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Wave Optics
Interference and Diffraction
Huygen´s Principle
Incomingwavefront
Often a plane wave
Law of refraction – revised
c = l f
Since c is constant
Wavelength , ln, and velocity oflight, vn, are changing in medium but frequency, f, stays unchanged
Phase difference
Number of wave lengths within length L
Phase difference
Diffraction
The smaller the slit the smaller is bending radius of created spherical wave
Young´s double slit experiment Bright and darkfringes
Path length difference
For bright fringes
For dark fringes
I = (E1 + E2 )², I0 = E0²
Intensity maxima at Intensity minima at
Intensity of double slit interference
Coherence : the two inerfering wave must be able to interfer,
i.e. wave fronts must have same wave length and same shape.
Temporal coherence length L =l²/2Dl
Spatial coherence length L =l/2Da a
ll==L
d
Rraum
Temporal coherence length
L =l²/(2Dl)
Spatialcoherence length
L =l/(2Da)
a
Michelson interferometer
object
Object thickness L, index nNumber of fringes induced by the object
Number of fringes without object
To measure L
Separate two subsequent Maxima
R
Interference at thin films
Constructive interference of r1 (phaseshift 0.5) and r2 (no phase shift)
n1 = n3 =1
at b
at a, c
n1/n2 interfacen2/n3 interface
Destructive interference of r1 (phase shift0.5) and r2 (no phase shift)
examples
Newton´s rings
Radius of the Nth ring is givenby
https://en.wikipedia.org/wiki/Newton%27s_rings#/media/File:Optical_flat_interference.svg
Refraction index for X-rays
l
iA
Zr
Nn A −−=−= 1
21 2
0
862
0
542
0
10..10"2
10..10)´(2
−−
−−
=
+=
k
k
k
kA
k
kk
k
kA
fA
rN
ffA
rN
l
l
< 1
Reflection and Refraction for X-rays
21
1
21
21
0
21
21
21
21
0
2
)sin(
)cos()sin(2
)sin(
)sin(
+
+
=
+
−
+
−=
E
E
E
E
b
r
• Snellius Law
• Fresnel formulas
1
2
2
1
cos
cos
N
N=
2
0E
ET b=
2
0E
ER r=
E0 Er
Eb
Using grazing angle q
Grazing incidence, varying
2
)1)(2
11()
2
11(
coscos
2
2
2
1
2
2
2
1
2211
+=
−−−
= nn
}2
2{
11
2
1
11
2
12
c
ci
−
−=
critical angle
Total external reflection
→Fresnel-Reflectivity
= 4.0...15.021 c
Fresnel equations: helpful approximations
2_2
2_1
)2
2
11(
)2
2
11(
2
2
12
1
1
2
1
11
1
11
2
11
2
11
−=
=
++
+−
−+
−−=
forr
forr
r
2_12
2
2_
2
2
)2
2
11(
2
2
2
1
1
1
11
1
1
1
2
1
11
1
2
11
1
=
+
++
−+
=
fort
fort
t
T=t² R=r²≈q-4
0,00 0,05 0,10 0,15 0,20
1
2
3
4
T(q
z)2
qz=4/lsin(a
i)
0,00 0,05 0,10 0,15 0,20
10-5
10-4
10-3
10-2
10-1
100
1/q4
qc
qz=4/lsin(a
i)
Re
fle
ctivity
11
Experimental set-up
Home equipment
X-raytube
Monochromator Detector
Sample
Knife edge
Layer thickness
2222 )2
(d
mcm
laa =−
Dt/t=Dai/ai
BN film on Silicon
Determination of density and mass
a 2=c
el
celr2
2
l
a =
ZN
A
A
elmass =
elelr
l
2
2 =
2a
si=7 1023cm-3, m =2.32gcm-3
LB30=4.6 1023cm-3, m =1.54gcm-3D/=2ai/ac
Organic film on silicon
Diffraction
Diffraction at a single slit
b = a/2 sin q
Condition for cancellation (minimum)
Generell:
a sin q = m l, m=1,2,3
Single slit interference - quantitative
Divide slit width, d, into m-1 virtual slits
