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JOURNAL OF COMPUTATIONAL PHYSICS 127, 363379 (1996)ARTICLE NO. 0181

Three-Dimensional Perfectly Matched Layer for the Absorption Electromagnetic Waves

JEAN-PIERRE BERENGER

Centre dAnalyse de Defense, 16 bis, Avenue Prieur de la Cote dOr, 94114 Arcueil, France

Received March 13, 1995; revised October 12, 1995

nates, the six components yield 12 subcomponennoted as Exy , Exz , Eyz , Eyx , Ezx , Ezy , Hxy , HxzThe perfectly matched layer is a technique of free-space simula-

tion developed for solving unbounded electromagnetic problems Hyx , Hzx , Hzy , and the Maxwell equations are repwith the finite-difference time-domain method. Referred to as PML, by 12 equations,this technique has been described in a previous paper for two-

dimensional problems. The present paper is devoted to the three-

dimensional case. The theory of the perfectlymatchedlayer is gener-

Exy

t

yExy (Hzx Hzy)

yalized to three dimensions and some numerical experiments areshown to illustrate the efficiency of this new method of free-spacesimulation. 1996 Academic Press, Inc.

Exz

t zExz

(Hyz Hyx)

z

1. INTRODUCTION

Eyz

t zEyz

(Hxy Hxz)

zThe perfectly matched layer [1] is a new technique of

free-space simulation developed for solving unbounded

Eyx

t xEyx

(Hzx Hzy)

xproblems with the finite-difference time-domain method[2, 3]. Referred to as PML, this technique is based on theuse of a layer especially designed to absorb the electromag-

Ezx

t

xEzx (Hyz Hyx)

xnetic waves without reflection from the vacuumlayer in-terfaces. In [1] the layer and the free-space simulation

technique were presented for two-dimensional problems.

Ezy

t yEzy

(Hxy Hxz)

yThe present paper is devoted to the three-dimensionalcase.

The first part of the paper generalizes to three dimen- Hxy

t *y Hxy

(Ezx Ezy)

ysions the theory of the PML medium [1]. It is shownthat the theoretical properties of the 2D medium arepreserved with the 3D one. The PML medium is then

Hxz

t *z Hxz

(Eyz Eyx)

zapplied to build an absorbing layer to simulate free spaceon the outer boundary of a 3D computational domain.

Hyz

t *z Hyz

(Exy Exz)

zThe second part of the paper is devoted to numerical

experiments. Comparisons with two previously used tech-niques of free-space simulation are provided. These ex-

Hyx

t *x Hyx

(Ezx Ezy)

xperiments show that the 3D PML method is very efficientand allows the absorption of the electromagnetic wavesto be widely improved.

Hzx

t *x Hzx

(Eyz Eyx)

x2. THEORY OF THE 3D PERFECTLY

MATCHED LAYER

Hzy

t *y Hzy

(Exy Exz)

y,

2.1. Definition of the PML Medium

In the PML medium, each component of the electro- where the parameters (x , y , z , *x , *y , *z ) are homneous to electric and magnetic conductivities.magnetic field is split into two parts. In cartesian coordi-

363

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364 JEAN-PIERRE BERENGER

SyExy0 H0 cos hz

Sz(E0 cos ex Exy0) H0 cos hy

S *x Hzx0 E0 cos ey

S *y (H0 cos hz Hzx0) E0 cos ex ,

where

Su 1 iu

, S *u 1 i

*u

(u x,y, z).

Inserting Exy0 from (3.a) into (3.b), ..., and Hzx0 frominto (3.l), we obtain a system of six equations in whis the ratio E0/H0:

FIG. 1. Propagation of a plane wave in a PML medium.

Zcos ex

Sz cos hy

Sy cos hz

If both x y z and *x *y *z 0 we can note Zcos ey

Sxcos hz

Szcos hx

that (1) yields the Maxwell equations. Thus, the absorb-ing medium defined by (1) holds as particular cases allthe usual media (vacuum, conductive media). In the 2D Zcos ez

Sycos hx

Sxcos hy

cases [1], due to the symmetry of the problem, (1)reduces to a set of four equations. Only one component

1

Zcos hx

S *zcos ey

S *ycos ezis split into two subcomponents, either the magnetic

component in the TE case or the electric component inthe TM case.

1

Zcos hy

S*x cosez

S *z cosex

2.2. Propagation of a Plane Wave in a PML Medium 1

Zcos hz

S *ycos ex

S *xcos ey .

