Wave Optics – Chapter 7 Study Guide

Foundations

Sections 7.1, 7.2, 7.3

Ray of light — the path followed by light in a uniform, homogeneous medium. It travels in a straight line until it meets another medium, where it reflects or refracts.

Reflection

Incident ray, reflected ray and normal lie in one plane.

i = r

Refraction (Snell)

Incident ray, refracted ray and normal lie in one plane.

n₁ sin i = n₂ sin r

How the theory of light evolved

Theory	Who	Idea	Verdict
Corpuscular	Descartes (1636), Newton	Light = hard, elastic, massless particles. Elastic collisions explain reflection; medium attraction explains refraction.	Predicts v_denser > v_rarer, which experiment contradicts.
Wave	Huygens (1668)	Light is a wave in a hypothetical medium, ether. Explains interference, diffraction, polarization, reflection, refraction.	Accepted. Gives v_denser < v_rarer. Ether never found (Michelson–Morley); Einstein (1905) removed the need for it.
Dual	Early 20th century	Light shows wave nature or particle nature depending on the situation. Particle of light = photon.	Modern view.

Ray optics (geometrical optics) — the study of optical phenomena assuming light travels in straight lines as rays. It works because the wavelength is much smaller than everyday objects and surfaces.

Wave optics — the branch of optics that uses the wave nature of light to explain phenomena.

Dual nature — light exhibits both particle and wave nature under different situations.

Light as an electromagnetic wave (7.3)

Light is a transverse electromagnetic wave. E and B are perpendicular to each other and to the direction of travel. It needs no medium, so it crosses vacuum.

n = c / vc = 3×10⁸ m/s. n(vacuum) = 1, n(air) ≈ 1.

Refractive index — ratio of the speed of light in vacuum to its speed in the medium.

Spectrum, increasing wavelength: γ-rays → X-rays → UV → Visible → IR → Microwaves → Radio. Visible light spans 400–700 nm; violet is shortest, red longest. Because n depends on wavelength, white light splits into a spectrum or rainbow (except at normal incidence).

Wavefronts and Huygens

Section 7.4

Primary source

Emits its own light: Sun, stars, flame, tube light, TV, firecrackers.

Secondary source

Reflects or scatters light from elsewhere: Moon, planets, humans, plants. Most everyday sources are secondary.

Wavefront — the locus of all points having the same phase at a given instant.

Source	Wavefront	Rays
Point source	Spherical	Radiate out, ⟂ to the surface (diverging beam)
Far from source	Plane	Parallel, ⟂ to the wavefront (parallel beam)
Linear source	Cylindrical	⟂ to the surface

A plane wave is a wave with a plane wavefront. A ray is always perpendicular to the wavefront.

Huygens' principle — each point on a wavefront acts as a secondary source emitting wavelets in all directions at the speed of light in the medium. The new wavefront is the envelope of the forward wavelets.

r = vTRadius of a wavelet after time T. Old wavefront → wavelets → envelope → new wavefront.

Backward wavelets are simply declared ineffective. The textbook calls Huygens' theory empirical: it was accepted because it works.

Reflection and Refraction

Sections 7.5, 7.6 · Both are long-answer derivations

Derive the law of reflection

Plane wavefront AB hits mirror MN. At t = 0, A touches the mirror; B has not.
B reaches C after time T, so BC = vT.
Points from A to C become secondary sources one after another.
At time T the wavelet from A has radius AE = vT, so AE = BC. The envelope is the reflected wavefront EC.
Triangles ABC and AEC are right-angled, share hypotenuse AC and have AE = BC, so they are congruent.
∠ACE = ∠BAC = i. Since AE ⟂ CE and AP ⟂ AC, ∠PAE = ∠ACE = r.
Therefore i = r. Incident ray, reflected ray and normal lie in one plane.

Derive Snell's law

Wavefront AB meets boundary MN. Speeds are v₁ (medium 1) and v₂ (medium 2).
B reaches C in time T: BC = v₁T. Meanwhile the wavelet from A travels AE = v₂T in medium 2.
Envelope of wavelets = refracted wavefront CE.
sin i = BC/AC = v₁T/AC and sin r = AE/AC = v₂T/AC.
Divide: sin i / sin r = v₁/v₂.
With v = c/n: v₁/v₂ = n₂/n₁.
Hence n₁ sin i = n₂ sin r.

Rarer → denser

v drops, n rises, i > r. Ray bends towards the normal.

Denser → rarer

Ray bends away from the normal.

What changes when light enters a new medium?

Speed v	Wavelength λ	Frequency ν
Changes	Changes	Unchanged

λ₂ = λ₁ (n₁/n₂) = λ₁ (v₂/v₁)From vacuum: λ = λ₀ / n. Higher n → shorter λ.

Lateral inversion — the apparent interchange of right and left in a mirror image. Image size equals object size.

Lab: refraction calculator

n₁

n₂

Angle i

Polarization

Section 7.7 · Proves light is a transverse wave

Unpolarized

E vibrates in all directions perpendicular to travel (Sun, bulb).

Plane polarized

E restricted to one direction.

Polarizer

Transmits E along one direction, blocks others. A polaroid is a plastic-sheet polarizer; the allowed direction is the polarizing axis.

Why transverse only? A transverse wave has many possible oscillation directions to restrict. A longitudinal wave (sound) oscillates along travel, so there is nothing to restrict.

Plane of vibration contains E. Plane of polarization is perpendicular to it. Parallel axes (θ = 0°) pass all light; crossed axes (θ = 90°) pass none.

