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1
Diffraction
1
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Diffraction“Any deviation of light rays from rectilinear path which is neither reflection nor refraction is knownas diffraction.” (Sommerfeld)
Types or kinds of diffraction:
1. Fraunhofer (1787-1826)
2. Fresnel (1788-1827)
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•In addition to interference, waves also exhibit another property – diffraction.
• It is the bending of the waves as they pass by some objects or through an aperture.
• The phenomenon of diffraction can be understood using Huygen’s principle
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Diffraction of ocean water wavesOcean waves passing through slits in Tel Aviv, Israel
Diffraction occurs for all waves, whatever the phenomenon.
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Huygens Principle : when a part of the wave front is cut off by an obstacle, and the rest admitted through apertures, the wave on the other side is just the result of superposition of the Huygens wavelets emanating from each point of the aperture, ignoring the portions obscured by the opaque regions.
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Secondary wavelets from apertures
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μv > μRRefraction
Deviation for blue is larger than that for red
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Deviation for red is larger than that for blue
Diffraction
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Fraunhofer diffraction
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Single slit diffractionPrincipal maximum
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First minimum
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Second minimum
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Young’s Two Slit Experiment and Spatial Coherence
If the spatial coherence length is less than the slit separation, then the relative phase of the light transmitted through each slit will vary randomly, washing out the fine-scale fringes, and a one-slit pattern will be observed.
Fraunhofer diffraction patterns
Good spatial coherence
Poor spatial coherence
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Diffraction from one- and two-slits
Fraunhofer diffraction patterns
One slit
Two slits
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Diffraction from small and large circular apertures
Far-field intensity pattern
from a small aperture
Far-field intensity
pattern from a large aperture
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Diffraction from multiple slits
Slit DiffractionPattern Pattern
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Superposition of large number of phasorsof equal amplitude a and equal successivephase difference δ. We have to find the resultant phasor.
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GP series
2/2/2/
2/2/2/
1
1
iii
ininin
i
in
eee
eeae
e
eaA
2/1
2/sin
2/sin
nie
na
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25
δ
δ
8δ
7δ
6δ
5δ
4δ
3δ
2δ
δδ/2
δ/2
δ/2
2/12/sin2 nienrA
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2/1
2/sin
2/sin
nie
naA
2
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Problem: Obtain intensity formula by integrationf
x sin θ is the path difference at a point x where the slit element dx is placed.
Integrate: a exp(iδ)
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For single slit path differencebetween the two ends of the slit
Δ = b Sin θ
Phase difference = 2π Δ/λ = nδ β = nδ/2 = πb sin θ/λ
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Intensity for single slit
I
β
β = π b sin θ / λ Minima at
β = + m π _
Maxima at
tan β = β
m = 1, 2, 3…
2
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For Principal maxima
0
sin1
( ) (0)I I
β = π b Sin θ / λ
2
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3
2sin ( cos sin )(0) 0
dII
d
For extrema of I()
tan β = β
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tan β = βMaxima
y=tan β
y= β
β
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For subsidiary maxima
1.4303 , 2.4590 , 3.470 ,.....
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Full width of central maximum
= λ / b
Δ x0 = f λ / bDiffraction envelope size
Point source
Smearing effectof diffraction
Δ x0
f
(First minima at sin ~ = /b)
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Double slit
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Phase=(2π/) * x sin
2
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Double slit intensity pattern for d=5b
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Single slit diffraction pattern X double slit interference pattern
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b sin θ = m λ
d sin θ = n λ
Diffraction Minima at
Interference Maxima at
/m = 0
Condition for Missing orders
When the above two equations are satisfied at the same point in the pattern (same θ), dividing one equation by the other gives the condition for missing orders.
bm
nd
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Missing orders 5, 10, 15, 20….
d/b = 5
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When we use the double-source equation to find locations of
bright spots, we find that there are some places where we
expect to see bright spots, but we see no light. This is known
as a missing order, and it happens because at that location
there's a zero in the single slit pattern.
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Remember!
If the zero in the single slit pattern, and a zero in the double slit
pattern coincides, it is not called a missing order, as there is no
order to be missing!
Also, if there is a local peak in the single slit pattern, and a zero in
the double source pattern, there will still be a zero (remember, we
multiply the functions!) - this is also not a missing order.
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N slit grating
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Normal incidence
Transmissiongrating
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= Number of slits
We know that the amplitude due to each single slit:
= δ /2
)2/sin(
)2/sin(
N
aA
sin
a
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Intensity pattern
for small b
InterferenceDiffraction
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sin2(x) sin2(5*x)
Behavior of
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sin2(5*x) / sin2(x)
sin2(10*x) / sin2(x)
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When = 0, or mπ (m = 0, ±1, ±2,…)
md sin (m = 0, ±1, ±2,…)
The intensity is maximum when this condition is satisfied. These are called the Primary maxima.m gives the order of the maximum.The intensity drops away from the primary maxima.The intensity becomes zero (N-1) times between two successive primary maxima.There are (N-2) secondary maxima in between.As N increases the number of secondary maxima increases and the primary maxima becomes sharper.
NN
sin
sin
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Intensity
I
γ
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N=15
The intensity becomes zero (N-1) times between two successive primary maxima.There are (N-2) secondary maxima in between.
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For two wavelengths
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d/b = 4
5th Order6th Order
7th Order
1st Order
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Principal maxima
Minima
When = 0, or mπ (m = 0, ±1, ±2,…)
md sin (m = 0, ±1, ±2,…)
Because at minima we should have, sin 0
If maximum is at m and Δw is the width, then the intensity should be zero at (m + Δw )
Nmd wm
))(sin(
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π + π/Nπ - π/N
I
γ
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Oblique incidence
Maxima
m
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Nm
d
Nm
d
wm
wm
1)sin(
)sin(
Let us estimate the width of the mth order Principal maximum
dNmw
wm
cos
,1with )sin( Expand w
sin (A+B) = sin A cos B + cos A sin Bd sin θm = mλ
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Dispersive power of grating is defined as
Width of principal maxima
When does the principal maxima get sharper?
md sinDifferentiate:
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Chromatic resolving power of a grating
= m N
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Chromatic resolving power of a prism
-
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Reflection grating
Transmission grating
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Bragg’s law X-ray diffraction from crystals:2d Sin θ = n λ
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30 40 50 60 70 80 90 1000
2000
4000
6000
8000
650oC
550oC
450oC
250oC
as-quenched
bulk Cu (222
)(3
11)
(220
)
(200
)
(111
)
Inte
nsity
(cps
)
2 (degree)
X-ray diffraction from (95% Cu 5% Co) crystals:
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68
Visibility of the fringes (V)
max min
max min
I IV
I I
Maximum and adjacent minimum of the fringe system
s
wacV sin
Reference: Eugene Hecht, Optics, 4th Ed. Chapter 12
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s
YaII o
2cos4
o
w
w
o dYYYs
aAYI
2/
2/
2 )(cos)(
)(cos2
oo YYs
aAdYdI
s
wacV sin