Lenses

We start from the definition of angular magnification and reach the two standard formulas, one for each way of using a magnifying glass.

Setting up. A simple microscope is a single convex lens of short focal length f. The object has height h, and the eye is held close behind the lens. D is the least distance of distinct vision (the near-point distance, 25 cm for a normal eye). The object is placed between the lens and its focus, so the image is virtual, erect and enlarged.

Step 1 — the angle without the lens. To see the object as big as possible with the bare eye, you bring it to the near point, a distance D away. The angle it then subtends is small, so

Case 1 — final image at the near point (maximum magnification).

Step 2 — where the object must be. The image must be at the near point, so with the sign convention v = -D. Let the object distance be u = -u_0.

Step 3 — the angle with the lens. The eye is at the lens, so it receives the rays at the angle the object subtends at the lens:

Step 4 — form the ratio, by the definition of magnifying power:

Step 5 — cancel h and simplify the divided fraction:

Step 6 — now find u_0 from the lens formula \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}, putting v = -D and u = -u_0:

Step 7 — clean up the signs:

Step 8 — move 1/D across:

Step 9 — multiply every term by D, because Step 5 needs D/u_0:

Step 10 — simplify the last term and use Step 5:

Case 2 — final image at infinity (relaxed eye).

Step 11 — where the object must be. For the image to go to infinity, the object must sit exactly at the focus, so u_0 = f.

Step 12 — the angle with the lens is then

Step 13 — form the ratio again:

Step 14 — cancel h:

Result.

What it means. A shorter focal length gives a bigger magnification, which is why a magnifying glass is a strongly curved little lens. The near-point setting gives exactly one extra unit of magnification, but the eye must accommodate, which tires it; the infinity setting is more comfortable for long use.

Conditions used. The eye is close to the lens, all angles are small (so that \tan\theta \approx \theta in Steps 1, 3 and 12), and D = 25 cm for a normal eye.

We start from Snell's law at one curved boundary and reach the equation that a single spherical refracting surface obeys. This one result, used twice, gives the whole lens theory.

Setting up. Light travels from a medium of index n_1 into one of index n_2 across a spherical surface of radius R, with pole P and centre of curvature C. A paraxial ray from the axial object O meets the surface at A, a small height h above the axis, and after refraction crosses the axis at I. Let \alpha, \beta, \gamma be the (small) angles that AO, AC and AI make with the axis. Note AC is the normal at A, since a radius is always normal to a sphere.

Step 1 — find the angle of incidence. In triangle AOC, the angle at C is the exterior angle rule applied the other way round: the exterior angle at A (which is i) equals the sum of the two opposite interior angles:

Step 2 — find the angle of refraction. In triangle AIC, \beta is the exterior angle, so it equals the sum of the two opposite interior angles r and \gamma:

Step 3 — make r the subject:

Step 4 — write Snell's law at A:

Step 5 — use the paraxial approximation. A is close to the axis, so i and r are small and \sin\theta \approx \theta:

Step 6 — substitute Steps 1 and 3:

Step 7 — express the three angles as ratios. For a small angle, angle \approx opposite side over adjacent side, and A is practically at P. With the Cartesian convention (u negative for a real object):

Step 8 — substitute these into Step 6:

Step 9 — cancel h, which multiplies every term. (This is the important moment: the answer does not depend on the ray's height, which is why a sharp image forms at all.)

Step 10 — collect the v and u terms on the left and the R terms on the right:

Step 11 — combine the right-hand side over the common denominator R:

Result.

What it means. One equation covers every single curved refracting boundary — the front of a lens, the back of a lens, the cornea of an eye — provided you put in the right n_1, n_2 and the right sign for R. Applied twice in succession it gives the lens maker's formula.

Conditions used. Paraxial rays only (Steps 5 and 7), a surface of small aperture, and all distances measured from the pole P using the new Cartesian convention.

We start from the single-surface equation just derived, apply it twice, and reach the formula that connects a lens's focal length to its shape and its glass.

Setting up. A thin lens of material index n sits in air. Its faces have radii R_1 (the first surface the light meets) and R_2. A real object is at u; the first surface alone would form an image at v_1, which the second surface then uses as its object to form the final image at v.

Step 1 — apply the single-surface equation to the first face. Light goes from air into glass, so n_1 = 1, n_2 = n, radius R_1, image distance v_1:

Step 2 — apply it again to the second face. Now light goes from glass into air, so n_1 = n, n_2 = 1, radius R_2, and the object is I_1. Because the lens is thin, the distance of I_1 from the second surface is still v_1:

Step 3 — add equations (1) and (2). We add rather than subtract because the n/v_1 terms have opposite signs and so will cancel:

Step 4 — cancel the two n/v_1 terms on the left. The intermediate image has now disappeared from the problem entirely:

Step 5 — rewrite (1-n) as -(n-1) so that the same factor appears in both terms:

Step 6 — take out the common factor (n-1):

Step 7 — now specialise to an object at infinity. Then 1/u = 0, and by the definition of focal length the image forms at v = f:

Step 8 — write it out:

Step 9 — recover the thin lens formula. The right-hand side of Step 6 is now known to equal 1/f, so

Result.

What it means. A lens's power comes from two things only: how different its glass is from the surroundings (n - 1) and how sharply its faces are curved. Make either bigger and f gets shorter. Notice also that a lens immersed in a liquid of the same index (n - 1 \to 0) has f = \infty — it becomes invisible and does nothing.

Conditions used. The lens is thin (Step 2 needs the two surfaces to be at effectively the same place), rays are paraxial, light is monochromatic, and the surrounding medium is air.

We start by tracing the light through the two lenses one at a time, and reach the focal length of the pair treated as one lens.

Setting up. Two thin lenses of focal lengths f_1 and f_2 are placed in contact, so the distance between them is zero. An object is at u. The first lens forms an intermediate image at v_1; the second lens then uses that image as its own object and forms the final image at v. Let F be the focal length of the equivalent single lens.

Step 1 — apply the lens formula to the first lens:

Step 2 — apply it to the second lens. Its object is I_1, and because the lenses are in contact the object distance for the second lens is the same v_1:

Step 3 — add (1) and (2). Again we add because the 1/v_1 terms carry opposite signs:

Step 4 — cancel the two 1/v_1 terms:

Step 5 — compare with the lens formula for the equivalent lens, \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{F}. The same object gives the same final image, so the left-hand sides are identical:

Step 6 — restate in terms of power, since P = 1/f:

Result.

What it means. Powers add; focal lengths do not. Two +2 D lenses in contact make a +4 D lens. Because a diverging lens has negative power, a +5 D lens combined with a -2 D lens gives +3 D — this is how an optician builds any prescription out of a small stock of lenses. The magnifications multiply, not add, because each lens magnifies what the previous one produced.

Conditions used. Both lenses are thin and in contact (zero separation), rays are paraxial, and the two lenses share the same principal axis. If they are separated by a distance d the extra term below is needed.