CBSE Class 12 Physics Wave Optics Notes Set 03

1. Wave Theory of Light

Newton described light as a stream of tiny particles (corpuscles). In 1678 Christiaan Huygens put forward the wave theory, in which light travels as a wave. The two models make opposite predictions about the speed of light in a denser medium such as water:
1. The corpuscular model predicts that light travels faster in a denser medium.
2. The wave model predicts that light travels slower in a denser medium.
Measurements of the speed of light in water (Foucault, 1850) showed that it is slower than in air, supporting the wave theory. Later, Young's interference experiment (1801) and Maxwell's electromagnetic theory firmly established that light is a transverse electromagnetic wave.

2. Wavefront

A wavefront is the locus of all points that oscillate in the same phase. The speed with which a wavefront moves outward from the source is the speed of the wave. The energy of the wave travels perpendicular to the wavefront, so a ray is always normal to the wavefront.

(i) Spherical wavefront: produced by a point source; the wavefronts are concentric spheres.
(ii) Cylindrical wavefront: produced by a linear source such as a narrow slit.
(iii) Plane wavefront: a small part of a spherical or cylindrical wavefront very far from the source looks flat. Light from the Sun or a distant star reaches the Earth as plane wavefronts, and the rays are parallel.

3. Huygens' Principle

Huygens' principle is a geometrical method for finding the new position of a wavefront from its known position at an earlier time.
1. Every point on a given wavefront acts as a fresh source of secondary wavelets, which spread out in all directions with the speed of the wave in that medium.
2. After a time τ, each secondary wavelet is a sphere of radius vτ.
3. The forward envelope (common tangent surface) of all these secondary wavelets gives the new wavefront at time τ.
The backward wave is ignored. Huygens simply assumed that the amplitude of the secondary wavelets is maximum in the forward direction and zero in the backward direction; a more detailed wave theory justifies this.

4. Refraction of a Plane Wave (Proof of Snell's Law)

Let a plane wavefront AB travel in medium 1 (speed v₁) and strike the surface PP′ of medium 2 (speed v₂) at an angle of incidence i. The end A touches the surface first. In the time τ that the disturbance from B takes to reach C, a secondary wavelet from A spreads into medium 2 to a radius AE = v₂τ, while BC = v₁τ. The tangent CE from C to this wavelet is the refracted wavefront, and the refracted ray makes an angle r with the normal.

From triangles ABC and AEC (common side AC):
sin i = \( \frac{BC}{AC} = \frac{v_1 \tau}{AC} \) and sin r = \( \frac{AE}{AC} = \frac{v_2 \tau}{AC} \)
\( \frac{\sin i}{\sin r} = \frac{v_1}{v_2} \)
Since n₁ = c/v₁ and n₂ = c/v₂, this gives n₁ sin i = n₂ sin r, which is Snell's law. The ratio \( \frac{v_1}{v_2} = \frac{n_2}{n_1} = n_{21} \) is the refractive index of medium 2 with respect to medium 1.

Important results:
1. The frequency of light does not change on refraction; it is fixed by the source.
2. Since v = νλ, the wavelength changes in proportion to the speed: \( \frac{\lambda_1}{\lambda_2} = \frac{v_1}{v_2} \). In a medium of refractive index n, λ_medium = \( \frac{\lambda_{\text{vacuum}}}{n} \).
3. If v₂ < v₁ (light entering a denser medium), r < i and the ray bends towards the normal.
4. When light goes from a denser to a rarer medium (v₂ > v₁), r > i. At the critical angle i_c, given by sin i_c = \( \frac{n_2}{n_1} \), the angle of refraction becomes 90°, and for larger angles there is total internal reflection.

5. Reflection of a Plane Wave

A plane wavefront AB strikes a reflecting surface at angle i. In the time the disturbance from B reaches C (distance vτ), a secondary wavelet of radius vτ grows from A in the same medium. The tangent from C to this wavelet is the reflected wavefront CE. Triangles ABC and CEA are congruent (right angles at B and E, common hypotenuse AC, and BC = AE = vτ), so the angle of incidence equals the angle of reflection (i = r). The speed, frequency and wavelength are all unchanged on reflection.

6. Behaviour of Plane Wavefronts in Optical Devices

(i) Thin prism: the lower (thicker) part of the wavefront travels a longer path in glass and is delayed, so the emerging wavefront is plane but tilted.
(ii) Convex lens: the central part of the wavefront passes through the thickest glass and is delayed most, so a plane wavefront emerges as a converging spherical wavefront that meets at the focus.
(iii) Concave mirror: a plane wavefront is reflected as a converging spherical wavefront that focuses at the focal point.
The total time taken by light from a point on the object to the corresponding point on the image is the same along every ray.

7. Coherent and Incoherent Addition of Waves

Superposition principle: at a point where two or more waves meet, the resultant displacement is the vector sum of the displacements due to the individual waves.

Path difference and phase difference: a path difference Δx corresponds to a phase difference
φ = \( \frac{2\pi}{\lambda} \Delta x \)

Coherent sources emit waves of the same frequency with a constant phase difference (in phase or with a fixed lag). Two coherent waves of equal intensity I₀ with phase difference φ give a resultant intensity
I = 4I₀ cos²\( \left(\frac{\varphi}{2}\right) \)
1. φ = 0, ±2π, ±4π, … (path difference nλ): I = 4I₀, constructive interference.
2. φ = ±π, ±3π, … (path difference (n + ½)λ): I = 0, destructive interference.

