Wave Optics: NEET notes

1. Light as a wave

NCERT §10.1

• Two pictures of light competed for a long time. Descartes, in 1637, treated light as a stream of particles (the corpuscular model) and obtained Snell's law from it. Newton developed the idea in his book Opticks, and the model is usually credited to him because that book was so widely read.
• In 1678 the Dutch physicist Christiaan Huygens proposed instead that light is a wave. Both models could account for reflection and refraction; they disagreed about the speed of light after refraction.
• For a ray that bends towards the normal, the particle model needs light to go faster in the second medium, while the wave model needs it to go slower.
• Foucault settled the question in 1850 by measuring light to be slower in water than in air, as the wave model says.
• Thomas Young's interference experiment of 1801 put the wave nature of light on firm ground. For about forty years after it, interference and diffraction experiments kept turning up results that only waves could explain.
• The wavelength of visible light is tiny: yellow light has a wavelength of roughly 0.6 μm, that is 6 × 10⁻⁷ m. Next to the size of ordinary mirrors and lenses this is so small that light seems to move in straight lines.
• Geometrical (ray) optics is the limit in which the finite wavelength is ignored altogether. A ray is then the path along which energy travels as the wavelength tends to zero.
• One objection remained: a wave was thought to need a medium, yet light crosses empty space. Maxwell answered it. His equations predicted electromagnetic waves travelling at a speed very close to the measured speed of light, so light is an electromagnetic wave: changing electric and magnetic fields that sustain each other and need no medium.

2. Wavefronts and Huygens principle

NCERT §10.2

• Drop a pebble into still water and circular ripples spread out. All points on one ring are the same distance from where the pebble fell, so they rise and fall together.
• A wavefront is a surface joining points that vibrate in the same phase: a surface of constant phase. The speed at which a wavefront moves outward is the wave speed, and the energy of the wave flows at right angles to the wavefront.
• A point source radiating equally in every direction gives spherical wavefronts. Far from the source, a small patch of such a sphere is almost flat, and the wave there is a plane wave with plane wavefronts.
• Huygens principle is a geometrical rule for getting a later wavefront from an earlier one. Every point on a wavefront acts as a source of secondary wavelets, which spread in all directions at the speed of the wave.
• To find the wavefront a time τ later, draw spheres of radius vτ centred on points of the present wavefront, where v is the wave speed in the medium. The common tangent to all these spheres, the forward envelope, is the new wavefront.
• For a spherical wavefront centred on O the new wavefront is again a sphere centred on O. For a plane wavefront the new one is a parallel plane further on. Rays are lines drawn at right angles to the wavefronts.
• The construction also yields a backward envelope, a backwave, which is never observed. Huygens removed it by assuming that the wavelets are strongest straight ahead and vanish backwards. That was an ad hoc fix; the real justification comes from a more complete wave theory.
• Wavefront shapes to recognise (Exercise 10.2): light spreading from a point source has spherical wavefronts; light leaving a convex lens with a point source at its focus has plane wavefronts; the part of a distant star's wavefront that reaches the Earth is plane.

