Ray and Wave Optics

1Spherical mirrors and the sign convention

One formula covers every mirror: 1/v + 1/u = 1/f, with f = R/2 and m = −v/u. What makes it hard is not the algebra but the bookkeeping. Measure every distance from the pole, take the direction of the incoming light as positive, and heights above the axis as positive. A real object then always has u negative; a concave mirror, whose centre of curvature sits on the incoming side, always has R and f negative.

Figure. Concave-mirror landmarks from 1/v + 1/u = 1/f: a distant object images near f, and an object at the centre images at the centre. Magnitudes shown; signs (real image v < 0) stay in the worked example.

How it works

1. Fix the originPole at zero, positive along the light. Nothing is measured from the focus.
2. Sign, then substitutePut in u, f and R with their signs already attached, then solve for v.
3. Read the sign backv negative means the image is in front of the mirror, where light really goes: real.

2Refraction and Snell's law

At a boundary, n₁sinθ₁ = n₂sinθ₂. Going into a denser medium the ray bends toward the normal; coming out, away from it. What actually changes inside the medium is the speed, v = c/n, and with it the wavelength, λ′ = λ/n. The frequency is set by the source and never changes, which is why light does not change colour on entering glass.

Figure. The incident and refracted rays are drawn at their true angles to the normal. Notice how much the ray straightens on entering the glass — a 60° approach becomes a 35° one, and that compression is the whole of Snell's law.

How it works

1. Draw the normalBoth angles are measured from the normal to the surface, never from the surface itself.
2. Denser bends inwardLarger n on the far side means a smaller angle there — the ray hugs the normal.
3. Divide by nSpeed and wavelength both drop by a factor n; frequency is untouched.

3Total internal reflection

Push the refraction angle to 90° and the refracted ray disappears along the surface. The incidence angle that does this is the critical angle, sinθ_c = 1/n, where n is the denser medium's index relative to the rarer one. Beyond it, none of the light escapes: it is all reflected back, and reflected perfectly, with no metal coating to tarnish. Two conditions, both required — the light must start in the denser medium, and the angle must exceed θ_c.

Figure. Three rays leave the same point in the glass at 25°, 41.8° and 60°. The first escapes, bent away from the normal. The second is already grazing the surface — that is the definition of θ_c. The third never leaves.

How it works

1. Denser to rarerOnly then can the refraction angle be the larger of the two and reach 90° first.
2. At θ_c it grazesThe refracted ray runs along the boundary; the transmitted beam is already vanishing.
3. Beyond θ_c, nothing leavesAll the energy goes into the reflected ray, obeying the ordinary law of reflection.

4Refraction at a single spherical surface

Before a lens has two surfaces it has one, and one curved boundary between media of index n₁ and n₂ obeys n₂/v − n₁/u = (n₂ − n₁)/R. Same origin as always — the pole of the surface, distances positive along the light — with n₂ the index of the medium the refracted ray ends up in. It is n₂ and n₁ that appear at a single boundary, not the (n − 1) of the whole lens; that familiar factor only emerges when you apply this relation at both faces of a thin lens and subtract.

Flatten the surface, R → ∞, and the right-hand side dies, leaving n₂/v = n₁/u — the apparent-depth rule. An object at real depth d under a medium appears at d × (n₂/n₁): a pool looks shallower from above (n₂ < n₁), and a bird looks higher seen from underwater (n₂ > n₁).

Figure. Single-surface formula: n₂/v − n₁/u = (n₂ − n₁)/R. The curvature term is the surface power; rearrange and the image term is the object term plus that power. No arc is drawn — the algebra is the figure.

How it works

1. Pole is the originMeasure u, v and R from the vertex of the surface, positive along the incoming light — never from the centre of curvature.
2. n₂ sits over vThe refracted ray ends in medium n₂, so n₂/v pairs with the far side and n₁/u with the object side.
3. Flatten to checkSet R → ∞ and the surface is a plane: n₂/v = n₁/u, which is apparent depth = real depth × n₂/n₁.

