Illustrated Physics
physics, drawn
Series

Refraction — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Refraction from n = 1 into n = 1.5. A ray crossing a boundary between media of refractive index 1 and 1.5, bending by the amount Snell's law requires.

    The bend at the boundary, and what it is really about

    Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all.

    part 1 · optics
  2. Where the refracted ray comes from, drawn with a compass. Wavevectors in units of the vacuum wavenumber, for light arriving at 30° from a medium of index 1.5 at a medium of index 1. Every direction available in the first medium lies on the circle of radius 1.5 and every direction available in the second on the circle of radius 1; the horizontal axis lies in the interface. The boundary cannot change the component along itself, because the two sides have to agree on the phase at every point of the interface, so the refracted wave is fixed by the vertical line at 0.7500 — and where that line cuts the smaller circle is the refracted direction, 48.59° from the normal. Nothing about least time or about wavefronts enters, and the ratio of sines is what the construction reads when the two radii are written as indices. The normal component is not conserved and is not meant to be: it goes from 1.2990 to 0.6614, which is the whole of the difference between the two rays.

    The law that only asks about one component

    A boundary between two media cannot change the part of a wave that runs along it, because both sides have to agree on the phase at every point of the surface at every instant. That single restriction produces the refracted ray, the critical angle, the evanescent field and every order of a diffraction grating, out of one drawing made with a compass.

    part 2 · optics
  3. A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.

    The angle that is two angles

    Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

    part 3 · optics
  4. Two intersections, and the rule that picks one. The phase-matching construction for light arriving at 40° from a medium of index 1 into one of index -1. The circles are each medium's own relation between wavevector and frequency; the vertical line is the tangential wavenumber, which the boundary conserves. The line crosses the second circle twice, and the construction alone does not say which point is the answer — the rule that does is that energy must travel away from the interface. For a positive index the energy runs along the wavevector and the upper point is taken. For a negative one the energy runs against it, so the lower point is taken, the wavevector points back toward the boundary, and the ray leaves at -40.0° — on the same side of the normal as it arrived. Nothing in the drawing has changed except which intersection is circled.

    The ray on the wrong side of the normal

    The phase-matching construction draws a circle and a line, and the line crosses the circle twice. Every earlier construction silently took the upper intersection. Which one is physical is decided by where the energy goes rather than by where the wavevector points, and in a medium whose group velocity opposes its phase velocity the answer is the other one — so the refracted ray leaves on the same side of the normal it arrived on, a flat slab focuses, and a lens can beat the diffraction limit until loss stops it.

    part 4 · optics
  5. Snell's law with space and time exchanged. Two constructions on the same diagram of frequency against wavenumber, with the light lines of a medium of index 1 and of index 1.5. On the left, a boundary in space: the wave crosses a still surface, the frequency is conserved, and the horizontal line at the incident frequency meets the new medium's line at a wavenumber 1.5 times larger — the familiar shortening of the wavelength. On the right, a boundary in time: the whole medium changes at once, the wavenumber is conserved, and the vertical line at the incident wavenumber meets the new medium's line at a frequency 0.667 times the old one. The vertical line also meets the new line's negative-frequency branch, which is a wave running backwards: a reflection in time. A spatial boundary reflects into the same frequency and a temporal one into the same wavelength.

    The reflection that needs no surface

    Change the refractive index of a whole medium at one instant and a wave already travelling through it splits in two, one part running on and one running back, though there is no surface anywhere for it to reflect from. A boundary in time is Snell's law with space and time exchanged: the wavelength is kept and the frequency changes, momentum is conserved and energy is not.

    part 5 · optics
  6. A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

    The crystal made of moments

    A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

    part 6 · optics
  7. A mirror for X-rays that works only at a glance. The fraction of 8 keV X-rays reflected by a smooth surface of gold and of silicon, against the grazing angle between the ray and the surface, in milliradians, with absorption included (solid) and without it (dashed). Below the critical angle √(2δ) — 10.02 mrad (0.57°) for gold, 3.88 mrad (0.22°) for silicon — the X-rays are totally reflected from outside the material, because its index is below the vacuum's; above it almost nothing is. Absorption rounds the edge and keeps the plateau a little below one. A mirror for X-rays is therefore a surface met at less than a degree, and it reflects well only while it is met that way.

    The index that falls below one

    Glass bends light because light is slower in it. X-rays are not slower in glass, or in anything: every material's refractive index for them is slightly less than one, by a few parts in a million. Everything about X-ray optics follows from that small number with the wrong sign. A lens that focuses them must be thinner in the middle than at the edges, and so weak that dozens are stacked in a row. A mirror reflects them only if they arrive at a glancing angle of less than a degree — totally, from the outside, the reverse of the way a diamond traps light — which is why the mirrors of an X-ray telescope are long, nested barrels rather than dishes.

    part 7 · optics
  8. A signal that fades as the fourth power of distance. Received power, in decibels relative to free space at one metre, against distance on a logarithmic scale, for a 900 MHz transmitter 30 m up and a receiver 1.5 m up over flat, perfectly reflecting ground, with the ground's reflection taken as −1; dashed, free space alone, falling 20 dB per decade. Close in, the direct and reflected waves beat as their path difference passes through whole wavelengths. Beyond the break distance 4hₜhᵣ/λ = 540 m, the path difference 2hₜhᵣ/d is less than half a wavelength and shrinking, and the reversed reflection cancels the direct wave more and more completely: the power falls 40 dB per decade, as d⁻⁴, 10 dB below free space at ten times the break and 30 dB below at a hundred times. The ground, which was supposed to add a second path, takes the signal away; raising either antenna, which lengthens the path difference, is the remedy, and is why masts are tall.