Let us consider a propagating plane wave whose electricand magnetic fields form the angles ex , ey , ez and hx ,

From (5), it is easy to show thathy , hz with the coordinate axis (Fig. 1). Let E0 , H0 betheir magnitudes and let Exy0 , Exz0 , Eyz0 , ..., Hzx0 , Hzy0 be

cos ex cos hx cos ey cos hy cos ez cos hz 0 the magnitudes of the 12 subcomponents. If this wavepropagates in the PML medium, the 12 subcomponentscan be written as

Sx

cos ex

Sy

cos ey

Sz

cos ez 0

Exy Exy0 ei(txyz) (2.a)

S *xcos hx

S*ycos hy

S*zcos hz 0

Exz Exz0 ei(txyz) (2.b)

2

SxS*x

2

SyS *y

2

SzS*z

Hzx Hzx0 ei(txyz) (2.k)

Equation (6) means that the electric and magneticHzy Hzy0 e

i(txyz). (2.l)are perpendicular in a PML medium. From (2) wwrite kx/, ky/, and kz/, so that (7.c)dispersion relation of the PML equations (1), connecInserting (2) into (1) yields the 12 equations

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ABSORPTION OF ELECTROMAGNETIC WAVES

to the wave numbers kx , ky , kz . As expected, ifx y propagating wave (2) is of the following form, in wc ( )1/2:z and *x *y *z 0 this equation reduces to the

dispersion relation of the Maxwell equations. Multiplyingthen (5.b) by cos hz Sz/S *z and (5.c) by cos hy Sy/S *y ,

0 ei(t(x cosxy cosyz cosz)/c) e(xcosx/c)x e(ycosy/c)y e(zcosubtracting the resulting equations, and taking into account

(7.b), one can show that

Thus, if the conductivities satisfy the matching imance condition, the phase propagates perpendiculathe (E, H) plane, as in a vacuum, and the magn Sx

SzS *z

cos ey cos hz SyS *ycos ez cos hy

Sx

S *xcos2 hx

Sy

S *ycos2 hy

Sz

S *zcos2 hz

Z. (8)of the wave is governed by the last three exponterms. This formula is analogous to that of thedimensional case [1]. The coordinates x, y, z are emded in separate absorbing terms, allowing the waSimilar formulas can be obtained for and from (5),be damped along one or two directions by selectior from (8) by circular permutations of the coordinates.adequate set of conductivities. If y z 0 a Inserting then and into (5.f) and taking into accountpropagating perpendicularly to the x axis is not abso(6), the impedance Z isas when y 0 in the 2D case. We can note tha3D theory includes the 2D theory [1] as a specialZ / (1/ G), (9)For instance, in the TE case: considering the andefined in [1] we have ex /2, ey ,where0, ez hx hy /2, so that (10) and (12)the corresponding equations of [1].

G wywz cos2 ex wzwx cos

2 ey wxwy cos2 ez

wx cos2 hx wy cos

2 hy wz cos2 hz

2.3. Transmission of a Wave through(10)

PMLPML Interfacesand

We will consider two PML media separated by an face perpendicular to the x axis (Fig. 2). In each m

wu Su

S *u

1 iu/

1

i*u /

(u x,y, z). (11) let us denote by (e , e) and (h , h) the angles thelectric and magnetic fields form with the (y, z) planthe y axis (Fig. 2). The angles ex , ..., hz defined

From (8), (9), and (11), we have previous section are related to the new angles by thlowing relations which are valid for both the electrimagnetic fields (u e or h):

GSx

wz cosey coshz wy cosez coshywx cos

2 hx wy cos2 hy wz cos

2 hz(12.a)

cos ux sin u

GSy

wx cosez coshx wz cosex coshzwx cos

2 hx wy cos2 hy wz cos

2 hz(12.b)

cos uy cos u cos u

cos uz cos u sin u .

G

Szwy cosex coshy wx cosey coshx

wx cos2

hx wy cos2

hy wz cos2

hz. (12.c)

Inserting (14) into (10) and (12) yields , , , G astions ofe , e , h , h . We now consider the transmissIf the couples of conductivities (x , *x ), (y , *y ), (z ,two polarizations of an incident wave: first, the tran*z ), satisfy the usual matching impedance conditionelectric polarization; second, the transverse magnet(/ */), we have wx wy wz 1 so that G 1.larization.Then, the impedance Z equals that of a medium (, ).

In this case, denoting by x , y , z the angles that the

Transverse Electric Case TEperpendicular to the (E, H) plane forms with the coordi-nate axis, in (12) the numerators in the ratios equalrespectively cos x , cos y , cos z , and the denominators In this case, the magnetic field lies in a plane para

the interface. We have h 0, h e/2, since elequal one. As a result, from (2), (4), and (12), in sucha matched PML medium, any subcomponent of the and magnetic fields are perpendicular from (6), and

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FIG. 2. Interface lying between two PML media.

where is the angle that the normal to the (E, H) plane S *y1S *z1

tan h1 S *y2S *z2

tan h2 . forms with the normal to the interface. Inserting (14) into(10) and (12) yields

This relation shows that sin h1 sin h2 and cos cos h2 in the following two cases:

G

Sywz sinsin h

wy cos2 h wz sin

2 h(15.a)

*y1 *z1 and *y2 *z2 . If, moreover, y1 y2 z2 , this case is two-dimensional. By rotating t

G

Szwy sin cosh

wy cos2 h wz sin

2 h(15.b)

z) plane of coordinates, (15), (16), and (17) yield thresults of [1].

