The Physics
Opus in profectus

Spherical Lenses

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If you are a vertebrate with eyes, then you have lenses. The same goes for octopus, squid, and cuttlefish (which are all cephalopods). Common optical devices that use lenses include eyeglasses, contact lenses, magnifying glasses, cameras, projectors, telescopes, binoculars, and microscopes. All sorts of things will act like lenses, even things that aren't normally thought of as lenses. As long as they're transparent and curved, they satisfy the definition and will behave in a manner similar to the devices commonly thought of as lenses. Car headlights and taillights are covered with lenses. (They're even called lenses in auto parts catalogs.) A drop of water is a lens. So are fishbowls, light bulbs, and drinking glasses. In general, a lens is any piece of transparent material with at least one curved surface.

The origins of the word lens can be traced back to the Seventeenth Century. Scientists at the time wanted a word to describe the shape of the glass pieces used in microscopes, telescopes, magnifying glasses, and reading glasses. For some unknown reason they decided against the strictly descriptive term "biconvex" (curved outward on two sides) and instead chose to name them for the small, flat beans they resemble — lentils. Lens lentis is the Latin word for lentil. It's good word for English speakers since few of us associate lenses with lentils and not all lenses are lentil shaped.

What makes a lens different from any other transparent object is its ability to focus light. A focus is a meeting point. The origins of this word are also Latin. Focus is the Latin word for fireplace. Like the word lens, the word focus first started to appear in scientific writing in the Seventeenth Century. Many people are familiar with the ability of magnifying glasses to concentrate sunlight to the point where it can burn paper. It is uncertain whether the scientist of this time were referring to the fire produced by this method or the fact that the fireplace was the meeting point (or focus) of the home at the time.

In any case, it's only half true that a lens can concentrate sunlight to a point that might then be used to a start a fire. Only a converging lens will do that — a lens that converges parallel rays of light down to a point. Some lenses do just the opposite — they diverge parallel rays of light from a point. Thus lenses are divided into two major categories called converging lenses and diverging lenses, respectively. The point to which these rays converge or from which they appear to diverge is called the focus or focal point and is indicated with the symbol F.

This behavior is best illustrated with pictures.

Rays of light parallel to the principal axis converge on the focus after passing through a converging lens.

Rays of light parallel to the principal axis appear to diverge from the focus after passing through a diverging lens.

The way to distinguish among the two types of lenses is to look at the relative thickness of two parts — the center and the edges. Converging lenses are thicker in the middle than they are at the edges, while diverging lenses are thicker at the edges than they are in the middle. Converging lenses include those that are biconvex (curved outward on both sides), plano-convex (flat on one side and curved outward on the other side), and convex meniscus (curved inward on one side and outward on the other side more strongly). Diverging lenses include those that are biconcave (curved inward on both sides), plano-concave (flat on one side and curved inward on the other side), and concave meniscus (curved inward on one side and curved outward on the other side less strongly).

Nearly every lens has a line of symmetry down its center (which may or may not be its geometric center, but usually is). Given a standard, simple lens like eyeglasses, contact lenses, or the lens in a movie projector, there is an obvious axis of symmetry about which the lens could be rotated and not have any effect on the image produced. Such an axis is called the principal axis. To keep life simple, when parallel rays are shown entering a lens, they are often all drawn parallel to the principal axis. Such rays are said to be paraxial. As I have just said, this is merely done for convenience. Actually, for an ideal lens, parallel rays always converge on or diverge from a point. (The aberrations of a real lens from an ideal lens are dealt with in another section of this book.) For a lens with spherical symmetry, the collection of focal points for any group of parallel rays form a focal plane. This behavior is best illustrated with a series of pictures.

Rays of light parallel to each other converge on the focal plane after passing through a converging lens as illustrated in this diagram.

Oh yeah, is that all?

Oh no. Welcome to the real world.

ray diagrams

at least two rays


Geometric derivation of the magnification equation.

Similar triangles. The magnification equation.

M = hi = di

Geometric derivation of the spherical mirror equation.

Magnification equation, similar triangles.

M =  hi  =  di  =  ƒ
ho do do − ƒ

cross multiply, distribute, collect like terms.

di(do − ƒ)  =  dof
dido − dif  =  dof
dido  =  dif + dof

Divide by didof.

dido  =  dif  +  dof
didof didof didof

Simplfy. The thin lens equation (physicists form).

1 = 1 + 1

"If I do, I die"

It is often more convenient to work with this equation if we give special names to the inverse quantities. The inverse of focal length is the refracting power (usually just called power) of the lens and the inverse of the object and image distances are called vergences. Using these new terms, the lens equation can be stated more compactly in words and symbols. "The power of a lens is the sum of the object and image vergences." Since it is a linear relationship, it is also much easier to handle mathematically.

The thin lens equation (optometrist form).

P = Vo + Vi

The SI units of each of these quantities is the inverse meter [m−1], which is given the special name diopter [D]. The power of eyeglasses and contact lenses are most commonly expressed in this unit. The greater the power a pair of eyeglasses or contact lenses has, the worse is the vision of the person prescribed them. Converging lenses, like those normally found in reading glasses, have positive refracting powers. Diverging lenses, like those worn by people with myopia or near-sightedness, have negative refracting powers. A flat piece of glass, which is essentially the same as no lens at all, has a power of zero. The eye has a power of 50 D when focusing on distant objects.

Splain Luthy

Sign conventions in geometric optics
characteristic +
focal length converging diverging
image type real virtual
orientation upright inverted


focus by moving the lens back and forth

focus by changing the focal length

Defects of the human lens
vision defect common name cause correction
myopia near sight long eyeball or
over curved cornea
diverging lens
hyperopia far sight short eyeball or
under curved cornea
converging lens
astigmatism unequally
curved cornea
barrel shaped lens
presbyopia old age sight lack of accommodation converging lens
(reading glasses)
emmentropia normal vision appropriately sized eyeball
and properly curved cornea
none needed,
you lucky bastards

lens tests: big eyes/small eyes, motion shifts

lens combinations

Every ray from an object (O) will continue undeviated if it passes through the center of a lens. This one path will be the same for each lens individually as well as for the combination of lenses. (The lenses are assumed to be so close together that their centers essentially coincide.)

The ray through the focus of the first lens (F1) will emerge parallel to the principal axis. The image formed will lie at the intersection of the two rays drawn. Since it is not the final image, it is indicated with the symbol I*.

The real image formed by the first lens serves as a virtual object for the second lens. Since it isn't the original object, it is indicated with the symbol O*. The ray parallel to the principal axis (indicated by a dashed line) will actually strike the second lens before it has a chance to form an image. When it does, it is refracted through the focus of the second lens (F2). The intersection of this ray with the first ray drawn through the center of the two lenses shows the location of the final image formed.

The combination of lenses acts as a single lens. We can locate the focus of this lens combination (F12) by tracing the emergent ray parallel to the principal axis back to the original object. The place where this ray intersects the principal axis is the location of the combined focus.

In physicist form…

1  =  1  +  1
f2 do* di
1  =  1  +  1
f2 di* di
1  =   − 
1  −  1
 +  1
f2 f1 do di
1  =  1  +  1  −  1
f2 do di f1
1  +  1  =  1  +  1
f1 f2 do di
1  =  1  +  1
f1+2 f1 f2

In optometrist form…

P2 = Vo* + Vi

P2 = −Vi* + Vi

P2 =  − (P1 − Vo) + Vi

P2 = Vo + Vi − P1

P1 + P2 = Vo + Vi

P1+2 = P1 + P2