Europhotonics

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Mie theory for describing particle scattering

C.L.A.R.T.E / Institut Fresnel, U.M.R. 6133

http://www.fresnel.fr/perso/stout/

Institut Fresnel, CNRS Aix Marseille Université,

Domaine Universitaire de Saint Jérôme,

13397 Marseille, France2018

Geometric optics approximation:

λ → 0

2

How do we know light is a wave ?

Huygens, Young, Fraunhofer, Fresnel’s answer :

Diffraction

3

Young’s double slit experiment is even more convincing

that light is a wave !

4

Quantum theory got started with light!

Ultra-violet “catastrophe”

5

Black body radiation Planck (1900)

Light became a particle ? “Lichtquanta”

Photo-electric Effect (Einstein 1905)

6

Quantum mechanics ?

Particles or waves

7

Are the `photons’ particles as well ?

answer : Apparently so, but are we sure ?

8

Is this really a proof that photons are particles ?(Semi-classical picture explains most phenomenon)

3 × 103 photons 1.2 × 104 photons

9.3 × 104 photons2.8 × 107 photons

9

Alain Aspect

In the 1980’s shown that light does indeed obey quantum mechanical

laws!(Wave particle duality, entanglement, Bell inequalities, quantum cryptography,…)

Is this the second quantum revolution?

Nevertheless the theory of light as an

electromagnetic wave is extremely successful

Explains most of the physics of the Fresnel Institute !

(“photonics”)

True ‘Quantum Optics’ is difficult !11

Wave equations - Fresnel coefficients

s-polarization

p-polarization

𝑟s = 𝑟⊥ =𝑛1cos𝜃i − 𝑛2cos𝜃t𝑛1cos𝜃i + 𝑛2cos𝜃t

𝑡s = 𝑡⊥ =2𝑛1cos𝜃i

𝑛1cos𝜃i + 𝑛2cos𝜃t

𝑟p = 𝑟∥ =𝑛2cos𝜃i − 𝑛1cos𝜃t𝑛2cos𝜃i + 𝑛1cos𝜃t

𝑡p = 𝑡∥ =2𝑛1cos𝜃i

𝑛2cos𝜃i + 𝑛1cos𝜃t

Fresnel coefficients plane waves, planar interface,

transparent media

𝑛 ω = 휀𝑟 ω 𝜇𝑟 ω

12

Light in the “real world” can be viewed as a wave phenomenon

… but complicated, and 3D !

13

Simplification : piecewise continuous media

Constitutive parameters

𝑫ω Ԧ𝐫 = ി휀 ω ∙ 𝑬ω Ԧ𝐫

𝑯ω Ԧ𝐫 = ി𝜇 ω ∙ 𝑩ω Ԧ𝐫14

Lorenz(1890)-Mie(1908)-Debye(1909) theory ?

Gustav Mie

(1868-1957)

Ludvig Lorenz (1829–91)

“Light scattering and reflection by a transparent sphere (surface)”in Oeuvres scientifiques de L. Lorenz. revues et annotées par H. Valentiner.

Tome Premier, Libraire Lehmann & Stage, Copenhague, 1898, p 403-529.

15

Scattering from a homogeneous sphere

Incident plane waves :

Particles in shaped beams :

Lorenz-Mie-Debye theory:

“Generalized” Mie theory :

16

F

Radiation forces

A photon transports both energy and momentum

E = cp

Peter Debye (1909) – radiation pressure

Radiation pressure

F

17

Optical Tweezers

High numerical aperture optics :

18

Mie theory from nano to milli-metric particles

Rayleigh scattering Radiation pressure

Rainbows, cloudsGlory – backscattering

19

Scattering and absorption by

nano-particles (plasmonics)

Mie(1908) - Contributions to the optics of turbid media, particularly of colloidal metal solutions

Stain glass ~ 17th century Lycurgus cup ~ 4th century

Gold “Ruby” glass

20

400 500 600 700 8000

1

2

3

4

5

6

Silver particle in glass R 100 nm

400 500 600 700 8000

2

4

6

8

Silver particle in glass R 50 nm

400 500 600 700 8000

5

10

15

20

Silver particle in glass R 20 nm

400 500 600 700 8000

2

4

6

8

10

12

14

Silver particle in glass R 5 nm

Silver spheres in glass

(Localized surface plasmon resonances)

s/(pR2) s/(pR2)

s/(pR2) s/(pR2)

21

sscat/(pR2)

sabs/(pR2)

sext /(pR2)

ANTARES / 2010

Dark field imaging of plasmonic particles

(imaging “below the diffraction limit”)

22

Interaction of light with subwavelength structures

Bohren, C. F., & Huffman, D. R. (2008). Absorption

and scattering of light by small particles. John Wiley

& Sons.

Lines of energy

flux Lines of energy flux

Weak interaction

𝜆D

𝜆

Resonant optical interaction

Ԧ𝐒ext + Ԧ𝐒inc = Ԧ𝐒tot − Ԧ𝐒scat23

D < 𝜆

Extinction cross sections

24

The Ket |Ψ is defined as a 6 component vector of

the (real valued) electromagnetic field

Γ0 𝑡 − 𝑡′ ≡ി휀 𝒓, 𝑡 − 𝑡′ 0

0 −ി𝜇 𝒓, 𝑡 − 𝑡′

𝑖𝜕

𝜕𝑡∞−𝑡𝑑𝑡′Γ0 𝑡 − 𝑡′ | Ψ 𝑡′ = 𝕃| Ψ 𝑡 + 𝑗𝑠 𝑡

𝑗𝑠 𝑡 ≡ΤԦ𝒋𝑠 𝒓, 𝑡 𝑖𝜖0

𝟎

In-medium Maxwell equation

| Ψ ≡𝑬 𝒓, 𝑡

𝑯 𝒓, 𝑡

𝕃 ≡ 𝑖𝑐0 𝛁 ×𝛁 × 0

Space-time evolution of `classical’ electromagnetic fields

25

Time-harmonic vs Fourier Transform

𝑗𝜔 𝑡 = ReΤԦ𝒋 𝒓 𝑖𝜖0

𝟎𝑒−𝑖𝜔𝑡

𝑬 𝒓,−𝜔∗ = 𝑬∗ 𝒓, 𝜔

𝑯 𝒓,−𝜔∗ = 𝑯∗ 𝒓, 𝜔

Ԧ𝒋 𝒓,−𝜔∗ = Ԧ𝒋∗ 𝒓,𝜔

Ψ𝜔 𝒓, 𝑡 = Re𝑬 𝒓

𝑯 𝒓𝑒−𝑖𝜔𝑡

Ψ 𝒓, 𝑡 = න

−∞

∞

𝑑𝜔𝑬 𝒓,𝜔

𝑯 𝒓,𝜔𝑒−𝑖𝜔𝑡

Time-harmonic formalism

Temporal Fourier transform

𝜔ി휀 𝒓, 𝜔 0

0 −ി𝜇 𝒓,𝜔

𝑬 𝒓,𝜔

𝑯 𝒓,𝜔= 𝑖𝑐

0 𝛁 ×𝛁 × 0

𝑬 𝒓,𝜔

𝑯 𝒓,𝜔+

ΤԦ𝒋𝜔,𝑠 𝒓, 𝜔 𝑖𝜖0

𝟎

Resonant States (Quasi-normal modes)Treatment of open resonant systems

𝜔𝛼ി휀 𝒓, 𝜔𝛼 0

0 −ി𝜇 𝒓, 𝜔𝛼− 𝑖𝑐

0 𝛁 ×𝛁 × 0

𝑬𝛼

𝑯𝛼

= 0

RS/QNM solutions use outgoing boundary conditions!

