Illustrated Physics
physics, drawn
Concept

Laser — where it appears

A source of light amplified by stimulated emission in a resonator, producing a beam that can fill a single spatial mode. Its brightness per mode is enormous, and passive optics can never merge two such beams into a brighter one.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

The temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative.

Hotter than any temperature there is

A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

thermodynamics · Third law The brightness N lasers can be made into. The greatest radiance — power per unit area per unit solid angle — that N identical single-mode lasers can be combined into by lossless optics, as a multiple of one laser's, against N, for four arrangements. Laid side by side and passed through any lenses and mirrors, they fill N modes and are never brighter than one: the étendue of the combination is at least N times one beam's. Two can share a mode if their polarisations are crossed, so polarisation doubles it once. Beams of N different wavelengths can be overlapped into one spatial mode by a grating, and beams with their phases locked together can be made into one mode outright; either way the radiance grows as N, because the beams are no longer N copies of the same mode.

The beams that add only if they differ

Take two identical lasers and try to merge their beams into one beam twice as bright. No arrangement of lenses, mirrors and prisms will do it: the brightness of light cannot be increased by passive optics, and two copies of the same beam can only ever be put side by side. The escape is to make the beams not copies. Give them crossed polarisations and two will merge. Give each a different colour and a grating will stack dozens into a single beam. Lock their phases so that they stop being separate beams at all, and they add — but only as well as the gaps between them allow and only while their phases are held to a twentieth of a wavelength.

optics · Etendue The hyperfine structure, read off a room-temperature cell. The difference between the probe's absorption with the pump off and on, which removes the Doppler hump and leaves only the Doppler-free features, against detuning. The three transitions, labelled by the excited level F′, sit at −423.6, −266.65 and 0 MHz, and the three crossovers midway between them at −345.1, −211.8, −133.3 MHz. In this model, which empties the ground level only by exciting it, the crossover between F′ = 2 and 3 is the second strongest feature and the weak transitions to F′ = 1 and 2 are small. In real rubidium spectra the crossovers are often the strongest of all, because excited atoms can fall to the other ground level and be lost to the transition altogether — optical pumping, which the model leaves out and which deepens every hole the pump burns. The positions do not depend on any of that: the separations between the features are the hyperfine splittings of rubidium's excited state, read to a fraction of a megahertz in a glass cell on a bench.

The lines a warm vapour hides

Shine a tunable laser through a cell of rubidium vapour and the absorption is one smooth hump half a gigahertz wide, though the atoms' own lines are eighty times narrower and three of them lie inside it. The atoms are moving, each sees the light Doppler-shifted by its own speed, and their lines are smeared together. Send a second, stronger beam back the other way and narrow dips appear in the hump at exactly the frequencies of the hidden lines — because only atoms standing still can be talked to by both beams at once. The vapour stays at room temperature and the motion is not removed; it is selected against.

quantum · Atomic spectra The cavities that hold a beam. Every two-mirror cavity as a point (g1, g2), with g = 1 − L/R for each mirror. The shaded region, between the axes and the hyperbolas g1 g2 = 1, holds a ray for ever; everywhere else a ray is walked out. The commonest designs sit on the boundary: two flat mirrors (1, 1), the confocal cavity with the mirrors' foci together (0, 0), the concentric cavity with their centres together (−1, −1), and a flat mirror facing a curved one at its own radius (1, 0). A boundary cavity is stable in principle and marginal in practice, which is why the confocal one, sitting at the corner where the two shaded regions touch, is usually built a little inside it.

The two mirrors that hold a beam

A laser is two mirrors facing each other, and the light in it makes thousands of round trips before it leaves. Whether a ray stays between the mirrors for ever or is walked out after a few bounces depends on a single number, built from their spacing and their curvatures, and the answer has a sharp edge: two flat mirrors sit exactly on it, which is why the first lasers were so hard to align. The criterion is the one that decides whether a pumped swing grows, whether a wave crosses a crystal, and whether a particle beam stays in an accelerator.

optics · Imaging An electron that runs into light and loses its energy to it. The energy of an electron of 511 MeV (γ = 1000) as it passes head-on through a 10-cycle pulse of 0.8 µm light, as a fraction of its starting energy, against the number of laser cycles it has crossed, for normalised field strengths a₀ = 30 and 100: with radiation reaction (solid) and without (dashed). Without it, the electron is shaken sideways while in the light and leaves with exactly the energy it brought, as a plane wave requires. With it, each half-cycle near the pulse's peak radiates away a slice: the electron leaves with 76 per cent of its energy at a₀ = 30 and 22 per cent at a₀ = 100, the rest gone as gamma rays.

