• The motivation of the work

On the general solution of the Maxwell equations and the impact on the

absorbing boundary conditions

J.-P. BERENGER

Centre d’Analyse de Défense 16 bis, Avenue Prieur de la Côte d’Or

94114 Arcueil, France jpberenger@gmail.com

Outline

• The motivation of the work

• The solutions of the Maxwell equations revisited in a vacuum

• The derivation of a relationship between the size of the radiating structures and the rate of decrease of the

surrounding evanescent fields

• The consequence for the optimization and design of the

absorbing boundary conditions

The motivation of the work

• The Operator ABCs (Mur, Higdon) reflect in totality evanescent waves -> must be placed far from scatterers.

• The PML ABC is better because of a unique reason: it can absorb evanescent waves:

.

. The CFS-PML: almost perfectly

™ Evanescent waves play an essential role in the effectiveness (or non effectiveness) of Absorbing Boundary Conditions.

™ A good knowledge/understanding of the evanescent waves present in problems of interest is needed to design, optimize,

render reliable, the PML ABC. Or to imagine novel ABCs.

Evanescent waves in some problems

a x

c x c y

t j

j e e

e ^χ

χ ω ω ω

ψ

ψ

−

±

= ω

χ ω^cutoff

a c n

cutoff

ω = π

with:

™ In waveguides: well described in textbooks, for various kind of guides (rectangular, circular, …):

Parallel plate guide:

™ Evanescent waves also can be found for other problems:

Scattering from corrugated surfaces (J.A.

Kong, p. 227)

θ

2 b

1 sin

sinh

2

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ +

±

= ω

θ π

χ a

c x n

y c jc

t

j e e

e ^χ

χ ω ω ω

ψ

ψ

Interaction of an incident wave with a scatterer (1)

™ Scattered field expressed as a sum of an infinity of terms in special cases (sphere, cylinder).

™ No general theory:

1/ what kind of evanescent field around the object?

2/ what is the characteristic length of decrease of the evanescent waves?

1/ and 2/ are essential for designing ABCs

Incident wave

Scattered field

Scattered field

Object

Interaction of an incident wave with a scatterer (2)

Propagative field

Propagative field Evanescent region

Object

w w

w

w

w Evanescent region

What is known:

Evanescent waves Traveling waves

Frequency

w f c

0 =2

Below resonance frequency the field is evanescent.

1

The evanescent field strongly decrease upon a length of the order of the object size.

2

Interaction of an incident wave with a scatterer (3)

The behavior of evanescent fields can be empirically observed by means of experiments with analytical ABCs.

dHig

dHig Higdon ABC

Object = 300-cell thin plate

Î Empirical recommendation for use of Mur or Higdon ABCs:

Object-ABC separation ~ Object size

Interaction of an incident wave with a scatterer (4)

™ In view of optimizing the CFS-PML the form of the evanescent waves has been postulated on an intuitive basis:

c d

e

ωsinh

−

w p c χ ω¹ sinh ≈

1- Same form as in a waveguide 2 - The magnitude is negligible

at d = w from the structure

c w

e

e

=

d

(with p = 1 or 2)

This yields the evanescence coefficient:

™ Using this empirical waveform, a dramatic reduction of

the CFS-PML thickness can be achieved.

Interaction of an incident wave with a scatterer (5)

- An object 1500 FDTD cells in length.

- PML-Object separation = 2 cells.

- Experiments with CFS-PML 3 and 4 cells in thickness.

11

1500 cells 150 cells

1 Ei

Hi

Good results with 4 cell thick PML – Acceptable with 3 cell thick PML

Interaction of an incident wave with a scatterer (6)

™ Thus, there is a need of theoretical basis and better understanding on what is present around scattering objects.

™ This has been done by:

1 - Revisiting the solutions of the Maxwell Equations so as to clarify the evanescent wave “concept”.

2 - Trying to connect the decrease of the evanescent fields to

the size of the object .

The solutions of the Maxwell equations in 2D (1)

y H t

E_{x z}

∂

= ∂

∂

∂ ε0

x H t

E_{y z}

∂

−∂

∂ =

∂ ε0

x E y E t

H_{z x y}

∂

−∂

∂

= ∂

∂

∂ μ0

) (

0

y k x k t j

j x y

e

e ^{− +}

=ψ ^ω ψ

ω θ c cos k_x =

ω θ c sin k_y =

) sin (cos

0

y c x

t j

j

e

e

ω ω

ψ

ψ ⁼

2 2

2 2

y

x

k

c = k + ω

θ

x y

Propagation Constant phase Constant magnitude

(Equation of dispersion)

The solution is then expressed as:

Ex Ey case Hz

=> a plane wave propagating in direction θ.

