direction normal to the interface. As such, the PML medium has been touted as the best absorbing boundary condition for numerical solution of partial

Transcription

1 Perfectly Matched Layers in the Discretized Space: An Analysis and Optimization y W. C. Chew and J. M. Jin Center for Computational Electromagnetics Electromagnetics Laboratory Department of Electrical and Computer Engineering University of Illinois Urbana, IL 68 Abstract The perfectly matched layer (PML) has recently been introduced by Berenger as a material absorbing boundary condition (ABC) for electromagnetic waves. Recently, it has been pointed out that this absorbing boundary condition is the same as coordinate stretching in the complex space. In this paper, the corresponding coordinate stretching is analyzed in the discretized space of Maxwell's equations as described by the Yee algorithm. The corresponding dispersion relationship is derived for a PML medium and then the problem of reection from a single interface is solved. A perfectly matched interface is shown not to exist in the discretized space, even though it exists in the continuum space. Numerical simulations both using nite dierence method and nite element method conrm that such discretization error exists. A numerical scheme using the nite element method is then developed to optimize the PML with respect to its parameters. Examples are given to demonstrate the performance of the optimized PML and its application to the nite element solution of scattering problems.. Introduction The perfectly matched layer (PML) as a material absorbing boundary condition (MABC) has been recently introduced for electromagnetic waves by Berenger (994). This material absorbing boundary condition holds great promise for truncating the mesh in the numerical solution of the partial dierential equations of wave scattering. Various workers have sought a dierent interpretation of this material absorbing boundary conditions (Chew and Weedon, 994; Katz et al., 994; Navarro et al., 994; Mittra and Pekel, 995; Sacks et al., 995; Chew et al., 995; Gribbons et al., 995). Work on it has also been extended to three dimensions. Recently, Chew and Weedon (994) have interpreted the perfectly matched layers as coordinate stretching in frequency domain. Since the stretching variables are complex, they necessitate the splitting of Maxwell's equations in the time domain. The PML is shown to be of very wide bandwidth, and is capable of absorbing static electromagnetic elds as well (Chew et al., 995). A perfectly matched interface is an interface between two half spaces, one of which is lossy, but the interface does not reect a plane wave for all frequencies and all angles of incidence. The loss of the wave in the lossy half space is in the y This work was supported by Oce of Naval Research under grants N and N , and the National Science Foundation under grants NSF ECS and ECS WCC thanks Eric Michielssen for identifying some inconsistencies in his program. This paper was presented at the 995 URSI Meeting, Newport Beach, and published in Electromagnetics, vol. 6, no. 4, pp.35-34, 996.

2 direction normal to the interface. As such, the PML medium has been touted as the best absorbing boundary condition for numerical solution of partial dierential equations. In this paper, the PML medium will be analyzed in the discretized space as described by the Yee algorithm the most popular nite dierence time domain algorithm for solving Maxwell's equations in the discretized space (Yee, 966). The dispersion relationship will be derived for a plane wave in a discretized PML medium. Then the plane wave reection of the single interface problem between two PML media will be solved. It will be shown that a perfectly matched interface does not exist between two PML media in the discretized space. This will be corroborated by numerical simulation using nite dierence method as well as nite element method (Chew, 99; Jin, 993). After that, a numerical scheme will be developed to optimize the PML with respect to its parameters. Numerical examples will be given to demonstrate the performance of the optimized PML and its application to the nite element solution of scattering problems.. Formulation The split Maxwell's equations in the time domain are (Chew and Weedon, 994) a t H sx + x H sx =?@ x^x E; () a t E sx + x E sx x^x H: () Another four equations could be derived by replacing x above by y or z. Using the notation in (Chew, 994), the above can be rewritten as (^a x ^) ~ ^H l? + ^ sx;m+ x^ ^H l+ =?@ ~ sx;m+ x^x E ~ l m; (3) m+ (~a x ~) m ^@t ~E l sx;m + ~ x ~ ~E l sx;m = ^@ x^x ^H l? : (4) m m+ In the above, m + = (m + ; n + ; p + ); and m = (m; n; p) denote the locations of the Yee's nite dierence time domain (FDTD) grid points (Yee, 966; Taove, ~ ~ x are forward dierence approximation of t x operators, and ^@ t and ^@ x are backward dierence approximation of the same operators. ^H denotes a back vector while ~E denotes a fore vector. Also, when a fore vector is associated with the node m, the respective eld components are associated with the respective branches forward to the node. For instance, the x component of ~E m is associated with the location (m + ; n; p). A similar denition applies for a back vector. For instance, the x component of ^H m+ is associated with the locations (m; n + ; p + ). The tilde and hat over a x,,, and x indicate that they are to be evaluated at the location of the fore vector and back vector, respectively, they are associated with. The above is similar to the use of Yee algorithm for implementing () and () by the nite dierence method (Chew, 994). The equations above correspond to the lossy wave equation. However, when the term associated ~ t or ^@ t is zero, the equations become diusive (Chew, 995). The time steps for the terms associated with x and x are chosen to ensure stability when the equations become purely diusive. A sucient condition for stability is to ensure that the time step satises the CFL condition (Chew, 995). Next, a time harmonic plane wave is assumed such that ~E l m = ~E m e?i!lt ; ^H l? m+ = ^H m+ e?i!lt ; (5)