The light that takes back an electron's energy

An electron at rest in the most intense laser light ever made feels its own radiation as a force a millionth of the light's push — the textbook reason radiation reaction never matters. Send the electron into the light at nearly the speed of light and the arithmetic turns over. In its own frame the light is stronger by twice its Lorentz factor, the power it radiates grows as the square of that, and a pulse a few tens of femtoseconds long can strip it of most of its energy as gamma rays. That is where the classical theory of a radiating charge, argued over for a century, finally has to answer to experiment — and where it starts to fail for quantum reasons.

relativity · Relativistic dynamics A hydrogen atom tilted by a laser field. The potential energy of hydrogen's electron along the direction of a laser's electric field at the field's peak, in atomic units (one unit of energy is 27.2 eV, one of length 0.053 nm), for peak intensities of 3 × 10¹³ and 10¹⁴ W/cm² and at the barrier-suppression intensity, 1.37 × 10¹⁴; the dashed line is the bound level, 13.6 eV below the ionisation limit. The field tips the Coulomb well into a barrier on one side. At 3 × 10¹³ W/cm² the barrier is 15 atomic units thick at the level — 0.8 nm — at 10¹⁴, 6; at the suppression intensity its top has fallen to the level and the electron can leave without tunnelling at all. The barrier is real only for the fraction of the laser's period that the field stays near its peak.

The atom a light wave opens like a wall

Red light cannot free an electron from hydrogen: each photon carries 1.55 electronvolts and the electron is bound by 13.6. Make the light intense enough, though, and the atom ionises anyway — by absorbing nine photons at once, or, at higher intensity, by having its potential tipped over by the light's electric field until the electron tunnels out through the side. Which of the two happens is decided by a single ratio, found by Leonid Keldysh in 1964: how long the electron needs to get through the barrier, against how long the field holds still. The tunnelling side of that ratio is where attosecond physics begins.

quantum · Tunnelling How narrow a laser's line can be, against its power. The Schawlow–Townes limit on linewidth, πhνΓ²n/P with Γ the cavity linewidth and n the inversion factor, against output power on logarithmic axes, for a helium–neon laser with a 30 cm cavity of finesse about 300 (Γ = 1.6 MHz) and for a semiconductor laser a third of a millimetre long (Γ about 30 GHz, n = 2), the latter also with Henry's linewidth enhancement 1 + α² for α = 4 (dashed). At a milliwatt the helium–neon laser's limit is 0.0025 Hz — millihertz, a coherence length of tens of millions of kilometres — and the semiconductor laser's 7.2·10⁵ Hz, 1.2·10⁷ Hz with the enhancement. Both fall in proportion to one over the power: more photons in the mode make each spontaneous kick a smaller fraction of the field. The square of the cavity linewidth is why a short, lossy diode cavity has a line ten billion times broader.

The width a perfect laser still has

A laser's light is the nearest thing to a single pure frequency that anything makes, and it is still not one. Every photon spontaneously emitted into the lasing mode adds a kick of random phase, and because nothing in a laser cares what its phase is, nothing pulls the phase back: it wanders, and a wandering phase is a line of finite width. Schawlow and Townes worked out how wide in 1958, before any laser had been built. For a milliwatt helium–neon laser the answer is a few thousandths of a hertz, a coherence length a quarter of the way to the Sun — and no real laser of that kind has ever come close, because mirrors move. For a semiconductor laser the answer is megahertz, and the measured lines were wider still, by a factor nobody expected.

optics · Coherence The spectrum two rates would make. Balance absorption against spontaneous emission alone, with level populations in the Boltzmann ratio, and the radiation comes out as x³ times e to the minus x, Wien's law; add stimulated emission, and the same balance gives x³ over e to the x minus one, which is Planck's. The two agree on the right, where a photon is much more energetic than the thermal energy and stimulated emission is negligible, and disagree on the left, where only Planck's approaches the classical count of waves. At x = 1 Planck's is 1.58 times Wien's. Measurements around 1900 at long wavelengths had already found that Wien's law failed, so experiment required the third rate seventeen years before anyone saw it in a laser or a maser.

The third process equilibrium demands

In 1916, with general relativity just finished, Einstein asked how atoms and light could reach equilibrium at all. Absorption and emission, balanced against each other, gave the wrong spectrum. A third process was needed, in which light makes an excited atom emit a copy of itself. Nobody had seen it, and it took forty years to be put to use. It also fixes a ratio that goes as the cube of the frequency, which is why X-ray lasers are so hard.

quantum · Blackbody

Named alongside it

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

CoherencePopulation inversionAttosecond pulseThe Boltzmann factorBounded spectrumCoherence lengthCompton scatteringCritical fieldDetailed balanceDiffraction gratingDoppler broadeningEinstein coefficients

All concepts