3. Refraction of a plane wave

NCERT §10.3.1

• A plane wavefront AB strikes the boundary PP′ between medium 1 (speed v₁) and medium 2 (speed v₂) at an angle of incidence i. Let τ be the time for the end B of the wavefront to reach the boundary at C, so BC = v₁τ.
• In that same time the wavelet from A travels a distance AE = v₂τ into medium 2. The tangent plane CE drawn from C onto this wavelet is the refracted wavefront.
• Triangles ABC and AEC share the side AC. So sin i = BC/AC = v₁τ/AC and sin r = AE/AC = v₂τ/AC, which gives sin i/sin r = v₁/v₂.
• If the ray bends towards the normal (r < i), then v₂ < v₁: light is slower in the second medium. This is the wave-model prediction that Foucault's water measurement confirmed.
• With refractive indices n₁ = c/v₁ and n₂ = c/v₂, where c is the speed of light in vacuum, the same result reads n₁ sin i = n₂ sin r. This is Snell's law.
• If BC is one wavelength λ₁ in medium 1, then AE is one wavelength λ₂ in medium 2, because the crest from B reaches C just as the crest from A reaches E. So λ₁/λ₂ = v₁/v₂: entering a denser medium, speed and wavelength drop in the same ratio.
• The frequency ν = v/λ does not change. Atoms of the medium are set into forced oscillation by the incoming light and re-emit at the frequency that drives them, so reflected and refracted light keep the incident frequency (Example 10.1).
• Slowing down costs the wave no energy: the energy a wave carries depends on its amplitude, not on its speed (Example 10.1).
• Exercise 10.1: 589 nm light passes from air into water, n = 1.33. The reflected light keeps 589 nm, ν = 3 × 10⁸/(589 × 10⁻⁹) = 5.09 × 10¹⁴ Hz and 3 × 10⁸ m/s. The refracted light has the same ν, speed 3 × 10⁸/1.33 = 2.26 × 10⁸ m/s and wavelength 589/1.33 = 443 nm.
• Exercise 10.3: in glass of refractive index 1.5, v = 3.0 × 10⁸/1.5 = 2.0 × 10⁸ m/s. The refractive index depends on colour; it is larger for violet than for red, so violet light travels more slowly in a glass prism.

4. Rarer medium, reflection and lenses

NCERT §10.3.2, §10.3.3

• When the second medium is rarer (v₂ > v₁), the same construction works, but the wavelet from A now outruns point B. The refracted ray bends away from the normal (r > i), and n₁ sin i = n₂ sin r still holds.
• As i grows, r reaches 90° first. The angle of incidence at which r = 90° is the critical angle ic, given by sin ic = n₂/n₁.
• For any angle of incidence larger than ic there is no refracted wave at all; the light undergoes total internal reflection.
• Reflection: a plane wavefront AB meets a mirror MN at angle i. While B travels BC = vτ to the mirror, the wavelet from A grows to radius AE = vτ in the same medium. The tangent CE from C is the reflected wavefront.
• Triangles EAC and BAC are congruent: they share AC, have AE = BC and each has a right angle (at E and at B). Hence the angle of reflection equals the angle of incidence, which is the law of reflection.
• Thin prism: the lower part of a plane wavefront crosses the most glass and is held back the most, so the wavefront comes out tilted and the beam turns.
• Convex lens: the middle of the wavefront passes through the thickest glass and is delayed most. The emerging wavefront is dented at the centre, becomes spherical and converges to the focus F. A concave mirror likewise turns a plane wavefront into one converging to its focus; concave lenses and convex mirrors are explained the same way.
• Every ray from a point on an object to its image takes the same time. Through a convex lens the central ray has the shortest path, but it spends longest in slow glass, so it arrives together with the rays near the edge.

5. Coherent and incoherent addition

NCERT §10.4

• Interference rests on the superposition principle: where several waves meet, the resulting displacement is the vector sum of the displacements each wave would produce alone.
• Two needles dipping up and down in step in a water trough make two sets of ripples. If the phase difference between the two waves at any point stays constant in time, the sources are coherent.
• At a point P equally far from S₁ and S₂, the waves arrive in step: y₁ = y₂ = a cos ωt, so y = 2a cos ωt. Intensity goes as amplitude squared, so I = 4I₀, where I₀ is the intensity from one source alone. Every point on the perpendicular bisector of S₁S₂ gets 4I₀.
• A path difference of λ is a phase difference of 2π. A path difference of 2λ gives 4π, so the waves again arrive in step (constructive, 4I₀). A path difference of 2.5λ gives 5π, the displacements are opposite, and they cancel (destructive, zero intensity).
• Rule: path difference S₁P ~ S₂P = nλ gives constructive interference; (n + ½)λ gives destructive interference, with n = 0, 1, 2, 3, ...
• For a general phase difference φ, y = a cos ωt + a cos(ωt + φ) = 2a cos(φ/2) cos(ωt + φ/2). The amplitude is 2a cos(φ/2), so I = 4I₀ cos²(φ/2): maxima at φ = 0, ±2π, ±4π, ... and zeros at φ = ±π, ±3π, ...
• With coherent sources φ at each point is fixed, so the bright and dark places stay put: a steady interference pattern. If the phase difference changes rapidly and at random, the pattern shifts faster than it can be seen, and the time-averaged intensity is I = 2I₀ everywhere. Such sources are incoherent, and their intensities simply add, as when two lamps light a wall.
• Exercise 10.5: the intensity is K where the path difference is λ, so φ = 2π and K = 4I₀. Where the path difference is λ/3, φ = 2π/3 and cos²(π/3) = ¼, so I = 4I₀ × ¼ = K/4.

