Wave Interference and Diffraction
Part 2: Single Slit
Paul Avery
University of Florida
http://www.phys.ufl.edu/~avery/
avery@phys.ufl.edu
PHY 2049
Physics 2 with Calculus
PHY 2049: Chapter 36
1
Single Slit Setup
Interference
Slit
Diffracted light
Screen
Light waves
PHY 2049: Chapter 36
2
Single Slit Analysis
a
θ
Waves from slit interfere
a sin θ = mλ
Minimum
m = ±1, ±2, ±3, …
Unlike, 2 slit, this is condition for a minimum!
PHY 2049: Chapter 36
3
Example 1: a = 5λ
Min
sin θ = m ( λ / a ) = 0.2m
m
sinθmin
θmin
±1
±0.2
±11.5
±2
±0.4
±23.6
±3
±0.6
±36.9
±4
±0.8
±53.1
±5
±1.0
±90
Total of 5 mimima
PHY 2049: Chapter 36
4
Intensity vs Angle for a = 5λ
PHY 2049: Chapter 36
5
Calculating Intensity For Single Slit
ÎFollow
similar method to double slit
‹ Each
sub-element of slit acts as a source of waves
‹ Waves radiate equally in all directions
a
θ
Waves from slit interfere
PHY 2049: Chapter 36
6
Single Slit Intensity (2)
ÎAdd
amplitudes by integration over slit (0 < y < a)
‹ Include phase difference vs y: φ y = 2π y sin θ / λ
‹ (Phase
difference from path difference: 2π × # wavelengths)
E ( t ) ∝ ∫ cos(ω t + φ y ) dy
a
0
φ y = 2π y sin θ / λ
E ( t ) ∝ ∫ cos(ω t + 2π y sin θ / λ ) dy
a
0
∝
sin (ω t + 2π a / λ ) − sin ω t
sin θ
PHY 2049: Chapter 36
7
Single Slit Intensity (3)
ÎIntensity
is time average of amplitude squared
I tot = K
=
⎛ sin(ω t + φ ) − sin ω t ⎞
⎜
⎟
sin θ
⎝
⎠
2
K2
sin 2 θ
−2
K2
sin θ
2
sin 2 (ω t + φ ) +
2
K2
sin 2 θ
sin 2 ω t
sin (ω t + φ ) sin ω t
φ = 2π a sin θ / λ
We work this out on next page
PHY 2049: Chapter 36
8
Single Slit Intensity (4)
1
sin ω t =
2
1
2
sin (ω t + φ ) =
2
1
sin ω t sin (ω t + φ ) = cos φ
2
2
I tot =
K2
sin 2 θ
(1 − cos φ ) =
I tot = I max
sin 2 α
α
2
3 terms
2 K 2 sin 2 12 φ
Sum
sin 2 θ
α = π a sin θ / λ
PHY 2049: Chapter 36
9
Single Slit Intensity (5)
2
ÎSo the intensity is I tot = I max sin ( π a sin θ / λ ) / (π a sin θ / λ )
ÎMinima
2
occur when argument inside sin() is mπ
a sin θ = mλ
PHY 2049: Chapter 36
10
Height of Maxima for Single Slit
ÎMaxima
‹ In
addition to central max, intensity I0, which occurs at α = 0
ÎHeight
‹m
‹m
‹m
‹m
‹m
ÎSo
occur approximately when α = (m+1/2)π
=
=
=
=
=
of maxima ≈ I0/ (m+1/2)2π2
1
2
3
4
5
I1
I2
I3
I4
I5
=
=
=
=
=
0.045I0
0.016I0
0.0083I0
0.0050I0
0.0033I0
height of maxima shrink rapidly!!
PHY 2049: Chapter 36
11