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

A lens does one thing to a ray: it bends it, by an amount that depends on where the ray strikes. That is all. Everything a lens is famous for — projecting, magnifying, focusing, making distant things reachable — follows from that one action applied to every ray at once.

The remarkable part is the at once. Light leaves a single point on an object in every direction, spreading and diverging, and a lens gathers a whole cone of those rays and sends them back to a single point. Every ray that left together arrives together, having taken visibly different routes.

Three rays, and why exactly those #

Of the infinitely many rays leaving the arrow’s tip, three can be drawn without solving anything, because for each of them the outgoing direction is known from the definition of the focal point rather than from a calculation.

The parallel ray. A ray that arrives parallel to the axis leaves through the far focus. That is what a focal length is: the distance at which parallel light converges.

The central ray. A ray through the middle of the lens carries straight on. The two faces are parallel there, so the ray emerges displaced but undeviated, and in a thin lens the displacement is negligible.

The focal ray. A ray passing through the near focus emerges parallel to the axis, which is the previous rule run backwards — optics is time-reversible, and every statement about rays has a mirror-image statement obtained by reversing them.

Three rays leave the same point and meet again at another point. That meeting is the image, and the whole construction is a claim that all the other rays meet there too. They do, and the reason is not obvious from the picture: it is a consequence of the specific way a lens’s bending power varies with height, which is engineered — a lens is ground into the shape that makes this true, and shapes that fail to make it true are what aberration means.

One equation, and a runaway #

The construction can be replaced by arithmetic. With $u$ the object distance, $v$ the image distance and $f$ the focal length,

$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f},$$

and the magnification is $-v/u$, the minus sign carrying the inversion.

The reciprocals make the behaviour much less intuitive than a linear relation would be, and the interesting region is close to the focus.

Far away, $1/u$ is nearly zero and $v \approx f$: everything distant images at the focal plane, which is why a camera focused at infinity has its sensor exactly one focal length behind the lens, and why focusing on a mountain and focusing on a star are the same adjustment.

Approach the focus and the image distance climbs steeply, through $2f$ when the object is at $2f$, and then to infinity as the object reaches $f$. An object at the focus produces no image anywhere: the rays leave parallel and never meet. That is a projector’s condition run backwards, and a lighthouse’s condition run forwards.

Inside the focus, $1/v$ turns negative. The rays leaving the lens are still diverging, so they never cross, and the image is on the same side as the object, upright and enlarged. Nothing arrives there — the light only appears to come from it — which is what makes it virtual. A magnifying glass held close to a page is a lens used in exactly this regime, and the fact that the image cannot be caught on a screen is the operational difference between a virtual image and a real one.

The same equation for a mirror #

The construction was derived from refraction at two curved surfaces, and it ought to be specific to lenses. It is not.

A concave mirror obeys $1/u + 1/v = 1/f$ with $f$ equal to half the radius of curvature. The rays are different — one reflects off the centre symmetrically, one arrives parallel and leaves through the focus — but the relationship between object, image and focus is identical.

This is not a coincidence and it is not deep. Both devices do the same thing to a wavefront: they add a delay that varies as the square of the distance from the axis, converting a diverging spherical wave into a converging one. A lens does it by making light take longer through the thick middle; a mirror does it by making the middle further away. Anything that imposes that quadratic delay will image, which is why a curved radio dish, a curved acoustic reflector and a zone plate made of nothing but opaque rings all obey the same formula.

Mirrors have one decisive advantage: they have no dispersion, because reflection does not depend on frequency. Every large telescope built since the nineteenth century is a reflector, and chromatic aberration is the reason. Some prism-based instruments get the reflection for free by arranging for the angle to exceed the critical one, which gives a mirror with no coating to tarnish — the reason good binoculars use glass prisms rather than silvered surfaces.

Where the bending actually happens #

The construction draws rays kinking at a single plane, which is a convenience rather than a description.