6. Young's double-slit experiment

NCERT §10.5

• Two sodium lamps lighting two pinholes give no fringes. Light from an ordinary source suffers sudden phase jumps in times of about 10⁻¹⁰ s, so two independent sources never keep a fixed phase relation. They are incoherent, and their intensities add on the screen.
• Thomas Young's way round this: light from one bright pinhole S falls on two pinholes S₁ and S₂ placed very close together in an opaque screen. Both are fed by the same S, so every phase jump in S reaches S₁ and S₂ alike and the two stay locked in phase. They behave as coherent sources, like the two needles in the water trough.
• The waves from S₁ and S₂ overlap on a screen GG′ and produce alternate bright and dark bands, called fringes. The analysis of the previous lesson fixes where they fall.
• Let d be the separation of S₁ and S₂, D the distance to the screen and x the distance of a point from the centre of the pattern. Bright fringes lie at x = nDλ/d, with n = 0, ±1, ±2, ...; the n = 0 fringe at the centre is the central bright fringe.
• Dark fringes lie at x = (n + ½)Dλ/d, with n = 0, ±1, ±2, ...
• Consecutive bright fringes, and consecutive dark ones, are the same distance apart, β = Dλ/d: the fringes are equally spaced. A longer wavelength or a more distant screen spreads them out; moving the slits further apart packs them closer.
• The computer-drawn pattern of Fig. 10.13 uses d = 0.025 mm, D = 5 cm and λ = 5 × 10⁻⁵ cm. Its spacing is Dλ/d = (0.05 × 5 × 10⁻⁷)/(2.5 × 10⁻⁵) = 1 × 10⁻³ m, that is 1 mm.
• Exercise 10.4: d = 0.28 mm, D = 1.4 m, and the fourth bright fringe is 1.2 cm from the centre. From x₄ = 4Dλ/d, λ = x₄d/(4D) = (1.2 × 10⁻² × 0.28 × 10⁻³)/(4 × 1.4) = 6.0 × 10⁻⁷ m = 600 nm.
• Exercise 10.6: with 650 nm and 520 nm together, the third bright fringe of 650 nm is at x = 3 × 650 nm × D/d = 1950 nm × D/d. Bright fringes of the two colours first coincide where n × 650 = m × 520; the smallest match is 4 × 650 = 5 × 520 = 2600 nm, so at x = 2600 nm × D/d (fourth of 650 nm on the fifth of 520 nm).

7. Diffraction at a single slit

NCERT §10.6, §10.6.1

• Look closely at the edge of the shadow of an opaque object and you will find alternate bright and dark bands just inside and outside it. This is diffraction, which every kind of wave shows: sound, light, water waves and matter waves.
• The wavelength of light is much smaller than most obstacles, so diffraction of light goes unnoticed in daily life. It still sets the limit on how finely the eye, a microscope or a telescope can separate close objects, and it makes the colours seen on a CD.
• Hearing someone talk from round a corner surprises nobody. Light also spreads a little beyond narrow holes and slits into the region that should be shadow; Newton and others noticed this before Young.
• Set-up: a parallel beam falls normally on a slit LN of width a. M is its midpoint, and the line from M at right angles to the slit meets the screen at C. The lines from different parts of the slit to a point P on the screen are close to parallel and make an angle θ with MC.
• Method: split the slit into many narrow strips and treat each as a source of secondary wavelets. The incoming wavefront is parallel to the slit, so all the strips start in phase; their contributions are added at P with the proper phase differences.
• Result: a bright central maximum at θ = 0; zero intensity at θ ≈ nλ/a with n = ±1, ±2, ±3, ...; and secondary maxima near θ ≈ (n + ½)λ/a, each weaker than the one before as n grows.
• The central maximum reaches from −λ/a to +λ/a, twice the angular width of each of the other bands. A narrower slit or a longer wavelength spreads the whole pattern wider.
• For example, with λ = 600 nm and a = 0.1 mm the first minimum is at θ = (6 × 10⁻⁷)/(1 × 10⁻⁴) = 6 × 10⁻³ rad; on a screen 1 m away that is 6 mm from the centre, and the central band is 12 mm wide.

