Nonlinear and Nano-Optics

The other half of the subject leaves light alone and builds the material. Two observations in 1987, by Eli Yablonovitch and Sajeev John, pointed out that a dielectric patterned periodically at the scale of a wavelength does to photons what a crystal lattice does to electrons: bands, and with enough index contrast a complete gap in which no propagating mode exists in any direction. Inside a photonic bandgap, light cannot travel, so a line defect becomes a waveguide that turns a right angle without loss, and a point defect becomes a cavity the size of a wavelength.

John Pendry went further and asked what properties could be synthesised rather than found. A lattice of thin wires behaves as a medium with negative electric permittivity; a lattice of split metal rings has a magnetic resonance and can present negative permeability. Victor Veselago had worked out in 1968 what a material with both would do: refract to the wrong side of the normal, reverse the Doppler shift, and focus with a flat slab. In 2000 David Smith's group built one for microwaves, and in 2006 a structure that routed microwaves around a central region, leaving it in a shadowless hole.

The claims outran the physics for a while. Pendry's perfect lens was supposed to recover evanescent waves and so beat the diffraction limit outright; absorption in the resonators limits how much of that survives. Later analysis showed that a passive linear cloak cannot hide an object across a wide band of frequencies at all. What is left is substantial — engineered index, flat metasurface optics now shipping in sensors — and narrower than the first announcements.

How intense does light have to be before an atom stops being a small perturbation on it? Compare the laser's field to the field the electron already feels. In hydrogen, the electron sits at the Bohr radius $a_0 = 5.29 \times 10^{-11}$ m, where the proton's field is

$$E_{\text{at}} = \frac{e}{4\pi\varepsilon_0 a_0^{2}} = 5.14 \times 10^{11} \text{ V/m}.$$

A light wave of amplitude $E$ carries intensity $I = \tfrac{1}{2}\varepsilon_0 c E^{2}$, so matching the atomic field takes

$$I = \tfrac{1}{2}\left(8.854\times10^{-12}\right)\left(3.00\times10^{8}\right)\left(5.14\times10^{11}\right)^{2} \approx 3.5\times10^{20} \text{ W/m}^{2} = 3.5\times10^{16} \text{ W/cm}^{2}.$$

That is the dividing line. Below it, light nudges electrons; above it, light dominates the nucleus's hold on them. Chirped pulse amplification clears the line by six orders of magnitude: focused petawatt pulses reach $10^{22}$ W/cm², where the electron's oscillation velocity approaches $c$ and the physics becomes relativistic.

Just above the line, something more useful than destruction happens. The field pulls an electron out of the atom, accelerates it away, then — half a cycle later, as the field reverses — drives it back into its parent ion, where it recombines and dumps all the kinetic energy it gathered as a single high-energy photon. This happens once per half-cycle, in phase across the gas, producing odd harmonics of the driving laser out to the hundredth order and beyond.

Why bother? Because of what short pulses require. The time-bandwidth relation for a Gaussian pulse is $\Delta t\,\Delta\nu \ge 0.441$, so a pulse lasting 100 attoseconds needs a spectral width of

$$\Delta\nu \ge \frac{0.441}{100\times10^{-18}} = 4.4\times10^{15} \text{ Hz}.$$

The entire visible spectrum, from 400 to 700 nm, spans only $3.9\times10^{14}$ Hz — a tenth of what is needed. A pulse must also contain fewer than one cycle's worth of ambiguity, so its carrier frequency has to exceed its bandwidth, which means

$$\lambda < \frac{c}{4.4\times10^{15}} = 68 \text{ nm}.$$

Attosecond pulses are therefore impossible in visible light as a matter of arithmetic. They must be made in the extreme ultraviolet, and high harmonic generation is the only practical way to get coherent light there on a tabletop.

What this buys is a shutter fast enough for electrons. The electron in the Bohr orbit travels at $\alpha c = 2.19\times10^{6}$ m/s around a circumference of $2\pi a_0 = 3.33\times10^{-10}$ m, giving a period of

$$T = \frac{3.33\times10^{-10}}{2.19\times10^{6}} = 1.52\times10^{-16}\text{ s} = 152 \text{ attoseconds}.$$

A 100-attosecond flash resolves a fraction of that orbit. It is the first time scale on which the motion of a bound electron is slow.

The pattern across this field is that each apparent barrier gives way to a nonlinearity, and then a new barrier appears one level down. Abbe's diffraction limit stands for linear optics, and Stefan Hell's depletion microscopy walks around it by saturating a transition, so that the effective spot shrinks as the square root of intensity — the technique described under cell biology, whose resolution is now limited by how many photons a fluorophore emits before it bleaches rather than by the wavelength. The amplifier damage threshold that capped pulse energy gave way to chirped pulse amplification, and the limit became the gratings. The electron's motion, once unresolvable, became measurable in attoseconds, and the new difficulty is conceptual: asking how long an electron takes to tunnel turns out not to have a well-defined answer, which is an unusual place for an optics experiment to end up.

Light began this thread as something to explain, in classical optics. It ends as the most precisely controlled thing in physics, and the instrument with which most of the rest is now measured.