Wave Optics

In wave optics, light is treated as a wave spreading out from its source. A wavefront is the locus of all points in a medium that are vibrating in the same phase at a given instant. The direction of propagation of light (a ray) at any point is always perpendicular to the wavefront at that point.

The shape of a wavefront depends on the source. A point source in a homogeneous medium produces spherical wavefronts, since every point at the same distance from the source has travelled the same optical path and is in the same phase. A line source produces cylindrical wavefronts. When the source is extremely far away, a small patch of the (nearly spherical) wavefront looks essentially flat, and we call it a plane wavefront — light reaching the Earth from a distant star is the standard example.

Huygens' Principle is a geometric construction used to find the position of a wavefront at a later instant from its position at an earlier instant. It has two parts: (a) every point on an existing wavefront becomes a source of secondary spherical wavelets, which spread out in the forward direction with the speed of light in that medium; (b) the new wavefront at a later time is the forward envelope, i.e. the common tangent surface, touching all these secondary wavelets.

Worked example (construction): A plane wavefront AB travels with speed v in a medium. To locate the wavefront after time τ, draw secondary wavelets of radius vτ centred on several points along AB. The new wavefront A′B′ is the plane tangent to all these wavelets, and simple geometry shows A′B′ is parallel to AB and displaced forward by exactly vτ — consistent with straight-line propagation of a plane wave at speed v. This same construction, applied to a curved (e.g. spherical) wavefront, correctly predicts that it keeps expanding while remaining centred on the source.

Huygens originally could not explain why the wavelets do not also produce a backward-travelling wave; this was later resolved by Fresnel using the idea that the amplitude of secondary wavelets is not the same in all directions (it is zero in the backward direction), a result that follows properly only from a more complete diffraction theory. For CBSE Class 12 purposes, it is enough to know that Huygens' Principle, despite this limitation, correctly predicts reflection, refraction and the general forward motion of light waves.

Law of reflection. Let a plane wavefront AB be incident obliquely on a plane reflecting surface MN, touching it first at A. While the secondary wavelet from B travels a distance BC = v·t to reach the surface at C, the wavelet from A (which started earlier) has already spread out into the medium as a hemisphere of radius AE = v·t = BC. The reflected wavefront is the tangent EC. Since the right triangles EAC and BAC share the hypotenuse AC and have AE = BC, they are congruent, so the angle between the incident wavefront and AC equals the angle between AC and the reflected wavefront. Converting to rays (perpendicular to the wavefronts) gives the familiar law: angle of incidence = angle of reflection, and the incident ray, reflected ray and normal all lie in the same plane.

Law of refraction (Snell's law). Now let the wavefront cross from medium 1 (speed v₁) into medium 2 (speed v₂). In the same time t that the wavelet from B travels BC = v₁t to reach the second medium, the wavelet from A spreads into medium 2 with radius AE = v₂t. From the geometry, BC = AC sin i and AE = AC sin r, where i is the angle of incidence and r the angle of refraction. Hence sin i / sin r = v₁/v₂, a constant for the two media, called the refractive index of medium 2 with respect to medium 1, n₂₁. Since n = c/v for each medium, n₂₁ = v₁/v₂ = n₂/n₁. Because frequency depends only on the source and stays unchanged as the wave crosses the boundary, while speed changes, the wavelength must also change: v = fλ gives λ₁/λ₂ = v₁/v₂. This immediately explains why light slows down and its wavelength shrinks on entering a denser medium (bending towards the normal), while its frequency and colour remain the same.

Worked example: A ray of light travelling in air (n₁ = 1) strikes a glass surface (n₂ = 1.5) at an angle of incidence of 30°. Find the angle of refraction.
Using sin i / sin r = n₂/n₁: sin r = sin i × (n₁/n₂) = sin 30° × (1/1.5) = 0.5/1.5 = 0.333.
r = sin⁻¹(0.333) ≈ 19.5°. As expected, the ray bends towards the normal on entering the denser medium.

By the principle of superposition, when two light waves overlap at a point, their displacements add. If the two waves maintain a constant phase difference at every point (i.e. they are coherent), the addition produces a stable pattern of alternately bright (constructive interference) and dark (destructive interference) regions — this phenomenon is called interference. Two independent sources are almost never coherent because their phases fluctuate randomly and independently; in practice coherent sources are made by splitting light from a single source into two parts, as Thomas Young did by illuminating two narrow, closely spaced slits with light from a single slit.

In Young's double-slit experiment (YDSE), two slits S₁ and S₂, separated by a small distance d, are illuminated by a common monochromatic source, and the resulting pattern is observed on a screen at distance D (with D >> d). At a point P on the screen at a small angle θ from the centre, the path difference between the two waves arriving at P is Δx = d sinθ ≈ d y/D, where y is the distance of P from the centre of the pattern (valid for small angles). Constructive interference (bright fringe) occurs when Δx is an integer multiple of the wavelength; destructive interference (dark fringe) occurs when Δx is an odd multiple of half the wavelength.