*y1 *y2 and *z1 *z2 . The transverse maG wywz sin2 wzwx cos

2 sin2 hwxwy cos2 cos2 h

wy cos2 h wz sin

2 h.

conductivities of the two media are equal. This cas(16) true 3D case. Using (15), both Equations (17) beco

At an interface between two media, the ratio of the trans- 11 Sy1Sz1 sin 1wy1 cos

2h1 wz1 sin2h1

1

G1mitted wave to the incident wave must be invariable whenmoving in the interface. As a result, the same demonstra-tion as that in [1] yields the SnellDescartes law connecting

22 Sy2Sz2 sin 2

wy2 cos2 h1 wz2 sin

2 h1

1

G2.

the 1 and 1 coefficients of the incident wave to the 2and 2 coefficients of the transmitted wave:

For the reflected wave, a similar demonstration 1 2 , 1 2 . (17) the corresponding angle equal to 1 . Let u

consider the incident, reflected, and transmitted Ei , Er, Et, Hi , Hr, Ht. Continuity of the componeUsing (15) in both media, the ratio of these two equa-

tions gives the interface gives

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Ei cos 1 Ercos 1 Etcos 2 (20.a)rp

wx1 wx2 wx1 wx2

.

Hi Hr Ht (20.b)

Transverse Magnetic Case TMThese equations are identical to those of the 2D case andyield the same reflection coefficient rp : The electric field lies in a plane parallel to the inte

We have e 0, h e /2, and h . From

(10), and (12),rp Z2 cos 2 Z1 cos 1

Z2 cos 2 Z1 cos 1. (21)

GSysin sin e

wz cos2 e wy sin

2 eWe now consider two media having the same transverse

conductivities, the same permittivity, and the same perme-ability, so that y , *y , z , *z , , of the two media are

GSzsin cos e

wz cos2 e wy sin

2 eequal. From (19) we have

G wzwx cos2 e wxwy sin

2 e

wx sin2 wy cos

2 sin2 e wz cos2 cos2 e

. sin 1G1

sin2

G2. (22)

From (27) and (17) which is valid for any polarizatiIf, moreover, the two media are matched ones, that is (x ,*x ), (y , *y ), and (z , *z ), satisfying the condition/ */ in each medium, we have G1 G2 1, so Sy1

Sz1tan e1

Sy2

Sz2tan e2 . that 1 2 from (22) and Z1 Z2 from (9). As a result,

(21) becomes

We have sin e1 sin e2 and cos e1 cos e2rp 0. (23)following cases:

The TE component is not reflected from an interface y1 z1 and y2 z2 . If, moreover, *y1 normal to x lying between two matched media having the *y2 *z2 this case yields the 2D TM case briefly desc

same y , *y , z , *z conductivities. This result generalizes in [1].the 2D case [1]. If and of the matched media are y1 y2 and z1 z2 . This is a true 3D case. not equal, rewriting (22) from (19) shows that rp is as an (27), Eqs. (17) becomeinterface lying between the usual (1 , 1) and (2 , 2)media, provided y2/y1 z2/z1 2/1 .

In the case of unmatched media having the same y , 11 G1 sin 1wz1 cos

2 e1 wy1 sin2 e1

22 G2 sin 2

wz2 cos2 e1 wy2 sin

2*y , z , *z , a simple formula can be obtained for thereflection coefficient, as in [1]. Using (9) and (22) the re-flection coefficient (21) becomes

As in the TE case, continuity of the components lyrp

sin 1 cos 2 sin 2 cos 1

sin 1 cos 2 sin 2 cos 1. (24)

the interface yields the usual rs reflection coefficien

Squaring (22), replacing G1 and G2 by their values fromrs

Z2 cos 1 Z1 cos 2Z2 cos 1 Z1 cos 2

. (16), we obtain

wx2 sin 1 cos 2 wx1 sin 2 cos 1 . (25)If the media have the same transverse conductiviti*y , z , *z , the same permittivity , and the same p

Formulas (24) and (25) are identical to those of the 2D ability , (30) becomescase [1], and so they yield the same reflection coefficientfor two unmatched media having the same transverse con-ductivities y , *y , z , *z : G1 sin 1 G2 sin 2 .

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If, moreover, the two media are matched ones, G1 G2 waves from vacuumlayer interfaces. As in [1], only pgating waves were considered above, nevertheless1, 1 2 , Z1 Z2 , so thatan absence of reflection from interfaces is also trueevanescent waves (see the Conclusion).rs 0. (33)

As a final remark, in the definition of the PML me(1), electric and magnetic fields are symmetricalThe TM components are not reflected from an interfaceresult is that a dual formulation can be derivednormal to the x axis lying between two matched media(5). In dual formulas, with G changed to 1/G

having the same transverse conductivities y , *y , z , definition (9), the sign of (12) is changed and *z .S, w are replaced by h , e , S*, 1/w in (10) andIn the case of two unmatched media of the same trans-The TE and TM cases are interchanged, the duverse conductivities, it can be shown that (26) also holds(15) and (16) being (27) and (28) with e , w, repfor the TM coefficient rs . Using (9) and(32) one can obtainby h , 1/w and an opposite sign in (27), and vice rs equal to the opposite sign of (24). Squaring (32) yieldsfor the duals of (27) and (28).(25) with wx1 and wx2 interchanged. The modified equa-

tions (24) and (25) give then rs equal to the rp coeffi-

2.4. Perfectly Matched Layer for the Finite

cient (26).

