Lenses

In the last chapter a prism turned every ray through the same angle. A lens is cleverer: it turns each ray through an angle that depends on where the ray passes through it. Get that dependence exactly right and all the light from one point comes back together at another point. That is what it means to focus.

#From prisms to a lens

Imagine a column of prisms stacked base to base, each one a little stronger than the one inside it. Rays through the middle are bent a little; rays near the edge are bent a lot. If the bending grows in exact proportion to the distance from the centre, every ray crosses the axis at the same point. The slider sets how many prisms make up the column:

With only a few prisms, the rays passing through any one prism leave it parallel to each other, so they cross the axis all along a stretch, marked by the bar below the axis. Add more prisms and the stretch shrinks, and more and more of the light crosses close to one point. In the limit the faces of the column smooth out into two curved surfaces, the prisms become a lens, and all the light from a distant point meets at a single point: the focus.

#Plus and minus lenses

A lens that is thicker in the middle, like the one above, makes light converge. It is called a plus or convex lens. Turn the prisms round, with their bases outward, and you get a lens that is thinner in the middle and makes light diverge: a minus or concave lens. The slider runs from strong minus to strong plus:

Parallel light passing through a plus lens converges to a point behind it, the secondary focal point F′. Through a minus lens it diverges as if it had come from a point in front of the lens; that point, a virtual one, is the minus lens’s F′. Every lens also has a primary focal point F, the same distance away on the other side: light diverging from F, or heading toward it for a minus lens, leaves the lens parallel.

#Power

The distance from the lens to F′ is its focal length $f^{\prime }$. A strong lens bends light sharply and has a short focal length, so we measure a lens by its power, the reciprocal of its focal length:

Power is measured in dioptres. A +5.00 D lens focuses parallel light 20 cm behind it, a +10.00 D lens 10 cm behind it, and a −4.00 D lens spreads parallel light as if from a point 25 cm in front. It is the same unit as vergence, and that is no accident. Parallel light arrives at a lens with a vergence of zero and leaves converging toward a point $f^{\prime }=1/P$ away, that is, with a vergence of exactly P. A lens adds its power to the vergence of the light.

#The vergence equation

Now let the light come from a nearby point. It reaches the lens diverging, with a vergence $U=1/u$, where $u$ is the object’s distance (negative, because it is to the left). The lens adds its power, and the light leaves with a vergence

heading for an image at $v=1/V$. This is the same addition as at a single surface in Chapter 2, $L^{\prime }=L+F$, now for a whole lens. From here on we write U for the vergence arriving at a lens, V for the vergence leaving it and P for the lens’s power, which keeps the letter F free for its focal points. Drag the object along the axis, or use the slider, and watch the wavefronts change their curvature at the lens:

With the object 50 cm in front of this +5.00 D lens, the light arrives with $U=-2.00\text{D}$. The lens adds +5.00 D and the light leaves converging with $V=+3.00\text{D}$, toward an image 33.3 cm behind the lens. Bring the object closer and U grows more negative. At 20 cm, $U=-5.00\text{D}$ exactly cancels the lens, V is zero, and the light leaves parallel: the image is at infinity. Closer still, the lens can no longer make the light converge at all, and the image becomes virtual.

This equation is the workhorse of clinical optics. It is the familiar lens formula $1/v-1/u=1/f^{\prime }$ in disguise, but written in dioptres every step is a simple addition, and the sign of each vergence tells you at a glance whether the light is converging or diverging. Here it is with the lens power free to change as well:

Try a minus lens: it can only add negative vergence, so a real object always gives diverging light and a virtual image. Try a strong plus lens with a far object: the image sits close behind the lens, about one focal length away.

#Catching the image

When V is positive, the light really does come together at the image. A screen placed there shows a sharp point; anywhere else it shows a patch of light. Drag the screen:

The patch shrinks to a point only at the image, 40 cm behind the lens for this object. In front of the image the rays have not met yet; behind it they have already crossed and are spreading apart again. Either way the screen shows a blurred patch whose size grows with the distance from the image and with the width of the beam. Keep this picture in mind: the retina of an eye that is out of focus is just such a screen.

#Magnification

A real object is made of many points, and the lens forms an image of each one. Light from the tip of the object comes together at the tip of its image. Drag the tip of the object around:

The real image formed by a plus lens is upside down: rays from the top of the object cross the axis and end up below it. How large the image is compared with the object is the lateral magnification, which is simply the ratio of the two vergences:

A negative magnification means an inverted image. With the object at twice the focal length, 40 cm here, $U=-2.50\text{D}$ and $V=+2.50\text{D}$, so $m=-1$: an inverted image of the same size, also 40 cm from the lens. With the object farther away the image is smaller, as in a camera or an eye; with the object between F and twice the focal length it is larger, as in a projector.

