Wave Optics

In this chapter: wavefronts and Huygens principle, reflection and refraction of plane waves, the Doppler effect for light, coherent and incoherent addition of waves, Young's double-slit experiment and fringe width, single-slit diffraction, the validity of ray optics, polarisation, Malus's law and Brewster's law.

Wavefronts and Huygens principle

A wavefront is a surface of constant phase. Waves from a point source have spherical wavefronts; far from the source a small part of the sphere is effectively a plane wavefront. Rays are the lines perpendicular to the wavefronts and show the direction of energy flow.

Huygens principle: every point on a wavefront acts as a source of secondary wavelets that spread out in the forward direction at the speed of the wave. The common tangent (envelope) to these wavelets after a time t is the new wavefront.

Refraction and reflection of plane waves

Let a plane wavefront meet a surface separating medium 1 (speed v₁) from medium 2 (speed v₂). While one edge of the wavefront travels a distance v₁τ in medium 1, the wavelet from the other edge travels v₂τ in medium 2. Constructing the new wavefront by Huygens principle gives

This is Snell's law, and it shows that light travels slower in the denser medium, which the corpuscular theory had got wrong. The same construction for reflection gives i = r. Huygens' construction also explains how a prism, a convex lens and a concave mirror change the shape of a plane wavefront.

Doppler effect for light

When a source moves away from an observer, the observed frequency drops. For speeds small compared with c, Δν/ν = −v_radial/c. Light from receding galaxies is shifted towards the red end of the spectrum (red shift); an approaching source shows a blue shift.

Coherent and incoherent addition

Two sources are coherent if the phase difference between them stays constant in time. When waves of intensities I₁ and I₂ with phase difference φ superpose,

• Constructive interference (bright): path difference nλ, phase difference 2nπ, I = 4I₀.
• Destructive interference (dark): path difference (n + ½)λ, phase difference (2n + 1)π, I = 0.

If the phase difference changes rapidly and randomly, cos φ averages to zero and the intensities simply add: I = I₁ + I₂. So two separate bulbs never show interference. Coherent sources are obtained by splitting light from one source, as the two slits in Young's experiment do. Energy is not destroyed in interference; it is redistributed from the dark fringes to the bright ones.

Young's double-slit experiment

Light from a narrow source falls on two narrow slits S₁ and S₂ separated by d. They act as coherent sources, and a pattern of bright and dark fringes forms on a screen at distance D (D ≫ d). For a point P at distance x from the centre O, the path difference is

• The central fringe (x = 0) is bright, since the path difference is zero.
• Bright and dark fringes are equally spaced and of equal width.
• If the whole apparatus is placed in a medium of refractive index n, λ becomes λ/n and β becomes β/n.
• With white light, the central fringe is white and the fringes on either side are coloured, with blue (shorter λ) closer to the centre than red.
• If the source slit is too wide, or S is moved too close to the slits, the fringes lose contrast and disappear.

Worked example: In a double-slit experiment, λ = 600 nm, d = 0.50 mm and D = 1.0 m. Find the fringe width, the distance of the third bright fringe from the centre, and the fringe width if the apparatus is immersed in water (n = 4/3).
Solution: β = λD/d = (600 × 10⁻⁹ × 1.0)/(0.50 × 10⁻³) = 1.2 × 10⁻³ m = 1.2 mm. Third bright fringe: x₃ = 3β = 3.6 mm. In water: β′ = β/n = 1.2 × 3/4 = 0.9 mm.

Diffraction at a single slit

When light passes through a single slit of width a comparable to λ, it spreads out and a pattern forms on a screen: a broad, bright central maximum with weaker secondary maxima on either side. Divide the slit into two halves: at the angle where the path difference between the edges is λ, each point in the upper half has a partner in the lower half with path difference λ/2, and they cancel. That is the first minimum.

Narrowing the slit makes the central maximum wider. Compared with double-slit interference, the diffraction fringes are of unequal width and intensity, whereas interference fringes (for narrow slits) are equally spaced and of nearly equal intensity. In fact, the double-slit pattern is an interference pattern modulated by the diffraction pattern of each slit.

Worked example: Light of wavelength 500 nm falls on a slit 0.2 mm wide, and the pattern is seen on a screen 1 m away. Find the distance of the first minimum from the centre and the width of the central maximum.
Solution: y₁ = λD/a = (500 × 10⁻⁹ × 1)/(0.2 × 10⁻³) = 2.5 × 10⁻³ m = 2.5 mm. Width of the central maximum = 2 × 2.5 = 5.0 mm.

Validity of ray optics

A beam of width a spreads by diffraction to a width of about λz/a after travelling a distance z. The spreading becomes comparable to a at the Fresnel distance, z_F ≈ a²/λ. For distances much less than z_F, diffraction can be ignored and ray optics works. For a 3 mm aperture and visible light (λ ≈ 500 nm), z_F is about 18 m.

Polarisation

Light is a transverse wave; in ordinary (unpolarised) light the electric field vibrates in all directions perpendicular to the direction of travel. A polaroid transmits only the component of E along its pass axis, giving plane polarised light. Only transverse waves can be polarised, so polarisation proves that light is transverse.

Light scattered at 90° by molecules of the atmosphere is polarised, which is why the blue sky seen through a rotating polaroid changes in brightness. Light reflected from a transparent surface is completely polarised at one angle of incidence, the Brewster angle, for which the reflected and refracted rays are perpendicular:

Worked example: Unpolarised light of intensity I₀ passes through two polaroids whose pass axes are at 60° to each other. Find the transmitted intensity.
Solution: After the first, I₀/2. After the second, (I₀/2) cos²60° = (I₀/2)(1/4) = I₀/8.

Common mistakes: (1) Using the bright-fringe condition for a single slit; a sin θ = nλ gives minima in diffraction but maxima in double-slit interference. (2) Forgetting that λ, not the frequency, changes in a medium. (3) Adding intensities of coherent waves directly; add amplitudes and include the phase. (4) Taking the central maximum width as λD/a; it is 2λD/a. (5) Applying Malus's law to unpolarised light; the first polaroid simply halves the intensity.

JEE and NEET focus

• Huygens construction for refraction and the result that light is slower in a denser medium.
• Resultant intensity for a given phase or path difference, and the ratio of maximum to minimum intensity.
• Fringe width in YDSE and how it changes with d, D, λ and the medium.
• Positions of minima and width of the central maximum in single-slit diffraction.
• Malus's law with two or three polaroids, and Brewster's angle.

Practice questions