The roof that splits a binocular's image

A binocular has to do something a telescope does not: it has to show the world the right way up and the right way round. A simple telescope’s objective lens makes an inverted image, and turning it over needs either more lenses, which make the instrument long, or reflections. The classic solution, Ignazio Porro’s prisms of the 1850s, folds the light through two right-angled prisms set crosswise, and gives binoculars their familiar dog-leg shape. The other solution puts a roof on a prism: two faces meeting at a right angle, like the ridge of a house, which turn the image over sideways in a single reflection pair and allow a straight, slim instrument.

Both rely on total internal reflection, which the angle past which light cannot leave found to reflect every photon beyond the critical angle, with no silvering and no loss. And for most of the twentieth century roof-prism binoculars, for all their convenience, were known to be slightly less sharp than Porro binoculars of the same quality. Nothing was wrong with the glass or the polishing. The reflections were losing something that a lossless reflection was not supposed to lose.

A reflection that keeps the light and shifts the phase #

Total reflection returns all the light, but not unchanged. The wave reflected from beyond the critical angle has its phase shifted, and the shift depends on its polarisation. For light polarised across the plane of incidence (s) and in it (p), the lags are

$$\delta_s = 2\arctan\frac{\eta}{n\cos\theta}, \qquad \delta_p = 2\arctan\frac{n\eta}{\cos\theta}, \qquad \eta = \sqrt{n^2\sin^2\theta - 1}.$$

The reflection that happens where the glass is not traced these phases to the evanescent wave that reaches beyond the surface before the light turns back: the wave lingers outside for a moment, and how long depends on its polarisation. The same evanescent wave, reaching a tenth of a micrometre into the medium beyond, is what the mirror that lights a tenth of a micrometre used to illuminate only the molecules touching a microscope slide; here it is the reason a perfect reflection is not a neutral one.

The difference between the two lags rises from zero at the critical angle, peaks at a little over 46° for crown glass, and falls back to zero at grazing incidence. The retarder with no crystal in it put that difference to use: Fresnel’s rhomb reflects light twice at the angle where the difference is 45°, turning linear polarisation into circular with no birefringent crystal. In a binocular the same difference is a nuisance. It is harmless in itself — a uniform change of polarisation across the whole beam does nothing to the image — and becomes harmful only when different parts of the beam are changed differently.

Two halves that meet the faces in opposite orders #

A roof divides the beam into two halves along its edge. A ray in one half strikes the left face first, then the right; a ray in the other half strikes the right face first, then the left. After both reflections every ray is travelling in the same direction, and geometrically the two halves are a perfect image of each other.

Polarisation is another matter. Each face resolves the light into its own s and p and shifts them by different phases. If the ray travels square to the roof’s edge, both faces share the same plane of incidence, the two reflections act on the same pair of polarisations in the same way, and the order makes no difference. If the ray is tilted out of that plane — as it is in every roof prism that folds the beam, where the light meets the roof faces at 60° rather than 45° — the two faces’ planes of incidence are different, the second reflection’s s and p are mixtures of the first’s, and the result of reflecting first from one face and then the other is not the same as the reverse.

The geometry fixes the angles. If the beam is tilted by an angle $\psi$ out of the plane square to the roof edge, each face is met at an angle $\theta$ with $\cos\theta = \cos\psi/\sqrt2$: 45° for a beam square to the edge, 60° for the Amici prism’s tilt of 45°. The planes of incidence of the two faces then differ by an angle that grows with the tilt, and at each face the light’s polarisation is split along that face’s own s and p. Tracing a ray means following its field as a vector through the two reflections, resolving it into the new s and p at each face and applying that face’s two phase lags — and doing it once in each order.

The two halves of a horizontally polarised beam leave the roof polarised almost at right angles to each other, as thin ellipses tilted opposite ways. Each half has lost no light. But two beams in different polarisation states cannot interfere completely, and an image is made by interference: the sharp core of a star’s image is where the light from every part of the aperture arrives in step. If the halves cannot add their amplitudes fully, the core weakens and the light spreads.