𝜔𝛼 Complex valued even if constitutive parameters are real

and frequency independent !

26

Resonant states (Quasi-Normal Modes)

associated with complex frequencies:

Scattering losses

t

𝑬𝑞 𝒓, 𝑡

Exponentially decreasing

in time

𝑬𝑞 𝒓, 𝑡 = 𝑬𝑛 𝒓 𝑒−𝑖𝜔𝑛′ 𝑡𝑒𝜔𝑛

′′𝑡

𝑬𝑞

Open cavities for light :

𝜔𝑞 = 𝜔𝑞′ + 𝑖𝜔𝑞

′′ , 𝜔𝑞′′< 0

27

Material (joule) losses

Resonant state (QNM) index

q = …,-2,-1,0,1,2,…

𝑟 ≫ 𝜆 ⇒ 𝑬𝑞 𝒓, 𝑡 ∝1

𝑘𝑟𝑒𝑖

𝜔𝑞

𝑐𝑟−𝑐𝑡

𝑘 =2𝜋

𝜆

Far-field domain

Divergence of resonant state (QNM) fields

r

Exponentially

diverging

Outgoing boundary conditions

𝑬𝑛 𝒓 = 𝑬𝑛𝑒𝑖𝜔𝑛𝑐 𝑟

= 𝑬𝑛𝑒𝑖𝜔𝑛′

𝑐 𝑟𝑒−𝜔𝑛′′

𝑐 𝑟

𝜔𝑛′′ < 0 𝑒−

𝜔𝑛′′

𝑐 𝑟 ⟶∞⟶

28

𝑬𝑛(r)

Far-field domain

“Exponential catastrophe”

Lamb, Horace. Proceedings of the London Mathematical Society 1.1 (1900)

𝜔𝑛 = 𝜔𝑛′ + 𝑖𝜔𝑛

′′ , 𝜔𝑛′′ < 0

We have recently been linking Resonant State

Expansions (RSEs) with another centenary theory

29

Incident -`excitation’ field : Outgoing scattered field

Lorenz - Mie - Debye theory

(1890) (1908) (1909)

Exact solution !

Problem studied

Fields expanded on the multipolar basis

𝑵𝑛,𝑚 𝑴𝑛,𝑚

Mie, G. Ann. Phys. 25, 377 (1908).

30

휀𝑏 = 1

We study spherical high refractive index resonators (also called Mie resonators)

Optical response can be analytically predicted using Mie theory

휀𝑠

Excitation fieldT-matrix Scattered field

S-matrix Outgoing fieldIncoming field

31

For spherically symmetric scatterers S and T are diagonal

with coefficients and 𝑇𝑛𝑒𝜔

𝑇𝑛ℎ

𝜔

Scattering operators

𝑇 =𝑆 − 𝐼

2

𝑆𝑛𝑒𝜔

𝑆𝑛ℎ

𝜔

S-Matrix for studying energy conservation, symmetry, limit

phenomenon, ...

The S–matrix relates the outgoing part of the total field to the incoming part

Spherically symmetric particles :

Energy conservation :

S is unitary for lossless scatterers at real frequencies

𝑆 †. 𝑆 = ӖI ⇒1

2𝑇 † + 𝑇 = 𝑇 †. 𝑇

𝑎𝑛,𝑚+,𝑒

= 𝑆𝑛𝑒𝑒𝑛,𝑚−,𝑒

𝑎𝑛,𝑚+,ℎ

= 𝑆𝑛ℎ𝑒𝑛,𝑚−,ℎ

𝑆 = ӖI+2 𝑇

𝑬tot 𝑘𝒓 = 𝑬exc 𝑘𝒓 + 𝑬scat 𝑘𝒓 = 𝑬+

𝑘𝒓 + 𝑬−

𝑘𝒓

=σ𝑛,𝑚∞ 𝑎𝑛,𝑚

+,ℎ𝑴𝑛,𝑚

+𝑘𝒓 + 𝑎𝑛,𝑚

+,𝑒𝑵𝑛,𝑚

+𝑘𝒓 + 𝑎𝑛,𝑚

−,ℎ𝑴𝑛,𝑚

−𝑘𝒓 + 𝑎𝑛,𝑚

−,𝑒𝑵𝑛,𝑚

−𝑘𝒓

𝑆𝑛𝑒

= 1 , 𝑆𝑛ℎ

= 1

𝑎 + ≡ 𝑆 . 𝑎 −

32

Physical response of the Sn functions lie along the real frequency axis

𝑆𝑛 𝑧 = 𝐴𝑛𝑒−2𝑖𝑧 ෑ

𝛼=−∞

∞𝑧 − 𝑧𝑧,𝛼𝑧 − 𝑧𝑝,𝛼

𝑧 ≡ 𝑘𝑅

𝑧 = 𝑘𝑅

𝜎1𝑒

𝜋𝑅2

6

𝑘𝑅 2 Unitary limit

33

zeros

poles

Electric dipole 𝑆1(𝑒)

𝑧

34

We want to find the residue factor directly !

𝑆𝑛 𝑧 = 𝐴𝑛𝑒−2𝑖𝑧 + 𝐴𝑛𝑒−2𝑖𝑧 𝛼=−∞

∞𝑟𝑛,𝛼

𝑧 − 𝑧𝑛,𝛼

𝑟𝑛,𝛼 =2𝑖𝑒2𝑖𝑧𝑛,𝛼

𝒩𝛼2

𝒩𝛼2 = න ε 𝒓 𝑬𝛼

2 − 𝜇 𝒓 𝑯𝛼2 𝑑𝒓

34

න0

∞

𝑟2𝑗𝑛 𝐾𝑟 𝑦𝑛 𝑘𝑟 𝑑𝑟k=1.37

K=2.96 +i 0.457 i

Killing Mie softly - Regularization !

𝑆𝑛 𝑧 = 𝐴𝑛𝑒−2𝑖𝑧 + 𝐴𝑛𝑒−2𝑖𝑧 𝛼=−∞

∞𝑟𝑛,𝛼

𝑧 − 𝑧𝑛,𝛼

Lorentzian form of response functions

𝑆𝑛 𝑧 = 𝐴𝑛𝑒−2𝑖𝑧 ෑ

𝛼=−∞

∞𝑧 − 𝑧𝑧,𝛼𝑧 − 𝑧𝑝,𝛼

36

𝑟𝑝,𝑛,𝛼 = 𝑧𝑝,𝑛,𝛼 − 𝑧𝑧,𝑛,𝛼 ෑ

𝛽≠𝛼,−∞

∞𝑧𝑝,𝑛,𝛼 − 𝑧𝑧,𝑛,𝛽

𝑧𝑝,𝑛,𝛼 − 𝑧𝑝,𝑛,𝛽

Residue factor ‘product rule’

Cross sections in Mie theory using the S-functions

Extinction :

Scattering :

Absorption :

