Method of Scalar Potentials for the Solution of Maxwell’s Equations in Three Dimensions

Nail A. GumerovInstitute for Advanced Computer Studies University of Marylandhttp://www.umiacs.umd.edu/~gumerov

In collaboration with Ramani Duraiswami

CSCAMM Seminar on March 8, 2006

In literature one can find opinions that the Debye potentials are good only for solution of problems for Maxwell’s equations with spherical boundaries…

We show how to construct an arbitrary solution and obtain an efficient computational algorithm…

Outline

IntroductionA bit on Maxwell’s equationsScalar (Debye) potentialsTranslation operatorsApplication example: Multiple scattering from spheres

Boundary conditions for scalar potentialsMie scatteringMultiple scatteringFast Multipole Method (FMM)

Conclusion (Future work)

Introduction (Lots of Applications)

Boundary Value Problems for

Maxwell’s Equations

• Antennas and Radars• Optical Instruments• Imaging• Communications• Electronic Devices• Environmental (Buildings)• MEMS• Photonic Structures• Nanomanipulations• Crystallography• Etc.

Introduction (Wavelength is a Scale)

Problems can be classified based on the ratio

of the typical problem length scale Lto the wavelength λ

k = 2π/λ − wavenumber

E.g. Optics: λ ∼ 500 nm:

0.011 nm

101 μm

10,0001 mmkLL

High Frequencies (Geometrical Optics)

Moderate Frequencies (Diffraction)

Low Frequencies (Electrostatics)

Region of ourcurrent interest

Asymptotic Theory, kL >> 1

Asymptotic Theory, kL << 1

Maxwell’s EquationsNo charges and currents: Electric field vector

Magnetic field vector

Circular frequency

Magnetic permeability

Electric permittivity

Speed of light

Constrained Vector Helmholtz Equations

Since

Eight equations,but six unknowns!

Scalar (Debye) Potentials

(prove can be found elsewhere, e.g. in our book)

Debye’s decomposition

Radiation ConditionsSilver-Müller radiation condition:

Sommerfeld radiation condition:

Incident field

Scattered field

Scatterer

Both conditionsrepresent the same physical fact which follows, from the Huygens Principle.

Expansions over Spherical Wave Functions

Spherical Bessel functions

Spherical Hankel functions

Spherical harmonics

Associated Legendre functions

Scalar:

Vector:

Function and Operator Representations

Scalar case:

1). Differential Operators:

For example, Sparse matrix

Depends on the Basis

Function and Operator Representations (2)Scalar case:

2). Translation Operators:

3). Rotation Operators:

Properties specific for Helmholtz equation:

Dense matrices

(Also commute with differential operators).

4). Rotation-Coaxial Translation Decomposition:

Program That Was Performed for Vector Basis Functions

Based on Properties of Vector Spherical Harmonics and Vector Addition Theorems:

Vector Differential Operators;Vector Translation Operators.

O.R. Cruzan, Translational addition theorems for spherical vector wave functions, Q. Appl. Math., 20, 33-39 (1962).

W.C. Chew and Y.M. Wang, Efficient ways to compute the vector addition theorem, J. Electromagnetic Waves and Applications, 7(5), 651-665 (1993).

Y.-L. Xu, Calculation of the addition coefficients in electromagnetic multisphere-scattering theory, J. Comp. Physics, 127, 285-298 (1996);erratum, J. Comp. Phys. 134, 200 (1997).

Some publications:

Why Scalar Potentials, Not Vector Basis Functions?

Scalar relations simpler…Scalar addition theorems simpler…Operations with scalar basis functions are simpler…Translation theory of scalar Helmholtz equation is well developed…If solution of Maxwell’s equations can be simply constructed from two solutions of scalar Helmholtz equation, why not to go in this way?

Answer: Simplicity.What’s the problem?

Some Problem…

Original decomposition:

Translation:

The Debye decomposition is not invariant with respect to translations!

How to get Debye’s decomposition in arbitrary reference frame?

Conversion Operators

Linear relations (operators):

So, representations:

Symmetry (due to similarly we can consider the magnetic field):

Derivation:

Diagonal operator in the bases of spherical scalar wavefunctions!

Represented by sparse matrices!

Define:

Conversion Operators are Represented by Sparse Matrices

Rotation-Coaxial Translation Decomposition

Debye’s decomposition is invariant with respect to rotation:

Conversion operators are very simple for coaxial translations:

Relation Between Representations of Electric Field Vector and Scalar Potentials

Given potentials, find field

Given field, find potentials

Proportional to H

Easy expansions of typical EM fields, e.g.:

Plane waves (via the Gegenbauer expansions);Dyadic Green’s function (electric and magnetic dipoles).

Multiple Scattering from Spheres

Incident Wave

ε, μ

εq, μq

Ein

Hin

kin

Scattered Waves

Escat Hscat

ScatterersTransmission conditions:

Radiating functions

J.H. Brunning and Y.T. Lo, Multiple scattering of EM waves by spheres, parts I and II, IEEE Trans. Antennas Propag., AP-19(3), 378-400 (1971).

Y.-L. Xu and R.T. Wang, Electromagnetic scattering by an aggregate of spheres: Theoretical and experimental study of the amplitude scattering matrix, Phys. Rev. E., 58(3), 3931-3948(1998).

Boundary Conditions for Scalar Potentials:

T-matrix for a Single Sphere (Reproduce Mie Solution)

Lorenz-Mie Coefficients(T-matrix entries)

Spherical Ricatti-Bessel functionsSpherical Ricatti-Hankel functions

Refraction index

Multiple Scattering

Some Computational Results(Theory Validation)

aq

bq

1). Truncation number selection

2). A posteriori error check

3). Comparison with available theoretical and experimental data.

