The condition a lens must meet

A spherical mirror cannot focus because rays striking further from the axis cross it nearer the mirror. The paraboloid cures that exactly: every ray from infinity crosses at one point, by the definition of the shape, and there is nothing left to improve.

Then move the source a fifth of a degree off the axis and the image falls apart.

The blur that appears is coma: an asymmetric flare, sharp at one end and spread at the other, that grows linearly with the distance off axis and quadratically with the aperture. It is the aberration that limits a fast telescope, and it is present in a mirror that has no spherical aberration whatever.

The condition #

Abbe’s statement is about the relationship between where a ray enters and how steeply it converges.

Consider a system that images an axial point perfectly — every ray from the object arrives at the image. Now ask for the image of a point a small distance $\eta$ off axis to be sharp as well. Fermat’s principle requires that all routes from the off-axis object to its image have the same optical path length, and expanding that requirement to first order in $\eta$ gives

$$\frac{h}{\sin u} = \text{constant}$$

where $h$ is the ray’s height in the entrance pupil and $u$ the angle it makes with the axis on the way to the image. For an object at infinity the constant is the focal length, so the requirement is $h = f\sin u$.

That is the sine condition, and it is worth noticing what it is not. The paraxial relation between height and angle is $h = f\tan u$, and the two agree only to first order. A system built so that its rays obey the tangent relation exactly — which is what a paraboloid does — violates the sine condition by the difference between the two functions.

A system meeting both requirements at once, stigmatic on axis and obeying the sine condition, is called aplanatic, and the word is worth having because such systems are the ones that image a field rather than a point.

Why one surface cannot do it #

The reason a paraboloid fails is a counting argument, and it is the useful way to think about the whole subject.

A single reflecting surface has one shape function to choose. Demanding stigmatism on axis uses it up completely: the requirement that all path lengths from infinity to a point be equal fixes the surface to be a paraboloid, with no freedom left. The sine condition is a second requirement on the same rays and there is nothing left to satisfy it with.

Give the system two surfaces and there are two shape functions, and both conditions can be met. That is the Ritchey–Chrétien design: two hyperboloids, chosen so that the pair is stigmatic on axis and obeys the sine condition, at the price that neither mirror alone is any good. It is what the Hubble telescope is, and what nearly every large professional telescope has been since about 1930.

The classical Cassegrain — a paraboloid primary and a hyperboloid secondary — is stigmatic on axis and does not satisfy the sine condition, so it has the same coma as the paraboloid alone. The change from one to the other is entirely about which conditions the two available freedoms are spent on.

Where the condition comes from, in one line #

The derivation is short enough to give, and it is the reason the condition is a theorem rather than a design heuristic.

Take a system that images the axial object point $O$ to the axial image point $I$ perfectly, so every path from $O$ to $I$ has the same optical length. Now consider a second object point $O'$ a small distance $\eta$ from $O$, perpendicular to the axis, and ask for its image $I'$ at $\eta'$ from $I$.

A ray leaving $O'$ instead of $O$ has its path lengthened at the start by $-n\eta\sin u_o$, where $u_o$ is the angle it leaves at, because the extra distance is the projection of $\eta$ on the ray. Arriving at $I'$ instead of $I$ shortens it by $n'\eta'\sin u_i$. For the image at $I'$ to be sharp, the total path length must be the same for every ray, which requires

$$n\eta\sin u_o = n'\eta'\sin u_i$$

for all rays, and hence $\sin u_o/\sin u_i$ constant — which for an object at infinity is $h = f\sin u$.

The whole argument is Fermat’s principle applied twice: once to fix the axial image and once, to first order in the displacement, to fix its neighbour. There is no approximation about small angles anywhere in it, which is why the sine appears and the tangent does not.

It also produces a second reading of the same equation. $n\eta\sin u$ is, up to a factor, the étendue of the bundle, so the sine condition says that an aplanatic system conserves étendue ray by ray rather than merely in total — and the brightness no lens can increase is the same statement made about a whole beam. An instrument that violates it is one whose rays are not carrying their share.

The rule of thumb it produces #

The practical form of all this is a formula for how large a field a given telescope has, and it is worth stating because it is unforgiving.

Coma’s angular blur for a paraboloid is approximately

$$\theta_{\text{coma}} \approx \frac{\eta}{16 F^2}$$

where $\eta$ is the field angle and $F$ is the focal ratio. The $F^2$ is what hurts. A slow instrument at $f/10$ has a comatic blur sixteen times smaller than a fast one at $f/2.5$ at the same field angle, so the usable field of a fast Newtonian is small in a way that surprises people who chose it for its speed.

Setting the coma equal to the diffraction limit gives the field over which the instrument is diffraction-limited, and for a fast paraboloid that is a couple of minutes of arc — smaller than the moon by a factor of fifteen. Everything outside it is comatic, and the fact that the mirror is perfect on axis is no consolation.

