Optics — Reflection, Refraction, Lenses and Wave Optics
Optics splits cleanly into two halves that answer different questions. Ray optics treats light as travelling in straight lines and tells you where an image forms; wave optics treats it as a wave and explains the things rays cannot — interference fringes, diffraction, polarisation. Almost all lost marks in ray optics come from the sign convention rather than from the physics, so this chapter fixes that first. The wave half then rests on a single idea: path difference converts to phase difference, and everything from Young's fringes to thin-film colours follows from it.
Refraction, Total Internal Reflection and Lenses
Snell's law is n₁ sin i = n₂ sin r, with n = c/v the refractive index. Light entering a denser medium slows and bends towards the normal. Note what does not change on refraction: the frequency is fixed by the source, so it is the wavelength that changes, λ_medium = λ_vacuum/n.
🎯 Total internal reflection, and the three things it explains
Going from denser to rarer, the refracted ray bends away from the normal, and at the critical angle given by sin C = 1/n it grazes the surface. Beyond C there is no refracted ray at all and the light is entirely reflected — hence total internal reflection, which requires both conditions: dense to rare, and i > C. Three familiar phenomena follow. Optical fibres trap light by repeated total internal reflection along the core. A diamond sparkles because its high refractive index of 2.42 gives a critical angle of only about 24°, so light entering it bounces many times before escaping. And a mirage appears because hot air near the road is optically rarer, so light from the sky bends progressively until it undergoes total internal reflection and reaches your eye from below.| Item | Result |
|---|---|
| Power of a lens | P = 1/f (in metres), in dioptres; negative for a concave lens |
| Lenses in contact | P = P₁ + P₂, so 1/f = 1/f₁ + 1/f₂ |
| Lens maker's formula | 1/f = (n − 1)(1/R₁ − 1/R₂) |
| Myopia (short sight) | Corrected by a concave lens |
| Hypermetropia (long sight) | Corrected by a convex lens |
| Prism | A + δ = i + e; at minimum deviation i = e and the ray is symmetric |
⚠️ A lens in water is weaker; a lens of lower index than its surroundings reverses
The lens maker's formula contains (n − 1) only when the lens sits in air; in general it is (n_lens/n_medium − 1). Put a glass lens (n = 1.5) into water (n = 1.33) and that factor drops from 0.5 to about 0.13, so f nearly quadruples and the lens becomes much weaker — which is why you cannot see clearly underwater without goggles. Push it further: if the surrounding medium is denser than the lens, the factor goes negative and a convex lens starts to diverge. An air bubble in water is exactly this case — it is a convex shape of lower index, so it behaves as a diverging lens, and that is a favourite question because the shape suggests the opposite.Wave Optics — Interference, Diffraction and Polarisation
In Young's double-slit experiment two coherent beams overlap and produce alternate bright and dark bands. A path difference Δx corresponds to a phase difference 2πΔx/λ, so bright fringes need Δx = nλ and dark fringes (n + ½)λ. The fringe width is β = λD/d, where d is the slit separation and D the distance to the screen.
| Change | Effect on β = λD/d |
|---|---|
| Slits moved closer (d smaller) | Fringes get wider |
| Screen moved further (D larger) | Fringes get wider |
| Red light replaced by blue | Fringes get narrower, since λ is smaller |
| Whole apparatus immersed in water | λ falls to λ/n, so fringes narrow by the factor n |
| One slit covered | Interference vanishes; a single-slit diffraction pattern remains |
🎯 Why two bulbs never interfere, and what coherence means
Interference needs a constant phase relationship between the two beams. Two independent bulbs emit in random, rapidly changing phase, so the pattern that would form is reshuffled billions of times a second and averages out to uniform illumination. That is why Young used one source and split it into two — both slits then carry the same phase fluctuations and their difference stays constant. It is also why lasers make interference easy to demonstrate. Note that energy is not destroyed at a dark fringe: it is redistributed into the bright fringes, so the total across the screen is exactly what arrived.🧠 Interference against diffraction — how to tell which is which
Both produce bright and dark bands, and questions rely on the confusion. Interference needs two or more sources and gives fringes of equal width and equal intensity. Diffraction happens at a single aperture or edge and gives a broad central maximum roughly twice as wide as the others, with intensity falling off sharply on either side. So count the bands: uniform and evenly spaced means interference; one dominant central band means diffraction. Diffraction also requires the aperture to be comparable to λ, which is why sound diffracts around a doorway easily (λ of metres) while light does not (λ of hundreds of nanometres) — and that difference is why you can hear someone around a corner but not see them.ℹ️ Polarisation — the one thing that proves light is transverse
Interference and diffraction show that light is a wave but say nothing about its direction of vibration. Polarisation does, because only a transverse wave can be polarised — sound, being longitudinal, cannot be. Ordinary light vibrates in all planes; a polaroid transmits one, halving the intensity, and a second polaroid at angle θ transmits I₀cos²θ by Malus's law, so two crossed polaroids at 90° transmit nothing at all. Light is also completely polarised on reflection at the Brewster angle, where tan θ_B = n and the reflected and refracted rays are perpendicular — which is why polarising sunglasses cut glare from a road or water surface so effectively.Key takeaways
- Cartesian convention: a real object distance is always negative; f > 0 converges, f < 0 diverges.
