Lens Systems

Few optical systems consist of a single thin lens. A trial frame holds several lenses at once, a spectacle lens works together with the eye behind it, and every real lens has two surfaces and some thickness. Fortunately one method handles all of them: follow the light through the system one step at a time, keeping track of its vergence.

#Lenses in contact add

Put two thin lenses right against each other. The light leaving the first one enters the second immediately, before its vergence has any chance to change, so the second lens simply adds its power on top of the first. Try two lenses together:

Together they act as a single lens of power

A +3.00 D and a +2.00 D lens together behave exactly like a single +5.00 D lens; a +3.00 D and a −3.00 D lens cancel out completely. This is why the weak lenses of a trial frame can simply be added, and why clinicians think of lens powers, rather than focal lengths, as the natural quantity to work with. Focal lengths don’t add; dioptres do. With strong lenses even the few millimetres between the cells of a trial frame begin to matter, as the next section shows, which is why the strongest sphere goes in the back cell, nearest the eye.

#Lenses apart: one step at a time

Now separate the lenses. Light leaves the first lens with some vergence $V_1$ and travels a distance $d$ before it reaches the second lens. On the way its vergence changes, just as in Chapter 1 (where we wrote it with L):

and then the second lens adds its power, $V_2=U_2+P_2$. Here are two +5.00 D lenses with parallel light coming in. Change their separation:

The first lens alone would focus the light 20 cm behind it. With the lenses 10 cm apart, the light reaches the second lens halfway to that focus, converging more strongly than when it left: $U_2=5/(1-0.1\times 5)=+10.00\text{D}$. The second lens adds its +5.00 D, and the light comes to a focus $1/15$ m, or 6.7 cm, behind it.

Think of it as a relay: the image formed by the first lens is the object for the second. Separate the lenses by more than 20 cm and the first image forms in the gap and the light then diverges from it toward the second lens, so $U_2$ is negative. Bring them closer than 20 cm and the second lens intercepts the light before the first image can form; that image becomes a virtual object, and the light arriving at the second lens is converging. At exactly 20 cm the first image falls right on the second lens, and the second lens can no longer move it at all.

#Moving a lens changes its effect

Often a lens is there to put its focal point in a particular place. In the next chapter, for instance, a spectacle lens must place F′ at the far point of the eye behind it. If the lens is moved, F′ moves with it, and to keep the focus where it was the lens needs a different power. Move this lens along the light:

A lens of power P, moved a distance d in the direction the light travels, needs a new power

to keep the same focal point. A plus lens moved toward its focus must become stronger; a minus lens moved in the same direction, away from its virtual focus, must become weaker. This change is called effectivity. It is small for weak lenses and short distances, but not for strong ones: a −10.00 D spectacle lens 12 mm in front of the eye does the same job as a −8.93 D contact lens on the cornea, and a +10.00 D spectacle lens the same as a +11.36 D contact lens.

#The equivalent lens

Seen from outside, any combination of lenses bends parallel light like one single lens. Its power, the equivalent power, follows from stepping along the system, and for two thin lenses it is given by Gullstrand’s equation:

But where is this equivalent lens? Not at either real lens. Extend the incoming and outgoing parts of a ray until they meet: the plane where they cross is the principal plane H′, and that is where the equivalent lens would have to stand.

With the lenses 10 cm apart, $P_e=5+5-0.1\times 25=+7.50\text{D}$, and H′ lies 6.7 cm in front of the second lens, between the two. Bring the lenses together and H′ moves onto them, while the equivalent power becomes the plain sum, +10.00 D.

For spectacles and contact lenses, though, what usually matters is not the equivalent power but where the light actually comes to a focus, measured from the back of the lens. That is set by the back vertex power: the vergence of the light leaving the last surface when parallel light enters the first.

It is the back vertex power that a focimeter (lensmeter) measures, and the power a spectacle prescription specifies.

#Thick lenses

A real lens is a combination of exactly this kind: two refracting surfaces separated by glass. Its front surface has a power $P_1=(n-1)/r_1$ and its back surface $P_2=(1-n)/r_2$. Inside the glass, light travels a distance t, but because of the refractive index it changes vergence as if it had travelled only $t/n$, so

and, in the same way, $\text{BVP}=P_e/(1-(t/n)P_1)$. For a typical spectacle lens the thickness term, $(t/n)P_1{P}_2$, is small, so the equivalent power is close to the sum of the two surface powers. Take a lens with a +8.00 D front surface and a −3.00 D back surface, 4 mm thick, with $n=1.5$. The surfaces add up to +5.00 D and the equivalent power is +5.06 D, but the back vertex power, the one that is prescribed, is +5.17 D, 0.17 D more than the surfaces suggest and more than the ±0.12 D or so that a finished lens of this power may be out by. The difference grows quickly for strong plus lenses, which have steep surfaces and thick centres.

#Spherical aberration

So far every lens has been perfect: all the light from one point met again at exactly one point. That is true of ideal lenses, and of real lenses for rays close to the axis. The optics of the vergence equation is paraxial optics: exact only for rays near the axis. A real lens with spherical surfaces bends rays near its edge a little too strongly. Here is a strong glass lens traced exactly with Snell’s law at both surfaces. Widen the beam:

Rays through the edge of the lens cross the axis in front of the paraxial focus, and the wider the beam, the bigger the error. There is no longer a single sharp focus, only a smallest blur somewhere in between. This is spherical aberration. Narrowing the beam reduces it, which is one reason why a small pupil sharpens vision. Aspheric surfaces, which flatten toward the edge, can almost remove it, which is why intraocular lenses and the condensing lenses of indirect ophthalmoscopy are often aspheric. (Strong plus spectacle lenses are made aspheric too, but mainly to make them flatter and thinner and to keep the view through their edges sharp.) The cornea also flattens toward its periphery, which keeps the eye’s own spherical aberration small, though not zero.

Spherical aberration is one of several ways real lenses fall short. Because the refractive index depends on colour, a lens has a slightly different focus for each colour (chromatic aberration). Rays entering the lens obliquely focus to two lines instead of a point (oblique astigmatism, the main reason for the meniscus form), and the magnification can change across the field, so that strong plus lenses make straight lines bow inward (pincushion distortion) and strong minus lenses make them bow outward (barrel distortion).

#Recap

• Thin lenses in contact add: $P=P_1+P_2$.
• For separated lenses, step along: the vergence changes between them, $U_2=V_1/(1-d{V}_1)$, and each lens adds its power.
• Moving a lens a distance d along the light changes the power it needs: $P_{\text{new}}=P/(1-dP)$ (effectivity).
• Equivalent power $P_e=P_1+P_2-d{P}_1{P}_2$ acts at the principal plane; spectacles are specified by back vertex power.
• Real lenses have aberrations. Spherical aberration grows with the aperture and is reduced by aspheric surfaces.

#Test yourself

#Further reading

• A. H. Tunnacliffe, Introduction to Visual Optics. Association of British Dispensing Opticians.
• M. Jalie, Ophthalmic Lenses and Dispensing. Butterworth-Heinemann.
• R. B. Rabbetts, Bennett and Rabbetts’ Clinical Visual Optics. Butterworth-Heinemann.