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M.P. Vaughan
PY3101 Optics
Diffraction
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Overview
• Near and far-field diffraction• Diffraction through a single slit
• Diffraction limited imaging
• Multiple slit diffraction
• The diffraction condition
• Diffraction gratings• Reflection gratings
• Monochromators
• Diffraction around objects
2
Diffraction regimes: near and far-field
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
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What is diffraction?
• Diffraction is the ‘bending’ of waves around objects or through
apertures
• It is an interference effect
3
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Diffraction – water waves
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Augustin-Jean Fresnel
Augustin-Jean Fresnel (1788 –1827)
Augustin-Jean Fresnel
Extensive work on interference
and diffraction
The wave theory of light
4
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Double slit diffraction
Thomas Young (1773 – 1829)
Double-slit experiment
First conclusive evidence of the
wave nature of light
Thomas Young
Double slit model may be derived from the N-slit case putting N = 2, so not explicitly derived here.
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Fresnel-Poisson-Arago bright spot
Predicted by Poisson as a counter-example to try to disprove Fresnel.
5
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Light passing through a narrow aperture
Huygens-Fresnel Principle
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Light passing through a narrow aperture
Maximum possible path difference
.max DABBPAP ==−=∆
6
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Limiting cases: λλλλ >> D
∆max always less than λ – wavelets add constructively in all directions.
Emergent field looks like point source.
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Limiting cases: λλλλ << D
Both constructive and destructive interference outside shaded region
Wavelets add constructively in this region
7
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Fresnel and Fraunhofer diffraction
Near field (Fresnel diffraction)
Far field (Fraunhofer diffraction)
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• Fresnel diffraction• Diffraction pattern varies with increasing distance from aperture
• Fraunhofer diffraction• Diffraction pattern settles down to a constant profile
• Applies for when the radial distance from the aperture satisfies the Fraunhofer condition
(see later).
Fresnel and Fraunhofer diffraction
8
Fraunhofer diffraction: single slit
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Single slit diffraction
ELis the field
strength per unit length
EPis the total
field a the point P
9
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Single slit diffraction
Field at x
.dxEdEL
=
Contribution to field EPdue to dE
( )( )[ ] .sin dxxkrt
xr
EdE
L
P−= ω
Total field EP
( )( )[ ] .sin
2/
2/∫− −=D
D
L
Pdxxkrt
xr
EE ω
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Single slit diffraction
r(x) is given by the cosine rule
( ) ( )θπ −−+=2
222 cos2RxxRxr
x
or
( ) .sin2
1
2/1
2
2
−+= θR
x
R
xRxr
To find a closed form solution, we must approximate this expression.
10
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Taylor series expansion of r(x)
The Taylor series expansion for a function (1 + ξ)1/2 is
( ) K+−+=+82
112
2/1 ξξξ
Hence,
( )
++−= Kθθ 2
2
2
cos2
sin1R
x
R
xRxr
and
( ) .cos2
sin 22
K++−= θθR
kxkxkRxkr
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The Fraunhofer condition
The third term in the expression for kr(x) takes its maximum when x ± D/2 and θ = 0. That is
.48
cos2 2
2
2
22
2
R
D
R
kD
R
kx
λπ
θ =→
The condition that this term makes a negligible contribution to the phase is
.4 2
2
πλπ
<<R
D
11
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The Fraunhofer condition
Neglecting the factor of 4 in the denominator of the condition just found, it may be re-written as
.DR
D λ<<
This is the Fraunhofer condition for far field diffraction.
