The Ruling

A glass prism separates white light into colors because each wavelength travels through glass at a slightly different speed, bending at a slightly different angle at each surface. The prism is a shaped piece of material. Its function depends on its geometry — the angle between the entry and exit faces — and its composition — the refractive index, which varies with wavelength. A different glass separates differently. A different angle shifts the spectrum. The function is inseparable from the shape and substance of the element.

In the early 1820s, Joseph von Fraunhofer built an alternative. He cut fine parallel grooves into a flat glass surface — hundreds of lines per millimeter, each groove a fraction of a wavelength wide. When light struck the surface, each groove scattered it in all directions. At most angles, the scattered waves from neighboring grooves arrived out of phase and canceled. At specific angles, they arrived in phase and reinforced. The reinforcement angle depended on the wavelength. Different wavelengths emerged at different angles. The flat surface separated light into colors with a precision the best prism could not match.

Fraunhofer used his gratings to map the dark lines in the solar spectrum — the absorption features that now bear his name. His successors built ruling engines that cut grooves with diamond points, achieving spacings of a few hundred nanometers over surfaces tens of centimeters wide. Henry Rowland at Johns Hopkins, beginning in 1882, built concave gratings that combined dispersion and focusing in a single element, eliminating the need for additional lenses. The resolving power of a grating — its ability to distinguish two wavelengths nearly identical — depends not on the precision of any individual groove but on the total number of grooves. A thousand grooves resolve the sodium doublet. A hundred thousand resolve isotope shifts. The function is not in the material. It is in the pattern.

X-rays cannot be focused by conventional lenses. The refractive index of glass at X-ray wavelengths is so close to unity that a glass lens would need to be hundreds of meters long to produce a measurable focal length. The problem is not the light. It is that the mechanism — refraction — barely operates at these energies.

A Fresnel zone plate solves this through a different mechanism entirely. It is a flat disk patterned with concentric rings, alternating between transparent and opaque. The radii of the rings are proportional to the square root of their ring number — first ring at radius r, second at r times the square root of two, third at r times the square root of three. This specific spacing ensures that light passing through every transparent ring travels a distance to the focal point that differs by exactly one wavelength from the light through the next transparent ring. The waves arrive in phase. They constructively interfere. The flat pattern focuses.

No refraction occurs. The focusing is entirely diffractive — it emerges from the interference of waves passing through apertures at precisely computed positions. The focal length is determined by the spacing of the rings. The resolution is determined by the width of the outermost ring. A zone plate with an outer ring width of ten nanometers achieves ten-nanometer resolution — sufficient for X-ray microscopy at scales approaching individual protein molecules.

The zone plate replaced a mechanism that cannot work at short wavelengths with one that works at any wavelength, provided the pattern can be fabricated at the required scale. The flat disk is not a degraded lens. It is an alternative that operates on a different physical principle, and its capability scales with the precision of the pattern, not with the properties of the material.

In 2016, Federico Capasso's group at Harvard demonstrated a flat lens made of titanium dioxide nanofins — pillars roughly six hundred nanometers tall and sixty to two hundred fifty nanometers wide, arranged on a glass substrate. Each nanofin acts as a subwavelength phase shifter. Light passing through a nanofin of one width and orientation emerges with a different phase delay than light passing through a nanofin of a different width. By varying the dimensions and rotations of the nanofins across the surface, the researchers imposed a parabolic phase profile — the same phase profile that a curved glass lens imposes through its varying thickness.

The result was a flat element, thinner than a wavelength of visible light, that focused a beam to a diffraction-limited spot. The metalens performed identically to a conventional lens of the same numerical aperture. But the function was encoded in the geometry of the nanostructures, not in the curvature of a surface.

The metalens is not restricted to parabolic phase profiles. Any phase function that can be described mathematically can be implemented by computing the required nanofin geometry at each point. A cylindrical profile produces a line focus. A helical profile produces a vortex beam. An axicon profile produces a Bessel beam. The surface is a programmable phase mask — fabricated by the same lithographic processes used to manufacture semiconductors, at the same scale, and potentially at the same cost.

What Fraunhofer achieved with a ruled grating — replacing the material properties of a prism with the pattern of a surface — the metalens extends to any optical function. A conventional lens works because its shape imposes a phase delay that varies with position: thicker at the center, thinner at the edge. The phase delay is what focuses the light. The shape is merely one way to produce it. The grating showed that a flat pattern can separate wavelengths. The zone plate showed that a flat pattern can focus. The metalens showed that a flat pattern can do anything a shaped piece of glass can do, because the function was never in the shape. It was in the phase.