Summary and ConclusionsDifference Technique

The 3D PML technique is a straightforward generAny plane wave can be split into TE and TM polariza-tion of the 2D one presented in [1]. The Maxwell equ

tions. Thus, for any propagating plane wave we have theare solved by the FDTD method [2, 3] within a com

following theoretical properties at an interface normal totional domain surrounded by an absorbing layer wh

x lying between PML media of same and :an aggregate of PML media whose properties havepredicted in previous sections.if the transverse conductivities y , *y , z , *z are

In the six sides of the domain, the absorbing medequal, the reflection is given by (26).matched PML media of transverse conductivities eqif, moreover, all couples of conductivities satisfy thezero, for instance, (0, 0, 0, 0, z , *z ) media in the umatching impedance condition in both media, there is noand lower sides of the domain (Fig. 3). As a result, oureflection at any angle of incidence and any frequency.waves from the inner vacuum can penetrate without rtion into these absorbing layers.Obviously these properties are also valid at interfaces nor-

mal to y and normal to z, the transverse conductivities In the 12 edges, the conductivities are selected ina way that the transverse conductivities are equal being respectively z , *z , x , *x and x , *x , y , *y .

An important case occurs when onemedium is a vacuum, interfaces located between edge media and side mThis is obtained by means of two conductivities or a medium of real and . Since a vacuum can be

regarded as a (0, 0, 0, 0, 0, 0) PML medium, there is to zero and the other four equal to the conductof the adjacent side media, as shown in Fig. no reflection from an interface located between a vacuum

and a PML medium whose transverse conductivities equal a result, there is theoretically no reflection fromsideedge interfaces.zero and whose longitudinal conductivities satisfy the

matching condition. For instance, the reflection equals In the eight corners of the domain, the conductare chosen equal to those of the adjacent edges, sozero from an interface normal to y located between a

vacuum and a (0, 0, y , *y , 0, 0) medium, provided y/ the transverse conductivities are equal at the inte

between edge layers and corner layers. So, the refle0 *y /0 . Such a special case will be used in thePML technique to ensure zero reflection from the vac- equals zero from all the edgecorner interfaces.

Thus, as in the 2D cases [1], in the 3D case an absouumlayer interfaces. Actually, as remarked before, thiscase is 2D and the demonstration in [1] ensures zero layer can be built in such a way that there is no theor

reflection from any of the interfaces in the comreflection. In the implementation of the PML technique,the 3D theory will only be called for the small interfaces tional domain.

Numerical implementation of the PML layer inlocated inside the PML layer near the edges and thecorners (see the next section). FDTD domain is also a straightforward generalizat

the 2D case [1]. In the inner vacuum, the finite-diffeIn conclusion, the properties of the 2D PML medium[1] are preserved in the 3D case. This will allow an ab- equations are the usual [2, 3] discretizations of the Ma

equations. In the PML layer, there are 12 subcomposorbing layer to be built on the outer boundary of a compu-tational domain, without theoretical reflection of plane to be computed in place of the six components. Th

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FIG. 3. Upper-right part of a computational domain surrounded by the PML layer.

FDTD grid is unchanged, the only change is that two sub- uumlayer interfaces, as Eyx and Ezx in the interfacemal to x, the numerical derivatives involve one pocomponents are computed at each point of the grid, Exy

and Exz at Ex points, . . ., Hzx and Hzy at Hz points. Discreti- the vacuum and the other in the PML layer. Such dtives are obtained by using the component in the vazation of (1) is obvious, as in 2D. For instance, with the

usual notations of the FDTD scheme, Exy is computed by and the sum of the two subcomponents in the laydetailed for the 2D case in [1].the equation derived from (1.a),

Continuity of tangential components at vacuuminterfaces is ensured as with the Maxwell equatioEn1xy (i ,j, k)the normal derivatives computed using one point

ey(i1/2,j,k)t/Enxy(i ,j, k) vacuum and one point in the layer. Such a continuity numerical implementation is consistent with the theorcontinuity enforced in (20), and then with the refle

1 ey(i1/2,j,k)t/

y(i ,j, k)y coefficients (23) and (33).The theoretical properties of the PML layers were

[Hn1/2zx (i ,j , k) (34)onstrated above for continuous media. From thispoint of view, there is no reflection from all the inter Hn1/2zy (i ,j , k)in the computational domain. Unfortunately, that

Hn1/2zx (i ,j , k) the case in the FDTD implementation of the PMLnique. Due to the discretization of the PML equati

Hn1/2zy (i ,j , k)], certain amount of numerical reflection occurs from variations of conductivities at the interfaces. As in tcase [1], in order to reduce this reflection the conductwhere y depends on the location in the layer. With inter-

faces normal to y located at index JL1 and JL2 as in Fig. 3, x , y , z increase from a small value in the vacuuminterfaces to a great value on the outer boundariesy y1 if J JL1 , y 0 if JL1 J JL2 , and y y2

if J JL2 . means that the conductivity in side layers depends odistance from the interface, i.e., on one mesh indexFor computing the tangential subcomponents in the vac-

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example, y1 and y2 to be used in (34) depends on J. In as in the 2D case [1]. So, for a wave crossing theand reflected by the perfectly conducting conditioedge and corner layers, the variation of the conductivity

is set equal to that in the adjacent side layer, with the on its outer boundary, the apparent reflection is to that of the 2D layer. This is in agreement witconsequence that the conductivities in the PML layer de-

pend only on one mesh index, x x(i), y y(j), and fact that the side layers are 2D layers, as remabove. From [1], for a layer of thickness and longituz z(k). As an illustration, at any point located on the

right of the plane J JL2 in Fig. 3, we have y2 (i , j, conductivity (), depending on the distance frointerface, the apparent reflection is, as a function ok) y2(j).