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Far field approximations
Assuming that the Fraunhofer condition is valid, the third term in the expression for kr(x) may be neglected and we have
( ) .sinθkxkRxkr −≈
The 1/r(x) factor appearing in the integral for EPis less
sensitive to changes in r(x) than the phase and we may simply put
( ).11
Rxr≈
12
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Integrating over x
Using these approximations, the expression for the total field EPbecomes
To perform this integral, we note that
[ ] ( ){ }.Imsinsin sinθωθω kxkRtiekxkRt
+−=+−
[ ] .sinsin2/
2/∫− +−=D
D
L
PdxkxkRt
R
EE θω
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The total field EP
Integrating over the x-dependent part
where
.sin2
θβkD
=
,sin
sin
2/
2/
sin2/
2/
sin
ββ
θ
θθ
Dik
edxe
D
D
ikxD
D
ikx =
=
−−∫
Hence, the total field EPis
( ).sinsin
kRtR
DEE
L
P−= ω
ββ
13
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Intensity profile for a single slit
Averaging EP over time gives
The squared modulus of this will be proportional to the intensity, i.e.
.sin
2 ββ
R
DEE
L
P=
( ) ( ) .sin
0
2
ββ
θ II =
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Intensity profile for a single slit
14
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Intensity profile for a single slit
The zeros of the peaks occur at values of
where m is an integer. Hence, the first zeros around the central peak are given by
,sin2
πθβ mkD
==
.sinD
λθ =
Note that this result is only valid for λ < D. In other cases, there are no zeros from –π to π.
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Diffraction from a circular aperture
The analysis of Fraunhofer diffraction from a circular slit follows similar lines to that of a single slit. The result for the intensity is
( ) ( ) ( ),
sin20
2
1
=
c
cJ
IIβ
βθ
where
( ),2sin θβ kDc=
D is the diameter of the aperture and J1 is the first order Bessel function.
15
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Bessel functions
( ) ( ) .2
1 sin∫−−−=
π
π
ττ τπ
dexJxni
n
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Intensity profile for a circular aperture
16
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The Airy disc
Airy pattern by Sakurambo. A computer-generated image of an Airy disk.URL: http://en.wikipedia.org/wiki/File:Airy-pattern.svg
Airy disc
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The Airy disc
Laser Interference by Petrov Victor. Diffraction of red laser beam by a circular aperture.URL: http://en.wikipedia.org/wiki/File:Laser_Interference.JPG
17
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The Rayleigh criterion
The first zero of the intensity profile for diffraction from a circular aperture occurs at
.22.1sinD
λθ ≈
This represents the minimum angular separation that two points can be so that they may be separately resolved.
Using the small angle approximation, this becomes
.22.1D
λθ ≈
This is known as the Rayleigh criterion.
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Diffraction limited imaging
Intensity profiles for two resolvable distant point sources.
Merged intensity profiles for unresolvable distant point sources.
18
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Diffraction from a square aperture
Square diffraction (public domain). A computer simulation of the intensity pattern formed by a laser of 663 nm wavelength incident on a square aperture of 20 by 20 micrometer, visible on a screen placed 1 meter from the aperture. In reality the image measures 30 by 30 cm.URL: http://en.wikipedia.org/wiki/File:Square_diffraction.jpg
Double slit diffraction
19
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Double slit diffraction
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Intensity profile for a double slit
20
Multiple slit diffraction
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Multiple slit diffraction (transmission)
Assume incident wavevector on left-hand-side is parallel to z.
21
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Multiple slit diffraction
For N slits, the total contribution of the field E
Pis
[ ] .sinsin1
0
2/
2/∑∫
−
=
+
−+−=
N
n
na
Dna
L
PdxkxkRt
R
EE θω
Again, we make use of the earlier approximations for far field diffraction.
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Multiple slit diffraction
,sin
sin
1
0
21
0
2/
2/
sin
∑∑−
=
−
=
+
−
=
N
n
ni
N
n
na
Dna
ikx
Deik
e
ββ
θα
θ
Focussing on the x-dependent part of the integral and factorising as before, we find
The new factor is a geometric progression with common factor ei2α
∑−
=
=1
0
2 .N
n
ni
NeS
α
where
.sin2
θαka
=
22
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Multiple slit diffraction
Multiplying SNby ei2α
so
,1
22 ∑=
=N
n
nii
NeeS
αα
( ) ,11 22 αα Nii
NeeS −=−
which gives
( ) .sin
sin1
ααα N
eSNi
N
−=
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Intensity profile for multiple slits
The phase factor may be dropped from this expression. Note also that since
it is useful to incorporate a normalising factor 1/N into this ratio. Hence, the intensity takes the form
,sin
sinlim
0N
N=
→ αα
α
( ) ( ) .sin
sin
sin0
22
=ββ
αα
θN
NII
23
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Intensity profile for multiple slits
The diffraction condition
24
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Diffraction condition
Note that the condition for constructive interference is
.sin λθ ma =
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Diffraction condition
We can re-write this as
.sin2
mka
πθ =
But this is just,mπα =
( ) ( ) .sin
sin
sin0
22
=ββ
αα
θN
NII
which gives the condition for the local maxima of the intensity
25
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Diffraction – wavelength dependence
.sin λθ ma =
Red (longer wavelength) light is diffracted to a greater extent than
blue (shorter wavelength).