Why matching polarisation is the price of interference #

That two beams must share a polarisation to interfere was established by Fresnel and Arago in 1819, in experiments that helped prove light a transverse wave: two beams from the same source, polarised at right angles to each other, give no fringes at all, however perfectly their paths are matched. The reason is in the addition of fields. Interference is the cross term when two fields are added and squared, and the cross term of two fields at right angles is zero, because their product has no component to survive the square. Two beams partly alike in polarisation interfere partly, in proportion to the overlap of their polarisation states.

Why two lamps never interfere found the other way two beams can fail to interfere — by having unrelated phases. The roof’s halves fail by the first route. They come from the same light and their phases are perfectly related; they simply arrive in different polarisations, and the overlap of 0.392 is the fraction of their fields that can still add coherently. Unpolarised light does not escape: it is a mixture of every polarisation, and every component suffers the same partial mismatch, so the average is no better than any one.

A point that becomes a smudge #

The image of a point through a roof prism can be computed directly, by adding the two halves’ fields at the focus.

The coated roof gives the ordinary diffraction pattern of the full aperture, the narrowest image the objective can make. The bare roof gives a peak only seventy per cent as high, with a broader central lobe and shoulders where the dark rings should be, all in the direction across the roof edge. Along the edge nothing changes. A star seen through such a binocular is very slightly elongated and its light slightly spread; a fine pattern of lines parallel to the edge loses contrast. The effect is too small to see as a doubled image, and too large to ignore in a comparison of two instruments side by side, which is how it was noticed.

A geometry that decides the loss #

The loss depends on the angle at which the beam meets the roof, and the geometry decides that before any glass is chosen.

Square to the edge the halves agree, whatever the glass. As the beam tilts, the faces’ planes of incidence separate and the halves diverge, and by the tilt of an Amici prism — whose roof turns the beam through 90° and meets each face at 60° — nearly a third of the peak has gone. The glass’s index matters much less than the geometry. That is the precise sense in which Porro prisms are immune: each of their reflecting faces is met by a beam square to the face’s edge, so the two halves of each right-angled reflection are not distinguished, and no phase correction is needed.

The corner that sends light home met the same physics with three faces instead of two. A solid glass corner reflector divides the beam into six sectors, each meeting the three faces in a different order, and returns six different polarisations; its returned spot loses three-quarters of its peak. The roof is the simplest case of the same failure — two faces, two halves, two orders — and it is the case where the remedy became an industry.

A coating that makes glass behave like a mirror #

The cure is to make each roof face reflect every polarisation alike, as an ideal mirror does. A mirror changes the polarisation of a reflected beam only by the geometry of reflection, which commutes for the two halves; it is the different phases on s and p that do the damage. Silvering the roof faces achieves this, at the cost of a few per cent of the light on each reflection and of a coating that tarnishes. The better solution, introduced by Zeiss in binoculars in the late 1980s, is a phase-correcting coating: a stack of thin dielectric layers on each roof face, designed so that the total phase difference between s and p on reflection is close to a mirror’s across the visible spectrum, while the reflection itself remains total.

The diagnosis came long after the symptom. For decades the softness of roof-prism images was blamed on the difficulty of making the roof’s right angle exact, which is real; only when the point images of roof prisms were examined interferometrically, with the roof angle known to be good, did the split along the edge stand out as something no polishing could remove, and the explanation in the polarisation of total reflection followed. A phase-coated roof is now tested the same way, by checking that the image of a point no longer depends on the polarisation of the light that forms it.

The tolerance is generous. The peak falls slowly at first as the departure grows — a coating thirty degrees from a mirror’s behaviour on each face costs two parts in a thousand — so the coating does not have to be exact, only far closer to a mirror than bare glass is, which departs by 139° at 60°. The difficulty is doing it across the whole visible spectrum and over the range of angles in a real beam, since the coating’s phase depends on both wavelength and angle. Every good roof-prism binocular made now carries such a coating, and with it the roof’s disadvantage has largely disappeared; side-by-side comparisons that once favoured Porro binoculars now find little to choose between them.