3 where t is the time-stepping size. Then in (3) and (4), the dierence operators become (Chew, 994) ~@ t )?i~ t ; ^@t )?i^ t (6) where ~ t =!sinc( t )e?it ; ^ t =!sinc( t )e it ; (7) and t =! t =. Consequently, for a time harmonic eld, (3) and (4) become?i~ t a x + ^ x e?it m+ ^H sx;m+ =?@ ~ x^x E ~ m ; (8a) m+?i^ t a x + ~ x m E ~ sx;m = ^@ x^x ^H m+ m : (8b) The above could be written as i~ t ( ^Sx ^) m+ ^H sx;m+ ~ x^x E ~ m ; (9a)?i^ t ( ~ Sx ~) m ~E sx;m = ^@ x^x ^H m+ ; (9b) where S x = a x + i x =^ t. The ratio x =(^ t a x ) is like the loss tangent of the PML medium. In the above, the hat over S x and implies that they are to be evaluated at the same locations as the components of ^H, and similarly, for tilde over the same scalar quantities, they are to be evaluated at the same locations as the components of ~E Ṙepeating the above for y and z, and combining the resultant equations, one obtains i~ t ^ m+ ^H m+ = ~ re ~E m ; (a) where?i^ t ~ m ~E m = ^re ^H m+ ; (b) ~r e = ^x ^Sx + ^y ^Sy + ^z ^Sz ; ^re = ^x ~ Sx ^@x + ^y ~ Sy + ^z ~ Sz : () Equations () are valid for inhomogeneous media as well. In this case, one needs to be cautioned on the interpretation of S, = x; y; z. For a homogeneous medium, a plane wave solution may be assumed such that ~E m = ~E e ikr ; ^H m+ = ^H e ikr ; () where, k = ^xk x + ^yk y + ^zk z, and r = ^xx + ^yy + ^zz; x = m x ; y = n y ; z = p z. Using this in (), one arrives at ~ t ^H = ~K e ~E ;?^ t ~E = ^K e ^H : (3) where ~K e = ^x ~ K x S x + ^y ~ K y S y + ^z ~ K z S z ; ^K e = ^x ^K x S x + ^y ^K y S y + ^z ^K z S z ; 3 (4a) (4b)