The standard the aberrations have to be compared against is diffraction. Two point sources are resolved or not according to a limit set by the aperture alone, and an aberration blur smaller than that limit is invisible — so the design target is not zero aberration but aberration below the diffraction spot. That is what makes the sine condition a practical criterion rather than an ideal: it says when the residual has stopped mattering.

What coma looks like, and why it is the worst-behaved aberration #

The name comes from the shape: a comatic image is a small comet, with a bright head and a fan spreading away from it, and the fan always points either toward or away from the axis depending on the sign.

The asymmetry is what makes it so much more damaging than its size suggests. Spherical aberration and defocus produce blurs that are symmetric about the ideal image point, so the centroid of the light is still in the right place and a measurement of position is unbiased. Coma’s blur is not symmetric: the centroid is displaced from the head of the comet by about a third of the flare’s length, and the displacement grows with the field angle.

For astrometry that is fatal. A star measured near the edge of a comatic field is reported at a position that is systematically wrong, in a direction that depends on where in the field it fell, and no amount of averaging removes it because it is not noise. Every wide-field astrometric survey therefore uses an aplanatic design or a corrector, not because the images look better but because the positions are otherwise biased.

For photometry it is a different problem: the flare spills outside whatever aperture the photometry uses, so the measured brightness falls off toward the edge of the field in a way that has to be calibrated out. And for spectroscopy it is worse still, because the flare and the slit are not aligned in any consistent way.

The general point is one worth carrying past optics. A symmetric error degrades a measurement; an asymmetric one biases it, and the second is much harder to live with, because averaging more data reduces the first and not the second.

The correctors, and what they are correcting #

A coma corrector is a small lens assembly placed near the focus of a fast paraboloid, and its job is stated exactly by the condition above: it is not correcting the mirror’s figure, which is right, but adjusting the mapping between ray height and convergence angle so that the combination satisfies the sine condition.

That is why such a corrector is a couple of centimetres across and works for a mirror half a metre wide. It does not touch the wavefront’s overall shape; it applies a small height-dependent change of angle. The same is true of the Schmidt plate, invented for the opposite problem — a spherical primary with a stop at its centre of curvature has no coma at all, by symmetry, and enormous spherical aberration, and the plate removes the second without disturbing the first.

The Schmidt case is the cleanest demonstration in the whole subject that these two aberrations are independent. A sphere with the stop at the centre of curvature has, from any direction, exactly the same geometry — every direction is an axis — so there is no way for it to distinguish on-axis from off-axis and no coma is possible. It has spherical aberration to spare, and one thin aspheric plate fixes that. Two problems, two independent cures, and the reason a Schmidt camera photographs a field several degrees wide.

The paraxial construction locates an image and says nothing whatever about its quality. Every statement in this essay lives in the gap between where that construction puts the image and where the light actually goes — and the mirror equation, which is exact in the paraxial limit, is silent about the entire subject. That silence is not a defect of the equation; it is what “paraxial” means.

What the condition buys: one blur for the whole field #

The sine condition is usually presented as a way of removing coma, which is true and undersells it. What it actually secures is that the shape of the blur stops depending on where in the field the point was.

An aplanatic system’s image of a point near the axis is, to first order in the field angle, the same function translated — same size, same shape, same orientation. A system that violates the condition has an image whose flare grows and rotates as the point moves outward, so no two points in the field are imaged alike.

That distinction decides whether the instrument can be described at all. If the blur is the same everywhere, the image is the object convolved with a single point-spread function, and everything that follows from a convolution becomes available: a transfer function that says how much contrast survives at each spatial frequency, a single number for resolution, and the possibility of undoing some of the blur computationally. If the blur varies across the field, none of that holds — a deconvolution with one kernel sharpens the middle and smears the edges, and there is no such thing as “the” transfer function of the instrument.

The property has a name, isoplanatism, and the region over which it holds is an isoplanatic patch. It is why an aplanatic design is worth its extra surfaces even when the comatic flare would have been small enough to tolerate: it is the difference between an instrument that has a point-spread function and one that has a different one at every point.

The condition for the other direction, and why both cannot hold #

Abbe’s condition asks that a point displaced sideways from the axis be imaged sharply. There is an obvious companion question — what does it take to image sharply a point displaced along the axis? — and it has an answer of the same form, found by Herschel in 1821, half a century earlier.

Running the same path-length argument for a longitudinal displacement gives Herschel’s condition: $n\sin^2(u_o/2)\,/\,n'\sin^2(u_i/2)$ must be constant across the aperture, where Abbe’s demanded that $n\sin u_o / n'\sin u_i$ be constant.

Both are requirements on the same rays, and they can be divided one by the other. Using $\sin u = 2\sin(u/2)\cos(u/2)$, the quotient reduces to the demand that $\cos(u_o/2)/\cos(u_i/2)$ be constant too — and three constraints of that kind on one function force $u_o = \pm u_i$ for every ray. That fixes the magnification: the two conditions hold together only at $|m| = n/n'$, which for an instrument working in air at both ends means unit magnification and nothing else.

