Interference

Interference

1. What Interference Is

Interference = superposition of two (or more) coherent light waves giving a non-uniform intensity distribution — bright (constructive) and dark (destructive) fringes where $I = I_1+I_2+2√(I_1 I_2) cos φ$.

If waves are incoherent, $cos φ$ averages to 0 ⇒ $I = I_1+I_2$ (no fringes).

2. Conditions for Sustained Interference

• Coherent sources: constant phase difference $φ = δ_1-δ_2$ (not random) — obtained by dividing one wavefront/amplitude.
• Monochromatic (or quasi-monochromatic) — $\lambda$ same, otherwise pattern washes.
• Same polarization state (preferably).
• ** comparable $I_1,I_2$ ** — contrast $V =2√(I_1 I_2)/(I_1+I_2)$ maximal when $I_1≈I_2$.
• ** Superposition principle:** fields add vectorially.

3. Path Difference and Phase Difference

For two waves reaching $P$ via paths $r_1,r_2$:

For in-phase sources, $φ = (2π/λ) Δ$.

• Constructive (bright):

amplitude $A = A_1+A_2$, intensity $I_{max}=(√I_1+√I_2)^2$.

• Destructive (dark):

amplitude $|A_1-A_2|$, $I_{min}=(√I_1-√I_2)^2$; if $I_1=I_2$, dark is zero.

4. Young's Double-Slit Prototype

Slits $S_1,S_2$ distance $d$, screen distance $D$, position $y$ on screen: $Δ ≈ d sinθ ≈ d y/D$.

uniform spacing for monochromatic light.

5. Energy Conservation Note

Interference redistributes energy — $I_{avg}$ over region = $I_1+I_2$, no net creation.

6. Worked Examples

(a) Two coherent beams I1=4, I2=1: Imax=9, Imin=1 ⇒ contrast V=(9-1)/(9+1)=0.8 ✓

(b) Δ =3λ/2 ⇒ phase 3π ⇒ destructive (dark) ✓

(c) Young's double slit λ=600 nm, d=0.2 mm, D=1.5 m: β= λ D/d =4.5 mm ✓

(d) Need to show path difference vs phase: Δ=λ/4 ⇒ φ=π/2 ⇒ I=I1+I2 (intermediate) ✓

7. Why It Matters for Next Topics

Division of wavefront (Young) vs amplitude (Newton's rings, Michelson) are the two ways to get coherent beams from one source — next topics.

8. Exam Points

• Define interference and coherent sources — 3 marks, very frequent.
• Conditions for sustained interference — 5 marks, very frequent.
• Derive $Δ = nλ$ (bright) and $(2n+1)λ/2$ (dark) — 5 marks, almost certain.
• Young's fringe width $β=λD/d$ with $y_{bright}, y_{dark}$ — 5-7 marks, very frequent.