Mie-theory for a golden sphere

A story of wavesPART I

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Research question?

• Mie-theory = scattering theory• Scattering theory:– How does light react to particles?

?

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THE MAXWELLS“All of electromagnetism is contained in the Maxwell

equations”Richard P. Feynman

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The Maxwells

• The general (microscopic) Maxwell equations in SI and frequency domain:

Gauss’s law

Faraday’s law

Ampère’s law

Gauss’s law of magnetics

Continuity equation

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In a material

• Splitting physics:

– Bound = ions + electrons in crystal– Cond = electrons in conduction band– Ext = external added charges

– Free = charges not bound to the crystal lattice

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In a material

• Splitting physics (2):

– Bound = ions + electrons in crystal– Cond = electrons in conduction band– Ext = external applied currents

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In a material

• Polarization:

• Magnetization:

• Conductivity: Ohm’s law

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The Maxwells in a material

• Maxwell 1:

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The Maxwells in a material

• Maxwell 2: no change

• Maxwell 3: no change

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The Maxwells in a material

• Maxwell 4:

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The Maxwells in a material

• In summary we get:

• These are the macroscopic equations• If no external fields and currents => same as

vacuum

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• The relative permittivity:

• The relative permeability:

Different notations

Electric susceptibility

Only for metals!

Magnetic susceptibility

Relative permittivity

Relative permeability

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Homogenity and isotropy

• Relation in real space:

– Isotropy:

– Homogenity:

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In a material: nonlinear

• Actually:

• and

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In a material: nonlinear

• Nonlinear terms:– Displacement field

– Magnetizing field

NONLINEAR OPTICS:Second Harmonics,Third Harmonics, …

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THE VECTOR SPHERICAL HARMONICS“The vectors M and N are obviously appropriate for the

representation of the fields E and H, for each is proportional to the curl of the other.”

Julius A. Stratton

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Mie-theory: EM field around a sphere

• Solve the maxwell equations = find E and B• Mie did this by expanding in a good basis.– For spherical systems: – Choose boundary conditions– Solve a set of equations

• Use E and B to calculate the Poynting vector S• Use S to calculate the cross section

Vector Spherical Harmonics

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Vector spherical harmonics

• We combine the Maxwell equations:

Wave equations

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Vector spherical harmonics

• The vector wave equation:

• First we solve the scalar form in spherical coordinates:

• But we know the solution:

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Vector spherical harmonics

• Now we make vectors:– LMN method:

– Y method:

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Vector spherical harmonics: LMN

• Properties:

Transverse

Longitudinal

Never radial

This does not mean:

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Vector spherical harmonics: LMN

• What do they look like?– Combine:

– With:

LC of spherical Bessels and Neumanns

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Vector spherical harmonics: LMN

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What you should remember

• In a material:

• The VSH form a basis for the solution of these equations.

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THAT’S ALL FOR NOW FOLKS“The best is yet to come”

Francis A. Sinatra

Mie-theory for a golden sphere

A story of wavesPART II

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What you should remember

• In a material:

• The VSH form a basis for the solution of these equations.

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Vector spherical harmonics: LMN

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MIE-THEORY“Beiträge zur optik trüber medien, speziell kolloidaler

metallösungen”Gustav Mie

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Ok: we solved the Maxwells

• Now what?• We adapt to a specific system:

R

Plane wave

q

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The three waves

• We need three waves in order to solve this:– The incoming wave

– The field in the particle

– The field generated by the particle

?

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Boundary conditions

• In general we have:

• In our case we demand at R:

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A plane wave in VSH

• We consider the following incoming wave:

• Or expanded in VSH:

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INTERLUDIUMLONGITUDINAL AND TRANSVERSE

“We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is

the cause of electric and magnetic phenomena.”James C. Maxwell

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Light VS sound

• Longitudinal wave: e.g. sound

• Transverse wave: e.g. light

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The divergence

• Taking the divergence yields:

Transverse fields have zero divergence

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The curl

• Taking the curl yields:

Longitudinal fields have zero curl

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Helmholtz decomposition

• Von Helmholtz said:– “Any well behaving vector field can be decomposed in the

gradient of a scalar summed with the curl of a vectorfield.”

• Thus:– “Any vectorfield can be decomposed in a transverse and

longitudinal part.”

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Helmholtz decomposition

• Von Helmholtz said:– “Any well behaving vector field can be decomposed in the

gradient of a scalar summed with the curl of a vectorfield.”

• Thus:– “Any vectorfield can be decomposed in a transverse and

longitudinal part.”

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A plane wave in VSH

• We consider the following incoming wave:

• Or expanded in VSH:

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A plane wave in VSH

• Hence we get:

• Bohren & Huffman:

LC of spherical Bessels and Neumanns

Due to orthogonality polarization

“The desired expansion of a plane wave in spherical harmonics was not achieved without difficulty. This is undoubtedly the result of the unwillingness of a plane wave to wear a guise in which it feels uncomfortable; expanding a plane wave in spherical wave functions is somewhat like trying to force a square peg into a round hole. However, the reader who has painstakingly followed the derivation, and thereby acquired virtue through suffering, may derive some comfort from the knowledge that it is relatively clear sailing from here on.”

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The other waves

• The wave in the sphere:

• The scattered wave:

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The solutions

• Using the boundary conditions:

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The electric field

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BEYOND WAVES“What is needed now is some flesh to cover the dry bones of the

formal theory.”Bohren & Huffman

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The Poynting vector

• So we can calculate the E and B field for every frequency

• But that is not always convenient.• The Poynting vector = flow of energy• The Poynting vector:

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The Poynting vector

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The cross section

• Gives idea of interaction strength:

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The cross section

• How do we calculate it?– Consider a large sphere

around the nanoparticle

• What energy flows through? A

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The cross section

• Total = absorption:

• Scattering:

• Extinction:

The cross section is defined positve

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An example: golden sphere in vacuum

• The relative cross section:

Wavelength (nm)

R = 500 nm

R = 50 nm

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What did we learn?

• The connection between D, H, charges and currents.

• Vector spherical harmonics are solutions to the wave equations and form a basis.

• Solving a set of equations will solve the scattering problem of a sphere

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THANK YOU FOR YOUR ATTENTION