the pivot problem the pivot that is not the eye measured for a camera turning about its tripod screw, with one thing changed. There every ray passed the pivot by the offset times the sine of its angle off the frame’s axis, and the offset was one fixed length. Here the offset depends on the angle, because the point the pivot ought to be at moves along the axis as the ray moves across the field.

At half a metre every number doubles: 8.79′ at the paraxial pivot, 2.82′ at the best one. That is the one-over-distance law measured for a wrong pivot, and it has a new reading here. The walk is a fixed length set by the glass, so its parallax falls as one over the subject’s distance exactly as a misplaced pivot’s does, and a panorama of a distant landscape never shows it. A panorama of a room does, and so does any stitched close-up.

What closer frames buy back #

A misregistration along a seam has two ingredients, and they can be separated exactly. For a point on the seam, each frame sees it from a centre sitting some distance along its own axis from the pivot — the walk at that point’s field angle, measured from wherever the pivot is. The two axes are the frame spacing apart, so the two centres are separated sideways by that offset times $2\sin(s/2)$ for a spacing $s$, and the disagreement at distance $D$ is that separation over $D$. At 60° spacing, $2\sin 30°$ is exactly one: the top of the seam sees centres 1.28 mm from the paraxial pivot, and 1.28 mm at one metre is 4.39′. The level point sees centres 0.46 mm out, which is its 1.57′. The formula matches the traced misregistration at every spacing measured, to a fifth of a per cent.

That makes the remedy visible, and it is the one the essay on a panorama’s two halves of parallax found for the half across the seam: it falls as the sine of half the frame spacing, so more frames buy it down.

At 45° spacing the paraxial pivot leaves 2.85′; at 30°, 1.69′; at 20°, 1.06′; at 10°, 0.51′. The best pivot does about twice as well at each: 1.09′, 0.74′, 0.50′, 0.25′. Two things are falling together. The factor $2\sin(s/2)$ falls in proportion to the spacing, and the seam’s field angles fall with it, so the pupil has walked less by the time a ray reaches the seam.

What closer frames cannot remove is the part set by the frame’s own height. A point level with the lens sits $s/2$ off each axis, so as the frames close up its field angle goes to zero and so does its walk; at 10° spacing it is down to 0.007′. A point 40° up the seam is at least 40° off each axis however close the frames are, and there the pupil has walked 0.85 mm. At 10° spacing the top of the seam still sees centres 0.85 mm from the paraxial pivot, and the only thing keeping its misregistration small is the sine of a small spacing.

What a nodal slide is calibrated for #

The ordinary way to find a lens’s rotation point is to put two objects in line near the centre of the frame, rotate a little, and slide the camera until they stay in line. That procedure locates the crossing of chief rays near the axis — the paraxial pupil — because the objects are near the centre of the frame.

The seam of a panorama is not near the centre of either frame. For 60° spacing it sits 30° to 48° off each frame’s axis, where the pupil has walked most of a millimetre on this front group. So the slide setting found by the standard procedure is the correct setting for a direction the panorama never stitches on, and a better setting for the seam exists 0.87 mm away from it. The difference is small on this model and would be larger on a design with a stronger front element; the essay does not claim a figure for any real lens, only that the setting found at the centre of the frame is not the one the seam needs.

The better procedure follows directly. Find the rotation point by putting the two objects in line at the edge of the frame where the seam will fall, not at its centre, so that the slide is set for the field angles that actually overlap — and, for a tall frame, at the height along the seam where the worst misregistration would otherwise sit.

What this model does not have #

It is the front of a lens, not a lens. A real wide-angle design has several elements in front of the stop and several behind, and its pupil walk is the combined effect of all the glass in front. A single meniscus demonstrates that the walk exists and has the shape of a spherical aberration of the pupil; it does not predict the walk of any particular lens, which could be smaller if the designer corrected for it or larger in a more extreme retrofocus design.

It is a single wavelength. Glass bends different colours by different amounts, so each colour’s pupil walks slightly differently, and a panorama’s seam carries colour fringes the model does not compute.

And the concentric control is not a free design. A front element concentric on its stop bends no chief ray, which is why its pupil does not walk, and it is also why it contributes nothing to widening the field. The walk is the price of the bending that makes a lens wide. Lenses built for panoramic work can choose their elements to reduce it; nothing here measures how far that can go.

Still open: how far focusing carries the pupil #

Everything above holds the lens at one focus. The eye is a place, not a point noted that the entrance pupil also moves slightly with focus, and that this is why panorama heads are set up at the focus distance the panorama will be shot at. For the simplest kind of focusing — the whole lens moving away from the sensor as a unit — the displacement has a clean form, because the pupil travels with the glass. Focusing is a zoom measures that the lens stands 5.56 mm further from the sensor when a 50 mm lens focuses at half a metre, and the question still open takes that number to the rotation point: how far the entrance pupil moves relative to a camera body’s tripod mount across the focus range, what that does to a panorama calibrated at infinity and shot at a metre, and whether the change of rotation point with focus is larger or smaller than the walk with field angle that this essay measured.