Valid for any scatterer

𝜎scat,𝑛𝑒,ℎ

=𝜆2 2𝑛 + 1

8𝜋1 − 𝑆𝑛

𝑒,ℎ2

𝜎ext,𝑛𝑒,ℎ

=𝜆2 2𝑛 + 1

4𝜋Re 1 − 𝑆𝑛

𝑒,ℎ

𝜎abs,𝑛𝑒,ℎ

= 𝜎ext,𝑛𝑒,ℎ

− 𝜎scat,𝑛𝑒,ℎ

=𝜆2 2𝑛 + 1

8𝜋1 − 𝑆𝑛

𝑒,ℎ2

37

38

Wave equations in 3D

(spherically symmetric isotropic potentials)

Acoustic waves: Electromagnetic waves:

Schrödinger’s equation:

Optical fields (𝜇 = 1) :

𝛁 ∙1

𝜌 𝑟𝛁𝜓 −

1

𝐵 𝑟

𝜔

𝑐

2

𝜓 = 0

𝛻 × 𝛻 × 𝑬 − 휀 𝑟𝜔

𝑐

2

𝑬 = 𝟎

∆𝜓 +2𝑚

ℏ2𝐸𝜓 −

2𝑚

ℏ2𝑉 𝑟 𝜓 = 0

𝛁 ×1

𝜇 𝑟𝛁 × 𝑬 − 휀 𝑟

𝜔

𝑐

2

𝑬 = 𝟎

∆𝜓 + 휀 𝑟𝜔

𝑐

2

𝜓 = 0

Scalar “light” (acoustics : 𝜌 = 1 , 휀 𝑟 = 1/𝐵 𝑟 ):

39

Wave equations in 3D

(spherically symmetric isotropic potentials)

Acoustic waves: Electromagnetic waves:

Schrödinger’s equation:

Optical fields (𝜇 = 1) :

𝛁 ∙1

𝜌 𝑟𝛁𝜓 −

1

𝐵 𝑟

𝜔

𝑐

2

𝜓 = 0

𝛻 × 𝛻 × 𝑬 − 휀 𝑟𝜔

𝑐

2

𝑬 = 𝟎

∆𝜓 +2𝑚

ℏ2𝐸𝜓 −

2𝑚

ℏ2𝑉 𝑟 𝜓 = 0

𝛁 ×1

𝜇 𝑟𝛁 × 𝑬 − 휀 𝑟

𝜔

𝑐

2

𝑬 = 𝟎

∆𝜓 + 휀 𝑟𝜔

𝑐

2

𝜓 = 0

Scalar “light” (acoustics : 𝜌 = 1 , 휀 𝑟 = 1/𝐵 𝑟 ):

( ) ( ) ( ) ( )Separation of variables : , , rr r

Scalar Helmholtz equation

in a spherically symmetric potential:

M ’

O

z

x

y

rM

r

M ’’

40

∆𝜓 + 휀 𝑟𝜔

𝑐

2

𝜓 = 0

1

sin2 𝜃

𝑑

𝑑𝜃sin 𝜃

𝑑

𝑑𝜃+

𝑚2

sin2 𝜃𝜓𝜃 𝜃 = −𝑛 𝑛 + 1 𝜓𝜃 𝜃 ⟹ 𝜓𝜃 𝜃 ∝ 𝑃𝑛

𝑚 cos 𝜃 𝑛 = 0,1, … , 𝑚

𝑑2

𝑑𝜙2 + 𝑎 𝜓𝜙 𝜙 = 0 ⟹ 𝑎 = 𝑚2 , 𝜓𝜙 𝜙 ∝ 𝑒𝑖𝑚𝜙

𝑚 = −∞,… ,−1,0,1, …∞

𝑚 ∈ ℤ⟹

1

𝑟

𝜕2 𝑟𝜓

𝜕𝑟2+

1

𝑟2 sin2 𝜃

𝜕

𝜕𝜃sin 𝜃

𝑑𝜓

𝑑𝜃+𝜕2𝜓

𝜕𝜙2+

𝜔

𝑐

2

휀 𝑟 𝜓 = 0

Legendre Polynomials : Pn(x)

Associated Legendre functions : Pnm(x)

( )

( )

( ) ( )

( ) ( )

0

1

2

2

2

3

1

13 1

2

15 3

2

P x

P x x

P x x

P x x x

( ) ( ) ( ) ( )

( ) ( ) ( )

/ 22

/ 2

1 1

1 sin

mmmm

n nm

mm m

nm

dP x x P x

dx

dP x

dx

Legendre Polynomials:

Associated Legendre functions:

41

( ) ( ) ( )0 211

2 !

nn

n n n n

dP x P x x

n dx

(Rodrigues’ Formula)

1

sin2 𝜃

𝑑

𝑑𝜃sin 𝜃

𝑑

𝑑𝜃+

𝑚2

sin2 𝜃𝜓𝜃 𝜃 𝑒𝑖𝑚𝜙 = −𝑛 𝑛 + 1 𝜓𝜃 𝜃 𝑒𝑖𝑚𝜙

−𝑳2 𝜓𝜃 𝜃 𝑒𝑖𝑚𝜙 = −𝑛 𝑛 + 1 𝜓𝜃 𝜃 𝑒𝑖𝑚𝜙 ⟹ 𝜓𝜃 𝜃 = 𝑃𝑛𝑚 cos 𝜃 𝑛 = 0,1,2, … ,∞

Scalar Spherical harmonics

( )00

11

10

2 2

22

21

2

20

10

4

3sin

81

3cos

4

1 15sin

4 2

152 sin cos

8

5 3 1cos

4 2 2

i

i

i

n Y x

Y e

n

Y

Y e

n Y e

Y

p

p

p

p

p

p

( ) ( )4

*

, , , ,

0

, ,n m n md Y Y

p

Orthogonal basis:

Angular momentum operator: 𝑳

42

𝑌𝑛,𝑚 𝜃, 𝜙 =2𝑛 + 1

4𝜋

𝑛 − 𝑚 !

𝑛 + 𝑚 !

1/2

𝑒𝑖𝑚𝜙𝑃𝑛𝑚 cos𝜃 𝑛 = 0,1, … ,∞ 𝑚 = −𝑛,… , 𝑛

≡ 𝑒𝑖𝑚𝜙 ത𝑃𝑛𝑚 cos𝜃

1

sin2 𝜃

𝜕

𝜕𝜃sin 𝜃

𝜕

𝜕𝜃+

1

sin2 𝜃

𝜕2

𝜕𝜙2 𝑌𝑛,𝑚 𝜃, 𝜙

≡ −𝑳2𝑌𝑛,𝑚 𝜃, 𝜙 = −𝑛 𝑛 + 1 𝑌𝑛,𝑚 𝜃, 𝜙

Scalar Helmholtz equation

in a spherically symmetric potential:

M ’

O

z

x

y

rM

r

M ’’

