The half cycle a focus adds

Where rays stop being enough found that a lens cannot bring light to a point: the wave arriving at the focus is spread over a spot whose size is set by the wavelength and the angle of convergence, and inside that spot the picture of converging rays breaks down. The quarter cycle a turning point costs found something stranger at every place where rays bunch together, a caustic: the wave passing through one picks up a fixed extra phase that no ray calculation predicts, a quarter of a cycle at a fold, and it noted in passing that a beam passing through a focus — where rays converge from every side at once — gains half a cycle.

That half cycle is the Gouy phase, after Louis Georges Gouy, who measured it in 1890. It is small and easy to overlook, and it turns out to decide something every laser depends on: the frequencies at which light can bounce back and forth between two curved mirrors and reinforce itself. This essay is about where it comes from and what it does.

Half a cycle ahead #

Take a beam of light with a Gaussian profile, the shape a laser emits, focused to a waist of radius $w_0$. Its width grows away from the focus as $w(z) = w_0\sqrt{1 + z^2/z_R^2}$, where $z_R = \pi w_0^2/\lambda$ is the Rayleigh range, the distance over which the beam stays roughly as narrow as its waist. On the axis, the field of such a beam oscillates as

$$e^{i\left(kz - \psi(z)\right)}, \qquad \psi(z) = \arctan\frac{z}{z_R},$$

the phase of a plane wave, $kz$, minus a correction. The correction is the Gouy phase.

Far before the focus, $\psi$ is $-\pi/2$: the beam’s phase is a quarter cycle behind a plane wave’s. Far after, it is $+\pi/2$, a quarter cycle ahead. Through the focus the beam gains half a cycle, and nearly all of it within a couple of Rayleigh ranges either side. A beam focused in one direction only — by a cylindrical lens, to a line rather than a point — gains a quarter cycle, half as much, because it is confined in one transverse direction instead of two.

Gouy saw it directly. He reflected light from a curved mirror and a flat mirror placed side by side and let the two beams overlap beyond the focus of the curved one. The interference fringes in the overlap were shifted, compared with what the path lengths alone predicted, by half a fringe — the half cycle the focused beam had gained. Ray optics had no way to account for it, since every ray in the focused beam travelled exactly the path length it was assigned.

The Rayleigh range sets the scale, and it is short for any beam focused tightly. A beam focused to a waist of a micrometre has a Rayleigh range of a few micrometres, so its half cycle is gained over a region the size of a cell; the focus that is a slab, not a plane found the same length as a microscope’s depth of focus. A laser pointer’s beam, a millimetre across at its waist, has a Rayleigh range of several metres, and its Gouy phase accumulates across a room. The half cycle is the same in both; only the distance over which it is spread differs.

Crests that slip forward #

Drawn as crests, the effect is a slight stretching of the spacing near the focus. Far from the focus the beam’s crests are spheres centred on the focus, spaced by a wavelength like the plane wave’s. Near it they flatten into planes and spread slightly apart; the beam has fewer crests per unit length along its axis there than a plane wave does. Count crests from far before the focus to far after and the beam has half a crest fewer, so each of its crests ends up half a wavelength further forward than the plane wave’s corresponding one.

Nothing in that travels faster than light. The crests are a pattern, and the place where a pattern’s crest sits can move faster than any part of the wave carrying it, as the speed that carries no signal found for phase velocities generally. The energy of the beam passes through the focus at the speed of light, and the extra half cycle is a rearrangement of where the phase is, not a lead in when anything arrives.

A pulse that turns upside down #

The clearest demonstration is with a pulse rather than a steady wave. A pulse only a single cycle long — a burst of terahertz radiation, say, whose electric field swings up, down and back once — has a definite shape, and the Gouy phase shifts every frequency in it by half a cycle. Shifting a wave by half a cycle reverses its sign. A single-cycle pulse that rises then falls before the focus therefore falls then rises after it: it comes out of the focus upside down. Experiments with terahertz pulses in 1999 recorded exactly that inversion, the field at the detector reversing as it was moved from one side of the focus to the other, with the reversal happening over a Rayleigh range. For a long pulse of many cycles the same half-cycle shift moves the carrier inside an unchanged envelope and is invisible; for a single cycle, it is the whole shape.

