Lesson 11 of 11 · 15 min
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Huygens, Young and polarisation, full chapter
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Must-know facts
20 facts
- 01A wavefront is a surface of constant phase; energy flows at right angles to it, along the rays.
- 02A point source gives spherical wavefronts; far away a small patch is a plane wavefront.
- 03Huygens: every point of a wavefront is a source of secondary wavelets; the forward envelope is the new wavefront.
- 04Refraction by wavefronts: sin i/sin r = v₁/v₂ = n₂/n₁, which is Snell's law n₁ sin i = n₂ sin r.
- 05Bending towards the normal means slower light in the second medium, as Foucault found in 1850.
- 06On refraction speed and wavelength change in the same ratio; frequency never changes.
- 07The energy a wave carries depends on its amplitude, not its speed.
- 08Critical angle: sin ic = n₂/n₁; beyond it, total internal reflection.
- 09Coherent sources keep a constant phase difference and give a steady pattern.
- 10I = 4I₀cos²(φ/2): 4I₀ at maxima, zero at minima; incoherent sources give 2I₀ everywhere.
- 11Path difference nλ: bright; (n + ½)λ: dark. A path difference of λ is a phase difference of 2π.
- 12Independent sources jump in phase about every 10⁻¹⁰ s, so they cannot show interference.
- 13Young locked two slits in phase by feeding both from one pinhole.
- 14Young's fringes: bright at x = nDλ/d, dark at (n + ½)Dλ/d, equally spaced by β = Dλ/d.
- 15Single slit of width a: central maximum at θ = 0, minima at θ ≈ nλ/a, weaker maxima near (n + ½)λ/a.
- 16Interference and diffraction redistribute energy; they neither create nor destroy it.
- 17Light is transverse; natural light is unpolarised.
- 18One polaroid passes linearly polarised light at half the unpolarised intensity, whatever its orientation.
- 19Malus' law: I = I₀cos²θ; crossed polaroids pass nothing.
- 20Polaroid between crossed polaroids: I = (I₀/4) sin²2θ, largest at θ = π/4.
Common traps
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Formulas
8 to know
Refraction from wavefronts
sin i/sin r = v₁/v₂ = n₂/n₁
n = c/v; Snell's law n₁ sin i = n₂ sin r.
Wavelength in a medium
λ₁/λ₂ = v₁/v₂
Frequency is unchanged.
Critical angle
sin ic = n₂/n₁
Light going from the denser medium 1 to the rarer medium 2.
Two-source intensity
I = 4I₀ cos²(φ/2)
Phase difference φ = (2π/λ) × path difference; incoherent sources give 2I₀.
Young's fringes
x(bright) = nDλ/d, x(dark) = (n + ½)Dλ/d
Fringe spacing β = Dλ/d.
Single-slit minima
θ ≈ nλ/a
n = ±1, ±2, ...; secondary maxima near (n + ½)λ/a.
Malus' law
I = I₀ cos²θ
I₀ is the polarised intensity after the first polaroid, half the unpolarised intensity.
Polaroid between crossed polaroids
I = (I₀/4) sin²2θ
Maximum I₀/4 at θ = π/4.
Key terms
16 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.