8. Interference, diffraction and seeing them

NCERT §10.6.1, §10.6.2

• How interference differs from diffraction has been argued over since both were discovered. Richard Feynman's view in his lectures: there is no important physical difference. When only a few sources, such as two, combine, the result is usually called interference; when very many combine, it is more often called diffraction.
• In Young's experiment the pattern on the screen is really double-slit interference superposed on the single-slit diffraction from each slit.
• A home demonstration needs two razor blades and a clear glass bulb, ideally with a straight filament. Hold the blades with their edges parallel and a narrow gap between them, keep the gap parallel to the filament, and look through it with the gap right in front of the eye (with spectacles if you wear them).
• After adjusting the gap width and keeping the edges parallel, bright and dark bands appear. Apart from the central band, the band positions depend on wavelength, so they show colours. A red or blue filter makes them clearer, and the red bands are visibly wider than the blue ones.
• Here the glowing filament stands in for the single source slit S. The eye's lens brings the bands to a focus on the retina, so the retina serves as the screen.
• A double slit cut in aluminium foil with a blade repeats Young's experiment with the same bulb. In daylight, the Sun's small bright reflection in a shiny convex surface, such as a cycle bell, can be the source.
• Never use the Sun directly: it can damage the eye, and it would give no fringes anyway, because it spans about ½° in the sky, far too wide a source.
• Interference and diffraction only redistribute light energy. Wherever a dark fringe has less light, a bright fringe has more; no energy is created or lost, in line with conservation of energy.

9. Polarisation of light

NCERT §10.7

• Hold one end of a long horizontal string whose far end is fixed, and move your hand up and down periodically. A wave y(x, t) = a sin(kx − ωt) travels along +x, where a is the amplitude, ω = 2πν and k = 2π/λ.
• The string moves along y while the wave travels along x, at right angles: a transverse wave. Because the displacement is along y it is called y-polarised. Each point moves on a straight line, so the wave is linearly polarised, and since the string stays in the x-y plane it is also called plane polarised.
• Moving the hand in the x-z plane instead gives a z-polarised wave, z(x, t) = a sin(kx − ωt). Both are transverse and linearly polarised.
• If the plane of vibration is changed at random over very short intervals, the wave is unpolarised. Its displacement keeps changing direction, but always stays perpendicular to the direction of travel.
• Light is a transverse wave: its electric field always oscillates at right angles to the direction the light travels. Natural light, from the Sun or a sodium lamp, is unpolarised; its electric vector takes every direction in the transverse plane, rapidly and at random.
• A polaroid is a thin plastic-like sheet containing long-chain molecules aligned in one direction. It absorbs the part of the electric field along the molecules and lets through the part perpendicular to them. That direction is the pass-axis.
• So unpolarised light that passes through one polaroid comes out linearly polarised along the pass-axis, with half the incident intensity. Rotating that single polaroid leaves the transmitted intensity unchanged.
• Polarisation is special to transverse waves. Longitudinal waves such as sound in air show interference and diffraction too, but they cannot be polarised.

