Chapter 36: Diffraction
Diffraction and Huygens’
Principle
Diffraction from a Single
Slit
Intensity in the Single-Slit
Pattern
Double-Slit Diffraction
Diffraction Grating
x-Ray Diffraction
Resolving Power
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Diffraction: Huygens' Principle, Single-Slit, and Intensity Patterns, Study notes of Physics
The concept of diffraction, focusing on huygens' principle, single-slit diffraction, and the resulting intensity patterns. The central maximum, dark fringes, and the role of phase differences in the formation of these patterns. Students of physics will benefit from this resource as study notes, summaries, or as a supplement to lecture materials.
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Chapter 36: Diffraction
Diffraction and Huygens’Principle
Diffraction from a SingleSlit
Intensity in the Single-SlitPattern
Double-Slit Diffraction
Diffraction Grating
x-Ray Diffraction
Resolving Power
Diffraction from Sharp Edges
The Poisson’s Bright Spot
Diffraction and Huygen’s Principle
Slit is divided intomany imaginary strips
Waves spread out from eachstrips as wavelets creatinginterference patterns beyondand around sharp edges.The spreading out of wavesthru small apertures or bysharp edges is called diffraction
Similar to the two-sourceinterference pattern, thesewaves interfere as theyspread out and create thediffraction pattern.
Consider a simpler case: a single slit
Single-Slit Diffraction
Central Maximum (
= 0, straight ahead)
All waves from each wavelets travelthe same distance to the screen (faraway) and they arrive
in phase
constructive interference.There will be a
bright fringe
in the
middle
at
Side note: Poisson’s Spot
obstacle (ball bearing)
screen
Wave spreading around from the topwill travel the
same
distance as the
wave spreading around from thebottom.At the mid-point (
= 0), these
waves will interfere
constructively
and create a
bright
spot although it
is in the
shadow region
Single-Slit Diffraction: Dark Fringes
Second Order Minimum
a
a
Divide the wavelets into 4 groupsIf waves from each adjacent groups destructively interfere
, we will have
another dark spot on the screen at
2
1
4
3
2
r
r
r
r
sin
2
a^4
sin a 4
(path diff.)
sin
2
a
or
a
a
Single-Slit Diffraction: Dark Fringes For higher order minimum with larger angular distance
we can use the same argument by subdividing the slit intomore groups (6, 8, 10, etc.).This leads to the following general formula for the darkfringes:
sin
,^
a
m
m
Note:1.
m
= 0 is
not
the first minimum!
In fact, it is the location for the central max.
Secondary maximum occurs
near
^3
etc. but not exactly.
Phase Difference from Path DifferenceFor each pair of adjacent phasors, there is a path difference
l^
and this path
difference induce a phase difference
between these adjacent phasors.
2
2
sin
2
l^
l^
y
Considering the phasor sum of all
N
phasors, the
total
phase difference
is,
^
^
2
s
2
sin
2
s^
in
in
N
N
y
N
y
a
^
^
^
So, the total phase difference
is a function of the angular location
Summing Phasors to Calculate
E
p
Central Maximum (
= 0, straight ahead):
0
P E
N
E
E
All phasors are
in phase.
Summing Phasors to Calculate
E
p
First Order Minimum (
0
P E
st minimum condition when last phasor’s tip matches up exactlywith the first phasor’s end.
Note:
sin
sin
a
a
same condition as previously derived.
Intensity in Single-Slit PatternFor
,
N
y^
dy
we can find an expression of the intensity
I
in terms of
The polygon becomesan arc of a circle.
•^
C
is the center of the arc
- angle A and B are right angles• interior angle at D is 180
o^
D
C
For the circular section ACB,
0
0
(^
has to be in radian)
radius
angle
arc
lenght
radius
E
E
radius
Intensity in Single-Slit Pattern
With
,the intensity of the pattern as a function of
is,
2
sin a
^
^
2
0
sin
sin a sin
I^
I^
a
^
^
^
Intensity 0
a
2 2
a
sin
2
3 3
a
a
2 2
a
3 3
a