This is the solution considered (explicitly or implicitly) in textbooks.

into the system

The solutions of the Maxwell equations in 2D (2)

• However, more general solutions can be found from the equation of dispersion:

where:

) sin cos

( ) sinh

sin cos

cosh ( 0

θ θ

ω χ θ

χ θ

ψ

ψ e

e

⎢⎣ ⎤

⎡ − +

=

c Y c X

t j

e

e ^χ

χ ω

ψ

ψ

cosh 0

⎥⎦ −

⎢⎣ ⎤

= ⎡ −

θ

x y

Y

X

X = x cos θ+ y sin θ Y = y cos θ- x sin θ

θ χ θ

χ θ

χ, ) cosh cos sinh sin

( j

C = +

θ χ θ

χ θ

χ, ) cosh sin sinh cos

( j

S = −

) , (χ θ ωC k_x = c

) , (χ θ ω S k_y = c

So that the waveform read:

By means of the coordinate change in the figure: The solution propagates in direction X and its magnitude is evanescent in direction Y

The solutions of the Maxwell equations in 2D (3)

c Y c X

t j

e

e

χ ω

ψ

ψ

cosh 0

⎥⎦ −

⎢⎣ ⎤

=

The phase varies in X The magnitude varies in Y

• Finally, the general solution of the Maxwell Equations reads:

• Does this general form hold the known special cases? Yes

c x c y

t j

j e e

e ^χ

χ ω ω ω

ψ

ψ

a θ

x y

Y

X X = x cos θ+ y sin θ Y = y cos θ- x sin θ

as an example for waveguides:

Ψ is of the general from. The evanescence coefficient χ depends on the guide size, i.e. on the boundary conditions.

The solutions of the Maxwell equations in 3D (1)

• The Maxwell equations yield:

) cos sin

sin sin

cos ( 0

θ θ

ϕ θ

ω ϕ

ψ

ψ = e

e

2 2

2 2

2

z y

x k k

c = k + +

ω

θ ω ϕ

sin c cos

k_x =

θ ω ϕ

sin c sin

k_y =

ω θ c cos k_z =

The solution is usually given as:

ϕ θ

x

y z

A plane wave propagation in direction (ϕ, θ)

The solutions of the Maxwell equations in 3D (2)

More general solutions can be found from the equation of dispersion:

( )

[ ^{χ ϕ θ χ ϕ θ η ϕ η} ]

ω

= j

k_x c

( )

[ ^{χ ϕ θ χ ϕ θ η ϕ η} ]

ω

= j

k_y c

[ ^{χ θ χ θ η} ]

ω cosh cos

sinh sin sin

k_z = c −

It can be verified that:

- The set k_x, k_y, k_ysatisfies the 3D equation of dispersion.

- If χ = 0 the set reduces to the usual (non evanescent) case.

- If θ = π/2 and η = 0, the set reduces to the 2D case (where ϕ is denoted as θ) where η is an additional angle

¾

How to interpret the waveform? As in 2D, by means of a change of coordinates

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟ =

⎟⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

z y x M Z

Y X

where X, Y, Z is a system of perpendicular coordinates defined by the angles ϕ, θ, η.

The solutions of the Maxwell equations in 3D (3)

ϕ θ

y z

η

η

X Y

Z

π/2−θ

( )

[ ϕ θ η ϕ η ϕ θ η ϕ η θ η]

ω χ

θ θ

ϕ θ

ϕ ω χ

ψ

ψ

sin sin ) cos cos sin

cos sin ( ) cos sin sin

cos (cos sinh

cos sin

sin sin

cos cosh

0

z y

c x

z y

c x t j

j

e e e

+ +

− + +

−

−

+ +

=

¾

The resulting waveform reads:

The solutions of the Maxwell equations in 3D (4)

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

− +

−

+

−

−

−

=

η θ

η ϕ

η θ

ϕ η

ϕ η

θ ϕ

η θ

η ϕ

η θ

ϕ η

ϕ η

θ ϕ

θ θ

ϕ θ

ϕ

cos sin

sin cos

cos cos

sin sin

sin cos

cos cos

sin sin

cos cos

sin cos

sin cos

sin sin

cos cos

cos sin

sin sin

cos M

( )

[ ^{ϕ θ η ϕ η ϕ θ η ϕ η θ η}]