5Thin lenses and image formation

The thin lens obeys 1/v − 1/u = 1/f with m = v/u. Note the two differences from a mirror: a minus sign in the formula and no minus sign in the magnification. A converging lens has f positive, a diverging lens f negative. Solve, then read the signs: v positive means the image is on the far side where light genuinely arrives, so it is real; m negative means inverted.

Figure. The lens is drawn as a plane, the thin-lens idealisation. One ray goes in parallel and leaves through F′; the other cuts the optical centre and carries straight on. They meet below the axis at 60 cm, which is why the image is real, inverted and twice as tall.

The two rays that fix the image

1. Parallel in, focus outA ray arriving parallel to the axis leaves through the far focus F′.
2. Straight through the centreA ray aimed at the optical centre passes on undeviated — no bend at all.
3. Where they crossThat intersection is the image tip; the axis fixes the foot.

6What sets f — lens maker's formula and power

A lens has the focal length its glass and its two curvatures give it: 1/f = (n − 1)(1/R₁ − 1/R₂), with the same sign convention as everything else. Two consequences do most of the work in exams. First, the factor is (n − 1), so it is the contrast with the surroundings that matters — immerse the lens and n becomes n_lens/n_medium. Second, define power as P = 1/f in metres, in dioptres, and lenses in contact simply add: P = P₁ + P₂.

Figure. Lens maker: 1/f = (n − 1)(1/R₁ − 1/R₂). Each curved surface contributes a term; equal double-convex radii add. Power in dioptres is 1/f with f in metres.

How it works

1. Sign each radiusR is positive when the centre of curvature lies beyond the surface, along the light.
2. Multiply by (n − 1)More curvature or more index contrast, shorter f, stronger lens.
3. Add in dioptresPowers add for lenses in contact. Focal lengths do not.

7The prism: deviation and minimum deviation

A ray crossing a prism refracts twice, and the two bends add to a deviation δ = i₁ + i₂ − A, where A is the prism angle and the internal angles always satisfy r₁ + r₂ = A. Sweep the angle of incidence from grazing to grazing and δ traces a shallow valley with a single minimum. At that minimum the path runs symmetrically through the prism — i₁ = i₂ and r₁ = r₂ = A/2 — so the minimum sits where the path is symmetric, not where the ray goes straight through.

At minimum deviation the index follows from geometry alone: μ = sin((A + δ_m)/2) / sin(A/2), which is exactly how a spectrometer measures the refractive index of a glass. For a thin prism (A only a few degrees) the whole relation collapses to δ = (μ − 1)A, independent of the angle of incidence — the form used all through dispersion.

Figure. Deviation against angle of incidence is a shallow valley — the curve is drawn screen-down, so its lowest point on the page is the smallest deviation. δ falls to a single minimum and climbs again towards grazing incidence at either end. The minimum sits at the symmetric passage i₁ = i₂, and the same δ_m can be reached from two different angles of incidence on the two arms of the valley, which is why a spectrometer reading is taken right at the turning point.

How it works

1. Two refractions, one identityr₁ + r₂ = A always, and δ = i₁ + i₂ − A. Snell's law at each face ties the i's to the r's.
2. Symmetry marks the minimumδ is least when i₁ = i₂; the ray then crosses parallel to the base and r₁ = r₂ = A/2.
3. Read the index off δ_mμ = sin((A + δ_m)/2)/sin(A/2). For a thin prism this reduces to δ = (μ − 1)A.

8Dispersion and dispersive power

A glass has a slightly larger index for violet light than for red, so through a prism violet bends more, not less, and white light fans into a spectrum. For a thin prism each colour deviates by δ = (μ − 1)A, so the angular spread between the extremes is the angular dispersion θ = (μ_v − μ_r)A, while the mean ray deviates by δ_y = (μ_y − 1)A.

