Illustrated Physics
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Series

Superposition — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

    When two waves meet, they simply add

    Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.

    part 1 · waves
  2. The cross term, and the fact that it averages to nothing. The intensity of two waves of amplitude 1 and 0.7 added together, against the phase difference between them, in turns. A detector reads the square of the summed amplitude, which is the sum of the two intensities plus a cross term that swings between plus and minus twice the product. At no phase difference the reading is 2.89 and at half a turn it is 0.09; the flat line is what the two would give with no interference, 1.49, and it is exactly the average of the curve over a whole turn. Interference redistributes and does not create — which answers the question of where the energy goes at a dark fringe by saying that it never left.

    What adding does to the energy

    Waves add their amplitudes and detectors read squares, so two waves together do not deliver the sum of what each delivers. Where the two get dimmer, the natural question is where the energy went — and the answer depends entirely on whether the sources can feel each other.

    part 2 · waves
  3. How far the vacuum is from being a nonlinear medium. The size of the vacuum's departure from linearity, as a fraction, against the electric field it is subjected to — thirteen decades of field and twenty-eight of correction, both logarithmic. The scale is the Schwinger field, computed here from the electron's mass and the fundamental constants as 1.32e+18 volts per metre: the field at which a pair gains its own rest energy over a Compton wavelength, and therefore the field at which the vacuum stops being a passive backdrop. The marks are the strongest fields that exist. a laboratory magnet, 10 T is 2.3e-9 of it; a hydrogen atom's own field is 3.9e-7 of it; a 10²² W/cm² laser focus is 2.1e-4 of it; the Schwinger field is 1.0e+0 of it; a magnetar, 10¹¹ T is 2.3e+1 of it. So a laboratory is twenty-eight decades from making the effect large, and a magnetar's field is above the critical one — which is why the only places the vacuum's nonlinearity has been seen are the two where the fields are not human: the ultraperipheral collision of two heavy nuclei, and the surface of a neutron star.

    The one medium that was supposed to add exactly

    Superposition holds because an equation is linear, and every material stops being linear at some amplitude. Empty space was the exception: Maxwell's equations are linear exactly, and two beams cross with no interaction of any kind. Quantum electrodynamics says otherwise — light scatters light, and a strong field makes the vacuum birefringent — at a field of 1.3 × 10¹⁸ volts per metre, which no laboratory has come within four decades of.

    part 3 · waves
  4. Modes added in step: a train of pulses. The intensity of the sum of N waves of equal amplitude at equally spaced frequencies, all starting in phase, against time in units of the round trip — the inverse of the spacing — for N = 5, 10, 20. The sum is a train of pulses, one per round trip, each of peak intensity N² times one wave's and width about 1/N of the round trip. Between the pulses the waves cancel almost completely. The average intensity is N, as it must be — 5.01 for N = 5, 10.04 for N = 10, 20.19 for N = 20 — so the pulses do not create energy; they move it all into a fraction 1/N of the time, where it is N times more intense than it would be if spread evenly.

    The phases that turn a glow into pulses

    A laser oscillates on many frequencies at once, spaced by the time light takes to go round its cavity. Add those waves with random phases and the output is a steady glow with noise in it. Add exactly the same waves with their phases equal and the output is a train of pulses, each N² times brighter than one wave and a fraction 1/N of the round trip long. Nothing about the frequencies, the amplitudes or the average power changes — only the phases, and they decide everything a detector fast enough would see.

    part 4 · waves
  5. 3 beams crossing: change the phases and the pattern only slides. The intensity where 3 plane waves of one wavelength cross in a plane, their directions spread evenly round the circle, for three different sets of relative phases, each panel spanning 2.2 wavelengths. Darkest shading is brightest, in five steps of a fifth of the peak. The three patterns are the same lattice of bright spots moved sideways: for each of the altered sets there is a shift that reproduces the first pattern to 2.0 per cent of the peak. With 3 beams in two dimensions there are 2 relative phases and 2 directions to slide in, so every change of phase is a slide.

