Illustrated Physics
physics, drawn
Series

Dispersion — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Rays through a raindrop. Parallel rays entering a spherical drop at different heights, refracting in, reflecting once from the back, and refracting out. The outgoing rays crowd together near one particular direction, and that crowding is the bow.

    The angle the rainbow has to be, and why nobody chose it

    A rainbow is at forty-two degrees because a function has a minimum there. Nothing about water, light or weather picks the number — it falls out of running Snell's law three times through a sphere.

    part 1 · optics
  2. Cancelling a derivative, and what is left over. How far the focus of a 500 mm lens moves with wavelength, for a single crown element and for a cemented pair of crown and flint whose powers satisfy the achromatic condition. The singlet's focus runs over 18.1 mm across the visible — a fifth of a per cent of its focal length, and utterly ruinous at any useful aperture. The doublet's runs over 2.268 mm, some 8× less, and — this is the whole content of the figure — it is not flat. The condition sets the rate of change of power with wavelength to zero, so the curve is stationary rather than constant: it returns to the corrected focus at exactly 2 wavelengths — 486 nm and 656 nm, which are the two Fraunhofer lines the condition was written at — and departs from it everywhere else, most at 400 nm. That residual is the secondary spectrum, it has the same sign at both ends of the visible, and no pair of ordinary glasses removes it, because two conditions cannot be met with one free ratio.

    Two glasses that cancel a derivative

    A single lens focuses blue light closer than red, and the difference ruins the image. Cementing a second lens of another glass behind it fixes the fault at two wavelengths and at no others, because the condition sets a slope to zero rather than a value.

    part 2 · optics
  3. The wavelength at which a fibre stops smearing a pulse. Group-delay dispersion against wavelength for fused silica, computed by differencing Malitson's Sellmeier fit twice rather than read off a table, in picoseconds of spread per nanometre of source width per kilometre of fibre. The material curve crosses zero at 1273 nm — a property of the glass, fixed by where the ultraviolet and infrared absorptions balance, and not adjustable. Below it a fibre is normally dispersive and above it anomalously so, and at 1550 nm, where silica is most transparent, it is 21.9. The second curve adds a waveguide term, which is negative because a mode confined by a core spreads into the cladding differently at different wavelengths, and which depends on the fibre's geometry rather than on the glass; it is drawn here as a constant offset chosen to put the total zero at 1310 nm, which is what standard single-mode fibre is built to do. The consequence in a system is the thing worth carrying away: a source one nanometre wide sent 100 km through fibre at 1550 nm arrives 1833 picoseconds broader than it left, against 0.0 at the zero. That is the whole reason a network runs at one wavelength rather than another, and the reason the two quantities an engineer wants — lowest loss and zero dispersion — sit at different wavelengths and have to be reconciled by design rather than chosen.

    The wavelength a fibre does not smear

    Fused silica's index has a second derivative that passes through zero at 1,273 nanometres, and that is not a design choice — it is where the ultraviolet and infrared absorptions balance. A pulse sent at that wavelength arrives the shape it left. A pulse at the wavelength of least loss arrives 1,800 picoseconds wider after a hundred kilometres.

    part 3 · optics
  4. The deviation that has a bottom. The angle by which a 60° and a 90° ice prism bends a ray, against the angle at which the ray arrives, for an index of 1.31. Each curve is traced ray by ray through both faces and stops where it stops: outside the plotted range the ray meets the second face beyond the critical angle and never leaves. Both curves have a minimum, and the minimum is the point of the figure twice over. Its value — 21.84° for the 60° prism and 45.73° for the 90° prism — is where the sky puts a halo. And its flatness is why there is a halo at all: near a minimum the deviation changes only in second order, so a wide band of orientations all deliver light to nearly the same angle, and a cloud of randomly tumbling crystals piles up a bright ring there while sending the rest of the light nowhere in particular. The passage at the minimum comes out symmetric — in at 40.92°, out at 40.92° — which was found by searching the traced curve rather than assumed. The window of incidence that gets through at all is 76.5° wide for the 60° prism against 32.2° for the 90° one, a factor of 2.4, and that is why one of the two halos is common and the other is rare.