Divide one by the other and the prism angle cancels: the dispersive power ω = (μ_v − μ_r)/(μ_y − 1) is a pure number belonging to the glass alone, typically about 0.02–0.03. Because ω is a material constant, two different glasses can be combined to cancel dispersion while keeping deviation (an achromatic prism) or to cancel deviation while keeping dispersion (a direct-vision prism).

Figure. Crown glass from the worked example: violet sits above red by 0.010 in μ. Angular fan gaps are unresolvable on a ray diagram; the index bars are the differences that enter ω = (μ_v − μ_r)/(μ_y − 1).

How it works

1. Split the deviation by colourδ = (μ − 1)A, and μ_violet > μ_red, so violet is deviated most and red least.
2. Spread over meanω = angular dispersion / mean deviation = (μ_v − μ_r)/(μ_y − 1). The prism angle A cancels.
3. Pair two glassesChoose the second prism's angle so the two dispersions cancel (achromatic) or the two deviations cancel (direct vision).

9Microscopes and telescopes

Put two converging lenses in a row and you have either a microscope or a telescope; which one depends on the focal lengths, and the figure of merit is always the same. What is magnified is the angle the image subtends at the eye, not its linear size. A compound microscope multiplies two stages: the objective forms a real, enlarged image (linear magnification ≈ L/f_o, with L the tube length) and the eyepiece views that as a simple magnifier (angular gain D/f_e, D = 25 cm), so M ≈ (L/f_o)(D/f_e).

A telescope looks at objects so distant that linear size is meaningless, so it works purely in angles. In normal adjustment — final image at infinity — its angular magnification is M = f_o/f_e and its tube length is f_o + f_e. The two instruments pull the objective in opposite directions: a microscope wants both focal lengths as short as possible, a telescope wants a long objective and a short eyepiece.

Figure. Both instruments magnify angle, not linear size. A typical school telescope sits near M = f_o/f_e ≈ 10; a compound microscope stacks two factors and reaches ~100. The comparison table names the formulas; the bars name the scale.

How it works

1. Microscope: two magnifiersObjective gives ≈ L/f_o, eyepiece gives D/f_e; multiply for the total M ≈ (L/f_o)(D/f_e).
2. Telescope: compare anglesDistant object, so M = f_o/f_e in normal adjustment and the tube is f_o + f_e long.
3. Which focal lengthsShort and short for a microscope; a long objective and a short eyepiece for a telescope.

10Huygens' principle and wavefronts

Wave optics begins with a rule for advancing a wave. A wavefront is a surface of constant phase; Huygens' principle says every point on it acts as a source of secondary spherical wavelets, and the wavefront a moment later is the forward surface just touching all of them. The rays are simply the perpendiculars to the wavefronts, so this one construction reproduces the whole of geometrical optics.

It earns its place in two ways. Crossing into a slower medium the wavelets advance less far, so the front tilts, and working the geometry through gives sin θ₁/sin θ₂ = v₁/v₂ = n₂/n₁ — Snell's law, derived rather than assumed. And it pins down what interference will need: two sources are coherent when they keep a fixed phase relationship, which is why the next concepts split a single wavefront in two rather than using two independent lamps.

Figure. Wavefronts are surfaces of constant phase, drawn here as vertical lines, and the ray is their perpendicular. At normal incidence the fronts stay parallel to the boundary but crowd closer on entering the glass — the spacing here drops by the index n = 1.5, from λ to λ/n, because the wave slows while its frequency, set by the source, does not. The secondary spherical wavelets that generate each new front are arcs and cannot be drawn with this vocabulary.

How it works

1. Every point reseedsTreat each point of the current wavefront as a source of a secondary spherical wavelet travelling at the local wave speed.
2. Take the envelopeThe new wavefront is the forward surface tangent to all those wavelets one instant later.
3. Rays are the normalsDraw the perpendiculars; where the medium slows the wavelets, the front tilts, and that tilt is Snell's law.