From a theoretical point of view, an implementation of angle of incidence :variable conductivities does not modify the properties ofthe absorbing layer since a layer with increasing conductiv- R() [R(0)]cos

ity can be regarded as a juxtaposition of one-cell thicklayers separated by internal interfaces. In side layers, only R(0) exp

2

0c

0() d .

the longitudinal conductivity vary from one one-cell layerto the next, the transverse conductivities are unchanged(equal to zero). So, there is no reflection from internal In this paper, some numerical tests will be shown forinterfaces between two adjacent one-cell layers. Similarly, ous profiles of conductivity, parabolic conductivitiewith the implementation described above, there is no re- [1], or conductivities increasing geometrically as in [4flection from internal interfaces located inside edge and last profile being an optimum one to reduce the num

corner layers; in all cases the transverse conductivities do reflection in wavestructure interaction problems. Dnot vary through these interfaces. Considering, for in- on the exact implementation of the conductivity stance, an internal interface normal to z inside a corner mesh points can be found in [1, 4].layer, z vary through this interface, but the transverseconductivities x and y are identical on both sides of it, 3. NUMERICAL EXPERIMENTSsince they do not depend on z.

In all the media of Fig. 3, theoretically there are 12 Although based on the use of an absorbing layesubcomponents. In practice, the number of quantities to signed to remove reflection from the vacuumlayer be computed is less. In side layers, since the transverse faces, the PML technique is not a perfect method ofconductivities are equal, there are only 10 quantities to be space simulation for two reasons. First, the theoreticcomputed, four subcomponents merging into two usual flection of a plane wave is not equal to zero, due tcomponents. Considering, for example, the right-side layer

need of a truncation condition on the outer boundof Fig. 3, since x z 0 and only Eyz Eyx is requiredin (1.h) and (1.k), Eqs. (1.c) and (1.d) merge so that theusual Ey component can replace Eyz and Eyx . Similarly,due to*x *z , (1.i) and (1.j) merge, so that Hy can replaceHyz and Hyx . In the edge layers, four subcomponents couldalso merge into two components, if some conductivitieswere equal, for example, ify2 z2 and *y2 *z2 in thex-directed edge of Fig. 3. In practice, the conductivity variesin the layer and such conditions are not satisfied. Similarly,in the corner layers, if all the electric and magnetic conduc-tivities are equal, the medium becomes the classical

matched medium involving the six usual components. Inpractice, due to the variations of the conductivities the 12subcomponents must be computed separately. In conclu-sion, in side layers there are only 10 quantities to be com-puted, and elsewhere 12 quantities. Since most of the PMLcells are in side layers, the memory requirements of onePML cell can be roughly estimated to about that of onecell of vacuum.

In the absorbing layers, the magnitude of a plane waveis ruled by the three absorbing terms of (13). In sidelayers, two absorbing terms equal unity, so that the FIG. 4. One or two dipoles radiating in free space. Locations o

Q, line A, line B.magnitude is governed by only one exponential term,

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FIG. 5. One dipole located at the center of a 24 24 24-cell computational domain surrounded by various absorbing boundary conUpper part shows the electric field at point Q. Lower part shows the L2 error on the 20 20 20-cell grid surrounding the dipole. The lasin lower part has been computed within a 48 48 48-cell domain surrounded by the second-order operator.

the layer. With a perfect conductor as truncation condition, pared to the results obtained by other methods ofspace simulation, the matched layer [5, 6] and the firthe reflection coefficient is given by (35) which equals one

at grazing incidence. Second, the discretization of the PML orders of the Higdon operator given in [1]. This opeis a special case of the general operator described iequations results in the presence of numerical dispersion

and numerical reflection from the vacuumlayer inter- Its first order is equivalent to the first order of theway approximation of the wave equation [8].faces. Thus, as in [1] some numerical experiments are pre-

sented here in order to evaluate the actual possibilities ofthe PML technique in practical computations. Most tests

3.1. Radiating Dipolesof [1] cannot be performed in 3D since their costs wouldbe prohibitive, especially the tests involving a plane wave. We have first considered a computational domain

by 24 by 24 cells of vacuum, surrounded by either aWe show two tests. The first is related to the absorptionof the field radiated by either one or two dipoles. The layer or an absorbing operator. The sizes of the cell

5 by 5 by 5 cm and the time step was 83.333 ps. A vesecond is a wave structure interaction problem. For both,the results computed using the PML technique are com- dipole was located near the center of the domain,

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FIG. 6. One dipole located two cells from the corner of a 14 14 14-cell computational domain. The electric field at point Q is showntime domain (upper part) and in the frequency domain (lower part).

point (12, 12, 12 ). The electric dipole Pe was imple- Beside the computations performed using the bou

conditions to be tested, we computed a reference somented by the FDTD equation

within a domain of 150 by 150 by 150 cells, allow

boundary-free solution for more than 250 time stepsEn1z E

nz

t

0[curl H]n1/ 2

t

0x3

dPe(tn1/ 2)

dt, (37)

center part.

