ELECTROMAGNETIC PROPERTIES OF AGGREGATED SPHERES

REVISITED

Vadim A Markel

University of Pennsylvania,Philadelphia

Departments of Radiology and Bioengineering

vmarkel@mail.med.upenn.edu

http://whale.seas.upenn.edu/vmarkel

CO-AUTHORS• V.N.Pustovit (Jackson State Univ.)• S.V.Karpov (L.V.Kirenskiy Inst., Krasnoyarsk)• V.S.Gerasimov and I.L.Isaev (Krasnoyarsk Technical University)• A.V.Obuschenko (Moscow Inst. of Physics and Technology)

ACKNOWLEDGEMENTS• ARO• Russian Academy of Sciences• Russian Foundation for Basic Research

REFERENCES• “Electromagnetic Density of States and Absorption of Radiation by

Aggregates of Nanosphereswith Multipole Interactions,” PRB 70, 54202 (2004).

• “Local Anisotropy and Giant Enhancement of Local Electromagnetic Fields in Fractal Aggregates of Metal Nanoparticles,” preprint: physics/0507202 (2005).

Problem: calculate optical (more generally, electromagnetic) responses of large fractal aggregates of metal nanospheres.

MOTIVATION

• Giant enhancement of effective nonlinear optical susceptibilities

• Localization of electromagnetic energy• Optical memory• Soot particles often have fractal geometry

(of interest in atmospheric optics).

Theoretical and Computational Approaches

1) Dipole approximation- 3N equations (N – number of nanospheres)- Easy to generalize beyond quasistatics- Inacurate for touching spheres

2) Geometrical renormalization- An approximation- Corrects to some extent the deficiency of the dipole approximation- Still 3N equations

3) Coupled multipoles- L(L+2)N equations (L – maximum order of multipole moment included in computations)- Slow convergence with L for conducting spheres in close contact- Computational complexity grows as L6

General Approach (Extinction And DOS)

ee

tot

2

2

( )4 Im( )

( ) 1

4 ( ) 2( )3 ( ) 1

w dwkV z w

w dw

mzm

σε πλ

π λλλ

Γ= =

−

Γ =

+=

−

∫

∫

Definition of DOSΓ(w) = Σn |cn|2δ(w-wn)

W|n> = wn|n>

Operator W is, in general, infinite-dimensional.Dipole approximation: size 3NTruncated coupled multipoles: size L(L+2)N

Is W sparse? Yes, approximately. But scarcity factor is not that large and using sparse solvers (e.g., PARDISO) does not help much.

Numerical Methods of Calculating the Spectra

( )( )=Im , w dwz z X iz w

θ δΓ= −

−∫

Calculation of spectra is based on different approximations forthe expression

0

In the limit 0 this corresponds to direct calculation of ( ) because lim[ ( , )] ( )

XX X

δ

δ

θ δ π→

→

Γ= Γ

Numerical Methods of Calculating the Spectra (cont.)

2

20

21

1 22

2

Three approaches for calculating :a) By expressing as a sum

( )

b) By expressing as a continued fraction

( )=

c) By choosing an analytical model f

n

n

czz w

bzbz a

bz az

θθ

θ

θ

θ

=−

− −− −

−

∑

…or ( )wΓ

Numerical Methods of Calculating the Spectra (cont.)

a) Expressing as a sum requires diagonalization of a matrix of the size ( 2) where - number of spherical particles - maximum multipole order (i) Numerical complexity: O(

M NL LNL

θ= +

3

2

) (ii) Memory requirement: O( ) (iii) Dipole approximation: 1

MM

L =

Numerical Methods of Calculating the Spectra (cont.)