43

∆𝜓 + 휀 𝑟𝜔

𝑐

2

𝜓 = 0

1

𝑟

𝜕2 𝑟𝜓

𝜕𝑟2−𝑳2

𝑟2𝜓 + 휀 𝑟

𝜔

𝑐

2

𝜓 = 0

𝜓 𝑟, 𝜃, 𝜙 ≡ 𝜓𝑟 𝑟 𝑌𝑛,𝑚 𝜃, 𝜙 𝑳2𝑌𝑛,𝑚 = 𝑛 𝑛 + 1 𝑌𝑛,𝑚

𝑌𝑛,𝑚 𝜃, 𝜙 = 𝑒𝑖𝑚𝜙 ത𝑃𝑛𝑚 cos𝜃

𝑛 = 0,1,2, … ,∞𝑚 = −𝑛,… , 𝑛

𝑑2𝑢𝑛𝑑𝑟2

−𝑛 𝑛 + 1

𝑟2𝑢𝑛 + 휀 𝑟

𝜔

𝑐

2

𝑢𝑛 = 0𝑢𝑛 𝑟 ≡ 𝑟𝜓𝑟 𝑟

Scattering from a spherically symmetric potential

( Quantum mechanical analogy )

Schrödinger equation :

Scalar light :

sb

R

44

𝑑2

𝑑𝑟2𝑢𝑛 + 𝑘2𝑢𝑛 − 𝑉eff

opt𝑟, 𝜔

𝑛𝑢𝑛 = 0 𝑉eff

opt𝑟, 𝜔

𝑛≡𝑛 𝑛 + 1

𝑟2− 휀𝑠 − 휀𝑏

𝜔

𝑐

2

𝜃 𝑅 − 𝑟

𝑘2 ≡ 휀𝑏𝜔

𝑐

2

𝑑2

𝑑𝑟2𝑢𝑛 +

2𝑚

ℏ2𝐸𝑢𝑛 − 𝑉eff

Sch𝑟

𝑛𝑢𝑛 = 0 𝑉eff

Sch𝑟

𝑛≡𝑛 𝑛 + 1

𝑟2−2𝑚

ℏ2𝑒2

4𝜋𝜖0𝑟

Effective potential – Photonic “atom”

Scalar light :

45

Plasmonic resonances

“repulsive” potential :

휀𝑠 < 휀b

Whispering Gallery modes

“attractive” potential with radiative loss :

( ) ( )( ) 2

Sch

eff 2

0

2

2

4

1n n

r

m eV r

rp

휀𝑠 > 휀b

휀𝑠 > 휀b

“Morphologic” resonances

“attractive” potential :

𝑉effopt

𝑟𝑛≡𝑛 𝑛 + 1

𝑟2− 휀𝑠 − 휀𝑏

𝜔

𝑐

2

𝜃 𝑅 − 𝑟

𝑘2 ≡ 휀𝑏𝜔

𝑐

2

Separation of variables:

M ’

O

z

x

y

rM

r

M ’’

0,1, 2... ,...,n m n n

Change of variables:

Spherical Bessel function equation:

( )2 22

,2 2

210 0

r rr n mk

r Lk Y

r r r

( ) ( ) ( ),, , ,r n mr r Y

( )( ) ( )2

2

2

2

,1 0r

r r n mkd r

r n n r Ydr

46

Homogeneous media Helmholtz equation

∆𝜓 + 𝑘2𝜓 = 0

𝑟2𝑑2𝑧𝑛𝑑𝑟2

+ 𝑟𝑑𝑧𝑛𝑑𝑟

+ 𝑘2𝑟2 − 𝑛 +1

2

2

𝑧𝑛 = 0

𝑘2 ≡ 휀𝑏𝜔

𝑐

2𝑧𝑛 𝑟

𝑘𝑟 1/2≡ 𝜓𝑟 𝑟

Homogeneous Helmholtz equation in spherical coordinates

Spherical Bessel, Neumann and Hankel functions:

Bessel (1): Neumann (2): Hankel + (3):

47

ℎ0+

𝑥 = −𝑖

𝑥𝑒𝑖𝑥

ℎ1+

𝑥 = −𝑒𝑖𝑥1

𝑥+

𝑖

𝑥2

⋮

𝑦0 𝑥 = −cos 𝑥

𝑥

𝑦1 𝑥 =cos2 𝑥

𝑥2−sin 𝑥

𝑥

⋮

𝑗0 𝑥 =

𝑠=0

∞−1 𝑠

2𝑠 + 1 !𝑥2𝑠 =

sin 𝑥

𝑥

𝑗1 𝑥 =sin 𝑥

𝑥2−cos 𝑥

𝑥⋮

𝑟2𝑑2𝑧𝑛𝑑𝑟2

+ 𝑟𝑑𝑧𝑛𝑑𝑟

+ 𝑘2𝑟2 − 𝑛 +1

2

2

𝑧𝑛 = 0

𝑧𝑛 𝑟

𝑘𝑟 1/2≡ 𝜓𝑟 𝑟

𝜓𝑟 𝑟 =

𝑗𝑛 𝑥 ≡𝜋

2𝑥𝐽𝑛+ Τ1 2 𝑥

𝑦𝑛 𝑥 ≡𝜋

2𝑥𝑌𝑛+ Τ1 2 𝑥

ℎ𝑛+

𝑥 ≡𝜋

2𝑥𝐻𝑛+ Τ1 2

+𝑥 = 𝑗𝑛 𝑥 + 𝑖𝑦𝑛 𝑥

ℎ𝑛−

𝑥 ≡𝜋

2𝑥𝐻𝑛+ Τ1 2

−𝑥 = 𝑗𝑛 𝑥 − 𝑖𝑦𝑛 𝑥

Linearly independent solutions

Spherical Bessel functions (1) Spherical Neumann functions (2)

𝑗1𝑗2 𝑗3

𝑦1 𝑦2

Outgoing spherical Hankel functions (+) Incoming spherical Hankel functions (-)

ℎ0+

𝑥 = −𝑖

𝑥𝑒𝑖𝑥

ℎ1+

𝑥 = −𝑒𝑖𝑥1

𝑥+

𝑖

𝑥2

⋮

ℎ0−

𝑥 =𝑖

𝑥𝑒−𝑖𝑥

ℎ1−

𝑥 = −𝑒−𝑖𝑥1

𝑥−

𝑖

𝑥2

⋮

𝑗0𝑦0

𝑦3

𝑗0 𝑥 =

𝑠=0

∞−1 𝑠

2𝑠 + 1 !𝑥2𝑠 =

sin 𝑥

𝑥

𝑗1 𝑥 =sin 𝑥

𝑥2−cos𝑥

𝑥⋮

𝑦0 𝑥 = −cos 𝑥

𝑥

𝑦1 𝑥 =cos2 𝑥

𝑥2−sin 𝑥

𝑥

⋮

48

49

m=0 m=1

m=2

m=3

m=4

n=1

n=2

n=3

n=4

j1

j3

j2

Bessel functions :

Spherical Harmonics :

Regular partial wave basis :

( ), ,n mY

Ψ𝑛,𝑚1

≡ 𝑗𝑛 𝑘𝑟 𝑌𝑛,𝑚 𝜃, 𝜙

𝑗𝑛 𝑘𝑟

Partial wave basis (scalar waves)

Plane wave Coefficients : Regular partial wave :

Outgoing “partial” wave :

Plane wave expansion :