ω χ

θ θ

ϕ θ

ϕ ω χ

ψ

ψ

sin sin ) cos cos sin

cos sin ( ) cos sin sin

cos (cos sinh

cos sin

sin sin

cos cosh

0

z y

c x

z y

c x t j

j

e e e

+ +

− + +

−

−

+ +

=

c Z c Y

c X t j

j

e e e

e

0

χ ω χ ω

ω ω

ψ

ψ =

Using the matrix of the change of coordinates:

The waveform becomes:

The solutions of the Maxwell equations in 3D (5)

ϕ θ

x

y z

η η

X Y

Z

π/2−θ

c Y c X

t j

j

e e

e

χ ω ω ω

ψ

ψ =

The wave:

- propagates in direction X defined by ϕ and θ - is evanescent in direction Y defined by angle η

This is the general solution of the Maxwell

Equations in a vacuum (the general eigen-mode).

Radiated Evanescent Field (1)

™

We consider a scattering or radiating 2D object

c Y c X

t j

j

e e

e

χ ω ω ω

ψ

ψ =

™

We need to evaluate (to find some characteristics of) the

evanescence coefficient χ of the evanescent waves that surround the object:

-At low frequency, this is f << resonance = c / 2 w - the object is assumed as symmetric, of size w

- - -

-

+ + +

+ +

+ -

-

size w

Is χ connected with size w ???

Radiated Evanescent Field (2)

c y c x

t j

j e e

e E

E ^χ

χ ω

ω ω^{cosh sinh} 0

−

= −

propagation evanescence

x y

x E_y

w/2 - w/2

We consider an evanescent wave whose propagation is parallel to the object:

We are at low frequency => at y=y

the vertical electric field is of the form:

⎟⎠

⎜ ⎞

⎝⎛ +

⎟−

⎠

⎜ ⎞

⎝⎛ −

= 2 2

)

( w

x w f

x f x E_y

f(x) =

Function f(x) may be any function, what is important is the addition of two f(x) centred at –w/2 and w/2.

y = y₀

Radiated Evanescent Field (3)

Defining

∫

the E field in k domain read:

) 2 (

sin 2 )

( )

( ^{2 2} wk g k

j e

e k g k

E

kw w j

k j

y ⎟

⎠

⎜ ⎞

⎝

= ⎛

⎥⎦

⎢ ⎤

⎣

⎡ −

= ⁻

π/w 5π/w k

k sin(wk/2)

g(k)

m w

k_m

π

) 1 2

( +

Peak values at:

2 4 /

) 4 / ) sin(

( w

k w

k k w

g ⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

if f(x) =

w

k E_y(k)

π/w 5π/w

3π/w

Radiated Evanescent Field (4)

The evanescent waves

∫

=e ^ω e−^{ω χ} a χ e ^{ω χ} dχ y

x

E_y( , ₀) ^{j t csinh y0} ( ) ^{jccosh x}

must match the field

∫

= wk e dk

k g e

y x

E_y ^{j t jkx}

sin 2 ) ( )

,

( ₀ ^ω

The phase terms must match each other, i.e. k = ω coshχ/c. From this,

/ 2) and present peak values for:

m w

c ^m k^m

χ π

ω cosh = = (2 +1)

χ a(χ)

χ₂ χ₀

χ₁

Radiated Evanescent Field (5)

For the fundamental value we have:

w c

χ π

ω ₌

cosh

For a waveguide we know:

a

a c

χ π

ω ₌

cosh

Same evanescence coefficient χ with just

object size replaced with waveguide size.

Radiated Evanescent Field (6)

The decrease of the corresponding evanescent wave is then

y w c y c

c

e

e

sinh _{2 2} 1

2

2 −

− −

=

π χ ω

ω

The “fundamental mode” decreases by exp(-π) = 0.043 upon the size w of the object. The other modes decrease faster, [just replace π with

(2m+1) π in the above exponential].

On a pure empirical basis we postulated in the past a decrease as

w y y p

c

e

e

=

This is in accordance with the above derivation with just p = π.

wy

c y

e

e

χ π

ω ₋

− ^sinh

=

for f << resonance c/2w

Radiated Evanescent Field (7)

• For waves that propagate obliquely

θ ω χ

θ ω χ

θ ω χ

θ ω χ

ω ^{cosh cos cosh sin sinh cos sinh sin}

0

c x c y

c y j c x

t j

j

e e e e

e E

E =

propagation

evanescence

x y

θ

θ χ π

ω

) cos 1 2

(

cosh m w

c ^m = +

Phase matching in x

w y

c y

e

e

χ π ω

sinh − cos

−

=

Radiated Evanescent Field - Summary

w d

e d

E

α

= )

(

E Field

There exists other contributions that decrease more rapidly (high order modes and oblique waves)

Evanescent field decreases rapidly with distance d from a radiating structure.

