Light and Vergence

Hold a pencil at arm’s length and look at its tip. Light from the room strikes the tip, and the tip scatters some of it back out in every direction. A tiny part of that light enters your eye, and the eye bends it so that it comes together again at a single point on your retina.

Everything in these chapters, from spectacles and contact lenses to prisms and the eye itself, is about two questions: where does the light from a point go, and where does it come back together? To answer them we need two ways of picturing light, and one number.

#Points of light

Every point of a lit object acts as a tiny source of light. Light leaves it in all directions at once, much like the ripples that spread across a pond when a pebble drops in. The figure below shows those ripples around a single point source. You can drag the source around.

Each circle joins the places that light from the source reached at the same moment. A surface like this is called a wavefront. In three dimensions the wavefronts are spheres, nested like the layers of an onion and growing steadily outward. Real light waves are packed far more tightly than this, about two thousand to the millimetre, so the drawing shows only a few of them, widely spaced, to show their shape.

Notice that the wavefronts become less curved as they grow. Near the source they bend sharply; far from it, a short piece of one is almost straight. That simple observation turns out to be the most important idea in clinical optics.

#Rays

Wavefronts show the shape of the light. Often we care more about where it is going, and for that we draw rays: lines that point in the direction the light travels. A ray always crosses the wavefronts at right angles, so the rays from a point source spread out in straight lines, like the spokes of a wheel.

The two pictures describe the same light, and you can switch between them in the figure. Rays are the better tool for following light through lenses and prisms. Wavefronts are the better tool for measuring how strongly light is spreading out or closing in, which is what we need next.

A real object, such as a letter on a test chart, is a collection of countless points like this one, each sending out its own wavefronts. A lens or an eye forms an image of the object by bringing the light from each object point back together at a matching image point. So if we understand what happens to the light from one point, we understand the whole picture.

#How curved is the light?

Let’s look at the light through a window, an opening that lets through only part of each wavefront. The opening could be the pupil of an eye or a spectacle lens. Use the slider to move the source closer to the window or farther away.

Close to the source, the wavefront in the window is strongly curved: it is a piece of a small circle. As the source moves away the circle grows, and the piece of it that fits in the window flattens. From a source a few metres away, the wavefront across a pupil-sized window is very nearly flat. From a distant star it is flat for all practical purposes, and the rays through the window are parallel.

The natural way to measure how sharply something bends is its curvature: one divided by its radius. A wavefront with a radius of 25 cm, or 0.25 m, has a curvature of $1/0.25=4$ per metre. One with a radius of 2 m has a curvature of 0.5 per metre, and a flat wavefront, whose radius is infinite, has a curvature of zero.

#Vergence and the dioptre

In clinical optics the curvature of a wavefront has its own name, vergence, and its own unit. A curvature of one per metre is called one dioptre, written D. The wavefront in the window above, 25 cm from its source, has a curvature of 4 dioptres. In a moment we’ll give it a sign as well.

Vergence always describes the light at one particular place, which we call the measuring plane: the plane of a lens, of the cornea, or of a spectacle frame. If $l$ is the distance in metres from that plane to the point the light is spreading from, the vergence is

Light can do more than spread out. A lens or a curved mirror can make it converge: its wavefronts then curve the other way and shrink toward a point, where the light comes to a focus. Converging light can be just as curved as diverging light, so we need a sign to tell the two apart.

The convention used throughout clinical optics, and in these chapters, is that light travels from left to right. Distances are measured from the measuring plane: positive to the right, in the direction the light travels, and negative to the left. Diverging light comes from a point to the left of the plane, so $l$ is negative and so is its vergence. Converging light is heading for a point to the right of the plane, so its vergence is positive. Parallel light has a vergence of zero.

The slider below sets the vergence of the light at the dashed measuring plane. Drag it from one end to the other: the light goes from strongly diverging, through parallel, to strongly converging.

At −10.00 D the light diverges from a point just 10 cm before the plane; at −1.00 D, from a point a metre away. At zero the wavefronts are flat. Past zero they curve the other way, and at +10.00 D the light converges to a point 10 cm beyond the plane. A large vergence, of either sign, always means a nearby point; a small one, a distant point. Here is the vergence of the light that reaches the eye from some familiar distances:

#Vergence changes as light travels

Because vergence describes the light at a particular place, it changes as the light moves on. In the figure below you can drag the measuring plane along the beam, or use the slider, and watch its reading in the graph underneath. The buttons switch between diverging and converging light.

Moving away from a source, the wavefronts grow and flatten, so diverging light gets weaker: −10.00 D at 10 cm from the source, −5.00 D at 20 cm, −2.00 D at half a metre. Converging light does the opposite. As it closes in, its wavefronts shrink and steepen, and its vergence climbs without limit as it reaches the focus. Just past the focus the light is diverging again, now from the focus itself, so its vergence starts out hugely negative and then slowly weakens.

If the vergence at one plane is $L_1$, then at a second plane a distance $d$ metres farther along the light it is

For example, light diverging with a vergence of −2.00 D has, 10 cm farther on, a vergence of $-2/(1+0.1\times 2)=-1.67\text{D}$: it is now 60 cm from its source. Light converging with +5.00 D has, 10 cm farther on, a vergence of $5/(1-0.1\times 5)=+10.00\text{D}$: its focus is now only 10 cm ahead.

This step-along equation is nothing more than $L=1/l$ applied twice. The point the light comes from, or is heading for, stays where it is; only our distance from it changes. Simple as it is, it explains why a spectacle lens and a contact lens for the same eye need different powers, as we’ll see once we have lenses.

#Light inside other materials

So far the light has travelled through air. Inside glass, water or the eye, light moves more slowly, and for reasons that will become clear in the next chapter, vergence there is defined with an extra factor: the refractive index $n$ of the material.

In air $n=1$ and nothing changes. Inside the eye, where $n$ is about 4/3, light converging toward a point 22.2 mm ahead has a vergence of $1.333/0.0222\approx +60\text{D}$. That is almost exactly the focusing power of a human eye, and it is no coincidence: the eye’s job is to give the light inside it just the vergence it needs to converge onto the retina.

The step-along equation changes in the same spirit. Inside a material, light travelling a distance $d$ changes its vergence as if it had travelled $d/n$ in air, so $L_2=L_1/(1-(d/n)L_1)$.

#Recap

• Light from every point of an object spreads out in wavefronts. Rays show its direction and cross the wavefronts at right angles.
• Vergence measures the curvature of the wavefronts at a chosen plane, in dioptres: $L=1/l$ in air, and $n$ times the curvature, $L=n/l$, in a material of refractive index $n$.
• Light travels from left to right. Diverging light has negative vergence, converging light positive, parallel light zero.
• Vergence changes as light travels: $L_2=L_1/(1-d{L}_1)$ in air, with $d/n$ in place of $d$ inside a material. Diverging light weakens; converging light strengthens until its focus.

#Test yourself

#Further reading

• A. H. Tunnacliffe, Introduction to Visual Optics. Association of British Dispensing Opticians.
• A. R. Elkington, H. J. Frank and M. J. Greaney, Clinical Optics. Blackwell Science.
• American Academy of Ophthalmology, Basic and Clinical Science Course, Section 3: Clinical Optics and Vision Rehabilitation.