Physics 1
Lecture 10: Wave optics : interference and diffraction
Prof. Dr. U. Pietsch
Reflection andrefraction
n1
n2
Law of reflection
Law of refraction
Law of refraction – revised
c = l f
Since c is constant
Wavelength , ln, and velocity oflight, vn, are changing in medium but frequence, f, stays unchanged
Phase difference
Number of wave lengths within length L
Phase difference
Huygen´s Principle
Incomingwavefront
Often a plane wave
Christiaan Huygens1629 - 1695
Diffraction
The smaller the slit the smaller is bending radius of the created spherical wave
Young´s double slit experiment Bright and darkfringes
Path length difference
For bright fringes
For dark fringes
Fresnel bi-layer experimentwith x-rays
-0,2 -0,1 0,0 0,1 0,20,1
1
10
100
1000
7 keV
Inte
nsi
ty(1
0ch
annels
)
z [mm]
Measurement
Simulation
I = (E1 + E2 )², I0 = E0²
Intensity maxima at Intensity minima at
Intensity of double slit interference
Coherence : the two interfering wave must be able to interfere,
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
-0.10 -0.05 0.00 0.05 0.10
5
10
15
20
z [mm, relative to center]
Calculation of coherence properties from the
Fraunhofer X-ray Diffraction at Circular Aperture
R
Eds
R
Eds
II
IIV
=
=
398,12
398,12sin
minmax
minmax
averaged verticalsource size s:(46 +/- 6)µm
(for R=7,43m , distance ofa Be-window in the beam path (looks like a virtual
source))
7keV
-0.10 -0.05 0.00 0.05 0.10
100
1000
inte
nsity [counts
per
10
s]
Imin,2,r
Imin,1,r
Imin,2,l
Imin,1,l
Imax,2,l
Imax,2,r
Imax,1,r
z [mm, relative to center]
Imax,1,l
6 8 10 12 14 160.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1 source pinhole = 10 µm
calculation 10 µm Pinhole
source pinhole = 15 µm
calculation 15 µm Pinhole
source pinhole = 25 µm
calculation 25 µm Pinhole
vis
ibili
ty
energy [keV]
Beryllium-window
Pin-hole
R
30m
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
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
Top Related