Two kinds of results are reported in Fig. 5. The where x is the spatial increment. The results in this paper part shows the vertical field Ez at point Q(12, 22, 12were computed using the gaussian pulse, located 10 cells from the dipole (Fig. 4) and two

from the absorbing boundary of the 24 24 24 doThe results computed using the two operators and

Pe(t) 1010 exp

t 3T

T

2

(T 2 ns). (38)

PML layers are compared to the reference solutio

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FIG. 7. One dipole located two cells from the corner of a 14 14 14-cell computational domain. The electric field is shown along lineB, at frequency 10 MHz.

to the analytical solution of the problem. The PML counterpart in the reference domain, the quantity repin Fig. 5 islayers are denoted as in [1], for example, a PML(8-P-

0.01) layer is a layer of eight cells, a parabolic profileof conductivity, and a normal reflection R(0) equal to

L2 22i222j2

211/ 2k21/ 2

[Ez(i,j, k) Ezr(i,j, k)]2. 0.01%. As it appears in Fig. 5, the operator method cannot

allow an exact solution to be computed. Conversely, usingthe PML method the results are in very good agreementwith the analytical solution and are superimposed on Figure 5 shows five results computed within the 24

24-cell domain, plus an additional result computedthe boundary-free reference.The lower part of Fig. 5 gives the L2 norm of the error the dipole at the center of a domain of 48 48 48

surrounded by the second-order operator. These ron the 20 20 20 grid surrounding the dipole. Denotingas Ez(i, j, k) the field in the test domain and Ezr(i, j, k) its show that the absorption of the field radiated by the

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is widely improved by using the PML method. Comparing technique the results are worse than those of Fig. 5other result was computed using a 4-cell thick mato the operator method, the 8-cell and 12-cell PML layers

allow the peak of the error to be reduced by more than layer ML(4-P-1), with a parabolic profile of conduand a normal reflection of 1%. This result is also in 107. Such a reduction could not be achieved by increasing

the computational domain surrounded by an operator, as Conversely, by using PML layers the results are veryat least for the 22 ns shown in Fig. 6. In the threeshown by the case of the 48 48 48-cell domain whose

computational cost was greater than that of the 8-cell PML layers, the conductivity increases as a geometric prosion. As a function of the distance from the interfaccase. The results in Fig. 5 are in agreement with those

observed in the 2D case [1]. given byThe results in Figs. 6 and 7 were computed within a

vacuum reduced to 14 14 14 cells, with the dipolelocated only two cells from a corner, at point (2, 2, 2 ). ()

0c

2 x

lng

gN 1ln R(0)g/x,

The upper part of Fig. 6 shows the field observed at pointQ of Fig. 4, located at (2, 12, 2 ) in the reduced grid.Due to the reduction of the domain, with the operator where g is the ratio of the geometric progression,

FIG. 8. Two dipoles located near a corner of a 16 14 14-cell computational domain. Upper part shows the electric field at point Qfrequency domain. Lower part shows the electric field along line A, at frequency 10 MHz.

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the spatial increment, and N is the number of cells in the approximately of opposite signs. With Pe from (38dipoles werelayer. Such layers are denoted as PML(N-Gg-R(0)).

The lower part of Fig. 6 shows the field at point Q inthe frequency domain. To obtain this result we computed Pe1(t) Pe(t), Pe2(t) Pe(t t), the field up to 5000 ns and then performed a Fourier trans-form. Only the first-order operator is shown in this figure,

where t is an interval of time. If t 0, then Ethe second-order one has been found unstable after thou-

in the plane of antisymmetry (I 4). By an adesands of time steps. Two results related to the matched

choice oft, Ez may be small in this plane, at frequelayer method are shown, the first one computed with theof interest. The results in Fig. 8 were computed

above 4-cell ML layer and the second one with a thickert 0.01 ns. The upper part gives the field at po

12-cell ML layer. The Fourier transform of the reference(4, 12, 2 ) and the lower part gives the field

has been computed using a 22-ns boundary-free resultline A, at 10 MHz. We observe that the results a

and supposing that the field equals zero later than 22from the reference when using the operator o

ns. As expected, the results computed by the operatortechniques. With the 8-cell and 12-cell PML layer

and ML techniques differ from the reference at anyresults at point Q are correct above frequencies

frequency. Results from the 4-cell and 12-cell ML layersMHz and 5 MHz. With the 12-cell layer the 50 dB

are not quite different. This is due to the fact that theof Ez along line A is correctly computed at 10 reflection is mainly determined by the physical reflectionSuch a test illustrates the very good absorption o

from the vacuum-ML interface [1, 6]. With PML layers,radiated field by PML layers whose normal refle

the results are widely improved. They are superimposed R(0) were equal to 0.01%.on the reference above a frequency which decreases as

