Two-Source Constructive and Destructive Interference Conditions

Crests Troughs

S1S1 P S2S2 Crests Troughs

S1S1 P l1l1 l2l2 S2S2 Crests Troughs

S1S1 P l1l1 l2l2 S2S2 l = l 2 - l 1 Path length difference: t = t 2 - t 1 Travel time difference: t = ( l 2 - l 1 )/v Crests Troughs

S1S1 P l1l1 l2l2 S2S2 Crests Troughs

S1S1 P l1l1 S2S2 l2l2 Crests Troughs

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2

S1S1 P S2S2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A)How long ago, before this snapshot was taken, did a 1, b 1, c 1, d 1, e 1, f 1, g 1, h 1 leave source S 1 ? How long ago did a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 leave source S 2 ? Express all your results, here and in the following in terms of the period of oscillation, T ! Tabulate the results! Reminder: It takes 1 period for a crest or trough to travel 1 wavelength (B) Tabulate all pairs of crests and/or troughs which left their resp. sources simultaneously. (C) Do the results in (A) depend on l 1 or l 2 ? Q1

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 and l 1 are equal: l 2 = l 1 How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q2

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 exceeds l 1 by one wavelength, λ : l 2 = l 1 + λ How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q3

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 exceeds l 1 by two wavelengths, 2 λ : l 2 = l λ How long after a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q4

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 is shorter than l 1 by two wavelengths, 2 λ : l 2 = l λ How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q5

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 exceeds l 1 by one half-wavelengths, λ/2 : l 2 = l 1 + λ/2 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q6

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 is shorter than l 1 by three half-wavelengths, 3 λ/2 : l 2 = l λ/2 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q7

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 is shorter then l 1 by one quarter-wavelength, λ/4 : l 2 = l 1 - λ/4 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q8

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 (A) How long after a 1,will b 1, c 1, d 1, e 1, f 1, g 1, h 1 arrive at the detector, P? Tabulate! (B) Assume P is positioned so that l 2 is exceeds l 1 by two third-wavelengths, 2 λ/3 : l 2 = l λ/3 How long after (+) or before (-) a 1 will a 2, b 2, c 2, d 2, e 2, f 2, g 2, h 2 arrive at the detector, P? Tabulate! (C) Using (A) and (B), tabulate all pairs of crests and/or troughs, from either source, which arrive simultaneously at P. (D) Is there constructive or destructive interference at P? Or neither? Explain! Q9

S1S1 P l1l1 S2S2 l2l2 Crests Troughs a1a1 b1b1 c1c1 d1d1 e1e1 f1f1 g1g1 h1h1 l = l 2 - l 1 Path length difference: t = t 2 - t 1 Travel time difference: t = ( l 2 - l 1 )/v a2a2 b2b2 c2c2 d2d2 e2e2 f2f2 g2g2 h2h2 Q10 Summarize your results for constructive and destructive interference at P in terms of two simple mathematical conditions for the and, equivalently, for the

l = l 2 - l 1 = m λ Constructive Interference =Intensity Maximum: Path length difference: t = t 2 - t 1 = m T Travel time difference: l = l 2 - l 1 = (m+1/2) λ Destructive Interference =Intensity Minimum: Path length difference: t = t 2 - t 1 = (m+1/2) T Travel time difference: m= 0, +1, -1, +2, -2, …(m+1/2) = +1/2, -1/2, +3/2, -3/2, … where the period T is: T = λ / v

Interference Pathlength Geometry

S1S1 P l1l1 S2S2 l2l2 d d = source-to-source spacing, l 1 = distance from S 1 to P, l 2 = distance from S 2 to P. Suppose S 1 and S 2 are two small loudspeakers, placed 6.8m apart and you can move P to any location. What is the largest possible absolute value of the path length difference, Δ l =l 2 – l 1. Explain your reasoning! Q11.1

S1S1 P l1l1 S2S2 l2l2 d Suppose the two small loudspeakers, S 1 and S 2, spaced 6.8m apart, oscillate in phase, sending out sound waves of wavelength λ=2.2m. Constructive interference occurs at any location of P where Δ l = m λ. Here m can be any integer: 0, +1, -1, +2, -2, … ; and |m| is called the order of the interference maximum. What is the largest possible order of interference, |m|, that can be observed, for any location of P ? Q11.2

S1S1 P l1l1 S2S2 l2l2 d L O y Lengths and coordinates needed to describe the positioning of sources, S 1 and S 2, and detector, P: d = source-to-source spacing, L = distance from observation screen to line of sources. y = y-coordinate of P, with y-axis along the observation screen and origin O on midline between, S 1 and S 2 Q11.3

S1S1 P l1l1 S2S2 l2l2 Q11.3 (contd.) d d/2 L O y (A) Derive exact equations for l 1 and l 2, each expressed in terms of d, L, and the y-coordinate of P. Hint: Pythagoras! (B) From this, obtain an exact equation for the pathlength difference, Δ l, in terms of d, L and y (C) At home: Solve the equation from (B) for y, to express y in terms of of d, L, and Δ l. Very difficult!

S1S1 P l1l1 S2S2 l2l2 d L O y The result in Q11.3 (B) is greatly simplified if d << L, by the so-called Fraunhofer approximation: Δ l ≅ d sin Θ where tan Θ = y/L. Test this approximation against Q11.3 (B), for fixed d=5cm, fixed Θ=65deg, increasing values of L and y: Tabulate! Hint: Keep enough signif. digits! You’re subtracting 2 large numbers with a very small difference. Q12.1 Θ

S1S1 P l1l1 S2S2 l2l2 d L O y S 1 and S 2, the two loudspeakers, spaced 6.8m apart, oscillate in phase, sending out sound waves of wavelength λ=2.2m. The detector P is moved along the y-axis from y=-∞ to y=+∞, at L = 150m. (A)Find the angles Θ and y-locations of all intensity maxima on the y-axis. How many are there? (B) Find the angles Θ and y-locations of all intensity minima on the y-axis. How many are there? Q12.2 Θ

Multi-Slit Constructive Interference Pathlength Geometry Intensity Plots

Notation: Δ l = l k+1 – l k ≈same for k=1, 2, …,N-1. Maximally constructive interference occurs when Δ l = m λ with m integer Again, by geometry: Δ l ≅ d sin(Θ) assuming L>> Nd; and tan(Θ) = y/L Multi-Slit (N-Slit) Interference and Diffraction Grating (N>>1) P O y 2 Δ l ΔlΔl ΔlΔl Δ l 3 Δ l 2 Δ l 3 Δ l ΔlΔl

Principal Maxima: sin(Θ) = m λ/d with m integer Secondary Maxima

A diffraction grating placed parallel to an observation screen, 40cm from the screen, Is illuminated at normal incidence by coherent, monochromatic light (a laser beam). Assume Fraunhofer conditions (L>> Nd) are satisfied. (a)If the 1 st order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable? (b) If the 2nd order principal maximum is observed on the screen 30cm above the central maximum, how many principal maxima altogether, incl. central maximum, are observable? Find the angles, Θ, and y-coordinates of all principal maxima on the screen: Tabulate! (c) How would your answers change if the device had been a double-slit (N=2) or a quintuple-slit (N=5) instead of a diffraction grating? Q13