Geometrical Optics

Light’s wavelength is tiny compared with mirrors, lenses and the things we look at, so diffraction can usually be ignored and light treated as straight rays. That is geometrical optics, and it starts with the humble mirror.

The laws of reflection#

A ray of light striking a smooth surface is reflected so that

1. the angle of incidence equals the angle of reflection (both measured from the normal), and
2. the incident ray, the reflected ray and the normal all lie in one plane.

The image in a plane mirror#

Put a point source S in front of a flat mirror. A ray SA striking the mirror at right angles comes straight back. Another ray SB, striking at angle i, reflects along BC. Extend AS and BC behind the mirror: they meet at a point I. Using the law of reflection, the triangles SAB and IAB are congruent, so SA = AI. Every reflected ray, extended backwards, passes through the same point I.

An eye receiving the reflected rays can’t tell where they really came from; they seem to diverge from I. So I is the image of S. No light actually passes through I, so it is a virtual image.

Curved surfaces#

The same laws work for a curved mirror, point by point. At the point where a ray strikes, imagine the plane just touching the surface (the tangent plane) and draw the normal perpendicular to it. The angles of incidence and reflection are measured from that normal.

Look into a shiny spoon: one side shows you upside down, the other right way up but tiny. Curved mirrors can magnify, shrink and flip images, and a few words and one sign rule are enough to describe them all.

The parts of a spherical mirror#

A spherical mirror is a piece cut from a hollow sphere, usually glass silvered on one side. If light reflects from the hollow (inner) side it is a concave mirror; from the bulging (outer) side, a convex mirror.

• Centre of curvature C: the centre of the sphere. Its radius is the radius of curvature R.
• Pole P: the middle point of the mirror’s surface.
• Principal axis: the line through P and C.

Focus#

Send a narrow beam parallel to the principal axis onto a concave mirror, close to the pole. The reflected rays all cross the axis at (nearly) one point: the focus F. On a convex mirror the reflected rays spread out, but they seem to come from one point behind the mirror, its focus.

The distance PF is the focal length; the plane through F perpendicular to the axis is the focal plane.

Rays far from the axis miss the focus slightly, which matters for big mirrors (you’ll meet this again under aberrations).

Tracing an image#

To find the image of a point, follow two rays from it to the mirror. At each point of incidence the normal is the line to C; reflect each ray by the laws of reflection.

• If the reflected rays actually meet, they form a real image (it can be caught on a screen).
• If they diverge, they seem to come from a virtual image.

Real and virtual objects

Usually rays diverge from a point object in front of the mirror: a real object. But a lens could send converging rays toward the mirror; the point where they would have met, behind the mirror, then acts as a virtual object. In short: where the incident rays meet (or would meet) is the object; where the reflected rays meet (or seem to) is the image.

The sign convention#

Mirror formulas involve the object distance, image distance, focal length and radius. To make one formula work in every situation we use a coordinate sign convention:

• The pole is the origin and the principal axis is the x-axis.
• The positive x-direction is the direction of the incident light (usually left to right).
• u, v, R and f are the x-coordinates of the object, image, centre of curvature and focus: positive if on the positive side of P, negative otherwise.
• Heights above the axis are positive, below negative.

With the sign convention in hand, a single short equation tells you where any spherical mirror forms its image, and a second one tells you how big it is and which way up.

Deriving the mirror formula#

A point object O sits on the axis of a concave mirror. One ray goes along the axis, hits the pole P and comes straight back. Another ray OA hits the mirror at A and reflects through I, where the image forms. CA is the normal at A, so it bisects the angle OAI.

Call the angles that OA, CA and IA make with the axis α, β and γ, and let θ be the angle of incidence (= angle of reflection). The exterior angle of a triangle equals the sum of the opposite interior angles:

For a paraxial ray A is close to P, so the small angles are AP/PO, AP/PC and AP/PI:

With light from the left, O, I and C are all on the negative side: PO = −u, PI = −v, PC = −R. Substituting:

Focal length

A very distant object (u → ∞) sends parallel rays, which meet at the focus: v = f. Then 1/f = 2/R:

Extended objects and magnification#

For an object AB standing on the axis, find the image of the tip B with two convenient rays, then drop a perpendicular to the axis:

• a ray parallel to the axis reflects through F (or away from F for a convex mirror);
• a ray aimed at C hits the mirror normally and comes straight back;
• a ray through F reflects parallel to the axis;
• a ray striking the pole reflects symmetrically about the axis.

Comparing the right-angled triangles formed by the ray that strikes the pole gives the lateral magnification:

A swimming pool always looks shallower than it really is, and a straw in a glass looks bent at the surface. Both are images made by refraction at a flat surface.

Snell’s law with refractive indices#

When a ray crosses from medium 1 into medium 2, the angles obey

The image formed by a flat surface#

An object O lies in medium 1 (index μ₁) below a flat surface; medium 2 (index μ₂) is above. A ray OP straight up crosses without bending. A ray OA striking the surface at a small angle i bends to angle r. Traced back, the two rays meet at I, a virtual image of O.