The consequence is a genuine impossibility rather than a difficulty. No optical system that magnifies can image a three-dimensional region sharply. It may be aplanatic, in which case it images a surface well and loses sharpness immediately in front of and behind it, or it may satisfy Herschel’s condition and image a line along the axis well while losing the field — and there is no design, however many surfaces it is given, that does both.

Every microscope is built on the first choice, which is why depth is recovered by moving the focus and stacking rather than by capturing it, and why the depth of field of a high-aperture objective is measured in fractions of a micrometre. It is not a limitation of the glass. It is two path-length conditions that cannot both be true.

What it costs #

The sine condition constrains a microscope objective more than a telescope. For an object at a finite distance the constant in $h/\sin u$ involves both conjugates, and satisfying it at high numerical aperture is the whole difficulty of designing an objective. A modern one has a dozen elements and most of them are there for the field rather than for the axis.

And it produces the reason immersion works twice. The condition in a medium of index $n$ is $nh/\sin u$ constant, so raising the index raises the achievable aperture — which is why an immersion objective resolves better — and it also means the correction has to be redone if the immersion medium changes. An objective designed for oil is not merely degraded in water; it is a different instrument.

Satisfying it costs light and space. Every additional surface reflects, absorbs and scatters, and a two-mirror aplanat has a hole in the middle of its primary and a secondary obstructing the aperture. The obstruction moves energy from the central peak of the diffraction pattern into the rings, which lowers contrast on extended objects — so the aplanatic design wins on field and loses on contrast, and which matters depends on what is being looked at.

The condition is about geometry and says nothing about colour. An aplanatic system built of glass is aplanatic at one wavelength, and the index of every glass depends on the wavelength, so a refracting aplanat has to satisfy the condition and be achromatic at the same time — two constraints on the same small number of freedoms. That difficulty is one of the strongest arguments for mirrors, which have no dispersion at all and therefore satisfy their conditions at every wavelength simultaneously.

And nothing here fixes astigmatism or field curvature. An aplanat is corrected for two aberrations out of the classical five. The Ritchey–Chrétien has astigmatism and a curved focal surface, which is why its detectors are either small or curved and why a third element is added when a really wide field is wanted.

A refracting design pays a cost a reflecting one does not. Two glasses’ focal lengths run differently with wavelength, and a doublet brings two wavelengths together and leaves the rest scattered about — so a refracting aplanat has to satisfy the sine condition and the achromatic one simultaneously, with the same two surfaces. A mirror has no dispersion at all, which is why every large telescope is one.

The sine condition and the diffraction bound are two halves of one statement about apertures. An aperture can collect a certain range of angles; the sine condition says what a system must do to image all of them to the same place, and the diffraction limit says how small a spot that range of angles can produce. Numerical aperture appears in both, with the immersion index inside it, which is why oil buys resolution and why it buys nothing if the sine condition is not met.

What the picture cannot show #

The trace is meridional and two-dimensional. Real coma is a three-dimensional figure — a family of overlapping circles growing along the axis of the flare — and the blur plotted here is the spread of the meridional fan, which is the right order and not the right shape. A proper spot diagram needs rays from the whole pupil.

Diffraction is absent from the trace and present in reality. The blurs computed here are geometric, and below about the diffraction limit they are not what is seen. Whether a given amount of coma matters is always a comparison against that limit, and the comparison depends on the aperture, which the geometric trace does not know about.

The sine condition is a first-order statement in the field angle. It guarantees a sharp image of points near the axis, which is what it was derived for. A wide field needs the higher-order conditions too, and satisfying all of them simultaneously is not possible with finitely many surfaces — which is why lens design is an optimisation rather than a construction.

And the two surfaces drawn are ideal. Every real mirror has a figure error, a support-induced deformation and a thermal gradient, and on a large telescope those are comparable with the aberrations discussed here. The design condition sets what is achievable; what is achieved is a manufacturing question.

The ladder from here #

Later rungs on this anchor: the five Seidel aberrations as one expansion, and why they are the complete list at that order; the aplanatic points of a sphere, which are the two conjugate positions where a single spherical surface is both stigmatic and aplanatic and which every immersion objective’s front element uses; field curvature and the Petzval sum, a constraint that survives every other correction; the wavefront description, in which all of this becomes a set of polynomial coefficients and the geometric picture is dropped; and adaptive correction, where the aberrations are measured and cancelled in real time rather than designed away.

The neighbouring ladders are what a lens is doing, where the paraxial construction is set up, and the mirror that cannot focus, whose defect the paraboloid cures and whose cure this essay is the price of. The image as a diffraction pattern is the limit all of these aberrations are measured against.