- Mirror 1/v + 1/u = 1/f with m = −v/u; lens 1/v − 1/u = 1/f with m = v/u.
- A convex mirror or concave lens can only ever give a virtual, erect, diminished image — use that as a check.
- On refraction the frequency is unchanged; the wavelength becomes λ/n.
- TIR needs dense→rare AND i > C, where sin C = 1/n — hence fibres, a diamond's sparkle and mirages.
- A lens in water is much weaker; a lens of lower index than its surroundings reverses its action.
- Fringe width β = λD/d — wider with larger D, narrower with blue light or in water.
- Interference gives equal fringes and needs coherence; diffraction gives a broad central maximum from one aperture.
Practice questions (8)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
A convex mirror can form an image that is:
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Answer: B — Virtual, erect and diminished
A convex mirror diverges the reflected rays, so for any real object it can only produce a virtual, erect, diminished image — which is exactly why it is used as a rear-view mirror, giving a wide field of view. This is a useful check on algebra: any calculation claiming a real image from a convex mirror has a sign error.When light passes from air into glass, which quantity remains unchanged?
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Answer: C — Frequency
Frequency is fixed by the source and cannot change at a boundary, so with v = fλ falling in the denser medium it must be the wavelength that falls, to λ/n. Assuming the wavelength is invariant is the standard error here, and it matters directly for fringe questions, where immersing the apparatus in water narrows the fringes by the factor n.Total internal reflection can occur when light travels:
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Answer: B — From a denser to a rarer medium beyond the critical angle
Both conditions are needed: the light must go from denser to rarer, and the angle of incidence must exceed the critical angle given by sin C = 1/n. Going the other way there is always a refracted ray, so TIR is impossible. This effect is what makes optical fibres work and what gives a diamond its sparkle, since its critical angle is only about 24°.In Young's experiment the whole apparatus is immersed in water of refractive index 4/3. The fringe width:
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Answer: B — Decreases by a factor of 4/3
β = λD/d, and in water the wavelength falls to λ/n while D and d are unchanged, so β falls by the factor n = 4/3. The frequency does not change, so it is entirely the wavelength that drives the effect — a direct application of the fact that only λ changes on entering a denser medium.Two independent light bulbs placed side by side do not produce an interference pattern because:
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Answer: B — They are not coherent — their phase relationship changes randomly
Interference needs a constant phase relationship. Independent sources emit in randomly fluctuating phase, so any pattern reshuffles billions of times a second and averages to uniform brightness. That is why Young split a single source into two slits, and why lasers make the demonstration easy — identical wavelengths alone are not enough.An air bubble inside water behaves as:
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Answer: B — A diverging lens, since its refractive index is lower than the surrounding medium
The lens maker's formula carries (n_lens/n_medium − 1), which is negative here since air (1.0) is less dense than water (1.33). A negative focal length means divergence, so the convex shape misleads — what determines converging or diverging is the index relative to the surroundings, not the shape alone.A single-slit diffraction pattern differs from a two-slit interference pattern in that diffraction gives:
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Answer: B — A broad central maximum with intensity falling off on either side
Diffraction from one aperture produces a central maximum roughly twice as wide as the others, with intensity dropping sharply outwards; interference from two coherent sources gives fringes of equal width and equal intensity. Counting whether the bands are uniform or dominated by a central one is the quickest way to identify which pattern a question describes.Polarisation of light establishes that light waves are:
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Answer: B — Transverse
Only a transverse wave has a vibration direction perpendicular to propagation that can be selectively filtered, so only a transverse wave can be polarised — sound, being longitudinal, cannot. Interference and diffraction show that light is a wave but say nothing about the direction of vibration; polarisation is the specific evidence that settles it.