In conclusion to these experiments with radiatinthe thickness of the layer increases, about 100 MHz with

poles, reflection from the outer boundary can be wthe 4-cell layer and 5 MHz with the 8-cell layer. Such

reduced by using a PML layer in place of the maresults can be explained by following the way of [4]. In

layer [5, 6] or the one-way operator [7]. Such an achwavestructure interaction problems, it has been found

ment reduces the numerical noise in the FDTD comthat the domain of validity of the PML technique is

tional domain, resulting in a substantial increase obounded by a frequency fc which depends on the numeri- dynamic range of the computed results. The computacal conductivity implemented in the first row of the layer,

requirements of PML layers are greater than those denoted as n(0). From [4], other methods, due to their thickness and the 10

subcomponents per cell. But this does not mean th

PML method is less efficient, since such layers allofc n(0)

20 c

4 x1 ggN 1

ln R(0). (41)better results to be computed within a given computadomain. When comparable accuracies are considereother boundary conditions require far greater domai

Thefc frequencies of the three layers of Fig. 6 are respec- that the PML method is more efficient in terms of otively equal to 50.3, 2.25, and 0.105 MHz. Such frequencies

computational requirements. This is illustrated by thare in accordance with the results in Fig. 6. Thus, Fig.

48 48-cell-domain calculation in Fig. 5 whose me6 suggests that the analysis of [4] is also valid for radiating

and time requirements were about 2.5 times greateantenna problems. One might think that specifications

those of the 24 24 24-cell domain surrounded bof an optimized PML layer could be determined for

4-cell PML layer.antenna problems, as has been done for interactionproblems in [4].

Figure 7 shows the magnitude of the Ez component alonglines A and B (Fig. 4) located two cells from the absorbingboundary conditions, at frequency 10 MHz. As the opera-tor and ML techniques, the 4-cell PML layer does notallow correct results to be computed since 10 MHz is farbelow frequency fc . Conversely, with the 8-cell PML layerthe results are close to the reference and with the 12-cellone they are exact.

The results in Fig. 8 were computed within a 16 1414-cell domain with a couple of dipoles (Fig. 4) separatedby four cells (20 cm), located near a corner, at points

FIG. 9. The 100 20-cell plate and the incident wave.(2, 2, 2 ) and (6, 2, 2 ). The radiated fields were

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FIG. 10. Wavestructure interaction problem. Electric field at the corner of a 100 20-cell plate computed using a matched layer or an opositioned 50 cells from the plate (upper part), or a parabolic PML layer positioned only two cells from the plate (lower part).

3.2. WaveStructure Interaction aries were 50 cells from the scatterer. The results comwith the 4-cell and 12-cell ML layers are about ide

We have considered the scattering problem of Fig. 9.and very close to the result obtained by means o

The scatterer was a plate of 100 by 20 cells and zero thick-second-order operator. All the results are only appness. The size of the cubic cell was 5 cm and the incidentmate solutions. To obtain better results a larger do

plane wave propagating downward waswould be required with scatterer-boundary separatleast equal to the size of the scatterer. The lower p

E(t) Einc(et/100 et) (tin nanoseconds) (43) the figure shows the results computed using PML l

set only two cells from the plate, which was then wi104 24 4-cell vacuum. The layers are the same asThis problem is close to the 2D test presented in [1]. Figure

10 shows the vertical field at the corner of the plate (Fig. of [1], with parabolic profiles of conductivity and 4,and 15 cells. The results are like those of the 2D9). In the upper part, results computed using the matched

layer and operator methods are compared to a boundary- Using the 10-cell layer the solution is very close treference for the 300 ns in Fig. 10; using the 15-cellfree reference solution. The plate was within a 200 120

100-cell domain of vacuum, so that the absorbing bound- the result is superimposed on the reference. Thus, as t

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FIG. 11. Wavestructure interaction problem. Electric field at the corner of a 100 20-cell plate computed using geometrical PML layerspart illustrates the effect of varying the ratio of the geometric progression. Lower part illustrates the effect of varying the number of cells in the

PML layer, the 3D one allows a very accurate simulation of Figure 11 illustrates the consequences of varyinparameters g and N. In this test, the incident wavfree space using only a very small vacuum around thea unit step (the first exponential of (43) replacescattering structure.

one). The upper part shows some results computedA detailed analysis of wavestructure interaction prob-15-cell layers of various g (in each case (44) is satislems is reported in [4] for 2D problems. It has beenThe results are in agreement with [4]. By reducfound that the optimum profile of conductivity is thethe solution tends towards the reference, g equal tgeometric progression (40), with R(0) on the order ofensures a correct solution, as with the 2D 100-cell1%, g evaluated as a function of the number of cells inin [4]. The second part of Fig. 11 shows some rthe scatterer length, and N set in such a way that theobtained using PML layers of various N and, following time, tc , is at least 10 times the duration ofvarious n(0) in the first row of the PML meshthe computation Dc :corresponding tc are reported in the figure. With cell and 7-cell layers the solution departs fromreference before the respective tc times, as exptc

1

fc 10 Dc (44)

