Wave Interference and Diffraction Part 2 PHY 2049 Physics 2 with Calculus PHY 2049: Chapter 36 1 Single Slit Setup Interference Slit Diffracted light Screen Light waves PHY 2049: Chapter 36 2 Single Slit PHY 2049: Chapter 36 3 Single Slit Analysis a θ Waves from slit interfere a sin θ = mλ Minimum m = ±1, ±2, ±3, … Unlike, 2 slit, this is condition for a minimum! PHY 2049: Chapter 36 4 Example 1: a = 5λ Min sin θ = m ( λ / a ) = 0.2m m sinθmin θmin ±1 ±0.2 ±11.5 ±2 ±0.4 ±23.6 ±3 ±0.6 ±36.9 ±4 ±0.8 ±53.1 ±5 ±1.0 ±90 Total of 5 mimima PHY 2049: Chapter 36 5 Intensity vs Angle for a = 5λ PHY 2049: Chapter 36 6 Single Slit Intensity PHY 2049: Chapter 36 7 Calculating Intensity For Single Slit ÎFollow similar method to double slit Each sub-element of slit acts as a source of waves Waves radiate equally in all directions a θ Waves from slit interfere PHY 2049: Chapter 36 8 Single Slit Intensity (2) ÎAdd amplitudes by integration over slit (0 < y < a) Include phase difference vs y: φ y = 2π y sin θ / λ (Phase difference from path difference: 2π × # wavelengths) E ( t ) ∝ ∫ cos(ωt + φ y ) dy a 0 φ y = 2π ( y sin θ / λ ) E ( t ) ∝ ∫ cos(ω t + 2π y sin θ / λ ) dy a 0 ∝ sin (ω t + 2π a / λ ) − sin ω t sin θ PHY 2049: Chapter 36 9 Single Slit Intensity (3) ÎIntensity is time average of amplitude squared I tot = K = ⎛ sin(ω t + φ ) − sin ω t ⎞ ⎜ ⎟ sin θ ⎝ ⎠ 2 K2 sin 2 θ −2 K2 sin θ 2 sin 2 (ω t + φ ) + 2 K2 sin 2 θ sin 2 ω t sin (ω t + φ ) sin ω t φ = 2π a sin θ / λ We work this out on next page PHY 2049: Chapter 36 10 Single Slit Intensity (4) 1 sin ω t = 2 1 2 sin (ω t + φ ) = 2 1 sin ω t sin (ω t + φ ) = cos φ 2 2 I tot = K2 sin 2 θ (1 − cos φ ) = I tot = I max sin 2 α α 2 3 terms 2 K 2 sin 2 12 φ Sum sin 2 θ α = π a sin θ / λ PHY 2049: Chapter 36 11 Single Slit Intensity (5) 2 ÎSo the intensity is I tot = I max sin ( π a sin θ / λ ) / (π a sin θ / λ ) ÎMinima 2 occur (Itot = 0) when argument inside sin() is mπ a sin θ = mλ PHY 2049: Chapter 36 12 Height of Maxima for Single Slit ÎMaxima In addition to central max, intensity I0, which occurs at α = 0 ÎHeight m m m m m ÎSo occur approximately when α = (m+1/2)π = = = = = of maxima ≈ I0/ (m+1/2)2π2 1 2 3 4 5 I1 I2 I3 I4 I5 = = = = = 0.045I0 0.016I0 0.0083I0 0.0050I0 0.0033I0 height of maxima shrink rapidly!! PHY 2049: Chapter 36 13