Illustrated Physics
physics, drawn
Concept

Optical path length — where it appears

Geometric distance multiplied by refractive index, the quantity a ray makes stationary and the one interference actually compares. Two routes differing by a whole number of wavelengths of it arrive in phase, which is why a lens is shaped to make every route through it equal.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

Snell's law, found by searching. Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.

The path that does not change

Reflection, refraction and the angle of the rainbow are not three laws. They are one condition — that the optical path length is stationary — and the word stationary rather than shortest is the whole of what makes an elliptical mirror and a rainbow the same statement.

optics · Fermat Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator.

The ray that bends without a surface

Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

optics · Fermat Stationary, and not always shortest. Three mirrors through the same point, with the same source and the same detector, and the length of the reflected path against where on the mirror the ray strikes. The actual ray is the one at the centre in all three, because the tangent there is horizontal and the two angles are equal whatever the curvature. Its path is a minimum at 0.6× the ellipse's curvature, every path equal at 1× the ellipse's curvature, a maximum at 1.6× the ellipse's curvature. On the most curved mirror every path the light did not take is shorter than the one it did, so the principle cannot be stated as least time and survive; it has to be stated as stationary time, and the difference is not a technicality.

The path that takes the longest time

Light is said to take the quickest route. Put a source and a detector in front of a mirror curved a little more than the ellipse through them, and the route it takes is slower than every route beside it — by construction, not by exception. The principle was never about least; it was about stationary, and the difference is where optics stops being geometry.

optics · Fermat Abbe's ratio, measured on the traced rays. The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is exactly stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the offaxis figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

The condition a lens must meet

A paraboloid brings every parallel ray to exactly one point. Move the source a fifth of a degree off axis and the image is a fan rather than a point, and the reason is a condition Abbe wrote down that has nothing to do with the axis — perfection at one point buys nothing at the next one along.

optics · Imaging Two principles, two classes of path, two things left free. On the left, five curves from the same launch point to the same target: the true trajectory of a particle of energy 0.7 in a uniform field, and four deformations of it that share both ends. On the right, four quantities computed along that family and plotted as departures from their values on the true path. Maupertuis' abbreviated action ∫p·ds, computed at fixed energy, is stationary — flat at the centre. Hamilton's action ∫L dt, computed at fixed duration, is stationary too. The other two are not: the time a fixed-energy path takes changes at first order in the deformation, and so does the energy a fixed-duration path carries. That is the whole difference between the two principles. Each holds one of those quantities fixed and lets the other vary, and neither can hold both.

The principle that fixes the energy instead of the clock

There are two principles of least action, they compare different sets of paths, and they are not the same statement. One holds the duration fixed and lets the energy vary; the other holds the energy fixed and lets the duration vary — and written that way, mechanics turns into optics with a refractive index.

mechanics · Least action The surface that focuses 1 into 1.52 with no error at all. A Cartesian oval: the locus of points for which 1 times the distance from the object plus 1.52 times the distance to the image is a constant. Every ray drawn takes exactly the same optical path, so every one arrives at the image point — not nearly, and not for small angles, but exactly, for rays at any angle the surface reaches. There is no spherical aberration because there is no approximation: this is what Fermat's principle asks for, solved rather than expanded. The surface is not a sphere, not a conic in general, and not anything a grinding machine makes easily, which is most of why lenses are spherical and aberrated instead.

The surface that images one point exactly

Fermat's principle says every ray from an object to its image must take the same time. Written as an equation, that is a curve Descartes found — a surface with no aberration at all, at any angle. It exists, it is easy to compute, and it images exactly one pair of points and nothing else.

optics · Fermat Every ray from one point, meeting again at another. Rays from a point half a radius from the centre of Maxwell's fish-eye, n = n₀/(1 + r²/R²), traced in 11 directions round the full circle. Every one is an arc of a circle, and every one passes through the point on the other side of the centre at distance R²/r from it — 2.00 radii — which is the image. The worst miss is 8.8e-4 radii, the size of the integration's own error. Rays leaving in opposite directions, rays leaving at right angles, rays that go the long way round: all arrive. The faint circles are contours of the index, which falls from n₀ at the centre to n₀/2 at one radius and n₀/5 at two.

The medium that images every point

A single refracting surface can be shaped to image one point perfectly and no other. A medium whose index falls smoothly away from a centre can do better: in Maxwell's fish-eye every ray from any point, leaving in any direction, arrives at one image point, and every such path takes exactly the same time. It is a perfect instrument for all of space at once — and the image it forms is sharp everywhere and a faithful copy nowhere.

optics · Fermat Rays and the surfaces of equal optical path. Rays from a point source 20 mm in front of a curved glass surface (radius 4 mm, index 1.5), traced exactly, and the surfaces on which the optical path from the source — distance times index — takes the same value (curves, every 2.5 mm of path). In air the surfaces are spheres round the source. In the glass the rays from the edge of the surface cross the axis at 12.8 mm, well short of the paraxial focus at 20.0 mm, and the equal-path surfaces are no longer spheres. They are still at right angles to every ray: measured on a fan of 641 rays over 10 surfaces in the glass, the largest departure from 90° is 0.005°. Further in, the surfaces fold into cusps where neighbouring rays cross, and there a surface has no single normal.

The surface every ray is normal to

Light from a point, bent by any number of lenses and mirrors, comes out as a tangle of rays that no longer meet anywhere. Hidden in the tangle is a surface every ray crosses at a right angle: the set of points the light reaches at the same optical time. That a surface always exists is a theorem two centuries old, and it is the reason a modern wavefront sensor works — an array of tiny lenses measures only the direction each patch of light is travelling, and the measurements can be added up into a shape only because they are the slopes of one.

optics · Fermat Three shapes of one cell, delaying light by the same total. The phase delay of 633 nm light across the middle of a model cell holding 300 pg of dry matter, in three shapes — rounded up, spreading and flattened — each with the same contents dissolved in volumes of 942, 965, 982 μm³. The rounded cell delays the light by 3.41 rad at its centre, the flat one by only 0.55. But the delay summed over the whole image is the same for all three, 536 rad·μm², and it gives back 300.0 pg through the refractive increment α = 0.18 mL/g: a cell is weighed by the light it delays, whatever shape it takes.

The cell weighed by the light it delays

A living cell is nearly transparent, and the light that passes through it comes out almost as bright as it went in. What changes is its timing: the cell's proteins and nucleic acids raise its refractive index a little above the water around it, and the light is delayed by a fraction of a wave. Measure that delay at every point of the image and add it up, and the sum is the cell's dry mass — the weight of everything in it except water — whatever shape the cell takes and however much water it holds. The water drops out because the medium is water too; the shape drops out because the delay is an integral of concentration along the path. Light becomes a balance that weighs a living cell to a part in a thousand without touching it.

optics · Imaging

Named alongside it

The objects these essays reach for when they reach for this one.

Fermat's principleRefractive indexAberrationSnell's lawStationary pointImagingVariational principleWavefrontGradientReflectionSpherical aberrationAction

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