2

b) Expressing as a continued fraction requires matrix-vector multiplications for a matrix of the size ( 2) (i) Numerical complexity: O( ) (ii) Memory requirement:

KM NL L

KM

θ

= +

2 (iii) -the order approximation ( ) has the same first moments as the true density ( ), where

( )

K

n

nn

MK X

KX

X X

μ

μ

Γ

Γ

= Γ∫

exactly

dXBack to specific tasks

Sum Rules

2e

3

0

Let

Then 4 (for all materials/shapes)

( ) 4

- diagonal element of the electrostatic polarizability tensor

zz

zz

z X i

dXk

d

δε π

σ λ λ π α

α

∞

= −

=

=

∫

∫

Spectral Variable z=X-iδ for Drudean Materials

2p2

2p

F2F

FF

F

F

For a Drudean material 1( )

4Re 1 , where 3 3

8If , then 3

4 3

mi

X z

X

ωω ω γ

ωπ ω ωω

ω ωπω ωω

π γδω

= −+

⎛ ⎞= = − =⎜ ⎟

⎝ ⎠−

≈ ≈

≈

Long-Wavelength Limit

2F

If ( 0),4 4 , 03 3

( - electrostatic conductivity)

X

λ ωπ π ωγ ωδ

ω σσ

→∞ →

≈ ≈ = →

2e 2

In this limit 1/ , if 4 / 3 "band"441/ , if 4 / 3 "band"3

k δ

λ ππε πλ π

∈⎧⎛ ⎞= Γ ∝ ⎨⎜ ⎟ ∉⎝ ⎠ ⎩

Convergence with CF order (N=50)

Convergence of Extinction Cross Section with L (N=2)

Convergence of Extinction Cross Section with L (N=50 vs N=2)

Convergence of DOS with L: (N=2, Parallel Polarization)

23π

−43π

inf. cyl.w⊥

inf. cyl.w

0.1( )XΓ

X

Convergence of DOS with L: (N=2, Parallel Polarization)

inf. cyl.w⊥

23π

−83π

−

flat discw⊥

0.1( )XΓ

X

inf. cyl.w

43π

Comparison of DOS: Two Spheres and Cylinder (Parallel Polarization)

0.1( )XΓ

X

DOS for an Infinite Linear Chain

DOS for Linear Chains of Different Length (Parallel Polarization)

DOS for Linear Chains of Different Length (Orthogonal Polarization)

DOS for a lattice CCA cluster for Different L (D=1.8)

Convergence of the First few Moments of DOS with L

Rotationally Averaged DOS for Different Types of Aggregates

Rotationally Averaged DOS for Offlattice Fractal Aggregates with Different D

Local Anisotrropy Factor

Local Dipole Moments vs. Local Anisotropy Factor

L=16

h = 0.05R(surface layer)

Correlation of Local Dipole Moments and Local Anisotropy Factors

CONCLUSIONS• For touching metal spheres, very high orders of multipole

meoments must be taken into account• Correspondingly, internal fields inside the nanospheres

is highly inhomogeneous• This has important consequences for nonlinear

susceptibilities (more work must be done)• Optical properties of large fractal aggregates are much

more determined by local geometry than previously thought. Large scale structure plays, perhaps, a minor role.

• Lattice and off-lattice fractal aggregates have similar electromagnetic properties after rotationalo averaging.

Fractals: Dipole approximation vs L=64

0.2 ( )XΓ

X

Renormalized Dipole Approximation DOS vs L=64 DOS

X

0.2 ( )XΓ

From analogy with DDA: =1.612

From requirement that a chain of spheres has the same depolarization coefficients as an infinite cylinder: =1.688

Fr

ξ

ξ

om conservation of thesecond moment of DOS: =1.788ξ

Wavelength Dependence:Clusters with different D: Fe

-1e , mε μ

, mλ μ

Wavelength Dependence:Clusters with different D: Pd

-1e , mε μ

, mλ μ

Wavelength Dependence:Clusters with different D: Al

-1e , mε μ

, mλ μ

CCA Cluster vs. Two Spheres: Fe

, mλ μ

-1e , mε μ

CCA Cluster vs. Two Spheres: Pd

, mλ μ

-1e , mε μ