The cost of being confined #

The reason has nothing to do with lenses and everything to do with being narrow. The fan of plane waves inside every beam found that any beam of finite width is a superposition of plane waves travelling in a fan of directions, and the narrower the beam the wider the fan — the same trade-off as sharpness has to be paid for, with transverse width and transverse wavenumber in place of time and frequency. A plane wave travelling at an angle $\theta$ to the axis has the full wavenumber $k$ along its own direction, but only $k\cos\theta$ along the axis. Every oblique part of the beam advances along the axis more slowly, in phase, than the axial part would.

Averaged over the fan, the beam’s axial wavenumber falls short of $k$ by about $\langle k_\perp^2\rangle/2k$, and for a Gaussian beam of local width $w$ that is $1/(kw^2)$. The deficit is largest where the beam is narrowest, at the focus, where its fan of directions is widest.

Add up the deficit along the beam’s path and the total is exactly the Gouy phase. With $kw_0^2 = 2z_R$, the integral of $1/(kw^2)$ from far before to far after the focus is $\int dz/[z_R(1 + z^2/z_R^2)]$ — an arctangent, running from $-\pi/2$ to $+\pi/2$. A smaller axial wavenumber means fewer crests per metre along the axis, so each crest of the beam sits a little further along than the plane wave’s corresponding crest, and the gap grows as the deficit accumulates: that is the sense in which the beam draws half a cycle ahead. Watched at one point instead, the same beam’s oscillation runs half a cycle late; the two descriptions are one fact. The magnitude is not: confining a wave to a width $w$ costs it a phase per unit length of $1/(kw^2)$, and passing through a focus costs exactly half a cycle in all, however tight the focus, because a tighter focus is narrower for a shorter distance.

The Gaussian is the gentlest case, because its fan of directions is itself Gaussian and falls smoothly to nothing. A beam cut off by a sharp-edged aperture, like the uniform circle of light a lens forms from a distant star, has a fan with long tails — the rings the rings that belong to the edge traced to the aperture’s sharp boundary — and its on-axis phase through focus does not follow the smooth arctangent: it wobbles, and the wobbles are the interference of the edge rays with the central ones. The total over the whole passage is still half a cycle, since the far fields before and after the focus are still converging and diverging spherical waves; only the path between them differs.

That account also explains the line focus. Confinement in one direction instead of two gives half the transverse spread, half the deficit, and a quarter cycle.

Higher modes slip further #

A laser can emit not only the smooth Gaussian beam but patterns with nodes across them — two lobes side by side, four in a square, rings — and each has its own Gouy phase.

A Hermite–Gauss mode with $m$ nodes in one direction and $n$ in the other gains $(m + n + 1)$ times the fundamental’s Gouy phase. The more structure a beam has across its width, the wider its fan of directions for the same overall size, and the larger its axial deficit. Modes with the same total number of nodes slip together, so their relative patterns are preserved through a focus, while modes with different totals slip by different amounts and their superposition changes shape as it passes through.

That is the principle of the mode converters that turn one kind of laser beam into another. A pair of cylindrical lenses gives the two transverse directions different Gouy phases, so that a beam made of two Hermite–Gauss patterns emerges with one shifted by a quarter cycle against the other — and if the two were chosen right, the output is a beam whose phase winds round its axis, one of the vortex beams the dark lines a wave cannot avoid described.

What decides a laser’s frequencies #

The largest consequence is inside every laser. A laser cavity is two mirrors facing each other, usually curved so that a beam bouncing between them is refocused on every pass and does not spread away. Light resonates in the cavity when a round trip brings it back to its starting point with its phase changed by a whole number of cycles. For flat mirrors and a plane wave, that happens when a whole number of half-wavelengths fits between the mirrors, at frequencies spaced by $c/2L$, the free spectral range. The phases that turn a glow into pulses found those longitudinal modes locked together into a train of pulses.

But the beam in a cavity with curved mirrors is focused on every pass, and it gains its Gouy phase each time. A transverse mode gains $(m + n + 1)$ times the fundamental’s Gouy phase per pass, so its resonances are shifted relative to the fundamental’s by a fraction of the free spectral range that depends on how strongly the mirrors focus.

