Wave Optics

1. Introduction

Wave optics treats light as a wave. It explains phenomena such as interference, diffraction, and polarization that cannot be explained by ray optics.

2. Huygens' Principle

Every point on a wavefront acts as a source of secondary spherical wavelets. The envelope of these wavelets gives the new wavefront.

2.1 Reflection and Refraction Using Huygens' Principle

Both laws of reflection and Snell's law can be derived using Huygens' wave theory, considering different speeds in different media.

3. Interference

3.1 Conditions for Sustained Interference

1. Coherent sources (constant phase difference).
2. Same frequency and amplitude.
3. Narrow sources.
4. Small separation between sources.

3.2 Young's Double Slit Experiment

Path difference: Δx = d sin θ = d y/D. Constructive interference (bright fringe): Δx = nλ. Destructive interference (dark fringe): Δx = (2n+1)λ/2.

Fringe width: β = λD/d.

3.3 Intensity Distribution

I = I₁ + I₂ + 2√(I₁I₂) cos δ, where δ is the phase difference.

3.4 Coherent Sources

Two sources that maintain a constant phase difference. Achieved by dividing a single wavefront (Young's method) or by division of amplitude.

4. Diffraction

4.1 Single Slit Diffraction

Central maximum is brightest and widest (2λ/a). Secondary maxima are less intense.

Condition for minima: a sin θ = nλ (n = 1, 2, 3, ...) Condition for maxima: a sin θ = (2n+1)λ/2

4.2 Width of Central Maximum

Angular width = 2λ/a. Linear width = 2λD/a.

'Diffraction sets the limit of resolution for optical instruments. Two point objects are resolved when the central maximum of one falls on the first minimum of the other (Rayleigh criterion).'

5. Polarisation

5.1 Polarised Light

Light with electric field oscillations confined to one plane. Unpolarised light has oscillations in all planes perpendicular to propagation.

5.2 Methods of Producing Polarised Light

1. Reflection (Brewster's angle: tan i_B = n₂₁).
2. Refraction through polaroids (selective absorption).
3. Double refraction in crystals (calcite, quartz).

5.3 Brewster's Law

When light is incident at Brewster's angle, reflected light is completely polarised with E perpendicular to the plane of incidence.

5.4 Malus' Law

I = I₀ cos² θ, where θ is the angle between the polariser and analyser axes.

6. Worked Problems

Problem 1: In YDSE, the slits are 0.5 mm apart and the screen is 1 m away. The third bright fringe is at 3 mm from centre. Find λ. Solution: y_n = nλD/d ⇒ 3×10^{-3} = 3×λ×1/(0.5×10^{-3}). λ = 3×10^{-3}×0.5×10^{-3}/3 = 5×10^{-7} m = 500 nm.

Problem 2: In a single slit experiment, slit width is 0.1 mm and screen is 2 m away. First minimum is at 1 cm from centre. Find λ. Solution: a sin θ = λ ⇒ a(y/D) = λ ⇒ 0.1×10^{-3}×(1×10^{-2}/2) = λ = 5×10^{-7} m = 500 nm.

Problem 3: Unpolarised light passes through two polaroids at 60°. Find intensity ratio. Solution: After first polaroid: I₁ = I₀/2. After second: I₂ = I₁ cos² 60° = (I₀/2)(1/4) = I₀/8. Ratio = 1:8.

7. Common Mistakes

'Students often confuse interference and diffraction. Interference involves a few sources (usually two), while diffraction involves a continuous distribution of sources (a single aperture).'

'In Malus' law, the intensity after the first polaroid is I₀/2 (not I₀), because unpolarised light has equal components in all directions.'

8. ISC Exam Focus

9. Self-Test Questions

1. Derive the expression for fringe width in YDSE.
2. In YDSE, the slits are 1 mm apart and 1 m from the screen. If λ = 600 nm, find fringe width.
3. Explain diffraction of light at a single slit. Derive the condition for minima.
4. State and prove Brewster's law. What is the polarising angle for water (n = 1.33)?
5. Two polaroids are crossed (90°). A third polaroid is inserted at 45°. Find the transmitted intensity if incident intensity is I₀.

10. Comparison: Interference vs Diffraction

'Interference and diffraction are both wave phenomena, but they arise from different arrangements of sources. Both demonstrate the wave nature of light.'