Wave optics

Huygens' principle

A wavefront is a surface on which all points oscillate in the same phase. A point source gives spherical wavefronts; a very distant source gives plane wavefronts. Huygens' principle: every point on a wavefront acts as a source of secondary wavelets that spread out at the speed of the wave; the new wavefront is the forward envelope of these wavelets. Using it, the laws of reflection and refraction can be derived, and it shows that light travels more slowly in a denser medium (n = c/v).

Interference

When two waves overlap, the resultant displacement is the sum of the individual displacements (superposition). For a steady pattern, the sources must be coherent (constant phase difference). With path difference Δ:

• Constructive (bright): Δ = nλ.
• Destructive (dark): Δ = (n + ½)λ.

For two equal sources, I = 4I₀ cos²(φ/2).

Young's double-slit experiment

Light from a single source falls on two narrow slits a distance d apart, acting as coherent sources. On a screen at distance D, equally spaced bright and dark fringes appear.

• Position of the nth bright fringe: y = nλD/d.
• Fringe width: β = λD/d.

Example: λ = 600 nm, d = 0.3 mm, D = 1.5 m gives β = 3 mm. If the whole apparatus is immersed in a liquid of index n, λ becomes λ/n, so β decreases by the factor n.

Single-slit diffraction

Light bends around obstacles and openings whose size is comparable to its wavelength. For a slit of width a:

• Minima occur at a sin θ = nλ (n = ±1, ±2, …).
• The central maximum is the brightest and twice as wide as the others; its angular width is 2λ/a and its linear width on a screen is 2λD/a.
• Secondary maxima, roughly midway between minima, become fainter.

For λ = 500 nm, a = 0.2 mm and D = 2 m, the central maximum is 10 mm wide.

Key points

• Huygens: every point on a wavefront is a source of secondary wavelets.
• Bright: nλ; dark: (n + ½)λ.
• β = λD/d; decreases in a denser medium.
• Single slit: minima at a sin θ = nλ; central width 2λD/a.

Common mistakes

• Using a sin θ = nλ for maxima in single-slit diffraction; it gives minima.
• Forgetting to convert mm and nm to metres.