(Yellow arrow is incident light and specular reflection)
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Diffraction condition
Incident and diffracted wavevectors:
,zk k=
( ).cosˆsinˆ' θθ zxk += k
26
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Diffraction condition: off-axis incidence
.sin,sinmi
aOBaAO θθ ==
Diffraction gratings
27
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The grating equation
( ) .sinsin λθθ maim
=+
For off-axis transmission, the diffraction condition is now
This reduces to the case of normal incidence when .0=i
θ
This result may be further generalised by taking the incident angle around to the front of the grating – i.e. making the grating into a reflection grating.
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Reflection gratings
28
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Reflection gratings
Light strikes the reflection grating at an angle θ
i. For certain angles
θm, the diffraction condition will
be met:
The path lengths of rays from the incident
wavefront via the successive rulings of the
grating and leaving at the same angle must
differ only be integral multiples of the
wavelength λ.
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Derivation of the reflection grating equation
Incident wavefront AC
Path from A to wavefront BD
.sinm
aAB θ=
Path from C to wavefront BD
.sini
aCD θ=
( ).sinsinim
aCDAB θθ −=−Path difference
29
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The reflection grating equation
The diffraction condition for a reflection grating may then be expressed mathematically as
( ).sinsinim
am θθλ −=
This is known as the reflection grating equation.
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Dispersion
The dispersion of a grating is defined as
.λθ
θd
dD
m=
Differentiating the grating equations, we have
.cosm
m
ad
dm θ
θλ
=To be solved in the
Problem Set!
30
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Dispersion
Hence,
.cos
ma
mD
θθ =
Thus, we have greater dispersion for
• higher order m• smaller slit spacing a
To be solved in the
Problem Set!
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Number of orders
Recall that
So the highest order m is the largest integral value of
( ) .sinsin λθθ maim
=±
( ).sinsinim
aθθ
λ±
.2
max λa
m <
Hence
31
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Resolving power
The resolving power of a grating is defined as
,λλ∆
=R
where, via Rayleigh’s criterion, ∆λ is the minimum resolvable wavelength between the peaks of two wavelengths with midpoint λ.
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Common example of a grating
32
Monochromators
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The Czerny-Turner monochromator
collimator
diffraction grating
camera
33
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• Entrance slit
• Collimator (concave mirror)
• Diffraction grating
• Camera (concave mirror)
• Exit slit
Czerny-Turner monochromator
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Optical spectrum analyser (OSA)
Optical spectrum analyser (OSA) based on diffraction gratings
34
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Spectroscopy – blackbody spectra
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Spectroscopy – emission and absorption lines
samplelight in
emission
light out - absorption
bright lines
dark lines
35
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Spectroscopy – chemical analysis
Spectra provides `chemical fingerprint’
Emission spectrum of iron
Diffraction around objects
36
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Diffraction around objects
diffraction at edge of object
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Diffraction around objects
similar to diffraction through a slit
37
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Diffraction around objects
‘large’ object (compared to wavelength)
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Diffraction around objects
‘small’ object (compared to wavelength)
38
Other examples of diffraction
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Diffraction around a razor blade
39
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X-ray diffraction (non-optical)
(See PY3105)
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Related phenomena:
Thin-Film Interference
(See Chapter 9, PY3101)
40