fi 3221 electromagnetic interactions in...

16/02/2016

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Alexander A. Iskandar

Physics of Magnetism and Photonics

FI 3221 ELECTROMAGNETIC

INTERACTIONS IN MATTER

• Rayleigh

Scattering

• Scattering

quantities

• Mie

Scattering

SCATTERING OF LIGHT

Alexander A. Iskandar Electromagnetic Interactions in Matter 2

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2

Main

C. Bohren and D. Huffman, Absorption and Scattering of Light

by Small Particles, Wiley-VCH


REFERENCES

• Imagine a particle under illumination of

electromagnetic wave

WHAT IS SCATTERING ?


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• Electric charges inside the particle will experience

oscillatory motion, and hence generate secondary

EM wave



• This secondary wave is commonly known as

scattered wave



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• Some part of the electromagnetic energy can be

transformed into other forms (ex : thermal energy)

• This process is called absorption

ABSORPTION


• Rayleigh scattering applies when light is scattered

by small particles

• By saying ‘small’ we mean particles with size much

smaller than wavelength of the incident light

RAYLEIGH SCATTERING

𝑑 <1

10𝜆


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Small particles (a << l): quasi-static approximation

The fields inside (E1) and outside (E2) the sphere may

be found from the scalar potentials and

RAYLEIGH THEORY OF OPTICAL

SCATTERING

a

z

r

E0 (constant)

e1

e2 The initially uniform field

will be distorted by the

introduction of the sphere

q

1 1E

1 ,r q 2 ,r q

2 2E

0 cosi E r q (potential )


Laplace’s equations in source -free domains:

Boundary conditions:

A SPHERE IN A UNIFORM STATIC

FIELD: THE SOLUTION

2

1 0

2

2 0

r a

r a

1 2 r a

1 1 2 2r re e r a

1 2 q q r a

2 0lim cosr

E r q

21 0

1 2

3cos

2E r

e q

e e

2 0 cosE r q

3 1 20 2

1 2

cos

2a E

r

e e q

e e

Solution:


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Electric potential inside and outside the sphere:

A sphere in an static field is equivalent to an ideal

dipole

A SPHERE IN A UNIFORM STATIC

FIELD: THE SOLUTION

21 0

1 2

3cos

2E r

e q

e e

3 1 22 0 0 2

1 2

coscos

2E r a E

r

e e q q

e e

2 0e p E

3 1 2 1 2

1 2 1 2

4 32 2

a Ve e e e

e e e e

Dipole moment:

Dipole polarizability:


Total scattered power (linearly polarized light):

Scattering cross-section ss and scattering efficiency Qs :

Short wavelength light is preferentially scattered

RAYLEIGH SCATTERING (a << l)

2

2 4 6 22 1 20

00 1 2

1 42 sin *

2 3 2s sP r E H d k a E

e e e q q

e e

2

2 4 6 2 41 220

0 1 2

81

2 3 2s sP E k a

e ees l

e e

2

2 4 41 2

1 2

8

3 2s sQ a k a

e es

e e

(dimensionless number)


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• Scattering intensity of small particles is inversely

proportional with the fourth power of wavelength of

the incident light

𝐼𝑠~1

𝜆4

• This implies that for short wavelength the scattering

will be strong

• A similar derivation for a small cylinder yields

𝐼𝑠~1

𝜆3

RAYLEIGH SCATTERING

INTENSITY

2

1)(

3

82

24

n

nkaQsca

1

1)(

4 2

23

2

n

nkaQsca


BLUE SKY : RAYLEIGH SCATTERING

• It explains why the sky is blue


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How to analyse scattering by larger object?

Use integrated Poynting vector.


MIE SCATTERING

The fields in the region outside of the scatterer can

be written as


SCATTERING PARAMETERS

si EEE

2

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The Poynting vector in the region outside of the

scatterer can be written as



extscainc

ississii

sisi

SSS

HEHEHEHE

HHEE

HES

**

21*

21*

21

*

21

*

21

ReReRe

Re

Re

The power absorb by the scatterer can be evaluated as



extscainc

A

abs WWWdW aS

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The power absorb by the scatterer can be evaluated as

where

with



extscainc

A

abs WWWdW aS

A

extext

A

scasca

A

incinc dWdWdW aSaSaS

,,

**

21

*

21

*

21

Re

Re

Re

issiext

sssca

iiinc

HEHES

HES

HES

Obviously, for a closed surface are,



extscainc

A

abs WWWdW aS

0 A

incinc dW aS

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Obviously, for a closed surface are,

Hence,



extscainc

A

abs WWWdW aS

0 A

incinc dW aS

extscaabs WWW

absscaext WWW

A cross section is defined as the normalized power

with respect to incoming wave, hence

From this cross sections, we can define efficiencies

as,



i

extext

i

absabs

i

scasca

I

W

I

W

I

W sss ,,

exteffextabseffabsscateffscat QAQAQA sss ,,

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To find the exact wave solution, we write that all

fields as a linear combination of some orthogonal

base functions

The orthogonal base function, , is chosen

according to the (symmetry) of the problem. It could

be a combination of

spherical harmonics and spherical Bessel function for spherical

symmetric scatterer, or it could be

Bessel function for cylindrical symmetric scatterer.