For nearly flat mirrors the beam is barely focused, the Gouy phase per pass is tiny, and all the transverse patterns resonate at almost the same frequencies. For a confocal cavity, mirrors separated by their own radius of curvature, the round-trip Gouy phase is exactly $\pi$ for the fundamental, so the transverse modes fall exactly halfway between the longitudinal ones or on top of them — every pattern with an even number of nodes coincides with a longitudinal resonance, every odd one sits midway. That degeneracy is why confocal cavities are used as scanning spectrum analysers: any beam, whatever its shape, is transmitted at evenly spaced frequencies. Near the concentric limit the Gouy phase approaches $2\pi$ per round trip and the cavity becomes unstable; beyond it, no beam is refocused at all.

Laser designers choose their mirrors partly to place the transverse modes where they will not compete with the fundamental, and the calculation they make is the Gouy phase of the cavity’s beam. Gravitational-wave detectors take the same care with kilometre-long arm cavities: a higher-order mode that happened to resonate alongside the fundamental would steal power from it and add noise, and the curvatures of their mirrors are chosen, and corrected by heating them, to keep every unwanted pattern’s Gouy-shifted resonance away from the laser’s frequency.

Contrast that flips with focus #

Microscopy meets the Gouy phase every time it uses interference. In the techniques that detect a single tiny particle by interfering the light it scatters with light reflected from the glass beneath it, the particle’s scattered wave has passed through a focus and the reference has not, so their relative phase changes by half a cycle as the focus is moved through the particle. The particle’s image turns from dark to bright as the focus moves: its contrast flips sign through focus. Phase-contrast microscopy, the optical tweezers the light that pulls rather than pushes described, and every holographic method that compares a focused wave with an unfocused one have to account for it, and in each the size of the effect is fixed by how far the object sits from focus in Rayleigh ranges.

Where the half cycle shows up elsewhere #

In every focused field. The phase anomaly is not special to light. A focused sound wave gains it, which matters for focused ultrasound in medicine, where the arrival time of a pulse at the focus determines heating. A focused electron beam in a microscope gains it, and so does a focused beam of atoms in an atom interferometer.

In the far-field approximation. The factor $-i$ that appears in front of Fresnel’s and Kirchhoff’s diffraction integrals — a quarter cycle — is the Gouy phase of a wave diffracted from an aperture to the far field, where it has passed through the equivalent of a line or point focus. Every front is a source met that factor as a puzzle in Huygens’s construction; it is the half of the Gouy phase that a wave has gained by the time it reaches the far field.

At the edge of geometrical optics. The quarter cycle at a fold caustic, the half cycle at a point focus and the phase factors of semiclassical quantum mechanics, which count the turning points of a classical orbit, are one family. Each is the phase that a wave gains where rays converge and the ray picture fails, and each can be computed by asking how many directions the wave has been squeezed in.

Still open: what the phase is measuring #

The Gouy phase is understood completely as a consequence of the wave equation, and there is no dispute about its value. What has been argued, in a small literature since the 1990s, is how best to interpret it: as a consequence of the transverse spread of wavenumbers, as above; as a geometric phase acquired by a beam whose shape changes along its path; or as a manifestation of the uncertainty principle for the beam’s transverse position and momentum. Each reading gives the right answer and each suggests a different generalisation. The geometric reading is perhaps the most suggestive: a beam that is focused and spreads again has traced a closed loop in the space of possible beam shapes, from wide and converging through narrow to wide and diverging, and a phase acquired around a closed loop in a space of shapes is the kind of phase that turns up wherever a system is carried round a cycle slowly — in the polarisation of light following a twisted fibre, in the spin of a particle carried round a magnetic field. Whether that connection is a coincidence of mathematics or a sign of one underlying structure has been argued in both directions.

The generalisations are where the open questions are. Beams that do not have Gaussian profiles — sharply truncated beams, the Bessel and Airy beams the packet that bends with nothing pushing it described — have Gouy phases that do not follow the arctangent, and in some the phase evolves in ways still being characterised. Very tightly focused light, where the polarisation of different parts of the fan differs, has a Gouy phase that depends on polarisation. And short pulses, made of many frequencies, have a Gouy phase that differs for each, so that a pulse’s carrier slips against its envelope through a focus — a shift that matters for the timing of the attosecond pulses produced at the focus of intense lasers, and that is being measured directly. The half cycle itself is settled. A wave cannot be squeezed through a narrow place without paying for it in phase, and a focus is the narrowest place there is.