Incoherent sources (for example, two independent bulbs) have a phase difference that changes rapidly and randomly, about 10⁸ times per second or faster. The average of cos²(φ/2) over time is ½, so I = 2I₀ everywhere: the intensities simply add and no steady pattern is seen.

Unequal intensities: for two coherent waves of intensities I₁ and I₂,
I = I₁ + I₂ + 2\( \sqrt{I_1 I_2} \) cos φ
\( \frac{I_{\max}}{I_{\min}} = \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} = \frac{(a_1 + a_2)^2}{(a_1 - a_2)^2} \), where a₁ and a₂ are the amplitudes (I ∝ a²).
In interference, energy is not created or destroyed; it is only redistributed from the dark regions to the bright regions.

8. Young's Double-Slit Experiment (YDSE)

Thomas Young obtained two coherent sources by letting light from a single narrow slit S fall on two close, narrow slits S₁ and S₂. Any change of phase at S reaches both slits equally, so S₁ and S₂ stay coherent. Their waves overlap on a screen at distance D and produce alternate bright and dark bands called interference fringes.

For a point P at distance x from the central point O (with D ≫ d and D ≫ x), the path difference is S₂P − S₁P ≈ \( \frac{xd}{D} \). (Only the final results below are required; the derivation is not in the current syllabus.)
1. Bright fringes: x_n = \( \frac{n\lambda D}{d} \), n = 0, ±1, ±2, …
2. Dark fringes: x_n = \( \left(n + \frac{1}{2}\right)\frac{\lambda D}{d} \), n = 0, ±1, ±2, … (the first dark fringe is at \( \frac{\lambda D}{2d} \))
3. Fringe width (distance between two consecutive bright or dark fringes): β = \( \frac{\lambda D}{d} \)
4. Angular fringe width: θ = \( \frac{\beta}{D} = \frac{\lambda}{d} \)

Key points about the fringe pattern:
1. The central point O is equidistant from S₁ and S₂ (path difference zero), so the central fringe is bright.
2. All fringes are equally spaced and of equal width; β increases with λ and D and decreases with d.
3. If the whole apparatus is immersed in a medium of refractive index n, the wavelength becomes λ/n, so β′ = \( \frac{\beta}{n} \) (fringes come closer).
4. With white light, the central fringe is white (all wavelengths have zero path difference there), and the fringes on either side are coloured, with violet nearer the centre and red farther out. Only a few coloured fringes are visible.
5. The source slit S must be narrow and the slits S₁, S₂ must be close together; if S is made too wide, the fringes lose contrast and disappear.
6. If one of the two slits is covered, the interference fringes disappear and only the broad illumination from a single slit is seen.

9. Diffraction (Qualitative)

Diffraction is the bending of light around the edges of an obstacle or aperture and its spreading into the region of geometrical shadow. It is noticeable only when the size of the aperture or obstacle is comparable to the wavelength. Since the wavelength of light (about 4 × 10⁻⁷ m to 7 × 10⁻⁷ m) is very small compared with ordinary objects, diffraction of light is not usually noticed, while diffraction of sound (wavelength of the order of 1 m) is common.

Single-slit diffraction: when a parallel beam of monochromatic light falls on a single narrow slit, the screen shows a broad, bright central maximum, with alternate dark and bright bands on both sides. The secondary maxima become rapidly weaker away from the centre. Each part of the slit acts as a source of secondary wavelets, and the pattern results from the superposition of these wavelets.

Qualitative features of the single-slit pattern:
1. The central maximum is the brightest and about twice as wide as the other maxima.
2. A narrower slit gives a wider central maximum (more spreading).
3. A longer wavelength gives a wider central maximum; red light spreads more than blue light.
4. With white light, the central maximum is white and the side bands are coloured.

Interference and diffraction compared:

10. Polarisation

Light is a transverse electromagnetic wave: the electric field oscillates perpendicular to the direction of propagation. In ordinary (unpolarised) light from a bulb or the Sun, the electric field vibrates in all directions perpendicular to the direction of travel, changing randomly. Polarisation is the effect that shows light is transverse; longitudinal waves such as sound cannot be polarised.

(i) Polaroid: a thin sheet of long-chain molecules aligned in one direction. It transmits only the component of the electric field along its pass axis. Light coming out of a polaroid has its electric field vibrating in one direction only and is called plane polarised (or linearly polarised) light.
(ii) When unpolarised light of intensity I₀ passes through a polaroid, the transmitted intensity is \( \frac{I_0}{2} \), whatever the orientation of the polaroid.
(iii) Malus' law: when plane polarised light of intensity I₁ falls on a second polaroid (the analyser) whose pass axis makes an angle θ with the plane of polarisation, the transmitted intensity is
I = I₁ cos² θ
The intensity is maximum for θ = 0° and zero for θ = 90° (crossed polaroids). On rotating the analyser through 360°, the intensity goes to zero twice.
(iv) Uses of polaroids: sunglasses that cut glare, camera filters, window panes whose transmission can be adjusted, and liquid crystal displays.

11. Quick Formula Recap