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