Experiment 7
Young’s Double Slit Experiment
And Its Variation
Introduction
This week’s lab explores the wave nature of light as established by the Young’s
Double Slit experiment and a variation on this theme known as Lloyd’s Mirror
Experiment. The principle of these is illustrated in Fig. 13C, Jenkins and
White, reproduced below.
A
S1
B
S
S2
C
Figure 1: Experimental arrangement for Young’s double-slit experiment
P
y
S1
S
d
θ`
θ
P0
X
S2
∑
Figure 2: Path difference for Young’s double-slit experiment
A narrow source slit, S, is illuminated with monochromatic light of wavelength
λ. Cylindrical waves thus generated, in turn, illuminate two narrow slits S1
and S2 , located symmetrically about S, with the separation between S1 and S2
being d. See Fig. 13F (J&W) also reproduced above. The cylindrical waves
emerging from S1 and S2 are coherent, having been generated from the same
original wave-front at S (comment on the concept of coherence in your report).
These waves superimpose on the screen, labeled Σ, at a distance X from the
plane defined by S1 and S2 . Thus,
(S1 P )2 = X 2 + (y − d/2)2 .
(1)
(S2 P )2 = X 2 + (y + d/2)2 .
(2)
1
(S2 P )2 − (S1 P )2 = (y + d/2)2 − (y − d/2)2 .
(3)
[(S2 P ) + (S1 P )]∗ [(S2 P ) − (S1 P )] = 2yd.
(4)
(S2 P ) − (S1 P ) = 2yd/[(S2 P ) + (S1 P )] ≡ ∆,
(5)
∆ being defined as the path difference between the waves arriving at P. For
X >> d,
(S1 P ) + (S2 P ) ≈ 2X.
(6)
∆ = yd/X.
(7)
This path difference results in a phase difference, δ, between the two waves,
δ = (2π/λ)∗ ∆ = (2π/λ)∗ (yd/X).
(8)
If the two waves arriving at P have amplitudes a1 and a2 , then the net
intensity at P is,
I(P ) = a21 + a22 + 2a1 a2 cos δ.
(9)
If the slits S1 and S2 are evenly illuminated then,
a = a1 = a2 ,
(10)
and I(P ) = 2a2 (1 + cos δ).
(11)
The intensity is thus periodic in δ, with,
δ = 2mπ, m = 0, ±1, ±2, . . .
(12)
2
yielding maxima on the screen Σ with intensity 4a , corresponding to bright
fringes, and,
δ = (2m + 1)π, m = 0, ±1, ±2, . . .
(13)
yielding minima with intensity 0, corresponding to dark fringes. Finally,
(2π/λ)∗ (yd/X) = 2mπ
(14)
ym = mλ∗ (X/d)
(15)
is the position of the m
fringe separation, ℓ,
th
bright fringe. The fringes are equidistant with
ℓ = ym+1 − ym = λ∗ (X/d)
2
(16)
Procedure
Place the large single slit in front of the HeNe laser source. DO NOT LOOK
DIRECTLY INTO THE LASER! Place the double slits approximately 50
cm from the single slit. Adjust the width of the single slit until the spot widens
into the central maximum of the single slit diffraction pattern. Center this wide
band of light onto the slit set of interest. Place a mirror at the end of the bench
and turn it through 45 degrees so that the diffraction pattern is visible on the
wall. This experiment will use three different slit sets.
Set
C
B
B
Pattern
A
A
C
Slit Width
.04mm
.04mm
.08mm
Slit Separation
.125mm
.250mm
.250mm
Using the first set (C, A) you will measure the fringe separation and calculate
the wavelength. Compare this with the known wavelength (6328Å). You will
take data in the following manner:
• Place a small pencil mark at each of the fringes (you should be able to get
fringes from -10 to +10)
• Turn down the source (to reduce eyestrain)
• Measure the fringe separation in groups of ten (from say -10 to 0, -9 to 1,
and so on)
• You should have about ten data points each measuring 10ℓ
• Lastly, measure the separation across all twenty fringes at once
• Measure X
Now replace the slit set with (B, A) and repeat above. Also, notice how the
intensity of the fringes varies. This is due to the single-slit diffraction pattern
that each slit in the set produces (not the source slit!). Measure the width of
this central maximum.
Place the last group of slits. Measure the width of the central maximum.
Analysis
For each set, (C, A) and (B, A), the fringe separation will be calculated in two
ways. First average your measurements of 10ℓ and take the standard deviation
to find the uncertainty in the measurements. Next use your measurement of
20ℓ and your estimation of the read error inherent in the micrometer to determine ℓ and your uncertainty in ℓ. Which is more precise? Is there any other
advantage for one over the other? Now, using standard propagation of error
techniques calculate your experimental wavelengths and compare with the theoretical value (assume d to be exact). Which method is more accurate? From
your measurements of the central maximum determine the relationship between
X, b (slit width), λ, and y (the width of the central maximum).
3
Lloyd’s Mirror
M
a
b
S1
O
P
S2
c
e
N
Figure 3: Lloyd’s mirror
Fig. 13L (J&W), also reproduced above, shows how the double slit experiment
can be performed with a source S1 , (illuminated with monochromatic light)
and its virtual image S2 produced by a plane mirror. Note S1 and S2 are thus
‘coherent’ with each other.
Procedure
Remove the double slits and holder from the bench. Observe the single slit
diffraction pattern on the screen. Keep the pattern wide bright enough to be
easily seen. Place a plane mirror stand approximately 50 cm away from the
single slit. Turn the mirror micrometer clockwise until half of the zeroth order
is occluded. Slightly rotating the mirror will cause a bright spot to be seen,
rotate the mirror until the bright spot is just at the edge of the pattern. Careful
inspection will now reveal the Lloyd’s fringes. Place the 5cm achromatic lens
five cm beyond the mirror. This will enlarge the fringes and make them easier
to see. Now record the 10-fringe separation and find ℓ as you did in Part One.
Replace the Red laser source (6328Å) with each of the Green, Orange, and Yellow lasers in turn and repeat.
Analysis
The equation for the Lloyd’s Mirror is the same as that for the Young’s Double
Slit, namely,
ℓ = ym+1 − ym = λ∗ (X/d)
However, now d is the distance between the real slit and its image in the mirror
(Important - (If the ‘mirror’ is a glass plate (not a front alumize) mirror),)
how does this change the interference pattern?) Use the known reference laser
to determine the unknown wavelengths and give the uncertainty in your final
results.
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