    The lattice no phase can bend

    Cross a few laser beams of one colour and they paint a crystal of light in the space where they overlap. The beams' phases drift whenever a mirror trembles, and it would seem the pattern must tremble with them. Whether it does depends on a count. Three beams in a plane, or four in space, give a lattice whose shape no change of phase can alter — the phases can only slide it. One beam more and the phases decide the shape, and must be held still.

    part 5 · waves
  6. The path the tip of the field traces. The tip of the total field over one cycle, for four arrangements of coils, with each coil's axis as a faint line and the field at one moment as an arrow. One coil: forward-turning part 0.50, backward-turning part 0.50; Two coils, 90° and a quarter cycle apart: forward-turning part 1.00, backward-turning part 0.00; Three balanced coils: forward-turning part 1.50, backward-turning part 0.00; Three coils, the third at 70 per cent current: forward-turning part 1.35, backward-turning part 0.15. One coil traces a line: equal forward and backward parts, a field that pulses and does not turn. Two coils at right angles a quarter cycle apart trace a circle of radius 1, three balanced coils a circle of 1.5, and a weakened third coil an ellipse, the sum of a large forward circle and a small backward one.

    The field that turns with nothing turning

    A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

    part 6 · waves
  7. Light added to its own reflection. Green light of 550 nm in an emulsion of refractive index 1.5, falling on a metal mirror at the left and adding to its reflection, which comes back inverted. Thin lines: the total electric field across the first 750 nm at three moments; heavy line: its envelope, 2|sin kz|; dashed: the envelope of the magnetic field, 2|cos kz|. The sum does not travel. Its electric field is zero at the mirror at every instant and again every 183 nm, half a wavelength in the emulsion, with maxima between; the magnetic field is largest exactly where the electric field vanishes. A light-sensitive layer anywhere in that space is exposed in sheets parallel to the mirror, and which field it responds to decides whether the first sheet lies at the mirror or a quarter-wavelength above it.

    The photograph that keeps the spectrum

    Light falling on a mirror adds to its own reflection, and the sum does not travel: it stands, with sheets of darkness every half-wavelength. Otto Wiener photographed those sheets in 1890 and found the film clear where it touched the mirror, which settled which of light's two fields acts on matter. A year later Gabriel Lippmann recorded the sheets through the depth of an emulsion and developed them into stacked layers that, lit with white light, send back exactly the colours that wrote them. His photographs store the spectrum itself, so they can tell apart two yellows that every colour film and every eye records as one.

    part 7 · waves
  8. Where cancelling a 500 Hz tone works. Sound at 500 Hz (wavelength 69 cm) arriving from every direction at random, as it does in a room or a cabin, and a small loudspeaker 50 cm to the left of a microphone (black dot), driven to cancel the sound exactly at the microphone. Contours mark where the sum is 20 dB quieter than the original (red), 10 dB quieter (blue), unchanged (grey) and 6 dB louder (black), in a plane through both. The zone at least 10 dB quieter is 12.0 cm across, at right angles to the line from speaker to microphone — 0.18 of a wavelength — and beyond the unchanged contour the cancelling source only adds sound. Silence has been made at a point, not in a room.

    The silence that fits in a tenth of a wavelength

    Two sounds of equal strength and opposite phase add to nothing, so any noise can be cancelled by playing its inverse. At one point. A microphone and a loudspeaker can silence the air exactly where the microphone is, and nowhere else for certain: in the diffuse sound of a room the quiet zone is a few tenths of a wavelength across, under ten centimetres at a kilohertz, and beyond it the cancelling speaker only adds noise. That is why noise cancellation works in a headphone and not in a room, and why it works on the drone of an engine and not on a voice.

    part 8 · waves
  9. Two fine gratings that make a coarse one. Top: two gratings of dark bars laid over each other, one with a period of 10 units and the other of 11, both with bars half a period wide. Bottom: the average transmission of the pair, smoothed over a few periods. Where the bars of the two gratings coincide, light passes through the gaps between them; where the bars of one fill the gaps of the other, nothing passes. The two drift in and out of step once every 110 units — the moiré period, p₁p₂/|p₁ − p₂| — producing broad dark and light bands 11 times coarser than either grating. Neither grating contains a period of 110; their product does.

    The pattern two gratings make that neither contains

    Lay one fine net curtain over another and broad, swirling bands appear that are in neither curtain. Two sets of lines a tenth of a millimetre apart, differing by one part in a hundred, make bands a centimetre apart. This is moiré: the beat of two gratings in space, as two notes a little out of tune beat in time. It magnifies whatever makes the gratings differ — a twist, a stretch, a shift — by the ratio of the bands to the lines, so it is used to measure strain in bridges and to align the layers of microchips. And two sheets of carbon, twisted by just over one degree, make a moiré on which electrons stop moving and the material superconducts.

    part 9 · waves

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