50

Ψ𝑛,𝑚+

𝑘𝒓 = ℎ𝑛+

𝑘𝑟 𝑌𝑛,𝑚 𝜃, 𝜙

Ψ𝑛,𝑚 𝑘𝒓 ≡ Ψ𝑛,𝑚1

𝑘𝒓 = 𝑗𝑛 𝑘𝑟 𝑌𝑛,𝑚 𝜃, 𝜙𝑝𝑛,𝑚 = 4𝜋𝑖𝑛𝑌𝑛,𝑚∗ 𝒌

𝑒𝑖𝒌∙𝒓 =

𝑛=0

∞

𝑝𝑛,𝑚Ψ𝑛,𝑚 𝑘𝒓

Helmholtz equation in spherical coordinates

0,1, 2... ,...,n m n n

Scalar “partial” waves

Transverse vector “partial” waves :

Bouwkamp-Casimir (Jackson) approach

Longitudinal vector “partial” waves

magnetic

electric

51

Ψ𝑛,𝑚 𝑘𝒓 ≡ 𝑗𝑛 𝑘𝑟 𝑌𝑛,𝑚 𝜃, 𝜙

∆𝜓 + 𝑘2𝜓 = 0𝛁 × 𝛁 × 𝑬 − 𝑘2𝑬 = 𝟎 𝑘2 = 휀𝜇

𝜔

𝑐

2

𝑴𝑛,𝑚 𝑘𝒓 ≡𝛻 × 𝒓Ψ𝑛,𝑚 𝑘𝒓

𝑛 𝑛 + 1

𝑵𝑛,𝑚 𝑘𝒓 ≡𝛻 ×𝑴𝑛,𝑚 𝑘𝒓

𝑘

𝑳𝑛,𝑚 𝑘𝒓 =𝛻 Ψ𝑛,𝑚 𝑘𝒓

𝑛 𝑛 + 1

∆𝑨 + 𝑘2𝑨 = 𝟎 𝑨 = 𝑳 , 𝑴 ,𝑵

𝛁 𝛁 ∙ 𝑳 + 𝑘2𝑳 = 𝟎

∆ 𝒓 ∙ 𝑯 + 𝑘2𝒓 ∙ 𝑯 = 𝟎

∆ 𝒓 ∙ 𝑬 + 𝑘2𝒓 ∙ 𝑬 = 𝟎

𝛁 ∙ 𝑴𝑛,𝑚 = 𝛁 ∙ 𝑵𝑛,𝑚 = 0

Vector spherical harmonics

3 types of vector spherical harmonics :

52

𝑌𝑛,𝑚 𝜃, 𝜙 =2𝑛 + 1

4𝜋

𝑛 − 𝑚 !

𝑛 + 𝑚 !

1/2

𝑒𝑖𝑚𝜙𝑃𝑛𝑚 cos𝜃 ≡ 𝑒𝑖𝑚𝜙 ത𝑃𝑛

𝑚 cos𝜃

𝕃op ≡1

𝑖𝒓 × 𝛁Angular momentum operator :

𝑾𝑛,𝑚1

𝜃, 𝜙 ≡ 𝒀𝑛,𝑚 𝜃, 𝜙 ≡ ො𝒓𝑌𝑛,𝑚 𝜃, 𝜙

𝑾𝑛,𝑚2

𝜃, 𝜙 ≡ 𝑿𝑛,𝑚 𝜃, 𝜙 ≡𝕃op

𝑖𝑌𝑛,𝑚 𝜃, 𝜙 = 𝒁𝑛,𝑚 𝜃, 𝜙 × ො𝒓

𝑾𝑛,𝑚3

𝜃, 𝜙 ≡ 𝐙𝑛,𝑚 𝜃, 𝜙 ≡𝑟𝛁𝑌𝑛,𝑚 𝜃,𝜙

𝑛 𝑛+1= ො𝒓 × 𝑿𝑛,𝑚 𝜃, 𝜙

n=0,1,2,…,∞ m=-n,…,n

n=1,2,…,∞ m=-n,…,n

න0

4𝜋

𝑑Ω𝑾𝜈,𝜇𝑗 ,∗

𝜃, 𝜙 ∙ 𝑾𝑛,𝑚𝑘

𝜃, 𝜙 = 𝛿𝑗,𝑘𝛿𝑛,𝜈𝛿𝑚,𝜈

Vector spherical harmonics

53

𝒀𝑛,𝑚 𝜃, 𝜙 = ො𝒓𝑌𝑛,𝑚 𝜃, 𝜙

𝑿𝑛,𝑚 𝜃, 𝜙 = 𝒆𝑖𝑚𝜙 𝜽𝑖 ത𝑢𝑛𝑚 𝜃 − 𝝓 ҧ𝑠𝑛

𝑚 𝜃

𝒁𝑛,𝑚 𝜃, 𝜙 = 𝒆𝑖𝑚𝜙 𝜽 ҧ𝑠𝑛𝑚 𝜃 − 𝝓𝑖ത𝑢𝑛

𝑚 𝜃

n=0,1,2,…,∞ m=-n,…,n

n=1,2,…,∞ m=-n,…,n

ത𝑢𝑛𝑚 𝜃 =

1

𝑛 𝑛 + 1

𝑚

sin 𝜃ത𝑃𝑛𝑚 𝜃

ҧ𝑠𝑛𝑚 𝜃 =

1

𝑛 𝑛 + 1

𝑑

𝑑𝜃ത𝑃𝑛𝑚 𝜃

n=1,2,…, -n < m < n

Vector partial waves and

Vector spherical Harmonics

( ) ( ) ( )

( ) ( )n

n n n

n

x j

x

x xj x

xj x

Ricatti-Bessel functions

54

𝑴𝑛,𝑚1

𝑘𝒓 = 𝑗𝑛 𝑘𝑟 𝐗𝑛,𝑚 𝜃, 𝜙

𝑵𝑛,𝑚1

𝑘𝒓 =1

𝑘𝑟𝑗𝑛 𝑘𝑟 𝑛 𝑛 + 1 𝐘𝑛,𝑚 𝜃, 𝜙 + 𝜓′𝑛 𝑘𝑟 𝐙𝑛,𝑚 𝜃, 𝜙

𝛁 × 𝛁 × 𝑬 − 𝑘2𝑬 = 𝟎

𝛁 ∙ 𝑴𝑛,𝑚 = 𝛁 ∙ 𝑵𝑛,𝑚 = 0 𝛁 ×𝑴𝑛,𝑚 = 𝑘𝑵𝑛,𝑚 𝛁 × 𝑵𝑛,𝑚 = 𝑘𝑴𝑛,𝑚

Matrix expressions for field developments

Excitation field development

Scattering field development

𝑝 𝑛,𝑚 = 𝑛 𝑛 + 1 −𝑚

𝐄exc 𝑘𝒓 =

𝑛=1

∞

𝑚=−𝑛

𝑛

𝑒𝑛,𝑚ℎ𝑴𝑛,𝑚

1𝑘𝒓 + 𝑒𝑛,𝑚

𝑒𝑵𝑛,𝑚

1𝑘𝒓

𝐄exc = 𝑴 1 𝑘𝒓 , 𝑵 1 𝑘𝒓 . 𝑒

𝑬scat = 𝑴 + 𝑘𝒓 , 𝑵 + 𝑘𝒓 . 𝑓

𝑒1,1ℎ

𝑒1,0ℎ

𝑒1,−1ℎ

⋮

𝑒1,1𝑒

𝑒1,0𝑒

𝑒1,−1𝑒

⋮

≡ 𝑒

𝑝 = 1,2, … ,∞

55

T-matrix :

