Illustrated Physics
physics, drawn
Concept

Spatial frequency — where it appears

How many cycles of a pattern fit into unit distance, which is the variable an optical system really acts on. A lens lays them out as positions across its back focal plane, so an aperture is a low-pass filter and a resolution limit is a cut-off.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

What arrives in the back focal plane. A grating of pitch 1.2 µm illuminated at 550 nm, and the spectrum that appears in the objective's back focal plane. Each spatial frequency in the object leaves at its own angle, sinθ = mλ/d, and the bar heights are the Fourier coefficients of the object's transmittance. The aperture admits everything inside sinθ = 0.65, which here is orders -1, 0, 1 — 2 of them carrying information about the pitch. Orders outside it are drawn faint and are simply lost: they never reach the image plane, and no amount of magnification afterwards recovers them. This is where a microscope's resolution is decided — not at the image, not by the eyepiece, but by which of these bars the front of the objective is wide enough to catch.

The image that is a diffraction pattern twice

A lens does not project an object onto a screen. It takes the object's spatial frequencies apart, spreading them across its own back focal plane at an angle each, and then puts them back together — so an image is the object's spectrum, filtered by whatever the aperture admits, transformed back. With only the zeroth order through, the image is a uniform grey with no information in it at all.

optics · Imaging Nine planes behind a grating, with no lens anywhere. The intensity across two periods of a 20 µm grating, at nine planes between it and the Talbot distance z_T = 2d²/λ = 1.26 millimetres, illuminated at 633 nm. The top profile is the grating itself and the bottom is the plane at z_T, and they agree to 2.8e-14: free space has reproduced the object with no imaging element of any kind. The middle profile, at half the Talbot distance, is the object shifted sideways by half a period, to 1.1e-14. At a quarter of the way the grating's own period has vanished entirely — its amplitude there is 1.3e-16 — because the odd orders have all turned by the same right angle and the even ones have not. With this grating open for half of each period there are no even orders either, so the plane is uniform: 9.6e-3 at twice the frequency as well, and a screen there shows no grating at all. None of this is interference between two beams; it is the whole spectrum of the object arriving with the phases exp(−iπλzm²/d²), which are all multiples of 2π when z is z_T.

The grating that photographs itself

Put a grating in a beam of light and hold a screen behind it. At one particular distance the screen shows the grating again — sharp, at full contrast, right way up, with no lens anywhere in the apparatus. Half way there it shows the grating shifted sideways by half a period. A quarter of the way there, a half-open grating shows nothing at all.

optics · Diffraction Position written as frequency. Four tubes of water side by side in a 1.5 T magnet, their protons all precessing at 63.87 MHz, until a gradient of 10 mT/m is switched on along the row. Then the field, and so the precession frequency, rises steadily across the row, by 425.8 Hz for every millimetre. The signal the receiving coil picks up is a sum of waves at different frequencies, and its spectrum is a picture: the amount of signal at each frequency is the number of protons at the corresponding position. The axis below is position and the axis above it frequency, and they are the same axis. Nothing has been focused. The radio wave at 64 MHz has a wavelength of 4.7 m in air, and the tubes are centimetres apart.

The image made of frequencies

A hydrogen nucleus in a magnetic field precesses at a frequency proportional to the field — 63.87 megahertz in a 1.5 tesla scanner. Make the field rise steadily across the body and every position precesses at its own frequency, so the radio signal the body gives off is a chord whose spectrum is a map of where the protons are. A magnetic resonance image is never focused. It is computed, as a Fourier transform of a signal recorded one number at a time, and its resolution, a millimetre, is set by how strong the gradient is and how long the signal is listened to — not by the wavelength of the radio waves, which is several metres.

quantum · Spin A transparent cell, seen four ways. Brightness across the image of a transparent cell 20 μm across, which delays the light passing through its middle by 0.3 radians (and its nucleus by a third as much again) and absorbs none of it, imaged with green light (0.55 μm) through an objective of numerical aperture 0.25; brightness relative to the background. In focus with a plain objective (dashed) the cell is all but invisible: the brightness varies between 0.998 and 1.001. Defocused by 8 μm it shows, 0.96 to 1.06, but only at its edges and nucleus, as bright and dark fringes that do not follow its thickness. With a quarter-wave plate on the undiffracted light the cell appears in focus, its brightness following its thickness: with the plate one way it is brighter where it is thicker, up to 2.02 times the background; with the plate the other way, darker, down to 0.38.

The specimen that is invisible until its light is delayed

A living cell in water is nearly transparent: it delays the light passing through it by a fraction of a wavelength and absorbs almost none, and a perfect microscope shows it as nothing. The delay is there in the light, and the eye cannot see a delay. Frits Zernike's remedy was to delay a different part of the light — the part the cell did not scatter — by a further quarter of a cycle, at the one place in a microscope where it is separated from the rest. The invisible phase becomes brightness, the cell appears, and the price is a bright halo round everything large.

optics · Imaging

Named alongside it

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

Fourier transformCoherenceDiffractionInterferenceWavefrontApertureChemical shiftFocal planeFresnel diffractionGratingImagingLarmor precession

All concepts