    The surface that cancels whatever skims it

    Glass reflects four per cent of light falling straight on it, and water at radio frequencies two-thirds of the wave's amplitude. Tilt either towards grazing and the difference disappears: every surface, whatever it is made of, reflects a wave that only skims it completely, and turns it upside down. A wave travelling along a surface therefore meets its own reflection reversed and cancels on the surface itself. That one limit explains a dark fringe Humphrey Lloyd saw with a single mirror in 1834, why a mobile signal over flat ground fades with the fourth power of distance rather than the second, why a radar above the sea is blind at the horizon, and how a cliff in Sydney became the first radio interferometer.

    part 8 · optics
  9. The share of the reflection that stays a mirror. The fraction of the reflected power that goes into the mirror direction rather than being scattered, which falls exponentially with the square of the phase roughness g = 4πσ sin ψ/λ, against the grazing angle on a logarithmic scale, for surfaces whose height varies with a standard deviation σ of 0.05, 0.5 and 5 wavelengths. A surface rough by a twentieth of a wavelength is a good mirror at every angle: 67 per cent at normal incidence. One rough by half a wavelength keeps only 7·10⁻¹⁸ at normal incidence but 74 per cent at 5°. One rough by five wavelengths reflects nothing specularly until the angle is below about 0.9°, where g = 1, and is a mirror below a few tenths of a degree. The angle at which a surface turns from matt to glossy scales as the wavelength over the roughness; no surface is too rough to be a mirror for light that grazes it closely enough.

    The rough surface that is a mirror at a glance

    A sheet of office paper is matt from above and shines when sighted along its surface. A choppy sea scatters a radar beam from overhead and returns it like a mirror near the horizon. An X-ray mirror polished to a third of a nanometre is still rough to X-rays except at angles below a degree. All three are one rule: two rays reflected from points at different heights differ in path by twice the height times the sine of the grazing angle, so roughness is measured in wavelengths times that sine — and every surface, however rough, is a mirror for light that skims it closely enough.

    part 9 · optics
  10. A lens with its glass taken out, focusing the same. Two lenses of focal length 200 mm and aperture 100 mm in acrylic (n = 1.49), each with its flat face to a parallel beam and the exact aspheric curve that sends every ray to one focus: solid (top), 12.2 mm thick at the centre, and the same curve cut into 13 rings 1 mm deep with the glass between them removed (bottom), on a backing 0.4 mm thick. Thickness is drawn six times larger than height, and the distance to the focus is not to scale. Every ray leaves each lens at the same angle from the same height, because the angle a ray turns through at a surface depends only on the slope of the surface where it crosses, and every ring keeps the slope it had. What was removed was glass the rays travelled through without turning.

    The lens with its glass taken out

    Snell's law turns a ray at a surface by an angle set by the slope of the surface where the ray crosses, and the glass behind the surface does not enter it. So a lens can be sliced into rings, the glass between the slopes thrown away, and the rays will not notice. Fresnel did it for lighthouses in 1822. The waves notice — every step cuts a few hundred wavelengths out of them — unless the steps are made exactly one wavelength deep, and then the lens becomes something else: a diffractive element whose colours run backwards, and which can cancel a glass lens's colour with a twentieth of its power.

    part 10 · optics
  11. How long a lens must hold back its central light. The time by which a lens must delay light passing through its centre relative to light passing its edge, so that every part of a short pulse reaches the focus together, against the lens's radius, for focal lengths of 0.1 mm, 1.0 mm, 10.0 mm (solid), in femtoseconds on logarithmic axes: the extra path from the edge, √(R² + f²) − f, over c. The dashed line is what a flat structure 1 μm thick can supply if its building blocks span a group index of 1: 3.3 fs. A 1 mm focal-length lens reaches it at a radius of 45 μm; a 10 mm one at 141 μm. A glass lens of 10 mm focal length and 5 mm radius holds its centre back by 3937 fs, which it does by being 2.27 mm thicker in the middle. A flat lens has no room for that.

    The delay a flat lens has no room for

    A glass lens is thick in the middle for a reason that is easy to miss. To focus light of every colour at one point, a lens must do more than bend each colour by the right amount: it must make every part of a light wave arrive at the focus at the same moment, and the light passing the edge has further to go. The glass in the middle is what holds the central light back to wait for it. A lens made flat — a kinoform, a metasurface a micrometre thick — has nowhere to keep that delay. The geometry of the focus sets how much delay is needed, the thickness sets how much is available, and between them they cap how large an achromatic flat lens can be: tens of micrometres, not centimetres.

    part 11 · optics

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