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Announcements 3/21/11 Prayer Two labs this week (telescope,
interferometer) Two mathematical facts you should know:
(They follow quickly from eix = cosx + i sinx)
cos2
ix ixe ex
sin
2
ix ixe ex
i
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Reading Quiz
According to the book, a double-slit diffraction pattern (viewed on a screen far from the slits) looks like:
a. A series of equal amplitude peaks, equally spaced
b. A series of equal amplitude peaks, alternating between two spacing distances
c. A series of alternating amplitude peaks, equally spaced
d. A series of alternating amplitude peaks, alternating between two spacing distances
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Interference... Interference...
A single source
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Interference... Interference...
Two sources
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Double slit experiment
Exactly the same as the two speaker demo
Goal: what’s the shape of that curve? How can we predict where the maxima & minima will be?
aka “Young’s Double Slit”
screen here
max
min
max
min
inte
nsity
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Experimental challenge
How do you get two points sources of light that are oscillating in phase with each other?
a. How did we do it with sound?b. Options for light? (I can only think of two)c. What he did:
Image credit: Wikipedia
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How to solve the problem
Complex numbers!! The light from each slit travels a different
distancea. This creates a phase shiftb. Incorporate the phase shift into ei
First: what’s the phase shift for two waves oscillating in phase with a known PL? = ( PL / ) 360
= 2PL /
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What’s PL? Clicker Vote: What
should we measure the path length relative to?
a. The top slitb. The bottom slitc. Halfway between the
two slits
= 2PL /
Approximation #1: d is small enough that the two rays are parallel. Requires d << L.
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The Answer
Etot = Etop slit + Ebottom slit
= …
I ~ |E|2
I = …
Plot of I(y) for I0=1, =500 nm, L=1 m, d=1
m
How did I turn into y?
Approximation #2: is small enough that y/L. Requires y << L.
= 2PL /
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Plots
20
1
2cos sin
2
tan /
dI I
y L
2
0 cosdy
I IL
(approx. 1 only) (approx. 1 and 2)
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screen here
max
min
max
min
inte
nsity
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screen here
max
min
max
min
max
min
max
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How to predict max/min
Max: cosx = 1 …
Min: cosx = 0 …
20
2cos sin
2
dI I
20 cos
dyI I
L
brightsind m
1dark 2sin ( )d m
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What you need to know
How to solve the problem this way—works for any number/pattern of slits
a. You analyze three slits for HW 32-4b. Could be problem like this on exam
This only works for very narrow slits (width << separation between slits)
The two formulas above (on notecard, unless you can quickly derive them)
Conditions for max/min (on notecard, unless you can quickly derive them)
20
2cos sin
2
dI I
20 cos
dyI I
L
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Wrapping Up
Demo: Double-slit experiment
20
2cos sin
2
dI I
20 cos
dyI I
L
max : sind
m
1min : sin
2
dm
max :dy
mL
1min :
2
dym
L
The fast way to get these