Malus' law

First polarizer: I₁ = I₀ / 2Unpolarized light loses half.

I = I₀ cos²θE₂ = E₁ cosθ, and I ∝ E². I₀ here is the intensity entering the analyzer.

Lab: Malus' law

Angle θ

Brewster's law

Brewster angle θ_B — the angle of incidence at which reflected light is completely plane polarized. Reflected and refracted rays are then perpendicular (θ_B + θ_r = 90°).

Derive tan θ_B = n₂/n₁

Snell: n₁ sin θ_B = n₂ sin θ_r
θ_r = 90° − θ_B, so sin θ_r = cos θ_B
n₁ sin θ_B = n₂ cos θ_B
Divide by n₁ cos θ_B: tan θ_B = n₂/n₁

Example: air to n = 1.5 gives θ_B = tan⁻¹(1.5) = 56.31°. Polaroid sunglasses cut glare from non-metallic surfaces.

Scattering: sunlight hitting air molecules and dust changes direction; blue scatters most, so the sky is blue. Scattered light is partially polarized, and at a 90° scattering angle it is plane polarized.

Interference

Section 7.8 · Young first observed it for light in 1801

Interference — non-uniform intensity from the superposition of two or more light waves: bright (constructive, in phase) and dark (destructive, opposite phase) regions.

Coherent sources — same frequency and a constant phase difference. Two independent primary sources are not coherent, so we split one source into two.

Young's double slit (D ≫ d)

Path difference

Δl = yd / D

Bright fringe

Δl = nλ
yₙ = nλD / d

Dark fringe

Δl = (n − ½)λ
yₙ = (n − ½)λD / d

Fringe width

W = λD / dEqual for bright and dark. Centre (n = 0) is bright.

	Path diff.	Phase diff.	Intensity
Bright	nλ	2nπ	4I₀
Dark	(n − ½)λ	(2n − 1)π	0

φ = 2πΔl / λ, E = 2E₀ cos(φ/2), I = 4I₀ cos²(φ/2)

With unequal amplitudes, I_max ∝ (E₁ + E₂)² and I_min ∝ (E₁ − E₂)², so dark fringes are not fully dark.

Lab: Young's double slit

Wavelength

Slit gap d

Distance D

Conditions for a steady pattern — C M A D N P

Coherent sources	Monochromatic light	Equal Amplitudes
D ≫ d	Narrow slits	Same Polarization

A constant non-zero phase difference between sources shifts the whole pattern but leaves fringe width unchanged.

Other ideas

Lloyd's mirror

Direct ray plus ray reflected at grazing incidence, which seems to come from a virtual source. Real + virtual source are coherent.

Thin films

Soap bubbles and oil films: light reflected from the top surface interferes with light reflected from the bottom. Reflection from a denser boundary adds phase π (path λ/2).

Optical path

OPL = n·d

Extra optical path over vacuum: d(n − 1).

Transparent plate on one slit

Δ = (n − 1)t

The whole pattern shifts; fringe width stays the same.

Diffraction

Section 7.9

Diffraction — light spreads into the region where ray optics predicts a shadow. It is noticeable when the slit or obstacle size is about λ. It is interference of many waves from one wavefront.

Fraunhofer (Far)	Fresnel (Near)
Very large distances; plane incident wavefront; lenses used	Smaller distances; cylindrical or spherical wavefront; no lens needed

Single slit of width a (D ≫ a)

Centre P₀: all paths equal, so a central bright maximum.
Extreme-ray path difference is a sinθ. For the first minimum, halves of the slit cancel in pairs (λ/2 apart).

nth minimum

a sinθₙ = nλ
yₙ = nλD / a

Secondary maxima

a sinθ ≈ (n + ½)λ

Roughly midway between minima.

Fringe width

W = λD / a

Central maximum

W_c = 2λD / a = 2W

Twice as wide and the brightest.

Interference vs diffraction

	YDSE	Single slit
Waves	Two coherent	Many, from one slit
Width	λD/d (d = slit separation)	λD/a (a = slit width)
Central fringe	Same width as others	Twice as wide
Brightness	Nearly equal	Centre brightest, others weaker

Exam Toolkit

Revise here the night before

Definitions to write word-perfect

Wavefront — locus of all points having the same phase at a given instant.

Huygens' principle — each point on a wavefront is a secondary source of wavelets; the new wavefront is their forward envelope.

Primary source emits its own light; a secondary source reflects or scatters light from another source.

Malus' law — I = I₀cos²θ gives the intensity of linearly polarized light through a polarizer.

Coherent sources — same frequency, constant phase difference.

Formula sheet

Topic	Formula
Refractive index	n = c/v
Snell's law	n₁ sin i = n₂ sin r
Wavelength in medium	λ = λ₀/n; λ₂ = λ₁n₁/n₂
Wavelet radius	r = vT
Malus	I = I₀cos²θ; first polarizer I₀/2
Brewster	tanθ_B = n₂/n₁
YDSE	Δl = yd/D; W = λD/d; I = 4I₀cos²(φ/2)
Plate on slit	(n − 1)t
Single slit	a sinθ = nλ; W = λD/a; W_c = 2λD/a

Memory hooks

Reflection proof

Touch → wavelets → AE = BC → congruent triangles → i = r

Medium change

v ✓, λ ✓, ν ✗

Diffraction

Fraunhofer = Far, Fresnel = Near

Steady fringes

C M A D N P

Slit symbols

d = distance between slits; a = aperture width

Likely long answers

Reflection and refraction derivations, Malus' law, Brewster's law, YDSE fringe width