Wave Optics
'When light passes through a narrow slit, it SPREADS — not because it is obstructed, but because it is a WAVE.'
1. Chapter Overview
Wave optics treats light as a WAVE (specifically, an ELECTROMAGNETIC wave). This chapter covers: HUYGENS' PRINCIPLE (every point on a wavefront is a source of secondary wavelets — the foundation of wave optics), the WAVEFRONT (the locus of points in the same phase), COHERENT SOURCES (essential for observing interference), YOUNG'S DOUBLE SLIT EXPERIMENT (YDSE — the classic demonstration of interference), DIFFRACTION (the bending of light around obstacles), and POLARISATION (restricting the vibration of light to one plane).
2. Huygens' Principle
Statement
- 'Every point on a WAVEFRONT acts as a source of SECONDARY SPHERICAL WAVELETS. The new wavefront is the ENVELOPE of these wavelets.'
Reflection and Refraction Using Huygens' Principle
- Reflection: Angle of incidence = Angle of reflection (proved using wave theory).
- Refraction: Snell's law n₁ sin θ₁ = n₂ sin θ₂ — derived from Huygens' principle by comparing speeds in the two media.
- 'Huygens' principle explains BOTH reflection and refraction — the change in wave speed at the boundary causes the bending.'
3. Interference — Young's Double Slit Experiment
Conditions for Interference
- Sources must be COHERENT (constant phase difference). Obtained by splitting a SINGLE wavefront (YDSE).
Path Difference and Fringe Pattern
- Path difference: Δx = d sin θ ≈ dy/D for small angles.
- Constructive interference (bright fringe): Δx = nλ, n = 0, 1, 2, ...
- Destructive interference (dark fringe): Δx = (2n+1)λ/2.
- Fringe width: β = λD/d. 'The distance between two consecutive bright or dark fringes.'
Intensity Distribution
- I = I₁ + I₂ + 2√(I₁I₂) cos Δφ.
- For I₁ = I₂ = I₀: I_max = 4I₀, I_min = 0.
- 'Constructive interference → waves ADD. Destructive → waves CANCEL.'
Worked Example 1
Problem: In YDSE, 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: For bright fringe: y_n = nλD/d. y₃ = 3λD/d ⇒ λ = y₃d/(3D) = (3×10⁻³×0.5×10⁻³)/(3×1) = (1.5×10⁻⁶)/3 = 5×10⁻⁷ m = 500 nm.
4. Diffraction
Single Slit Diffraction
- 'When light passes through a slit comparable to its wavelength, it SPREADS — each point in the slit acts as a source.'
Conditions
- Minima: a sin θ = nλ (n = 1, 2, 3, ...). a = slit width.
- Angular width of central maximum: 2θ = 2λ/a.
- Width of central maximum: w = 2λD/a.
Difference Between Interference and Diffraction
| Aspect | Interference (YDSE) | Diffraction (Single Slit) |
|---|---|---|
| Source | TWO coherent sources | SINGLE slit (many point sources) |
| Fringe width | β = λD/d — ALL fringes EQUAL | Central max is TWICE as wide |
| Intensity | All maxima have SAME intensity | Central max is BRIGHTEST — decreases away |
| Minima condition | d sin θ = (2n+1)λ/2 | a sin θ = nλ |
5. Polarisation
What is Polarisation?
- Unpolarised light: Electric field vectors vibrate in ALL directions perpendicular to propagation.
- Polarised light: Electric field vibrates in ONLY ONE plane.
- 'Polarisation proves that light is a TRANSVERSE wave — longitudinal waves (sound) cannot be polarised.'
Methods of Polarising Light
- Polaroid: A sheet that transmits light vibrating in only one plane.
- Reflection: Light reflected at BREWSTER'S ANGLE is completely polarised perpendicular to the plane of incidence.
- Double refraction: Some crystals (calcite) split light into two polarised rays.
