From rays to waves
Geometrical optics predicts where rays travel, but wave optics explains patterns produced when coherent light overlaps, bends around apertures, or is filtered by direction of oscillation. A monochromatic wave may be written as an oscillation with wavelength λ, frequency f, and phase. Since c = fλ in vacuum, changing frequency changes wavelength. Coherence means a stable phase difference; it is essential for a stationary interference pattern.
Δ = r₂ − r₁ determines phase difference: Δφ = 2πΔ/λ.
Bright maxima occur when Δ = mλ.
Dark minima occur when Δ = (m + 1/2)λ.
Double-slit interference
For slit separation d and observation angle θ, Δ = d sin θ. Maxima satisfy d sin θ = mλ. On a distant screen at distance L and small angles, yₘ ≈ mλL/d, so adjacent bright fringes are separated by Δy = λL/d. Increasing wavelength or screen distance spreads the pattern; increasing slit separation compresses it. For equal-amplitude waves, intensity varies with phase rather than merely adding as two independent brightnesses.
Diffraction and polarization
A single slit of width a produces minima at a sin θ = mλ for nonzero integers m. At small angles, the central maximum has width approximately 2λL/a, twice the spacing from the centre to the first minimum. Diffraction becomes prominent when an opening is comparable with the wavelength. Polarization confirms that light is transverse: an ideal linear polarizer selects one electric-field direction. Malus law gives I = I₀ cos²φ for polarized light meeting an analyser at relative angle φ. Unpolarized light first becomes I₀/2 after one ideal polarizer.
Unifying idea: interference and diffraction follow superposition of amplitudes, while detectors measure intensity proportional to the time-averaged square of the resulting amplitude.