Mie Scattering (Single Sphere)

0.1 1 10

5

10

15

20

ka

p 0

10-2 10-4

10-6

10-8

Conventional

ka = 10k

p

Error in Boundary Conditions (21x20 surface grid)

EM Energy Density

Incident wave vector

Incident wave polarization

50

Conventional:

In theory:

Perfect conductor

Two Spheres

ka = 10

Error in Boundary Conditions (21x20 surface grid for each sphere)

EM Energy Density

50

Perfect conductors

1 1.5 2 2.5 3

4

8

12

16

20

δ

p 0

10-2

10-4

10-6

10-8

Conventional

ka=5

k

p

p

k

100 Random Spheres

kp

kamax= 10, kD = 139

0 10 20 30 40 5010

-8

10-6

10-4

10-2

100

Iteration #

Abs

olut

e er

ror i

n ex

pans

ion

coef

ficie

nts

N=100, ka=10

GMRES

0 1 2 3 4

x 104

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Node #

Rel

ativ

e E

rror o

n th

e B

ound

ary

(Ele

ctric

)

N=100, ka=10, #nodes=38200

Imaging plane

Log of EM Energy Density

ε int/ε =10+0.1i,μ int/μ =1.

p

200 400 600 800 1000

100

200

300

400

500

600

700

800

900

1000

0 1 2 3 4 5

x 104

10-8

10-7

10-6

10-5

10-4

Node #

Rel

ativ

e E

rror o

n th

e B

ound

ary

(Ele

ctric

)

N=125, ka=15, #nodes=47750

0 10 20 30 40 5010

-10

10-8

10-6

10-4

10-2

100

Iteration #

Abs

olut

e er

ror i

n ex

pans

ion

coef

ficie

nts GMRES

N=125, ka=15

Grid of 125 Spheresk

kamax= 15, kD = 210

Imaging plane

Log of EM Energy Density

ε int/ε =10+0.1i,μ int/μ =1.

Scattering Matrix Measurements and Computations

Amplitude Scattering Matrix,depends on the spherical polar angles θ and φ.(can be expressed via the far-field of the scalar potentials)

Scattering Matrix Measurements and Computations (Two Touching Spheres)

0 60 120 180

-2

0

2

4

θ ο

log 10

(i 11)

experimentXupresent

two sphere configuration, ka1=ka2=7.86

0 60 120 180-2

0

2

4

θ ο

log 10

(i 22)

experimentXupresent

two sphere configuration, ka1=ka2=7.86

Optical BK7 glass:

Compared with experiments and computations of Xu & Gustafson (1997-2003)

Scattering Matrix Measurements and Computations (15 Touching Spheres)

Compared with experiments and computations of Xu & Gustafson (1997-2003)

z0 60 120 180

-2

0

2

4

θ ο

log 10

(i 11)

experimentXu present

15 sphere configuration, ka1-3=7.86, ka4-15=5.03

0 60 120 180-2

0

2

4

θ ο

log 10

(i 22)

experimentXu present

15 sphere configuration, ka1-3=7.86, ka4-15=5.03

0 60 120 180

-2

0

2

θ ο

log 10

(i 12)

experimentXu present

15 sphere configuration, ka1-3=7.86, ka4-15=5.03

k

p

xSmaller spheres acrylic,Larger spheres BK7 glass.

Fast Multipole MethodUses hierarchical data structures (octrees) to build multipole and local expansions for each box occupied by scatterers;Uses translation operators to reexpand solutions;Can speed up matrix-vector product from O(N2) to O(NlogN);For the Helmholtz (or Maxwell) equations complexity strongly depends on the complexity of translation operations;For volume distribution of scatterers to provide O(NlogN) complexity the cost of single translation should be not more than O(p3);Computations of translation operators using Gaunt or Clebsch-Gordancoefficients (or 3-j Wigner symbols) requires O(p5) operations, and results in O(N5/3) algorithms (if matrices precomputed and stored this complexity reduces to O(N4/3)).Rotation-Coaxial Translation decomposition + sparse conversion operators provide O(p3) single translation cost and O(NlogN) algorithm for volume scattering;Method of scalar potentials brings an algorithm for solution of Maxwell’s equations which is approximately 2 times slower than solution of scalar Helmholtz equation (Maxwell=2Helmholtz).

Some results for the scalar Helmholtz equation

0.1

1

10

100

1000

10000

10 100 1000 10000Number of Scatterers

CP

U T

ime

(s) Total

Matrix-Vector Multiplication

External Loop

Internal Loop

y = ax

y = bx

y = cx1.25

FMM+FGMRES

0.1

1

10

100

1000

10 100 1000 10000 100000Number of Scatterers

CP

U T

ime

Per

Iter

atio

n (s

)

y=ax

Volume Fraction = 0.2, ka=0.5

Periodically-Random Spatial Distributionof Spheres of Equal Size

FMM

l = 2max

2

3

44

y=bx2

Direct

Conclusion (Future work)

Debye potentials can be efficiently translated, which results in an efficient algorithm for the Maxwell’s equations (it is cheaper to solve 2 Helmholtz equations than 3 or 6 Helmholtz equations+divergency free conditions);More research is needed for efficient use of diagonal forms of the translation operators in the method of scalar potentials (while this can be used in more or less straightforward way);Efficient forms of boundary integral equations for the method of scalar potentials should be derived to solve problems with arbitrary boundaries;Efficient preconditioners for the Maxwell equations are needed.

THANKS !