10. Malus' law and crossed polaroids

NCERT §10.7

• Place a second identical polaroid in the beam. Now rotating one of them has a striking effect: at one setting almost no light gets through the pair, and 90° from that setting nearly all the light leaving the first polaroid passes the second.
• Why: the light leaving the first polaroid has its electric field along that polaroid's pass-axis. If the second pass-axis makes an angle θ with the first, only the component E cos θ along it gets through.
• Intensity goes as the square of the field, so I = I₀ cos²θ. This is Malus' law, where I₀ is the intensity of the polarised light that left the first polaroid.
• θ = 0 passes all of I₀; θ = 90° (crossed polaroids) passes nothing; θ = 45° passes I₀/2. In one full turn of the second polaroid, through 2π, the intensity goes through two maxima and two minima.
• One polaroid halves unpolarised light, so a pair can set the transmitted light anywhere from 50% of the original intensity down to zero just by changing the angle between their axes.
• Example 10.2: a polaroid P₂ is rotated between two crossed polaroids P₁ and P₃. After P₂ the intensity is I₀cos²θ. The angle between P₂ and P₃ is π/2 − θ, so the light leaving P₃ is I₀cos²θ sin²θ = (I₀/4) sin²2θ.
• That is zero when P₂ lines up with either crossed polaroid (θ = 0 or π/2) and greatest, I₀/4, when θ = π/4. Inserting a third sheet lets light through a pair that on its own would block it completely.
• Polaroids control intensity in sunglasses and window panes, and are used in photographic cameras and 3D movie cameras.

Must-know facts

1. A wavefront is a surface of constant phase; energy flows at right angles to it, along the rays.
2. A point source gives spherical wavefronts; far away a small patch is a plane wavefront.
3. Huygens: every point of a wavefront is a source of secondary wavelets; the forward envelope is the new wavefront.
4. Refraction by wavefronts: sin i/sin r = v₁/v₂ = n₂/n₁, which is Snell's law n₁ sin i = n₂ sin r.
5. Bending towards the normal means slower light in the second medium, as Foucault found in 1850.
6. On refraction speed and wavelength change in the same ratio; frequency never changes.
7. The energy a wave carries depends on its amplitude, not its speed.
8. Critical angle: sin ic = n₂/n₁; beyond it, total internal reflection.
9. Coherent sources keep a constant phase difference and give a steady pattern.
10. I = 4I₀cos²(φ/2): 4I₀ at maxima, zero at minima; incoherent sources give 2I₀ everywhere.
11. Path difference nλ: bright; (n + ½)λ: dark. A path difference of λ is a phase difference of 2π.
12. Independent sources jump in phase about every 10⁻¹⁰ s, so they cannot show interference.
13. Young locked two slits in phase by feeding both from one pinhole.
14. Young's fringes: bright at x = nDλ/d, dark at (n + ½)Dλ/d, equally spaced by β = Dλ/d.
15. Single slit of width a: central maximum at θ = 0, minima at θ ≈ nλ/a, weaker maxima near (n + ½)λ/a.
16. Interference and diffraction redistribute energy; they neither create nor destroy it.
17. Light is transverse; natural light is unpolarised.
18. One polaroid passes linearly polarised light at half the unpolarised intensity, whatever its orientation.
19. Malus' law: I = I₀cos²θ; crossed polaroids pass nothing.
20. Polaroid between crossed polaroids: I = (I₀/4) sin²2θ, largest at θ = π/4.

Common traps

Formulas

Key terms

Wavefront

A surface on which every point vibrates in the same phase.

Plane wavefront

A flat wavefront, as from a very distant source or a point source at a lens's focus.

Secondary wavelets

Small waves imagined to spread from every point of a wavefront in Huygens' construction.

Huygens principle

The new wavefront is the forward envelope of secondary wavelets from the old one.

Refractive index

n = c/v, the ratio of the speed of light in vacuum to that in the medium.

Critical angle

Angle of incidence in the denser medium for which the refracted ray grazes the boundary.

Coherent sources

Sources whose phase difference stays constant in time.

Incoherent sources

Sources whose phase difference changes rapidly and randomly; their intensities simply add.

Fringes

The alternate bright and dark bands of an interference or diffraction pattern.

Fringe spacing

Distance between neighbouring bright (or dark) fringes, Dλ/d in Young's experiment.

Diffraction

Spreading of a wave around edges and through openings into the geometrical shadow.

Unpolarised light

Light whose electric vector points in random, rapidly changing directions across the beam.

Linearly polarised light

Light whose electric vector oscillates along one fixed line.

Polaroid

A sheet of aligned long-chain molecules that passes only the field component along its pass-axis.

Pass-axis

The direction, perpendicular to the aligned molecules, along which a polaroid transmits the electric field.

Malus' law

Polarised light through a polaroid at angle θ has intensity I₀cos²θ.