Multipole representation of the T-matrix

of a scatterer

Excitation field Scattering field

0

𝑬exc = 𝑴 1 𝑘𝒓 , 𝑵 1 𝑘𝒓 . 𝑒 𝑬s = 𝑴 + 𝑘𝒓 , 𝑵 + 𝑘𝒓 . 𝑓

𝑓 = 𝑡. 𝑒

𝑡𝑬s

𝑬exc

56

57

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

h h h

n s p n p n p

h h h

n s p n p n p

s

e e e

s n s p n p n p

s

e e e

n s p s n p s n p

j k R s j kR e h kR f

k R s kR e kR f

j k R s j kR e h kR f

k R s kR e kR f

( ) ( )

( ) ( )

n n

n n

x xj x

x xh x

Mie coefficients : an , bnHomogeneous sphere T-matrix is diagonal in multipole space

Ricatti-Bessel functions

s ss

k N

k N

Index contrast :

Continuity of Tangential E and H fields at the surface (r=R)

( ) ( )

( ) ( )

e e

p p

h

p

n

h

pn

a

b

f e

f e

R𝑬s 𝑘𝒓 = 𝑴 + 𝑘𝒓 , 𝑵 + 𝑘𝒓 . 𝑓

𝑬e 𝑘𝒓 = 𝑴 1 𝑘𝒓 , 𝑵 1 𝑘𝒓 . 𝑒 𝑬int 𝑘s𝒓 = 𝑴 1 𝑘s𝒓 , 𝑵 1 𝑘s𝒓 . 𝑠

𝑎𝑛 =𝜇𝑠/𝜇 𝜓′𝑛 𝑘𝑠𝑅 𝑗𝑛 𝑘𝑅 − 𝜌𝑠

2𝑗𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅

𝜇𝑠/𝜇 𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅 − 𝜌𝑠2𝑗𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅

𝑏𝑛 =𝜓′𝑛 𝑘𝑠𝑅 𝑗𝑛 𝑘𝑅 − 𝜇𝑠/𝜇 𝑗𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅

𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅 − 𝜇𝑠/𝜇 𝑗𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅

“Symmetrized” Mie coefficients

(Mie : Bohren & Huffman)

( ) ( )

( ) ( )

e e

p p

h

p

n

h

pn

a

b

f e

f e

𝑎𝑛 =𝜌𝑠𝜓𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅 − 𝜇𝑠/𝜇𝑏 𝜓′𝑛 𝑘𝑠𝑅 𝜓𝑛 𝑘𝑅

𝜌𝑠𝜓𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅 − 𝜇𝑠/𝜇𝑏 𝜓′𝑛 𝑘𝑠𝑅 𝜉𝑛 𝑘𝑅

𝑏𝑛 =𝜌𝑠𝜓′𝑛 𝑘𝑅 𝜓𝑛 𝑘𝑅 − 𝜇𝑠/𝜇𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅

𝜌𝑠𝜓′𝑛 𝑘𝑠𝑅 𝜉𝑛 𝑘𝑅 − 𝜇𝑠/𝜇𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅

𝜌𝑠 =𝑘𝑠𝑘=𝑁𝑠𝑁

𝑎𝑛 =휀𝑠/휀𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 𝑗𝑛 𝑘𝑅

휀𝑠/휀𝑏 𝑗𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅

𝑏𝑛 =𝜇𝑠/𝜇𝑏 𝑗𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑅 𝑗𝑛 𝑘𝑅

𝜇𝑠/𝜇𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅

Correctly symmetrized Mie coefficients

Dangerous but conventional “symmetrization” of the Mie coefficients

58

T-matrix of a sphereThe T-matrix of a spherically symmetric object is diagonal in multipole space

Mie Coefficients : 𝑎𝑛 , 𝑏𝑛

𝑓𝑝ℎ

𝑓𝑝𝑒

=

𝑡1ℎ

⋯ 0⋮ ⋱ ⋮

0 ⋯ 𝑡𝑛ℎ

0 ⋯ 0⋮ ⋱ ⋮0 ⋯ 0

0 ⋯ 0⋮ ⋱ ⋮0 ⋯ 0

𝑡1𝑒

⋯ 0⋮ ⋱ ⋮

0 ⋯ 𝑡𝑛𝑒

𝑒𝑝ℎ

𝑒𝑝𝑒

= 𝑡𝑒𝑝ℎ

𝑒𝑝𝑒

𝑡𝑛(𝑒)

= −𝑎𝑛 𝑡𝑛(ℎ)

= −𝑏𝑛

𝜌𝑠 =𝑘𝑠𝑘=𝑁𝑠𝑁

𝑝 𝑛,𝑚 = 𝑛 𝑛 + 1 −𝑚

𝑎𝑛 =휀𝑠/휀𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 𝑗𝑛 𝑘𝑅

휀𝑠/휀𝑏 𝑗𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅

𝑏𝑛 =𝜇𝑠/𝜇𝑏 𝑗𝑛 𝑘𝑠𝑅 𝜓′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑅 𝑗𝑛 𝑘𝑅

𝜇𝑠/𝜇𝑏 𝜓𝑛 𝑘𝑠𝑅 𝜉′𝑛 𝑘𝑅 − 𝜓′𝑛 𝑘𝑠𝑅 ℎ𝑛 𝑘𝑅59

Development of a vector plane wave

(Vector partial wave basis)

Plane wave coefficients :

Plane wave expansion :

60

𝑝𝑛,𝑚(ℎ)

= 𝑖𝑛4𝜋𝑿𝑛,𝑚∗ 𝒌i ∙ ො𝒆𝑖 𝑝𝑛,𝑚

(𝑒)= 𝑖𝑛−14𝜋𝒁𝑛,𝑚

∗ 𝒌i ∙ ො𝒆𝑖

𝑒𝑖𝒌∙𝒓ො𝒆inc =

𝑛=1

∞

𝑚=−𝑛

𝑛

𝑝𝑛,𝑚(ℎ)

𝑴𝑛,𝑚1

𝑘𝒓 + 𝑝𝑛,𝑚(𝑒)

𝑵𝑛,𝑚1

𝑘𝒓

Multipole representation of the T-matrix

of a scatterer

𝑡 =

𝑡1ℎ

⋯ 0⋮ ⋱ ⋮

0 ⋯ 𝑡𝑛ℎ

0 ⋯ 0⋮ ⋱ ⋮0 ⋯ 0

0 ⋯ 0⋮ ⋱ ⋮0 ⋯ 0

𝑡1𝑒

⋯ 0⋮ ⋱ ⋮

0 ⋯ 𝑡𝑛𝑒

𝑡 =

𝑡1,1ℎ,ℎ

𝑡1,2ℎ,ℎ

⋯

𝑡2,1ℎ,ℎ

⋱ ⋮

⋮ ⋯ 𝑡𝑛,𝑛ℎ,ℎ

𝑡1,1ℎ,𝑒

𝑡1,2ℎ,𝑒

⋯

𝑡2,1ℎ,𝑒

⋱ ⋮

⋮ ⋯ 𝑡𝑛,𝑛ℎ,𝑒

𝑡1,1𝑒,ℎ

𝑡1,2ℎ,𝑒

⋯

𝑡2,1𝑒,ℎ

⋱ ⋮

⋮ ⋯ 𝑡𝑛,𝑛𝑒,ℎ

𝑡1,1𝑒,𝑒

𝑡1,2𝑒,𝑒

⋯

𝑡2,1𝑒,𝑒

⋱ ⋮

⋮ ⋯ 𝑡𝑛,𝑛𝑒,𝑒

𝑓 = 𝑡. 𝑒

O

O

50

“Scattering” matrix

For a spherical scatterer, S3=S4=0 : and

ik

z

x

62

𝑆1 𝜃 = 0 = 𝑆2 𝜃 = 0 =1

2

𝑛=1

∞

2𝑛 + 1 𝑎𝑛 + 𝑏𝑛

𝑆1 𝜃 =1

2

𝑛=1

∞

2𝑛 + 1 𝑎𝑛𝜋𝑛 cos𝜃 + 𝑏𝑛𝜏𝑛 cos𝜃

𝑆2 𝜃 =1

2

𝑛=1

∞

2𝑛 + 1 𝑏𝑛𝜋𝑛 cos𝜃 + 𝑎𝑛𝜏𝑛 cos𝜃

Forward scattering :