MULTIPOLE EXPANSION METHOD

n

nnnnnn fcfbfa 3

)3(

2

)2(

1

)1( ˆ)(ˆ)(ˆ)()( erererrE

)()(ri

nf

Since we are working with a cylindrical shaped object,

it would be more convenient to use the curl operator

in the form of cylindrical coordinate

𝛻 × 𝐄 =1

𝜌

𝜌 𝜌𝜙 𝑧 𝜕

𝜕𝜌

𝜕

𝜕𝜙

𝜕

𝜕𝑧𝐸𝜌 𝜌𝐸𝜙 𝐸𝑧

= 𝑖𝜔𝜇( 𝜌 𝐻𝜌 + 𝜙 𝐻𝜙 + 𝑧 𝐻𝑧)

𝛻 × 𝐇 =1

𝜌

𝜌 𝜌𝜙 𝑧 𝜕

𝜕𝜌

𝜕

𝜕𝜙

𝜕

𝜕𝑧𝐻𝜌 𝜌𝐻𝜙 𝐻𝑧

= −𝑖𝜔𝜀( 𝜌 𝐸𝜌 + 𝜙 𝐸𝜙 + 𝑧 𝐸𝑧)


SCATTERING BY INFINITE CYLINDER

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Consider a problem as

depicted in the figure

A cylinder is illuminated by an

electromagnetic plane wave

The plane wave is propagating

on x-y plane toward negative

y-axis (no z-dependence)

The magnetic field is

polarized parallel with the z-

axis





From the Maxwell equations, we obtain

𝐻𝑧 =1

𝑖𝜌𝜔𝜇

𝜕(𝜌𝐸𝜙)

𝜕𝜌−𝜕𝐸𝜌

𝜕𝜙

𝐸𝜌 = −𝐸𝜙 =1

𝑖𝜔𝜀

𝜕𝐻𝑧

𝜕𝜌

Solving for the magnetic field,

𝜌2𝜕2𝐻𝑧

𝜕𝜌2 + 𝜌𝜕𝐻𝑧

𝜕𝜌+ 𝑘2𝜌2𝐻𝑧 = −

𝜕2𝐻𝑧

𝜕𝜙2

with

𝑘2 = 𝜇𝜀𝜔2

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Solving the previous partial differential equation by

separation of variable method, the radial part yield

𝜌2𝜕2𝑅𝑧(𝑘𝜌)

𝜕𝜌2+ 𝜌

𝜕𝑅𝑧(𝑘𝜌)

𝜕𝜌+ 𝑘2𝜌2 − 𝜈2 𝑅𝑧 𝑘𝜌 = 0

Which is the Bessel equation, with solution can be

a combination of :

1. Bessel function 𝐽𝜈(𝑘𝜌)

2. Neumann function 𝑌𝜈(𝑘𝜌)

3. Hankel function of the first kind 𝐻𝜈(𝑘𝜌)

4. Hankel function of the second kind 𝐻𝜈2(𝑘𝜌)



Later, it will be shown that we need only Bessel

function 𝐽𝜈 𝑘𝜌 and Hankel function of the first kind 𝐻𝜈 𝑘𝜌

The solution of the fields will take the following form

𝜒 𝜌, 𝜙 = 𝑀𝜈𝐽𝜈 𝑘𝜌 + 𝑁𝜈𝐻𝜈(𝑘𝜌) 𝑒𝑖𝜈𝜙

∞

𝜈=−∞

𝜒 𝜌, 𝜙 can be the electric field or the magnetic field

𝑀𝜈 and 𝑁𝜈 are the unknown coefficients yet to be

determined



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Since our problem is invariant in z direction, we may

consider only the cross-sectional situation of the

problem as depicted below

In region 1, there is an incoming plane wave

approaching a cylinder



∑ −1 𝜈𝐽𝜈 𝑘1𝜌 𝑒𝑖𝜈𝜙

Region 1

Region 2

The cylinder will generate secondary electromagnetic

field




∑𝑁𝜈𝐻𝜈 𝑘1𝜌 𝑒𝑖𝜈𝜙

Region 1

Region 2

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In region 2, a standing wave ensues