and : m/(m-1)≈ 1
Single slit interference - quantitative
m-1 virtual slits
E=\frac{ E_0 e^{ i(\omega t - k r)} }{(m-1)r }[ 1+e^{ i\triang \phi } + e^{ i 2\triang \phi } + e^{ i 3\triang \phi } +\cdots + e^{ i (m-1)\triang \phi } ]
I= E E^{ * }=\frac{ E_0^{ 2 } }{ r^{ 2}} \nlimes{\frac{ \sin^{ 2 } (m \frac{ 1 }{ 2}\triang \phi)}{ \sin^{ 2 }(\frac{ 1 }{2 }\triang \phi )(m-1)^{ 2 } }} =\frac{ E_0^{ 2 } }{r^{ 2 } } \frac{ \sin^{ 2 }(\frac{ m }{m-1 }\frac{ \pi d \sin \alpha }{ \lambda }) }{ (m-1)^{ 2 }\sin^{ 2 }(\frac{ 1 }{m-1 }\frac{ \pi d \sin \alpha }{ \lambda }) }
(m-1)^{ 2 }\sin^{ 2 }(\frac{ 1 }{m-1}\frac{ \pi d \sin \alpha }{ \lambda }) \approx(\frac{ (\pi d \sin \alpha))^{ 2 } }{(m-1)^{ 2 } \lambda^{ 2 } }(m-1)^{ 2 }
Intensity of single slit diffraction
Phase difference of two interfering waves
Minima at
1.Min at sin q =l/a
= 90° for l/a =1= 11.5 for l/a = 0.2= 5.7 for l/a = 0.1
Diffraction at circular aperture
compare
Resolvability of two neighbored apertures
Requested angular separation
Small angles
Double slit experiment (again)
Interference for zero slit width
Interference of single slit
a
a
d
Diffraction gratings
Width of the interference lines
First minimum occurs, if N is number of slits
Grating spectrometer
Solid state physics
Lecture 2: X-ray diffraction
Prof. Dr. U. Pietsch
The mean aim of Max von Laue (1912)X-rays are electromagnetic wave with wave length much smaller
than wave length of visible light. X-rays are diffracted a crystal lattice
1914
Nobelpreis fürPhysik
Original experiment von Laue, Friedrich und Knipping
X-ray tube
Photographic film
CollimatorCrystal
Displayed at Deutsche Museum in Munich
20.April 1913
Photographic film
Erster Kristall
Cu2SO4 ⋅ 5 H2O
First Laue Experiment
1912: Begin of modern Crystallography
X-rays are electromagnetic waves of very short wavelength ( ~ 1 Å = 10-10 m).
Crystals are periodic structures in 3D : interatomic distances are of similar order of magnitude as
x-ray wave length
X-ray diffraction is a method to determine the geometric structure of solids !
Explaination by interference at 3D lattice
Explanation of Laue pattern
recip. lattice vector
𝑎 = 𝑏 = 𝑐
ql
ql
ql
sin2
sin²²²
2
sin4²²²
2
d
lkh
a
a
lkh
=
++=
=++
Alternative description of Laue-pattern by W.H.Bragg und W.L.Bragg
Interference at dense backed
„lattice planes“
Nl=2dsin
Bragg equation
X-ray tube and tube spectrum
E = h v = e U
lmin= h c / e Ua
lmin = 12.4/Ua Brems-strahlung
Characteristicradiation
l=2d1 sinq1
l=2d2 sinq21
1
2
2
Measuring lattice parameters
reflected X-ray
incident X-ray
refracted and diffracted X-ray
2
reflected X-ray
incident X-ray
refracted and diffracted X-ray
2
reflected X-ray
incident X-ray
refracted and diffracted X-ray
2
[111]
[1-10]
[01-1]
out-of-plane diffraction
In-plane diffraction
[-101]
78 80 82 84 86 88 90 9210
0
101
102
103
104
105
106
107
inte
nsity
2 (deg)
333ZB
=0006w
D=30nm
Da/a=-0,003
InAs
GaAs
X-ray diffraction of InAs Nanowires on GaAs[111]
ESRF
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