Illustrated Physics
physics, drawn
Series

Periodic media — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The frequencies a repeat will not carry. The band structure of a medium made of quarter-wave layers of index 1 and 2, repeated for ever: frequency against Bloch phase across one cell, in the reduced zone. Inside a band the phase runs from 0 to π and the wave travels. Between bands there is no real phase at all, and the shaded strips are frequencies at which the medium supports nothing — not a weakly transmitted wave, no wave. Gap 1 runs from 0.784 to 1.216; Gap 2 runs from 2.784 to 3.216, in units of the quarter-wave design frequency. The first, measured off the drawn band edges, is 0.4327 wide against the 0.4327 of (4/π)·arcsin|r| — the same number computed from the Fresnel ratio of the two indices alone, agreeing to 2.6e-14 per cent. Every band edge sits where the phase is 0 or π, which is to say where the wave's own period fits the repeat a whole number of times: the gap is a property of the periodicity, and the materials only decide how wide it is.

    The gap a repeat opens

    Stack two transparent materials in alternating layers and there is a band of frequencies the stack will not carry — not weakly, not with loss, but not at all. Nothing has been absorbed and neither material has a resonance there. What forbids those frequencies is the repeat itself, and the width of the band has a closed form containing only the ratio of the two indices.

    part 1 · waves
  2. The one frequency a broken repeat lets through. Transmission through a quarter-wave stack whose repeat is broken once, on a logarithmic scale, against frequency in units of the quarter-wave design frequency. The perfect stack forbids this whole band; adding a single half-wave layer at the centre opens one line inside it, at exactly the middle of the gap, through which the stack transmits everything. With 4 pairs either side the line is 2.50e-3 wide, a quality factor of 400, on a background of 4.5e-4; With 6 pairs either side the line is 1.56e-4 wide, a quality factor of 6410, on a background of 6.3e-6; With 8 pairs either side the line is 1.00e-5 wide, a quality factor of 100000, on a background of 9.2e-8. The line narrows geometrically as the mirrors are made thicker, because the field inside the defect leaks out through a barrier whose transmission falls exponentially with its thickness — so the useful quantity of a band gap turns out to be not what it excludes but how well it can trap what a single flaw is allowed to hold.

    The mode that lives in the mistake

    A perfect stack of alternating layers refuses a whole band of frequencies — not weakly, not with loss, but not at all. Break the repeat once, by inserting a single layer of the wrong thickness, and exactly one frequency inside that band passes through the whole stack with a transmittance of one. The useful thing about a forbidden band turns out to be not what it excludes but what a single flaw is thereby allowed to hold.

    part 2 · waves
  3. The chain's dispersion, and the frequency it stops at. Frequency against wavenumber for a chain of equal masses joined by equal springs, in units where the spacing, the mass and the spring are one. At long wavelength the curve is a straight line through the origin — the chain behaves as a continuous string with a sound speed, and the departure from the line is second order in the wavenumber, which is why a lattice is invisible until the wavelength approaches the spacing. At the zone edge, where neighbouring masses move in exact opposition, the curve flattens: the frequency stops rising, the group velocity falls to zero, and the mode is a standing wave that carries nothing. Above that frequency there is no travelling solution at all.

    The frequency a lattice cannot carry

    A continuous string carries every note. A row of masses joined by springs does not: there is a highest frequency, set by nothing but the time one mass takes to be pushed back by its neighbours, and above it a disturbance does not travel at all.

    part 3 · waves
  4. Two states at zero that only one side of the chain has. Every energy level of a chain of 20 cells, 40 sites, whose couplings alternate between v inside a cell and w = 1 between cells, against the ratio v/w from 0 to 2. The bulk levels fill two bands, bounded by the dashed lines ±|v − w| and ±(v + w), with a gap between them that closes at v = w. For v < w two levels sit at zero energy, in the middle of the gap, drawn in the warning colour: at v/w = 0.5 they are within a millionth of the coupling of zero. For v > w the gap is empty — at v/w = 1.5 the nearest level to zero is 0.527. Nothing about the chain's middle distinguishes the two sides of v = w; the difference is at its ends.

    The end that knows how the middle was cut

    A chain whose links alternate, strong and weak, has the same bands whichever kind of link is counted as inside a cell. The infinite chain cannot tell the two choices apart. A finite chain can: cut it so that a weak link is outermost and each end holds a state at exactly zero energy, in the middle of the gap; cut it the other way and it holds none. What decides is not anything at the ends but a whole number counted from the bulk — how many times a loop winds round a point.

    part 4 · waves
  5. A gap that every angle from air falls into. The stop bands of a stack of quarter-wave layers of index 4.6 and 1.6 — tellurium and polystyrene, the pair of the first such mirror — against frequency, in units of the design frequency, and the parallel index β = n₀ sin θ₀ the light brings along the layers; TE polarisation on the right, TM on the left. Shaded regions are gaps. Light from air can only have β between −1 and 1, the vertical lines; within those lines the gaps for both polarisations overlap between f = 0.848 and 1.321, the band marked across the figure, so every angle of incidence and both polarisations are reflected: a relative width of 43.6%. The TM gap narrows as β grows and closes at the internal Brewster index, 1.51, which light from air cannot reach.