    The ring at twenty-two degrees

    A halo round the sun is a caustic in orientation rather than in space. Most of the ice crystals in a cirrus cloud send light nowhere in particular; the ones near minimum deviation all send it to nearly the same angle, because a minimum is flat — and the angle they pick has a red inner edge, which is the reverse of a rainbow.

    part 4 · optics
  5. A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

    The delay that is a random variable

    A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

    part 5 · optics
  6. Colours that leave a grating on different roads. A pair of parallel diffraction gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for three wavelengths, 760, 800 and 840 nm. The first grating sends each colour off at its own angle — 24.3° for 760 nm, 27.4° for 800 nm, 30.5° for 840 nm — and the second turns them all back parallel to the input, side by side. Between the gratings the red travels the longest road: for gratings 10 mm apart the slant paths are 10.97 mm, 11.26 mm, 11.61 mm. Measured to a common wavefront after the second grating, the longer road still wins, so red arrives last: the geometry has given the pulse the opposite ordering to the one glass gives it.

    The dispersion made of angles

    Glass delays blue light more than red, so a short pulse that passes through glass comes out stretched with its red end first. At the wavelengths where most short-pulse lasers work, no transparent glass does the opposite. Two diffraction gratings do it with no material at all. The first spreads the colours in angle, the second turns them back parallel, and in between the red has travelled the longer road. A spread in angle, however it is made, always adds this kind of dispersion, of the sign glass cannot give. It is how every high-power ultrafast laser squeezes its pulse back after amplifying it.

    part 6 · optics
  7. A filter made of glass powder and oil. The fraction of light transmitted straight through a 5 mm layer of N-BK7 grains packed to half its volume in the matching liquid, against wavelength, for grains of radius 5, 10 and 20 μm, from van de Hulst's extinction for large, weakly refracting particles. At the crossing the grains vanish and the layer is clear. A few tens of nanometres away each grain shifts the light passing through it in phase by enough to scatter a little of it out of the beam, and a thousand grains in a row scatter all of it: the passband is 51.7 nm wide at half height for 5 μm grains, 36.3 nm for 10 μm, 25.5 nm for 20 μm. Larger grains make a narrower filter, because the phase a grain imposes grows with its size; the same mismatch of index that a fine powder lets through, a coarse one scatters.

    The colour at which glass disappears

    Lower a glass rod into an oil of the same refractive index and it vanishes, except for a faint coloured outline. The outline is the clue. Glass and oil disperse light differently, so their indices can agree at only one wavelength; at every other the glass is still there, bending and scattering a little. Grind the glass to powder and pack it in the oil, and the mixture is clear at that one wavelength and opaque at the rest — a colour filter made of nothing but two transparent materials, whose colour moves by a whole hue for a few degrees of warming.

    part 7 · optics
  8. Fringes that bend into hooks beside a line. The interference fringes seen in a spectrograph behind an interferometer with 10 cm of sodium vapour at 10¹⁴ atoms per cubic centimetre in one arm and a glass plate in the other, against wavelength across the D₂ line (horizontal) and position along the slit (vertical, in fringe spacings). Far from the line the plate tilts every fringe evenly. Near the line the vapour's dispersion sweeps each fringe up on one side and down on the other, and every fringe turns back on itself in a hook: the turning points sit −100 and 100 picometres from the line centre, the same for every fringe. The line itself, a few picometres wide, is a dark gap between them; the hooks lie far outside it, in vapour that is transparent.

    The hooks that weigh an absorption line

    A strong absorption line is black at its centre, and how black it is says almost nothing about how many atoms made it — double them and the line hardly darkens. The atoms are still there, and they still bend light: on either side of the line, where the vapour is transparent, its refractive index rises and falls by an amount proportional to their number. In 1912 Dmitry Rozhdestvensky found a way to read that index straight off a photograph. Interference fringes passed through a spectrograph curl into hooks beside the line, and the distance between the hooks, squared, counts the atoms — whatever the line's shape and however saturated its core.

    part 8 · optics
  9. Lamp noise that a decay does not notice. 400 simulated measurements of an absorption of 1 × 10⁻⁷ per metre, with a source whose brightness jumps by 5 per cent between shots and a detector with 0.1 per cent noise. Read directly, as transmission through a metre of gas, each measurement scatters by 7 × 10⁻² per metre and the true value is five orders of magnitude below that. Read as a fitted decay from a cavity with 99.99 per cent mirrors, it scatters by 5 × 10⁻⁸, about a million times less. The slope of a logarithm has no information about how bright the pulse started, so a source's noise simply is not in the answer. Most of the method's sensitivity comes from this and only some from the long path.

    The absorption measured with a stopwatch

    The usual way to measure how much light a gas absorbs is to compare a beam before and after, and a very weak absorption vanishes in the flicker of the lamp. Trap a pulse between two excellent mirrors, and it bounces for ten kilometres inside a tube a metre long, leaking out a little at each bounce. How fast that leak dies measures the gas, and how bright the pulse began does not matter at all. A brightness measurement has become a measurement of time.

    part 9 · optics

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