For small angles, sin ≈ tan, so sin i ≈ PA/PO and sin r ≈ PA/PI, and Snell’s law gives

Looking from air (μ₂ = 1) at an object in a medium of index μ:

The object seems raised toward the observer by

Look up at the surface of a swimming pool from underwater and, beyond a certain angle, it shines like a silver mirror. Glass and water can reflect perfectly, without any silvering at all.

The critical angle#

When light passes from an optically denser medium (larger μ) into a rarer one (smaller μ), it bends away from the normal: the angle of refraction r is bigger than the angle of incidence i.

Increase i and r grows faster. At one angle of incidence, the critical angle θc, the refracted ray skims along the surface (r = 90°). For larger i, no angle r can satisfy Snell’s law: there is no refracted ray at all, and all the light reflects back into the denser medium. This is total internal reflection.

Usually the critical angle is quoted for light going from a medium into air. Putting μ₂ = 1 and r = 90° in Snell’s law:

Optical fibres#

An optical fibre is a very thin thread of glass or plastic, only micrometres across. Light sent into one end strikes the wall at a very glancing angle, far beyond the critical angle, and is totally reflected. It bounces along inside the fibre, even round bends, with almost no loss. A bundle of fibres is a light pipe.

Uses

• Seeing inside the body: a light pipe is passed down into, say, the stomach. Some fibres carry light in; others carry the image back out to the viewer.
• Communication: thousands of telephone signals, mixed into the light, can travel along one fibre at once, far more clearly than in ordinary cables.

A prism turns a beam of light through an angle, bending it toward its base. How much depends on the angle at which the light goes in, and there’s one special angle that gives the least bending of all.

Following a ray through a prism#

A prism’s cross-section is a triangle; two of its faces are the refracting surfaces, meeting at the angle of the prism A. A ray strikes the first face at angle of incidence i and refracts at angle r. Inside, it meets the second face at angle r′ and leaves at angle i′, the angle of emergence, provided r′ is not beyond the critical angle.

Without the prism, the ray would have gone straight on. The angle between that original direction and the final ray is the angle of deviation δ. Each face turns the ray, by (i − r) and (i′ − r′), so

The normals at the two faces meet at some point V. In the four-sided figure formed by the apex, the two points of incidence and V, two angles are right angles, so the angle at V is 180° − A. In the triangle formed by the ray inside and the two normals, r + r′ plus that same angle is 180°. So:

Minimum deviation#

As i changes, so does δ, and for one particular i it is smallest. At that point the ray passes through the prism symmetrically: i = i′ (and r = r′).

Why symmetric?

Light paths are reversible. Suppose the minimum happened with i bigger than i′. Send light backwards along the emerging ray: it retraces the path exactly, with the same deviation, which must also be the minimum. But now the angle of incidence is the old i′ and the emergence the old i, so incidence is smaller than emergence, which contradicts the assumption. The same argument rules out i < i′. If the minimum is unique, it must have i = i′.

Refractive index from minimum deviation#

At minimum deviation δm: r = A/2 and i = (A + δm)/2. Snell’s law μ = sin i / sin r then gives

Thin prisms

If A is small, δm is small too and the sines can be replaced by the angles:

Before lenses, master one curved boundary. The front of your eye, the side of a goldfish bowl and each face of a lens are all single spherical surfaces between two media.

Setting up#

Two transparent media, of refractive indices μ₁ and μ₂, are separated by a spherical surface with centre of curvature C. A point object O in medium 1 lies on the line through C, which meets the surface at P. We use the same sign convention as for mirrors: P is the origin, the axis is OPC, positive in the direction of the incident light; u, v and R are the x-coordinates of object, image and centre.

Deriving the formula#

The ray OP hits the surface normally and goes straight on. Another ray OD makes a small angle α with the axis and refracts along DI, where the image I forms. CD is the normal at D. With β the angle DCP and γ the angle DIC, the triangles give

For small angles Snell’s law becomes μ₁i = μ₂r, and α ≈ DP/PO, β = DP/PC, γ ≈ DP/PI. Putting these together:

With PO = −u, PI = +v, PC = +R:

Extended objects: magnification#

A ray from the tip Q of an extended object heading for C meets the surface normally and goes straight on, so the image of Q lies on the line QC. Similar triangles and the formula above give the lateral magnification:

A lens is just two spherical surfaces close together. Put the last lesson’s formula to work twice and out comes the rule every lens designer uses.

Thin lenses#

A lens is a piece of transparent material bounded by two spherical surfaces (or one spherical and one flat). It is thin if its thickness is small compared with the object and image distances. Its two surfaces have centres C₁ and C₂ and radii R₁ and R₂. The line through C₁ and C₂ is the principal axis, and the point P of the lens on it is the optical centre.

The foci#

A narrow beam parallel to the axis, near the optical centre, is refracted twice and emerges either converging to a point F₂ or diverging as if from F₂. The lens is then a converging (convex) or diverging (concave) lens.

If the medium is the same on both sides, the two focal lengths are equal. “Focus” and “focal length” will mean F₂ and PF₂.