𝜋𝑛 cos𝜃 ≡1

sin𝜃ത𝑃𝑛1 cos𝜃 𝜏𝑛 cos𝜃 ≡

𝑑

𝑑𝜃ത𝑃𝑛1 cos𝜃

Back scattering : 𝑆1 𝜃 = 𝜋 = −𝑆2 𝜃 = 𝜋 =1

2

𝑛=1

∞

2𝑛 + 1 𝑎𝑛 − 𝑏𝑛

𝐸∥,s𝐸⊥,s

=exp 𝑖𝑘𝑟

−𝑖𝑘𝑟

𝑆2 𝑆3𝑆4 𝑆1

𝐸∥,i𝐸⊥,i

Possible values of the Mie coefficients

limited by the underlying physics !

Im{a}

Re{a}0.5

1

i

i/2

-i/2

𝑎𝑛 , 𝑏𝑛

𝑎𝑛 𝜔 and 𝑏𝑛 𝜔

Mie Coefficients

Unitary ‘limit’ : 𝑎𝑛 = 1 , 𝑏𝑛= 1

Energy conservation circle

Re 𝑎𝑛 = 𝑎𝑛2

Re 𝑏𝑛 = 𝑏𝑛2

52

‘Invisible’ mode 𝑎𝑛 = 0 or 𝑏𝑛 = 0

𝑆1,2 𝜃 = 0 =1

2

𝑛=1

∞

2𝑛 + 1 𝑎𝑛 + 𝑏𝑛

𝑆1 𝜃 = 𝜋 = −𝑆2 𝜃 = 𝜋 =1

2

𝑛=1

∞

2𝑛 + 1 𝑎𝑛 − 𝑏𝑛

Ideal absorption ‘limit’ : 𝑎𝑛 =1

2, 𝑏𝑛=

1

2

Scattering Cross section

-1

-0.5

0

0.5

1-1

-0.5

0

0.5

1

-0.4

-0.2

0

0.2

0.4

-1

-0.5

0

0.5

1

Dipole scattering :

Differential scattering cross section

Multipole scattering :

ss

64

𝑺inc =1

2Re 𝑬∗inc × 𝑯inc =

1

2

휀𝑏𝜇𝑏

𝜖0𝜇0

𝑬inc2𝒌inc

𝑺s =1

2Re 𝑬∗scat ×𝑯scat = lim

𝑟→∞

1

2

휀𝑏𝜇𝑏

𝜖0𝜇0

𝑬scat2ෞ𝒓

𝑑𝜎𝑠𝑑Ω

𝜃, 𝜙 ≡ lim𝑟→∞

𝑟2𝑺s ∙ ො𝒓

𝑺inc ∙ 𝒌inc= lim

𝑟→∞𝑟2

𝑬scat2

𝑬inc2

𝜎𝑠 ≡ න0

4𝜋

𝑑Ω𝑑𝜎𝑠𝑑Ω

𝜃, 𝜙

𝒌inc

𝒌inc𝑬inc

Absorption cross section

Absorption cross section : a e ss s s

65

Ԧ𝐒tot =1

2Re 𝐄tot

∗ × 𝐇tot =1

2Re 𝐄inc

∗ × 𝐇inc +1

2Re 𝐄inc

∗ × 𝐇scat+𝐄scat∗ × 𝐇inc +

1

2Re 𝐄scat

∗ × 𝐇scat

𝜎𝑎 ≡ −𝑟2 Ԧ𝐒tot ∙ ො𝒓𝑑Ω

Ԧ𝐒inc ∙ 𝒌inc

Ԧ𝐒exc Ԧ𝐒scatԦ𝐒inc

𝑬tot = 𝑬inc + 𝑬scat

𝑺tot =1

2Re 𝑬tot

∗ ×𝑯tot

Extinction cross section

( )inc inc inc

1Re

2

S E H

“Authentic” optical theorem :

For a sphere :

( ) ( )1 20 0S S

66

=2𝜋

𝑘2

𝑛=1

∞

2𝑛 + 1 Re 𝑎𝑛 + 𝑏𝑛

𝜎ext =4𝜋

𝑘2Re 𝑆1,2 0°

𝜎ext ≡ − lim𝑟→∞

𝑟2 𝑺exc ∙ ො𝒓𝑑Ω

𝑺inc ∙ 𝒌inc

𝑺exc =1

2Re 𝑬inc

∗ × 𝑯scat+𝑬scat∗ ×𝑯inc = 𝑺tot − 𝑺scat − 𝑺inc

Extinction cross section

Extinction cross section :

exts E||E||2

67

𝑺ext + 𝑺inc = 𝑺tot − 𝑺scat

Cross sections in Mie theory

( ) ( ) ( ) ( ) ( )( )( )

( ) ( )( ), , ,

1 121

24 2 1Re Re

1 1

e e h h e h

p e n n n n n n

n

n n nt t t t t t

n n nk

ps s

Extinction :

Scattering :

Radiation pressure cross

section on a sphere :

Absorption : a e ss s s

68

( ) ( ) ( ) ( )( )2 2

†

2

2 2

21

21

22

2. 2 1 1

1 e h

nn ns

n

n

n

n af f nk k

bt tk

pps

Valid for any scatterer

𝜎𝑒 −1

𝑘2Re 𝑓†. 𝑝 = −

2𝜋

𝑘2

𝑛=1

∞

2𝑛 + 1 Re 𝑡𝑛𝑒+ 𝑡𝑛

ℎ= +

2𝜋

𝑘2

𝑛=1

∞

2𝑛 + 1 Re 𝑎𝑛 + 𝑏𝑛

400 500 600 700 800 900 1000

nm

1

2

3

4

R2 Silver sphere : R 5 nm

ANTARES / 2010

Ready for application !

400 500 600 700 800 900 1000nm

1

2

3

4

5

6

7

R2

Silver sphere : R 50 nm

400 500 600 700 800 900 1000

nm

2

4

6

8

10

12

R2 Silver sphere : R 25 nms/(pR2) s/(pR2)

s/(pR2) s/(pR2)

69

Low contrast dielectric physics : Ripples, Gallery modes

and the Extinction paradox

kR

s/pR2

s/b= 3

70

High contrast dielectric physics : Magnetic dipole modes

kR

s/pR2

s/b= 20

71

Plasmonic physics : Electric dipole modes

kR

s/pR2 s/b= -2.2

72

Gallery (i.e. surface) modes as “leaky” modes

73

magnetic n=6electric n=6

Gallery (i.e. surface) modes as “leaky” modes

44

75

What physics is hidden in the Mie coefficients ?

