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Note 35 Double Slit Interference
This is conceptually the same as interference for sound. The light source in these cases is usually
a slit or a hole. The light passing through the slit diffracts, spreads out, and interfere with each
other.
The two light wave of the same frequency (monochromatic light), when they are emitted with the
same phase (coherence) and they overlap in space (superposition), will interfere with each other.
If they are in phase at the destination, they will constructively interfere. If they have opposite
phase at the destination, they will destructively interfere.
source 1
destination
source 2
Path Length Difference Approach
The condition are also just like sound. It is based on the path length difference. The condition for
constructive interference is if the path length difference is an integer number of wavelengths.
L1 − L2 = mλ where m = 0, ± 1, ± 2, ...
This is the condition for destructive interference.
L1 − L2 = (m + 12)λ where m = 0, ± 1, ± 2, ...
When we deal with most kinds of light, we are talking about visible light. The wavelength of visible
light is in the range between 750 nm (red) and 300 nm (violet). That means the path length
differences are small for small m values.
The path length difference is also this, geometrically.
L1
L1
x
L2
L2
θ
θ
d
d
L
path length
difference
d sinθ
L1 − L2 = d sin θ
If the angle is small, then we can use the small angle approximation.
L1 − L2 = d sin θ ≅ d tan θ = d
x
L
page 1
The double slit interference conditions are the following. For constructive interference,
d
x
= mλ where m = 0, ± 1, ± 2, ...
L
For destructive interference,
d
x
= (m + 12)λ where m = 0, ± 1, ± 2, ...
L
Intensity Approach
The intensity of a sound wave depends on the amplitude of the sound wave squared.
Analogously, the intensity of light depends on the amplitude of the light wave squared. The
amplitude of light is measured by the electric field of the electromagnetic wave. The electric fields
from the two light waves at a certain location are
E1 = Eo sin(ωt) and E 2 = Eo sin(ωt + φ)
The phase difference between the two waves is caused by the path length difference. A phase of
2π is equivalent to one wavelength.
d sin θ = 0 ⋅ λ ⇒ φ = 0
d sin θ = 1 ⋅ λ ⇒ φ = 1 ⋅ 2π
...
d sin θ = m ⋅ λ ⇒ φ = m ⋅ 2π
The total electric field is
(
)
(
)
E = Eo ⋅ ⎡⎣ sin(ωt) + sin(ωt + φ) ⎤⎦ = E 0 ⋅ ⎡⎢ sin ωt + φ2 − φ2 + sin ωt + φ2 + φ2 ⎤⎥
⎣
⎦
(
) ( )
( ) (
)
(
) ( )
E = Eo ⋅ ⎡⎢ sin ωt + φ2 cos − φ2 + sin − φ2 cos ωt + φ2 + sin ωt + φ2 cos + φ2 + sin
⎣
(
( ) cos ( ωt + ) ⎤⎥⎦
φ
2
φ
2
) ( ) ⎤⎥⎦
E = Eo ⋅ ⎡⎢ 2sin ωt + φ2 cos
⎣
φ
2
The sine part of the electric field oscillates at that location over time, but the frequency is too high
to see anyway so we will just take an average. The cosine is the result of the path length
difference. When the magnitude of the cosine is 1, we get maximum amplitude or constructive
interference. This means
⎛φ ⎞
cos ⎜⎜ ⎟⎟⎟ = 1 ⇒
⎜⎝ 2 ⎟⎠
φ
= 0, ± π, ± 2π, ... ⇒ φ = 0, ± 2π, ± 4π, ... ⇒ ΔL = 0, ± λ, ± 2λ, ...
2
When the cosine is 0, we get zero amplitude for the sum or destructive interference. This means
⎛φ ⎞
cos ⎜⎜ ⎟⎟⎟ = 0
⎜⎝ 2 ⎟⎠
⇒
φ
π
3π
5π
1
3
= ± , ± , ± , ... ⇒ φ = ±π, ± 3π, ± 5π, ... ⇒ ΔL = ± λ, ± λ, ...
2
2
2
2
2
2
The intensity is the time average of the amplitude squared. The average of a sine-squared
function is just a constant. We lump them all into the amplitude Imax.
⎛φ ⎞
I = I max cos2 ⎜⎜ ⎟⎟⎟
⎜⎝ 2 ⎟⎠
This is called the double slit diffraction or interference pattern.
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The phase can be written as the path length difference converted into an angle. The conversion
factor is that there are 2π radians per every wavelength.
φ = ΔL ⋅
2π
2π
= d sin θ ⋅
λ
λ
⎛
⎛ x π⎞
⎛π d ⎞
π⎞
I = I max cos2 ⎜⎜ d sin θ ⋅ ⎟⎟⎟ ≅ I max cos2 ⎜⎜ d ⋅ ⎟⎟⎟ = I max cos2 ⎜⎜ ⋅ x ⎟⎟⎟
⎜⎝
⎜⎝ L λ ⎟⎠
⎜⎝ L λ ⎟⎠
λ ⎟⎠
This is what the function looks like for
I = cos2 ( 10x )
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Increasing the slit separation or decreasing the wavelength does this.
I = cos2 ( 15x )
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
Decreasing the slit separation or increasing the wavelength does this.
I = cos2 ( 5x )
1
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
page 3