1. Two-Source Constructive and Destructive Interference Conditions

2. Crests
Troughs

3. S1 P
Crests
Troughs S2

4. S1 P l1
Crests
Troughs S2 l2

5. l1 l2 l = l2-l1 t = t2- t1 t = (l2- l1)/v P S1 S2
Crests
Troughs S2 l2
l = l2-l1
Path length difference:
t = t2- t1
Travel time difference:
t = (l2- l1)/v

6. S1 P l1
Crests
Troughs S2 l2

7. P l1 S1
Crests
Troughs l2 S2

8. l1 l2 P S1 S2 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2 Crests
Troughs l2 a2 b2 c2 d2 e2 S2 f2 g2 h2

9. P S1 S2 Q1 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2 Crests
Troughs
How long ago, before this snapshot was
taken, did
a1, b1, c1, d1, e1, f1, g1, h1 leave source S1 ?
How long ago did
a2, b2, c2, d2, e2, f2, g2, h2 leave source S2 ?
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 l1 or l2 ? a2 b2 c2 d2 e2 S2 f2 g2 h2

10. l1 l2 P S1 S2 Q2 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 and l1
are equal:
l2=l1
How long after a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

11. l1 l2 P S1 S2 Q3 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 exceeds l1
by one wavelength, λ:
l2=l1 + λ
How long after a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

12. l1 l2 P S1 S2 Q4 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 exceeds l1
by two wavelengths, 2λ:
l2=l1 + 2λ
How long after a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

13. l1 l2 P S1 S2 Q5 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 is shorter
than l1 by two wavelengths, 2λ:
l2=l1 - 2λ
How long after (+) or before (-) a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

14. l1 l2 P S1 S2 Q6 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 exceeds l1
by one half-wavelengths, λ/2:
l2=l1 + λ/2
How long after (+) or before (-) a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

15. l1 l2 P S1 S2 Q7 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 is shorter
than l1 by three half-wavelengths, 3λ/2:
l2=l1 - 3λ/2
How long after (+) or before (-) a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

16. l1 l2 P S1 S2 Q8 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 is shorter
then l1 by one quarter-wavelength, λ/4:
l2=l1 - λ/4
How long after (+) or before (-) a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

17. l1 l2 P S1 S2 Q9 a1 b1 c1 d1 e1 f1 g1 h1 a2 b2 c2 d2 e2 f2 g2 h2
Crests
Troughs l2
(A) How long after a1,will
b1, c1, d1, e1, f1, g1, h1
arrive at the detector, P? Tabulate!
(B) Assume P is positioned so that l2 is exceeds l1
by two third-wavelengths, 2λ/3:
l2=l1 + 2λ/3
How long after (+) or before (-) a1 will
a2, b2, c2, d2, e2, f2, g2, h2
(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! a2 b2 c2 d2 e2 S2 f2 g2 h2

18. l1 l2 l = l2-l1 t = t2- t1 t = (l2- l1)/v P S1 S2 Q10 a1 b1 c1 d1 e1f1 g1 h1
Crests
Troughs l2
Summarize your results for constructive and
destructive interference at P in terms of two
simple mathematical conditions for the
and, equivalently, for the
l = l2-l1
Path length difference:
t = t2- t1
Travel time difference:
t = (l2- l1)/v a2 b2 c2 d2 e2 S2 f2 g2 h2

19. l = l2-l1= m λ t = t2- t1 = m T l = l2-l1= (m+1/2) λ
Constructive Interference
=Intensity Maximum:
Path length difference:
t = t2- t1 = m T
Travel time difference:
l = l2-l1= (m+1/2) λ
Destructive Interference
=Intensity Minimum:
Path length difference:
t = t2- t1 = (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

20. Interference Pathlength Geometry

21. Q11.1 S1 P l1 S2 l2 d
d = source-to-source spacing, l1 = distance from S1 to P, l2 = distance from S2 to P. Suppose S1 and S2 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, Δl =l2 – l1 .
Explain your reasoning!

22. Q11.2 P l1 S1 l2 d S2
Suppose the two small loudspeakers, S1 and S2, 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 Δl = 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 ?

23. Q11.3 P l1 S1 y l2 d O L S2
Lengths and coordinates needed to describe the positioning of sources, S1 and S2, 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, S1 and S2

24. l1 l2 d/2 d L d/2 P S1 y O S2 Q11.3 (contd.)
(A) Derive exact equations for l1and l2, 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, Δl, 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 Δl. Very difficult!

25. Q12.1 P l1 S1 y l2 Θ d O L S2
The result in Q11.3 (B) is greatly simplified if d << L, by the so-called Fraunhofer approximation:
Δl ≅ 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.

26. Q12.2 S1 P l1 S2 l2 d L O y Θ
S1 and S2, 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.
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?

27. Q12.3 S1 P l1 S2 l2 d L O y Θ
S1 and S2, the two loudspeakers, spaced distance d apart, oscillate in phase, sending out sound waves of wavelength λ. Detector P is moved along the y-axis from y=-∞ to y=+∞, at fixed distance L from the source line. The 1st intensity maximum above midpoint O is found at Θ1=24.8o.
How many intensity maxima are there, altogether, on the y-axis from y=-∞ to y=+∞?
How many intensity minima are there, altogether, on the y-axis from y=-∞ to y=+∞?

28. Q12.4 S1 P l1 S2 l2 d L O y Θ
S1 and S2, the two loudspeakers, spaced distance d apart, oscillate in phase, sending out sound waves of wavelength λ. Detector P is moved along the y-axis from y=-∞ to y=+∞, at fixed distance L from the source line. The 3rd intensity minimum above midpoint O is found at Θ5/2=53.8o.
Find Θm for all intensity maxima (m=0,+1,-1,…) on the y-axis from y=-∞ to y=+∞?
Find Θm+1/2 for all intensity minima (m+1/2=+1/2,-1/2,+3/2,-3/2,…) on the y-axis from y=-∞ to y=+∞?

29. Multi-Slit Constructive Interference Pathlength Geometry Intensity Plots

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

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

32. Q13
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.
If the 1st 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
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?