11Young's double slit and fringe width

Two slits cut from the same wavefront stay in step, so their light can interfere. At a point y up the screen the path difference is Δ = dy/D. Bright fringes sit where Δ = nλ, dark ones where Δ = (n − ½)λ, and consecutive fringes of the same kind are β = λD/d apart. Everything in the pattern is that one length scale: β widens if you lengthen λ or D, and narrows if you separate the slits.

Figure. The two gaps in the barrier are the slits, separated by d. Both send light to the same point P on the screen, but the lower one travels dy/D further. Whether P is bright or dark is decided entirely by how many wavelengths fit into that small extra length.

How it works

1. Split one sourceBoth slits are fed by the same wavefront, so their phase difference is fixed in time.
2. Path difference dy/DFor small angles the extra distance from the far slit is d sinθ ≈ dy/D.
3. Count in wavelengthsWhole numbers of λ give brightness, half-odd numbers give darkness, spaced β apart.

12Intensity, phase and shifting the pattern

Fringes are not on-or-off. Convert the path difference to a phase, φ = 2πΔ/λ, and the intensity of two equal slits varies smoothly as I = 4I₀cos²(φ/2) — four times one slit's intensity at the maxima, zero at the minima, and the average still 2I₀, so nothing is lost. Now slide a thin sheet of index n and thickness t over one slit. It adds (n − 1)t to that path, and the entire pattern slides by (n − 1)tD/d toward the covered slit. The spacing β does not change, because β never depended on the path difference at the centre.

Figure. Solid and dashed curves have identical shape and identical spacing; the dashed one is simply displaced. That is what a slab over one slit does — it re-centres the pattern without rescaling it.

How it works

1. Path to phaseφ = 2πΔ/λ. One whole wavelength of path is one full 2π of phase.
2. Add the amplitudesEqual slits give I = 4I₀cos²(φ/2), peaking at four times a single slit.
3. A slab re-centres itExtra path (n − 1)t moves the zero-order fringe; every fringe follows, β intact.

13Single-slit diffraction and resolution

A single slit of width a spreads light into a broad central band flanked by faint side bands. The dark bands sit at a sinθ = nλ with n = 1, 2, 3… — and n = 0 is excluded, because straight ahead every part of the slit arrives in phase. That condition looks exactly like the double-slit bright-fringe condition and means the opposite; nothing here costs more marks. The central maximum spans 2λ/a in angle, twice the width of any other band, and the same 1/a scaling sets the resolving limit of a circular aperture, θ ≈ 1.22λ/D.

Figure. The side bands are barely visible — the first carries under 5% of the central peak — which is why the pattern looks like a single bright band with faint echoes. Halve the slit and this whole picture stretches to twice the width.

How it works

1. Pair up the slitSplit the slit in two halves; at a sinθ = λ every point cancels its partner.
2. Minima, not maximaSo a sinθ = nλ locates darkness, with n = 0 excluded by construction.
3. Narrow slit, wide patternWidth ∝ 1/a. Squeeze the slit and the light spreads further, not less.

14Polarisation — Malus and Brewster

Unpolarised light is an even mixture of every transverse orientation, so the first polariser always transmits exactly half, whatever its angle. After that the light has a definite direction and Malus's law takes over: a second polariser at θ to the first passes I = I₀cos²θ. Reflection polarises too — at Brewster's angle, tanθ_B = n, the reflected beam is completely plane-polarised, and the reflected and refracted rays leave at right angles to each other.

Figure. The curve is flat near 0° and near 180° and steepest at 45°, so a polariser is least sensitive to misalignment where it passes most light. Note that it touches zero only at 90° — crossed sheets, and nothing else, give darkness.

How it works

1. First sheet halves itAveraging cos²θ over all orientations gives ½ — the angle of the first sheet is irrelevant.
2. Then MalusEvery sheet after the first transmits cos²θ of what reaches it, θ measured to the previous axis.
3. Reflection at θ_Btanθ_B = n; the reflected ray is fully polarised perpendicular to the plane of incidence.