Illustrated Physics
physics, drawn
Concept

The paraxial approximation — where it appears

The treatment of rays as making small angles with the axis, which makes lens formulae linear and hides every aberration at once. Every aberration is a term the approximation dropped, so a lens design is the business of managing what the first-order formulae cannot see.

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

A converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

What a lens is doing, and why three rays are enough

A lens bends every ray that reaches it. The construction uses three, because three are all that can be drawn without calculation — and any three that meet prove all the rest do.

optics · Imaging A sphere does not have a focus. Parallel rays reflected off a spherical mirror. Rays striking further from the axis cross it nearer the mirror, so there is no single point where all of them meet — the blur is spherical aberration, and a perfect sphere has it inherently.

The mirror that cannot focus, and the shape that can

A perfect sphere does not bring parallel light to a point. The blur is not a manufacturing defect — it is what the shape does, and the shape is used anyway, for a reason worth knowing.

optics · Imaging Image distance against object distance. Image distance in focal lengths against object distance in focal lengths. At exactly one focal length the image runs off to infinity; inside it the image distance goes negative, which means virtual.

The focus that is a slab, not a plane

A lens images one plane and no other, which would make every photograph and every micrograph almost entirely out of focus. What rescues them is a tolerance — and there are two of them, one from rays and one from waves, which give different answers and stop being interchangeable exactly where microscopes work.

optics · Imaging A wave that falls with nothing pulling it. The intensity of Berry and Balazs's Airy packet, |Ai(x − t²/4)|², drawn at six moments t = 0 to 5 in units where ħ = m = 1, each profile raised by its time. It is an exact solution of the free Schrödinger equation — no potential, no force — and equally of the equation that governs a light beam in the paraxial approximation, with t then the distance travelled. Its shape never changes and never spreads, and its main lobe, which starts at x = −1.02, moves along the parabola x = t²/4 (dashed): −0.77 at t = 1, −0.02 at t = 2, 1.23 at t = 3, 2.98 at t = 4, 5.23 at t = 5. A uniform acceleration of a half, with nothing accelerating it. The packet carries infinite energy, which is the price of the trick: its oscillating tail stretches to minus infinity, and the tail is what pushes the main lobe along.

The packet that bends with nothing pushing it

Every wave packet left to itself spreads, and every free particle moves in a straight line. In 1979 Berry and Balazs found a packet that does neither: it keeps its shape for ever, and its peak accelerates sideways along a parabola, in empty space, with no force acting. Nothing in it breaks Newton's laws — the packet's centre of mass stays exactly where it was, and only its bright head runs off, fed by a long dim tail — and the trick turns out to be the rainbow's: the bright lobe is the envelope of a family of straight rays. Thirty years later it was made with light, and then with electrons.

waves · Wave packets

Named alongside it

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

Focal lengthCausticMagnificationReal and virtual imagesAberrationAiry functionApertureAspheric surfaceConic sectionDiffraction limitDispersionEhrenfest theorem

All concepts