∑𝑁𝜈𝐻𝜈 𝑘1𝜌 𝑒𝑖𝜈𝜙

∑𝑀𝜈𝐽𝜈 𝑘2𝜌 𝑒𝑖𝜈𝜙

Region 1

Region 2

Applying the boundary conditions, we obtain the

following value for the coefficients 𝑀𝜈 and 𝑁𝜈

𝑁𝜈 = −1 𝜈−1𝑛𝐽𝜈

′ 𝑘1𝑎 𝐽𝜈 𝑘2𝑎 − 𝐽𝜈 𝑘1𝑎 𝐽𝜈′ 𝑘2𝑎

𝑛𝐻𝜈′ 𝑘1𝑎 𝐽𝜈 𝑘2𝑎 − 𝐻𝜈 𝑘1𝑎 𝐽𝜈

′ 𝑘2𝑎

𝑀𝜈 = −1 𝜈𝑛𝐽𝜈 𝑘1𝑎 𝐻𝜈

′ 𝑘1𝑎 − 𝑛𝐽𝜈′ 𝑘1𝑎 𝐻𝜈 𝑘1𝑎

𝑛𝐽𝜈 𝑘2𝑎 𝐻𝜈′ 𝑘1𝑎 − 𝐽𝜈

′ 𝑘2𝑎 𝐻𝜈 𝑘1𝑎

where 𝑛 =𝜀2

𝜀1



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Intensity of the scattered wave can be derived from

the definition of Poynting vector, more specifically,

the time-averaged Poynting vector

𝐒𝐬𝐜𝐚 =1

2𝑅𝑒 𝐄𝐬𝐜𝐚 × 𝐇

𝐬𝐜𝐚 (∗)

thus, the radial component of the Poynting vector is

𝐒𝛒𝐬𝐜𝐚 =

1

2𝑅𝑒 𝐄𝛟𝟏

𝐬𝐜𝐚 × 𝐇𝐳𝟏

𝐬𝐜𝐚 (∗)

Inserting the field components yield

𝐒𝛒𝐬𝐜𝐚 =

1

𝜔𝜀1𝜋𝑎∑ 𝑁𝜈

2𝝆


INTENSITY AND POWER OF THE

SCATTERED WAVE

To determine the power of the scattered wave, we

apply an imaginary surface around the cylinder and

calculate the surface integral below

𝑃𝑠𝑐𝑎 = 𝐒𝛒𝐬𝐜𝐚 . 𝑑𝐀

or,

𝑃𝑠𝑐𝑎 = 1

𝜔𝜀1𝜋𝑎∑ 𝑁𝜈

2𝝆 . 𝑅𝐿 𝑑𝜙𝝆

or,

𝑃𝑠𝑐𝑎 =2𝐿

𝜔𝜀1∑ 𝑁𝜈

2


INTENSITY AND POWER OF THE

SCATTERED WAVE

𝑎

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Scattering Cross Section 𝐶𝑠𝑐𝑎 is the ratio of 𝑃𝑠𝑐𝑎

with respect to the intensity of the incoming plane

wave

𝐶𝑠𝑐𝑎 =𝑃𝑠𝑐𝑎𝐼𝑖𝑛𝑐

Intensity of an incoming plane wave with the

amplitude of its magnetic field is unity can be written

as

𝐼𝑖𝑛𝑐 =𝜇0𝑐

2

Thus

𝐶𝑠𝑐𝑎 =4𝐿

𝑘1∑ 𝑁𝜈

2


SCATTERING CROSS SECTION

• Scattering Efficiency 𝑄𝑠𝑐𝑎 is defined as

𝑄𝑠𝑐𝑎 =𝐶𝑠𝑐𝑎𝐺

G is the particle cross-sectional area as viewed by

the incoming wave, thus for our case

𝐺 = 2𝑎𝐿

• This yields

𝑄𝑠𝑐𝑎 =2

𝑘1𝑎∑ 𝑁𝜈

2


SCATTERING EFFICIENCIES

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• Silver nanocylinder is illuminated by EM plane wave


EXAMPLES

• Silver nanosphere is illuminated by EM plane wave


EXAMPLES

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APPLICATIONS

𝜀2 = 4

𝜀1 = 𝑠𝑖𝑙𝑣𝑒𝑟

75 nm

100

nm


APPLICATIONS

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APPLICATIONS


APPLICATIONS

fi 3221 electromagnetic interactions in...

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