Whether an eye can tell #

A binocular’s objective, 42 millimetres across, could resolve two stars about three seconds of arc apart. At eight times magnification that becomes 25 seconds of arc at the eye, well below the minute of arc a typical eye resolves, so a hand-held binocular is limited by the eye rather than the glass, and the roof’s loss does not show as a failure to separate close points. How far apart two things have to be drew the distinction between resolution — whether two points can be told apart at all — and contrast, how strongly a pattern near the limit is reproduced. The roof attacks contrast: the spreading of each point’s light lowers the contrast of every fine pattern lying parallel to the edge, from the grain of tree bark to the feathers of a distant bird, and the eye, comparing two instruments side by side, sees one as crisper. For spotting scopes and astronomical instruments with high magnification, where the glass rather than the eye sets the limit, the loss is plain.

The Schmidt–Pechan and the other roofs #

The Amici prism, with its roof meeting the beam at 60°, is the simplest roof prism and the easiest to compute. Most modern roof binoculars use the Schmidt–Pechan pair, two prisms separated by a thin air gap, with the roof on one of them, or the Abbe–König prism, a single block with a roof; in each the beam meets the roof faces at different angles and the details of the loss differ, but the mechanism — two halves meeting two faces in opposite orders with planes of incidence that differ — is the same. The Schmidt–Pechan has an additional complication: one of its faces is met at less than the critical angle and must be silvered or given a dielectric mirror, so that it reflects by ordinary reflection rather than total.

There is also a second, purely geometric reason roof prisms are harder to make well: the roof edge must be a right angle to within a few seconds of arc. An error in the angle splits the image too, because the two halves are then turned in slightly different directions — an effect quite separate from polarisation, and the reason a roof prism is far more expensive to manufacture than a Porro prism of the same size. The polarisation problem, unlike the angle problem, cannot be polished away.

Roofs appear wherever an image must be turned over in a compact space: the pentaprism in a reflex camera’s viewfinder is a roof pentaprism, and the image it shows the eye suffers the same split, unnoticed because a viewfinder is not asked for the sharpness of a binocular. Total reflection inside a cut gemstone, which the window a diamond is cut inside followed, involves many faces met in many orders, and the polarisation of light leaving a diamond is scrambled in exactly this way — harmlessly, since a diamond is valued for the light it returns and not for the image it forms.

What the drawings leave out #

The figures trace rays through an idealised roof — two perfectly flat faces at exactly 90°, met at a single angle — with the incoming beam either polarised or unpolarised, and compute the image across the roof edge as the interference of the two halves of a one-dimensional aperture. A real binocular has a round aperture, so the edge divides it into two half-discs, and its image has a two-dimensional shape that the one-dimensional cross-section only samples; the peak ratio is similar. Real beams contain a range of angles across the field of view, and the loss varies across the field. The coatings are represented by a single number, their departure from a mirror’s phase behaviour, where a real multilayer has a departure that varies with wavelength and angle. The domain of the drawings is a single colour, glasses of index 1.517 and 1.72, and the central field of view.

Still open: how close a coating can get across the spectrum #

Phase coatings work well in the visible, where they were designed. As binoculars and spotting scopes are pushed to work in dim light, where the eye’s sensitivity shifts to the blue, and as roof prisms are used in instruments that work into the near infrared, the coatings have to hold their phase behaviour over a wider band. Multilayer designs that are both phase-correcting and broadband are an optimisation problem with no simple solution, and how close they can come to an ideal mirror over a factor of two in wavelength, while keeping the reflection total and the coating durable, is an active question for optical designers.

The physics is a short chain. Total reflection shifts s and p by different phases — up to 46° in crown glass — and a roof, whose two halves meet its faces in opposite orders with different planes of incidence, sends them out in different polarisations: met at 60°, as in an Amici prism, the halves overlap only 0.392 and a point’s image keeps 0.70 of its peak, while a beam square to the edge, as in a Porro prism, loses nothing; a coating that makes each face behave like a mirror within a few tens of degrees restores the image. The light is all there; what the roof takes away is the agreement between its halves.