Sign convention

As before: optical centre at the origin, principal axis as the x-axis, positive along the incident light; u, v, f, R₁, R₂ are x-coordinates; heights up are positive. With light from the left, f is positive for a converging lens and negative for a diverging lens.

The lens maker’s formula#

Let the lens (index μ₂) sit in a medium of index μ₁. At the first surface, light goes from μ₁ into μ₂ and forms an image at v₁. Since the lens is thin, take the origin at P for both surfaces:

That image is the object for the second surface, where light goes from μ₂ back into μ₁:

Adding the two eliminates v₁:

For u → ∞, v = f. In air (μ₁ = 1, μ₂ = μ):

Combining the two equations gives the lens formula:

A camera, a projector and a magnifying glass can all use the same convex lens. Only the object’s distance changes, and with it the size, position and orientation of the image.

Three easy rays#

To find the image of the tip of an object, trace any two of these rays:

1. Through the optical centre: straight on. Near its centre a thin lens is like a thin slab with parallel faces, so the ray is neither turned nor noticeably shifted.
2. Parallel to the axis: emerges through F₂ (convex), or as if from F₂ (concave).
3. Through F₁ (or heading for it, for a concave lens): emerges parallel to the axis.

Where the emerging rays cross (or seem to cross, traced back) is the image of the tip; drop a perpendicular to the axis for the whole image.

Magnification#

The ray through the optical centre makes similar triangles of object and image, with PO = −u and PO′ = +v:

A summary#

A concave lens always gives a virtual, erect, smaller image of a real object, lying between the lens and F.

Opticians don’t talk about focal lengths: a prescription says “+2.5” or “−1.75”. That number is the lens’s power, and powers of lenses in contact simply add.

Power of a lens#

A lens of focal length +25 cm has power 1/0.25 m = +4 D; one of −50 cm has power −2 D. A short focal length means a strong lens.

Two thin lenses in contact#

Lens L₁ forms an image of the object at v₁; this image is the object for L₂, which forms the final image at v. Since the lenses touch and are thin, the same distance v₁ applies to both:

So the pair behaves exactly like one equivalent lens:

How much a lens bends a ray#

A ray from an axial object O strikes a thin lens at height h and crosses the axis again at the image I. Its direction changes by δ = α + β, where α ≈ h/PO and β ≈ h/PI for small angles. With PO = −u, PI = v:

Two lenses a distance apart#

Separated lenses are not equivalent to one thin lens in general. But for a parallel incident beam they are. A ray at height h₁ is turned by δ₁ = h₁/f₁ at the first lens, travels distance d, reaches the second lens at height h₂ = h₁ − dδ₁, and is turned by δ₂ = h₂/f₂. An equivalent lens of focal length F would turn it by δ = h₁/F = δ₁ + δ₂. Substituting:

The equivalent lens sits where the incident ray and the final emergent ray, extended, cross: a distance dF/f₁ from the second lens, on the side the light comes from (18.23).

Our neat formulas assumed thin lenses, small apertures and rays close to the axis. Real instruments break these rules, and their images show it as blur, smears, coloured fringes and bent lines. These defects are called aberrations.

Spherical aberration#

A spherical mirror or lens brings only paraxial rays to a single focus. Rays striking further out, the marginal rays, focus at a slightly different point (for a concave mirror, closer to it). The image of a point becomes a blurred patch.

The rays crowd along a bright curve, the caustic (you can see one inside a coffee cup in sunlight). On a screen the image is a disc; moving the screen finds where the disc is smallest, the circle of least confusion. The distance between the paraxial and marginal foci measures the aberration.

Reducing it

• Parabolic mirrors focus all parallel rays at one point, and a lamp at the focus sends out a parallel beam: car headlights use them.
• Stops: an opaque sheet with a small hole lets only a narrow pencil of rays through. Sharper, but dimmer.
• Share the bending between both lens surfaces. A plano-convex lens imaging a distant object does much better with its curved side toward the object than its flat side.
• Combine convex and concave lenses, whose spherical aberrations are opposite, so they partly cancel.

For a single lens forming a real image of a real object, spherical aberration can be reduced but never removed completely.

Coma#

For a point object off the axis, rays through different rings of the lens form circles of different sizes whose centres are shifted. Together they make a comet-shaped smear: coma. Careful choice of radii and stops reduces it.

Astigmatism#

For an off-axis point, the image is also spread along the axis. Move a screen and you first get a short line image, then a different shape, staying in rough focus over a range. (This lens defect shares its name with, but is not the same as, the eye defect.)

Curvature of field#

The best image of a flat object is often formed not on a flat plane but on a curved surface. It is closely linked to astigmatism; both are reduced by stops in the right places.

Distortion#

If the magnification changes with distance from the axis, an extended object is imaged out of shape: straight lines come out curved.

Chromatic aberration#

The refractive index of glass varies with wavelength, so a lens has a slightly different focal length for each colour: longest for red, shortest for violet. In white light each colour forms its own image.

• Axial (longitudinal) chromatic aberration: the separation along the axis between the images for the extreme colours. A convex lens puts violet nearer than red (positive); a concave lens does the opposite (negative).
• Lateral chromatic aberration: since m depends on v, the images for different colours differ in size.