Another important result:

w c

χ π

ω ₌

cosh or, far below resonance: cons t A

w

c = =

= tan

sinχ π ω

(drawn to scale)

The fundamental “mode” decreases about:

Application to PML optimization

w c

o

ε0

π α =

c x vacuum

x

e x x

θ ε χ

σ

ψ

ψ( ) ( ) ^{0 cosh cos}

= −

ε ω α

σ s j

x x

x =1+ +

Normal PML

CFS-PML

Enormous absorption at low frequency

=> total reflection

=A χ ωsin

(Kuzuoglu and Mittra, 1996)

c x c x

f f vacuum

PML

x x x

e e

θ α χ

σ θ ω

ε χ σ

ψ ψ

ψ ^{α sinh sin} ₀ ^{1 ⎟⎟⎠sinh sin}

⎜⎜ ⎞

⎝

⎛ +

= ε =

π α

α = 2

<< f f

= A χ ωsin

Since the attenuation in independent of frequency

From this, the attenuation of the evanescent wave can be set equal to the attenuation of traveling waves (reasonable value, say 40-80 dB), by using the optimum α:

(differs from published values with π because p = 1 replaced with p = π)

Application to PML optimization

- An object 1500 FDTD cells in length.

- PML-Object separation = 2 cells.

- Experiments with CFS-PML 3 and 4 cells in thickness.

11

1500 cells 150 cells

1 Ei

Hi

Good results with 4 cell thick PML – Acceptable with 3 cell thick PML

Design of novel ABCs

-A Huygens analytical ABC to absorb the traveling waves

-A real stretch of FDTD mesh to absorb (annihilate) the evanescent waves.

A simple idea: the combination of:

The Huygens ABC

Zero field

Zero field Zero

field

Zero field

Any boundary condition

Source field

Outgoing field

Outgoing field

Huygens surface radiating field opposite to outgoing field

Introduced independently in special cases: Multiple absorbing surfaces (Sudiarta, 2003) and Teleportation (Diaz and Scherbatko, 2004). Later generalized as

Huygens ABC (Berenger, 2007).

The principle is simple:

radiating a wave

opposite to the outgoing field.

Combination of Huygens ABC with a stretched mesh

- A Huygens ABC to absorb traveling waves (highest frequencies).

- Outside the HABC only low frequency evanescent waves are present. For a scatterer of size W they decrease in function of distance as:

=> A very coarse mesh region (real stretch of coordinates) can be used to “absorb”

evanescent waves (i.e. to permit natural decrease). We can hope the needed coarse mesh be < 10-20 cells in thickness.

Coarse Mesh

w d

e E d

E( ) = ₀ ⁻

Frequency Traveling waves

Evanescent waves

Stretch of Mesh HABC (Higdon)

λ = 2 w

The proposed idea: ^HABC

(Higdon)

No ABC Coarse Mesh

Coarse Mesh

Coarse Mesh

Scatterer

Use of a stretched mesh outside the HABC

Δ_max

Fine mesh HABC (Extensions not drawn)

Coarse Mesh Mesh Transition

0 3 4 4+ng

Transition Coarse

Distance from Object in cells Spatial step

Δ

Object HSG

ng cells growing geometrically

- HSG 3 cells from object - 4 cells of constant size Δ

- ng cells that grow geometrically from Δto Δ_max

Characteristic length of decrease of surrounding evanescent waves = size w of object

=> Maximum step Δ_max probably of the order of size w.

An experiment with a stretched mesh

=> 17 stretched cells can replace 900 non-stretched cells

In this experiment the transition region is constant (ng = 8 cells) and the maximum step Δ_max varies from w / 10 to W / 1.

dPEC < 40 cells Ù900 non-stretched cells

HABC PEC

The size of the object is 300 FDTD cells

Summary

• The main feature (the rate of decrease) of the evanescent waves can be related to the size of the radiating object.

• This permits optimization of existing ABCs and design of novel ABCs.

• The theory remains to be re-written in a more rigorous form (the field = sum of evanescent waves). And

computational tests could be performed (finding the

development in evanescent waves on a surface and

verifying that it produces correct field elsewhere).