    The mirror that works from every direction

    A stack of alternating transparent layers reflects nearly all the light of one colour arriving straight on, and less as the light tilts, because tilting moves the forbidden band. A structure that repeats in only one direction ought therefore to be a mirror for only a range of directions. It is not, if the layers differ enough: light arriving from air cannot bring enough sideways momentum to escape the forbidden band at any angle, for either polarisation, and a flat stack of plastic and tellurium reflects every angle over a band of frequencies almost half as wide as its centre.

    part 5 · waves
  6. A seabed that reflects one wavelength. The fraction of a surface wave's amplitude reflected by a patch of sinusoidal sandbars 50 m apart and 0.8 m in amplitude, in 4 m of water, against the wavelength of the incoming wave, for 4 bars and 8 bars, from the shallow-water equations integrated through the patch. 4 bars reflect 56 per cent of the amplitude at 99.8 m; 8 bars reflect 86 per cent of the amplitude at 100.4 m, close to twice the bar spacing, where each bar's small reflection returns in step with the others. Away from that wavelength the reflections cancel, and the more bars there are, the narrower the band they reflect.

    The sandbars that reflect the sea

    A stack of alternating layers reflects the one colour whose half-wavelength matches the repeat. A row of sandbars on a gently sloping seabed is the same arrangement for ocean waves, and it does the same thing. Waves twice as long as the bar spacing are sent back out to sea, one small reflection per bar, all adding in step. A patch of eight bars can return most of the swell that meets it. The reflected wave, added to the incoming one, makes a standing pattern whose nodes are spaced exactly like the bars. Sand gathers in that pattern, so a field of bars can build its own next bar.

    part 6 · waves
  7. Light from one guide, spreading through a row of them. The power in each of a row of identical coupled waveguides after light launched into the central one has travelled 1.5, 3, 5 and 7 coupling lengths (1/C) along the row, drawn as stacked bars with the distance increasing upward: |Jₙ(2Cz)|², where n counts guides from the centre. The light does not spread into a single widening hump as it would in a uniform slab. It forms two beams that run outward at a constant rate, carrying most of the power in their leading edges near n = ±2Cz, and leaves the centre nearly dark: after 7 coupling lengths the central guide holds 2.9 per cent and the brightest guides, at ±12, hold 8.1 per cent each. This is discrete diffraction: the spreading of light that can only step between neighbouring sites.

    The light that spreads in two beams

    Shine light into one of a row of identical optical fibres laid side by side and close enough to leak into each other, and it does not spread out into a widening glow as it would in a slab of glass. It splits into two beams that run outward at a fixed speed, bright at their leading edges, leaving the middle almost dark. The reason is that light hopping between neighbours has a fastest possible sideways speed, and piles up at it. The same periodicity gives the row two tricks no uniform medium has: a direction in which a beam travels without spreading at all, and a direction in which diffraction runs backwards, so that a spread beam can be gathered up again.

    part 7 · waves
  8. A pile of glass plates, added two ways. The transmission of light at normal incidence through a pile of N glass plates (index 1.5) in air, on a logarithmic axis, against N. Dashed: Stokes's 1862 result for light whose reflections add as intensities, T = (1 − R)/(1 + (2N − 1)R) with R = 0.040 per surface, which falls only as 1/N: 0.107 at 100 plates, 0.029 at 400. Solid: the same pile with the reflections added as waves and every plate and gap of random thickness, as ⟨ln T⟩ over 200 random piles (red) and for one pile (grey). It falls in a straight line, by 0.0843 per plate, an exponential decay with a localisation length of 24 plates: 10⁻³ at 100 plates and 10⁻¹⁴ at 400. Interference with random phases does not average away; it traps.

    The pile of glass that stops every colour

    A sheet of window glass reflects about eight per cent of the light that falls on it. A pile of a hundred sheets, George Stokes calculated in 1862, should still let through a tenth, because each extra sheet only adds a little more reflection to the total. That is right for sunlight, whose colours are too mixed to interfere. For a single colour, with every sheet and every gap a slightly different thickness, the same pile lets through a thousandth, and a pile of four hundred a hundred-millionth of a millionth. Interference with random phases does not average away into the intensity sum. In one dimension it traps every wave, at every wavelength, however weak each reflection is.

    part 8 · waves
  9. The chirp a staircase returns. The pitch of the echo of a clap from the staircase, heard by a listener 10 m from its foot, against time since the clap: the strongest frequency in successive 24 ms slices of the synthesised echo (dots), and the rate at which the reflections arrive, together with twice that rate (lines). The fundamental falls from about 767 Hz to 476 Hz over a sixth of a second, with its second harmonic an octave above — a descending chirp, which to many listeners sounds like a bird. It is Bragg reflection: the stair is a grating, the first-order echo is the frequency whose half-wavelength equals the extra path from one step to the next, and as the line of sight climbs, that extra path grows and the reflected pitch falls.

    The chirp a staircase sings back

    A hand clap at the foot of the great stair of El Castillo at Chichén Itzá is answered by a chirp — a falling whistle that some have compared to the call of the quetzal bird. Nothing in the stone is tuned. Each of the ninety-one steps returns its own copy of the clap, the copies arrive as a train of clicks, and the ear hears a regular train of clicks as a pitch. The pitch falls because the steps are not evenly spaced as the listener sees them: low steps are seen almost level, a tread apart, and high ones almost along the slope, a whole step's diagonal apart. It is Bragg reflection from a grating whose period, seen from one spot, changes along its length.

    part 9 · waves

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