With the 10-cell layer, (44) is roughly satisfied (

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300 ns), the solution is correct for the whole time reported problems, as shown in this paper and in [1, 4, 9]. Sean increase of accuracy and dynamic range inside thein the figure. Thus, the results in Fig. 11 show that the

conclusions of the analysis [4] are also valid with the putational domain. This last improvement may belarge, too, in radiation problems as those shown i3D PML layer. Parameters g and N defining the layer

can be specified as functions of the number of cells in paper and in [1, 10, 11], or in other applications ain [12].the scatterer and the duration of the computation, as in

the 2D case. Starting from the material in [1], a lot of worbeen done by others about the PML method, resuThe results in Figs. 10 and 11 also show that the PML

method allows the computational requirements to be in various formulations and modifications of the PML media. These investigations also provide widely reduced. In each computation, estimates of the

memory requirement and computational time can be interpretations of the PML concept or extend its posties. Here we mention those we are aware of. Inobtained from the number of unknowns. Taking into

account that in PML cells there are 10 subcomponents the PML media are described in terms of strecoordinates. In [14, 15], discussions about the phper cell in side layers and 12 in edge and corner layers,

with the 10-cell PML the overall number of unknowns meaning of PML media can be found, along with attto reduce the PML equations to the Maxwell equawas about 1,400,000 instead of more than 14,400,000

with the operator or ML layer located 50 cells from It is shown that in side media of a computational doin which transverse conductivities equal zero, thethe plate. That yields an estimate of the ratio of the

computational requirements greater than 10. In accor- equations are equivalent to the Maxwell equations

an additional active current density. In [16], the dance with such prediction, for the 10-cell-PML andoperator calculations, in our computer the ratio of compu- method is extended to curvilinear coordinates. In [1

evanescent waves are addressed. In theory, such wtational times was equal to 10.4. Thus, solving the problemin Fig. 10 with the PML method, memory requirement are not reflected from vacuumPML interfaces. F

in [19] the PML layer is extended to lossy mediaand computational time were more than 10 times shorter.Moreover, the accuracy was better. If the scatterer- in [20] a formulation suitable to the finite-element m

is presented.boundary separation were equal to 100 cells, in orderto improve the solutions of the operator or ML methods, In conclusion, the PML technique is a very ef

method of free-space simulation, as shown here athe gain would be greater, on the order of a factorof 50. Thus, very large reductions of computational other papers referenced above. Nevertheless, due

numerical reflection, its implementation in the Frequirements can be expected from using the PMLmethod in wavestructure interaction problems. A more method is not perfect. For wavestructure inter

problems as those encountered in electromagnetic codetailed discussion related to two realistic scatterers canbe found in [9]. ibility, this reflection is analyzed in [4], allowing an

mized layer to be specified. But we think that thisis only a first step toward controlling and reducin4. CONCLUSIONnumerical reflection. Other research should be dothe future, in order to improve and optimize the FIn this paper, the absorbing PML medium [1] has been

described in the three-dimensional case. We have shown implementation of the PML method.that all the theoretical properties of the 2D medium [1]are also valid in the 3D case, especially the possibility

REFERENCESof zero reflection from interfaces lying between media ofdifferent conductivities. As a result, at least in theory, an

1. J. P. Berenger, J. Comput. Phys. 114, 185 (1994).

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3. A. Taflove, Wave Motion 10, 547 (1988).case.

4. J. P. Berenger, IEEE Trans. Antenna. Propag. 44(1), 110 (1As shown by two finite-difference experiments in this5. R. Holland andJ. W. Williams,IEEE Trans. Nucl. Sci. 30, 4583paper, despite a small amount of numerical reflection, the6. J.P.Berenger, in Actesdu Colloque CEM, Tregastel, France, 19actual simulation of free space by the PML method may

published).be far better than that obtained using methods such as7. R. L. Higdon, Math. Comput. 49, 65 (1987).the matched layer [56] or the one-way operator [7]. In8. B. Engquist and A. Majda, Math. Comput. 31, 629 (1977).practical applications of the finite-difference time-domain

method, two kinds of improvements can be expected. First, 9. J. P. Berenger, Ann. of Telecommun. 51(12), 39 (1996).a reduction of the computational burden. Such a reduction 10. D. Eclercy, Rapport de DEA, Universite de Limoges, France, 19

published).is very large when dealing with wavestructure interaction

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11. D. S. Katz, E. T. Thiele, and A. Taflove, IEEE Microwave Guid. 16. E. A. Navarro, C. Wu, P. Y. Chung, and J. Litva, Electro30(20), 1654 (1994).Wave Lett. 4(8), 268 (1994).

17. J. De Moerloose and M. A. Stuchly, IEEE Microwave Guid12. C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, and A. Taflove,Lett. 5(10), 344 (1995).IEEE Microwave Guid. Wave Lett. 4(10), 344 (1994).

18. B. Chen, D. G. Fang, and B. H. Zhou, IEEE Microwave Guid13. W. C. Chew and W. H. Weedon, Microw. Opt. Technol. Lett. 7(13),

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19. J. Fang and Z. Wu, IEEE Microwave Guid. Wave Lett. 5(114. R. Mittra and U . Pekel, IEEE Microwave Guid. Wave Lett. 5(3), (1995).

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(1995).15. C. M. Rappaport, IEEE Microwave Guid. Wave Lett. 4(8), 90 (1995).

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