Brewster's Law
- tan θ_B = μ (where θ_B = Brewster's angle).
- 'At Brewster's angle, the REFLECTED and REFRACTED rays are PERPENDICULAR to each other.'
Law of Malus
- I = I₀ cos² θ (where θ = angle between the polariser and analyser transmission axes).
- 'When two polarisers are CROSSED (θ = 90°), NO light passes through.'
6. Resolving Power
- Rayleigh's criterion: Two point sources are JUST RESOLVED when the central maximum of one falls on the FIRST MINIMUM of the other.
- Resolving power of a microscope: RP = 1/d_min = (2NA)/λ (NA = numerical aperture).
- Resolving power of a telescope: RP = 1/θ_min = D/(1.22λ) — depends on the DIAMETER of the objective.
7. Common Mistakes
- Coherent sources requirement: Two INDEPENDENT sources (two different bulbs) do NOT produce interference — they are INCOHERENT. Both slits must be illuminated by the SAME source.
- Interference vs diffraction: Interference uses TWO sources. Diffraction uses ONE extended source (the slit). The patterns look similar but have DIFFERENT intensity distributions.
- Brewster's angle: The reflected ray is 100% polarised, but the refracted ray is ONLY PARTIALLY polarised.
- Polarisation direction: A polaroid DOES NOT affect the DIRECTION of propagation — it only selects one plane of VIBRATION.
8. CBSE Exam Focus
- Huygens' principle — reflection and refraction using wave theory
- Interference — YDSE (β = λD/d), constructive/destructive conditions
- Diffraction — single slit, minima, width of central maximum
- Difference between interference and diffraction
- Polarisation — Brewster's law (tan θ_B = μ), Malus' law (I = I₀ cos² θ)
- Resolving power of microscopes and telescopes
9. Self-Test
Q1: In YDSE, λ = 600 nm, d = 0.2 mm, D = 1.5 m. Find the fringe width. A1: β = λD/d = (600×10⁻⁹×1.5)/(0.2×10⁻³) = (9×10⁻⁷)/(2×10⁻⁴) = 4.5×10⁻³ m = 4.5 mm.
Q2: A single slit of width 0.1 mm is illuminated by λ = 500 nm. The screen is at 1 m. Find width of central maximum. A2: w = 2λD/a = 2×500×10⁻⁹×1/(0.1×10⁻³) = (10⁻⁶)/(10⁻⁴) = 10⁻² m = 1 cm.
Q3: Find Brewster's angle for glass (μ = 1.5). A3: tan θ_B = μ = 1.5 ⇒ θ_B = tan⁻¹(1.5) ≈ 56.3°.
Q4: Two polarisers are at 60° to each other. If I₀ is incident on the first, find the transmitted intensity. A4: After first polariser: I₁ = I₀/2. After second (analyser): I₂ = I₁ cos² 60° = (I₀/2)(1/2)² = (I₀/2)(1/4) = I₀/8.
Q5: In YDSE, what happens to the fringe width if the whole apparatus is immersed in water (μ = 4/3)? A5: In water, λ' = λ/μ = λ/(4/3) = 3λ/4. β' = λ'D/d = (3/4)(λD/d) = (3/4)β. Fringe width DECREASES by factor of 3/4.
10. Conclusion
Wave optics reveals the TRUE NATURE of light:
- HUYGENS: 'The wavefront model — every point is a source of secondary waves. Reflection, refraction, and diffraction all explained.'
- INTERFERENCE: 'Light + Light can give DARKNESS — when the waves are exactly out of phase.'
- DIFFRACTION: 'Light BENDS around corners when the obstacle is small enough. The narrower the slit, the MORE the spreading.'
- POLARISATION: 'Light waves vibrate in all directions — or in just one. The ultimate proof that light is TRANSVERSE.'
'Wave optics does not replace ray optics — it SUPERSEDES it, explaining phenomena that the ray model cannot.'