Snell’s law is the only physics present. A ray meeting the front surface bends toward the normal; the normal points in a different direction at every height on a curved surface, so rays at different heights bend by different amounts; and the surface is ground so that the variation is exactly what is needed to bring them back together.

That is why the focal length depends on the index and the curvature and on nothing else, and it is why the lens equation is a geometric consequence rather than a law. Grind a different curve and the arithmetic changes; the physics does not.

What sets the focal length #

The lens equation treats $f$ as given. It comes from the glass and the grinding, through the lensmaker’s equation:

$$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right).$$

Two things are worth reading out of it. The factor $(n-1)$ says a lens works because its index differs from its surroundings — and a glass lens in a liquid of matching index is invisible and does nothing, which is a good demonstration and an unnerving one. Underwater, the eye’s own lens loses most of its power for the same reason, since the index difference between cornea and water is far smaller than between cornea and air. A diving mask restores vision by putting air back in front of the eye.

The curvature terms say power adds. Two thin lenses in contact behave as one with $1/f = 1/f_1 + 1/f_2$, which is why optometrists work in dioptres — reciprocal focal lengths in metres — and why a prescription can be filled by stacking. It is also the loophole the achromatic doublet exploits: combine two glasses so the powers add to something useful while the dispersions cancel.

Where the model stops #

The thin lens is an idealisation with four separate failures, and every one of them is a whole branch of lens design.

Thin. The model assumes the lens has no thickness, so both refractions happen at one plane. A real lens has two surfaces at a distance, and thick-lens optics replaces the single plane with two principal planes.

Paraxial. Every step assumed small angles — that $\sin\theta$ could be replaced by $\theta$, which is the same substitution that makes a pendulum simple, fails quadratically in the same way, and is applied here to the sines in Snell’s law specifically. Rays far from the axis strike a spherical surface at angles where the approximation breaks, and they cross the axis nearer the lens than the paraxial rays do. That is spherical aberration, and it is not a manufacturing defect: a perfect sphere has it inherently, which is why fine lenses are ground aspheric.

One wavelength. Covered above, and the reason a cheap lens fringes purple.

Rays at all. The deepest limit. Even a perfect lens cannot focus light to a point, because light is a wave and a converging wave of finite aperture produces a spot of finite size — the Airy disc, whose radius is roughly $1.22\lambda f/D$. That is interference, not imperfection, and no improvement in grinding reduces it. It sets the resolution of every telescope and microscope, it is why bigger apertures see finer detail, and it is the reason optical microscopy stalls at around 200 nanometres.

What an image is, exactly #

It is worth being precise about what has been constructed, because the everyday word does a lot of unnoticed work.

An image is a point-to-point correspondence. Each point of the object maps to one point where light from it reconverges. Nothing about the arrangement is picture-like; there is no screen required, no observer, and the light does not know it has formed an image. A real image exists in mid-air whether or not anything is there to catch it, and putting a hand into the space where a projector’s image forms proves it, brightly.

That correspondence is what the eye and the camera both need, and it is why the retina and the sensor sit where they do — one focal length back for a distant object, and further for a near one. It is also why an out-of-focus photograph is not a blurred picture of the object but a perfectly sharp record of a failed correspondence: each object point has spread into a disc, and every disc has been added together. Deconvolution can sometimes undo it, precisely because what happened was a definite mathematical operation rather than a loss of information.

The ladder from here #

Later rungs: the lensmaker’s equation derived from refraction at two spherical surfaces. Thick lenses and principal planes. The five Seidel aberrations and what each does to a star image. The achromat and the apochromat. The eye as an optical instrument, with a variable-focus lens and a fixed image distance — the opposite arrangement to a camera. Telescopes and microscopes as two-lens systems, and angular magnification as the quantity that matters when the object cannot be moved. The diffraction limit taken seriously. And the Fourier-optics view, in which a lens performs a mathematical transform and the focal plane holds the spatial frequencies of the object.

Lenses were in use for four centuries before anybody could say what a focal length was. Grinding glass until it works is an older technology than the theory of why it does.