( )

( )

( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

s n s n s n n s

s n s n s n n s

s n n s s n s n

s n n s s n

n

n

s n

a

b

k R kR kR k R

k R kR kR k R

kR k R k R kR

kR k R k R kR

Difficult to extract physical understanding from exact solutions !

76

Mie coefficients are the analogue of the Fresnel reflection coefficients

(spherical geometry)

||

(1, ) ( , )

(1, ) ( , )

(1, ) ( , )

(1, ) ( , )

ss

ss

ss

ss

R

R

2 2( , )k

k

Fresnel reflection coefficients

sN

N

k

k s

s sk N

k N

sNN

( ) ( )

( ) ( )

( )

( )

( )( ) ( )

( ) ( )

( ) ( )

( ) )

( )

(

sn n s

h

ns

n n s

sn n

n

n

n

n

se

ns

n n s

kR kR

t

kR kR

kR kR

t

kR

j kR

h kR

j R

Rh kR

k

k

TM

TE

𝜑𝑛+ 𝑥 ≡

𝑥ℎ𝑛+ 𝑥

′

ℎ𝑛+ 𝑘𝑅

𝜑𝑛2

𝑥 ≡𝑥𝑦𝑛 𝑥 ′

𝑦𝑛 𝑘𝑅

𝜑𝑛1 𝑥 ≡ 𝜑𝑛 𝑥 ≡

𝑥𝑗𝑛 𝑥 ′

𝑗𝑛 𝑘𝑅

77

Quasi-static limit : magnetic response ?

(electric)

(magnetic)( )

( ) ( )

( )

( ) ( )

1

2 10

2( )

2 1 !! 2 1 !!

( )

2 1

( 1)

( 1)lim

( 1)

( 1)!! 2 1 !!

h s bn

b s

kRe s b

n

b s

n

n

nt

n n

nt

n

i kR

n n

i kR

n nn

𝑡𝑛𝑒= −

𝑗𝑛 𝑘𝑅

ℎ𝑛+

𝑘𝑅

휀𝑠휀𝜑𝑛 𝑘𝑅 −𝜑𝑛 𝑘𝑠𝑅

휀𝑠휀𝜑𝑛

+𝑘𝑅 −𝜑𝑛 𝑘𝑠𝑅

𝑡𝑛ℎ= −

𝑗𝑛 𝑘𝑅

ℎ𝑛+

𝑘𝑅

𝜇𝑠𝜇𝜑𝑛 𝑘𝑅 −𝜑𝑛 𝑘𝑠𝑅

𝜇𝑠𝜇𝜑𝑛

+𝑘𝑅 −𝜑𝑛 𝑘𝑠𝑅

78

Quasi-static limit – electric dipole

Electric dipole term (n=1) dominates for small particle sizes :

lim𝑥→0

𝜑𝑛 𝑥 → 𝑛 + 1

lim𝑥→0

𝜑𝑛+

𝑥 → −𝑛

lim𝑥→0

𝑡𝑛𝑒𝑥 = lim

𝑥→0−

𝑗𝑛 𝑥

ℎ𝑛+

𝑥

휀𝑠휀𝜑𝑛 𝑥 −𝜑𝑛 𝜌𝑠𝑥

휀𝑠휀 𝜑𝑛

+𝑥 −𝜑𝑛 𝜌𝑠𝑥

→𝑖𝑥2𝑛+1 𝑛 + 1

2𝑛 − 1 ‼ 2𝑛 + 1 ‼

휀𝑠휀− 1

휀𝑠휀 𝑛 + 𝑛 + 1

lim𝑥→0

𝑗𝑛 𝑥 →𝑥𝑛

2𝑛 + 1 ‼

lim𝑥→0

ℎ𝑛 𝑥 →2𝑛 − 1 ‼

𝑖𝑥𝑛+1

𝑥 = 𝑘𝑅

lim𝑘𝑅→0

𝑡𝑛𝑒𝑘𝑅 → 𝑖𝑘𝑅3

2

3

휀𝑠 − 휀

휀𝑠 + 2휀

79

Polarizability approach (plasmonics)

( ) ( ) 3

16 et ik p

Electric dipole moment and polarizability,

( )0 exc p E s

( )

𝛼0 𝜔 ≡ lim𝑘𝑅→0

𝛼 𝜔 = 4𝜋𝑅3휀𝑠 − 휀

휀𝑠 + 2휀

Electric dipole polarizability can be deduced from Mie theory !

80

Polarizability approach

( ) ( ) 3

16 et ik p

Electric dipole moment and polarizability,

Infinite ? dielectric response

When 2s

No !

( )0 exc p E s

( )

Unitarity imposes :

( )0

0

3lim 42

s b

kRb s

R

p

𝛼 𝜔 =𝐴 𝜔

1 − 𝑖𝑘3

6𝜋𝐴 𝜔

lim𝜔→0

𝛼 𝜔 =𝛼0

1 − 𝑖𝑘3

6𝜋𝛼0

α 𝜔 ≤6𝜋

𝑘3

81

Unitary limit on cross sections

For a dipole resonance :

Flux conservation : cross section limit doesn’t depend on the

properties of the resonator !

𝜎ext =2𝜋

𝑘2

𝑛=1

∞

2𝑛 + 1 Re 𝑎𝑛 + 𝑏𝑛Unitarity

≤𝜆2

2𝜋

𝑛=1

∞

2𝑛 + 1 2

𝜎𝑑 ≤3𝜆2

2𝜋

𝜎𝑑𝑒

𝜋𝑅2≤

6

𝑘𝑅 2

Plasmonic physics : Electric dipole modes

kR

Log10[s/pR2 ] s/b= -2.2

82

1 2 3 4 5

-8

-6

-4

-2

2

6

𝑘𝑅 2𝜎1

𝑒

𝜋𝑅2

𝜎tot𝜋𝑅2

𝜎1𝑒

𝜋𝑅2≤

6

𝑘𝑅 2

lim𝑥→0

𝑡1𝑒𝑥 =→ 𝑖𝑥3

2

3

휀𝑠 − 휀

휀𝑠 + 2휀

83

Polarizability picture for electric dipoles

( )0 exc p E

skc

Energy conservation : Im α 𝜔 =𝑘3

6𝜋α 𝜔 2

lim𝜔→0

𝛼 𝜔 = 4𝜋𝑅3휀𝑠 − 휀

2휀 + 휀𝑠

Unitary limit : α 𝜔 ≤

6𝜋

𝑘3

α 𝜔u.l. 6𝜋

𝑖𝑘3

0.5 1 1.5 2

-3

-2

-1

1

2

3

Plasmonic physics : Electric quadrupole modes ?

kR

Log10[s/pR2 ] s/b= -1.53

84

10

𝑘𝑅 2

𝜎1𝑒

𝜋𝑅2

𝜎

𝜋𝑅2

𝜎2𝑒

𝜋𝑅2≤

10

𝑘𝑅 2

lim𝑥→0

𝑡2𝑒𝑥 =→

𝑖𝑥5

15

휀𝑠 − 휀

2휀𝑠 + 3휀