#Three easy rays

You don’t have to trace a whole fan of rays to find an image. Three rays from the tip of the object have paths that are known in advance, and any two of them are enough:

• A ray parallel to the axis leaves through F′, or, for a minus lens, as if it came from F′.
• A ray through the centre of the lens passes straight through without bending.
• A ray through F, or heading for F behind a minus lens, leaves parallel to the axis.

Where the rays meet is the image. Drag the object toward F and the ray through F meets the lens far from its centre: the lens is then drawn extended with a dotted line, as if it were as big as needed. A real lens may be smaller, but the image is in the same place.

#Virtual images and the magnifying glass

Bring an object inside F of a plus lens and the lens can no longer overcome the light’s divergence: V is negative and the light leaves still diverging, as if from a point farther away than the object. There is nothing to catch on a screen, but an eye behind the lens collects the light and sees a virtual image, upright and enlarged.

This is a magnifying glass. As the object approaches F, the virtual image moves away and grows. With the object exactly at F the light leaves parallel and the image is at infinity, which is the relaxed way to use a magnifier. We’ll come back to magnifiers in the chapter on optical instruments.

#Minus lenses

A minus lens always forms a virtual, upright and smaller image of a real object, somewhere between the lens and F′. Here is a −2.50 D lens with the same three rays:

The image is smaller, but it is also closer. Seen from the lens itself it would look the same size; spectacles minify because the eye sits a little behind the lens and sees the image from farther away. That is why minus spectacles make the world look slightly smaller, about 7% smaller with −5.00 D lenses worn 15 mm from the eye, while contact lenses, right on the eye, hardly do. And to someone facing the wearer, the eyes behind strong minus lenses look small.

#Lenses are prisms too

We built a lens from prisms, and the idea works in reverse. Away from its optical centre, every part of a lens acts as a prism, and its strength grows in proportion to the distance from the centre. This is Prentice’s rule:

where $c$ is the distance from the optical centre in centimetres, $P$ the power in dioptres and $\Delta$ the prism in prism dioptres.

A plus lens is thickest at its centre, so its prisms have their bases toward the centre. A minus lens is thickest at its edge, so its bases point away from the centre. Drag the point the eye looks through:

Looking 1 cm below the centre of a +4.00 D lens is like looking through a 4 Δ prism, base up. This is how spectacles can include prism without any prism being ground into them: the lens is simply mounted off centre. It is also how unwanted prism creeps in when spectacles are badly centred, which matters most for strong lenses.

#Anisometropia and reading

Prentice’s rule matters most when the two eyes need different lenses, which is called anisometropia. To read, the eyes look down through their lenses, about 10 mm below the optical centres, and each lens then acts as a vertical prism. If the powers differ, so do the prisms:

With +1.00 D on the right and +4.00 D on the left, the eyes read through 1 Δ and 4 Δ of base-up prism: a vertical imbalance of 3 Δ. Most people tolerate an imbalance of up to about 1 to 1.5 Δ; more than that can cause discomfort or double vision when reading. The usual remedies are a slab-off (bicentric grinding) of the more minus lens, which adds base-up prism to its reading area; separate reading glasses; or contact lenses, which move with the eyes and so induce no prism.

#Recap

• A lens behaves like a set of prisms whose strength grows with distance from its centre, so it can focus light.
• Plus lenses converge light, minus lenses diverge it. Power is $P=1/f^{\prime }$, in dioptres.
• A thin lens adds its power to the vergence of the light: $V=U+P$, and the image is at $v=1/V$.
• Real images (V positive) can be caught on a screen; virtual images (V negative) can only be seen through the lens.
• Magnification is $m=U/V$; a negative value means an inverted image.
• Prentice’s rule: $\Delta =cP$. Plus lenses have their bases toward the optical centre, minus lenses away from it.

#Test yourself

#Further reading

• A. R. Elkington, H. J. Frank and M. J. Greaney, Clinical Optics. Blackwell Science.
• M. Jalie, Ophthalmic Lenses and Dispensing. Butterworth-Heinemann.
• M. P. Keating, Geometric, Physical, and Visual Optics. Butterworth-Heinemann.