Gratings & diffractionFrom lines to a focus
A mologram arranges bound molecules into a diffraction pattern. Their scattered fields reinforce in selected directions. A focusing pattern also accounts for the path from each position to the focus. This lesson builds those ideas from individual scatterers.
Objective. See how ordered lines diffract light, and how curving them makes a focus.
Recommended first: Coupling.
One scatterer produces a weak field
A molecule illuminated by the guided field scatters a weak optical field into multiple directions. Its angular distribution depends on polarization and the surrounding interfaces. The grating signal arises when fields from many ordered scatterers combine coherently.
Many lines interfere: the grating equation
Put scatterers in a row, evenly spaced by a period Λ. In a given direction, the wavelet from each line travels a slightly different distance to a far detector. So each arrives with a slightly different phase.
This introductory model assumes equally illuminated scatterers, normal incidence and a wavelength λ in the observation medium. The observation angle θ is measured from the normal; m is the diffraction order. A guided-wave mologram also includes the phase accumulated by the illuminating mode, as introduced on the coupling page.
Step from one line to the next and the extra path adds a fixed phase φ = (2π/λ)·Λ·sinθ. When φ is a whole number of turns, the wavelets have the same phase and reinforce. In other directions, they partly or fully cancel. That condition, Λ·sinθ = mλ, is the grating equation. The resulting intensity has peaks in selected directions.
Each arrow is one line's wavelet; they are laid tip to tail. Sweep the detection angle. When the per-line phase step is zero the arrows line up and the resultant (orange) is long: that is a diffraction peak. Just off it, the arrows curl into a circle and the sum collapses.
More lines, sharper peaks
Two lines give a broad, gentle interference. Add more and the peaks where everything agrees get narrower and brighter. A small change in angle now throws the many wavelets out of step much faster. A 400 µm mologram with a period of a few hundred nanometers contains about a thousand periods, so its diffracted direction is very well defined. Ordered contributions can produce a strong peak relative to randomly distributed scatterers; actual contrast depends on the optical background.
Change how many lines contribute. The curve is the diffracted intensity around one peak. Few lines: a wide bump. Many lines: a narrow spike with weak side lobes. Each curve is normalized to its own peak; absolute peak intensity would scale as N² for equal scatterers. This is why a large mologram has a precise direction.
Curve the lines and it focuses
A grating of straight, evenly spaced lines sends light off in a direction: the diffracted rays leave parallel. Now vary the spacing across the pattern, packing the lines closer toward the edges. Each part of the grating then bends light by a slightly different amount. Chosen correctly, every diffracted ray is aimed at one point.
The grating has become a lens, a diffractive lens or Fresnel zone plate, and the light converges to a bright spot at its focal length. This is exactly what a mologram does. That is why the technique is called focal molography: the binding pattern is shaped so the diffracted light focuses to a single camera spot.
Increase the chirp, the change in spacing from center to edge. At zero the diffracted rays leave parallel. As the chirp grows they bend toward a common focus, closer as the chirp increases. This is a ray schematic, not a computed mologram pattern. The real pattern compensates both the guided incident phase and the path to the focus.
Why a focus, and not just a bright direction?
Focusing concentrates coherent light into a small detector region. The fraction collected depends on the diffraction pattern, imaging optics, and camera sampling. The next module turns those lines into binding sites, so patterned binding changes the focus. Patterned non-specific binding can also contribute.
Worked example. Streptavidin binds at repeated ridge positions. The pattern compensates the illumination phase and the path to the focus so coherent contributions reinforce there.
Predict before revealing the explanation. If two equal scattered fields arrive half a cycle apart, is their combined intensity twice that of one?
Reveal the explanation
No. Their fields cancel at that observation point in the ideal case. Add fields first, then square the resultant magnitude.