The spacing between successive bright (or dark) fringes, called the fringe width β, works out to β = λD/d, and is uniform across the pattern (for small angles). Using the phase difference φ = (2π/λ)Δx, the resultant intensity at any point is I = I₁ + I₂ + 2√(I₁I₂) cosφ. For two sources of equal intensity I₀ each, this simplifies to I = 4I₀cos²(φ/2), giving a maximum of 4I₀ and a minimum of zero. Notice that the average intensity over the whole pattern is still 2I₀ (= I₁+I₂), so energy is only redistributed by interference, not created or destroyed — energy conservation is not violated.

Worked example: In a YDSE set-up, d = 1 mm, D = 1 m and λ = 600 nm. Find the fringe width and the position of the 5th bright fringe.
β = λD/d = (600 × 10⁻⁹ m)(1 m) / (1 × 10⁻³ m) = 6 × 10⁻⁴ m = 0.6 mm.
Position of the 5th bright fringe: y₅ = nβ = 5 × 0.6 mm = 3 mm from the centre.

Diffraction is the spreading of light as it passes through a narrow opening or around an edge whose size is comparable to the wavelength of light. Unlike Young's double-slit interference, which combines light from two separate, widely-spaced coherent point sources, single-slit diffraction arises from the superposition of secondary wavelets originating from every point across the same slit — effectively, interference among infinitely many coherent sources spread continuously over the slit width.

When a plane wavefront of wavelength λ falls normally on a narrow slit of width a, the diffraction pattern observed on a distant screen consists of a broad, intense central maximum flanked symmetrically by much fainter secondary maxima, separated by minima. To locate the minima, the slit is imagined as divided into pairs of strips; the condition for the m-th minimum is a sinθ = mλ (m = ±1, ±2, …; m ≠ 0). Secondary maxima occur roughly midway between successive minima, near a sinθ = (m + ½)λ, and are far weaker than the central maximum — the first secondary maximum has only about 1/22 of the central maximum's intensity.

Because the first minima occur at sinθ = ±λ/a, the central maximum is twice as wide (in angle) as any secondary maximum: its angular width is Δθ = 2λ/a, and its linear width on a screen at distance D is w = 2λD/a. A narrower slit (smaller a) produces a wider, more spread-out diffraction pattern — showing that diffraction effects become more pronounced as the opening size approaches the wavelength of light.

Worked example: A parallel beam of light of wavelength 600 nm falls normally on a slit of width 0.3 mm, and the pattern is observed on a screen 1.5 m away. Find the distance of the first minimum from the centre, and the width of the central maximum.
sinθ₁ = λ/a = (600 × 10⁻⁹)/(0.3 × 10⁻³) = 2 × 10⁻³ (small angle, so θ₁ ≈ sinθ₁ in radians).
y₁ = Dθ₁ = 1.5 × 2 × 10⁻³ = 3 × 10⁻³ m = 3 mm.
Width of central maximum = 2y₁ = 6 mm.

Ordinary light from sources such as the sun or a bulb is unpolarised: its electric field vibrates randomly in all directions perpendicular to the direction of travel. That light can be polarised at all is direct evidence that light is a transverse wave (a longitudinal wave, like sound, cannot be polarised, since its vibrations are already confined to the direction of propagation).

A Polaroid is a synthetic material containing long-chain molecules aligned in one direction, which absorbs the component of the electric field parallel to the chains and freely transmits the perpendicular component. It thus has a special direction called its pass axis. When unpolarised light passes through a single Polaroid, exactly half its intensity is transmitted (I = I₀/2), regardless of the Polaroid's orientation, and the transmitted light is now plane polarised, vibrating only along the pass axis.

If this plane-polarised light of intensity I₀ is then passed through a second Polaroid (called the analyser) whose pass axis makes an angle θ with the first Polaroid's pass axis, only the component of the electric field along the analyser's axis is transmitted. Since intensity is proportional to the square of amplitude, the transmitted intensity follows Malus's Law: I = I₀cos²θ. When the two axes are parallel (θ = 0), all the polarised light passes through; when they are crossed (θ = 90°), no light is transmitted.

Light can also become polarised on reflection. At a special angle of incidence called the polarising angle or Brewster's angle θB, the reflected ray is entirely plane polarised (with vibrations perpendicular to the plane of incidence), and the reflected and refracted rays are exactly perpendicular to each other. Brewster's Law states tanθB = n₂₁, where n₂₁ is the refractive index of the reflecting medium with respect to the medium of incidence.

Worked example: Unpolarised light of intensity 20 W/m² is incident on a Polaroid, and the transmitted light then passes through a second Polaroid whose axis is at 30° to the first. Find the final transmitted intensity.
After the first Polaroid: I₁ = I₀/2 = 20/2 = 10 W/m² (now plane polarised).
After the second Polaroid (Malus's Law): I₂ = I₁cos²30° = 10 × (0.75) = 7.5 W/m².

Polarisation has many practical applications in everyday devices: polarised sunglasses cut out glare from horizontally polarised light reflected off water or roads, LCD screens use polarisers to control transmitted light, and polarising filters are used in photography and 3D cinema glasses.