4 and ~K = k sinc( )e i ; ^K = k sinc( )e?i ; = x; y; z (5) where = k =, and is the space discretization size. Combining the equations in (3), one has? t ^H = ~K e ^K e ^H = ^K e ( ~K e ^H )? ~K e ^K e ^H (6) where, t =!sinc( t ). Since from (3), ~K e ^H =, and hence, More specically, the above reduces to where K = t, K = k sinc( ), = x; y; z. Equation (8) can be satised by t = ~K e ^K e : (7) K = K Sx x + K Sy y + K Sz z ; (8) K x = KS x sin cos ; K y = KS y sin sin ; K z = KS z cos : (9) Notice that S x ; S y, and S z are complex so that K x ; K y, and K z are complex even if the rest of the numbers are real. In other words, the loss of the medium can be controlled independently in the x, y, and z directions by various choices of the complex stretching variables S. One should note the parallel of this analysis compared to the continuum space case (Chew and Weedon, 994). The above represents the theory of Yee algorithm in a homogeneous PML medium. To establish the stability of the method, one can solve for t from (8) given the value of K x, K y, and K z, and hence the value of!. If the roots are such that! is in the upper half complex plane, then the FDTD scheme is unstable. 3. Single Interface Reection Problem The previous section discussed the PML theory for homogeneous media in a discretized space. Next, we like to formulate the theory in the discretized space for a single-interface problem consisting of a planar interface separating two homogeneous half spaces. To this end, a TE incident wave is assumed in region and the waves in regions and are written as (Chew and Weedon, 994) ~E = ~E e ikir + R T E ~E r e ikrr ; (a) ~E = T T E ~E t e iktr ; (b) where k i = ^xk ix + ^yk iy? ^zk iz, k r = ^xk rx + ^yk ry + ^zk rz, R T E and T T E are reection and transmission coecients, respectively. Phase matching requires that k ix = k rx = k tx, k iy = k ry = k ty. Therefore, k iz = k rz = k tz. Without loss of generality, we can let ~E r = ~E t = ~E. Requiring that the tangential E eld be continuous across an interface at z = ; implies that + R T E = T T E : () 4

5 Using Equation (), we have In the above, ^H = ~ K ie ~ E ~ t e ikir + R T E ~ K re ~ E ~ t e ikrr ; (a) ^H = T T E ~ K te ~E ~ t e iktr : (b) ~K ie = ^x S x K x e ix + ~K re = ^x S x K x e ix + ~K te = ^x S x K x e ix + ^y S y K y e iy? ^y S y K y e iy + ^y S y K y e iy? ^z S z K z e?iz ; ^z S z K z e iz ; ^z S z K z e?iz ; (3a) (3b) (3c) where K i = k i sinc( i ), i = k i, i = ;, = x; y; z. By phase matching, K x = K x, K y = K y, x = x, and y = y. From Equation (), it can be shown K ^@ z ^z ^H m+ = z ~E s (4) i~ t S z where ~E s indicates components transverse to ^z. The above implies that ^z ( ^H p=? ^H p=? ) = m K z z ~E s i~ t S z p= where p is the discrete variable in the z direction, and p = is the same as z =. The above is the boundary condition for the magnetic eld for a material discontinuity at z =. Extracting the tangential components of (a) and (b), and inserting into (5), we have? S z? Kz e?iz? R T E K z e iz + K z S z T T E = The above could be rearranged to yield K z? K z e?iz + R T E e iz =?T T E S z S z Solving () and (7) for R T E and T T E yields " K z e iz S z ik z z S z p= K? i z z S z (5) T T E : (6) p= # : (7) R T E = K ze?iz =( S z )? A K z e iz =( S z ) + A ; (8) T T E = K z=( S z ) e iz + e?iz ; (9) K z e iz =( S z ) + A 5

6 where K z = k z sinc( z ), and A = K ze iz S z? ik z z S z p= : (3) The reection and transmission coecients for TM waves can be similarly derived. For consistency check of the solution above, when the single interface is removed by setting the two media to be the same, and with the fact that?ik z z = exp(?i z )? exp(i z ), the reection coecient becomes identically zero, and the transmission coecient equals one. With the reection and transmission coecient for a single interface derived, one can use a recursive scheme as in (Chew, 995) to derive the reection coecient of a multilayered PML medium. The solution is also consistent with the direct use of nite dierence method whereby the problem is rst reduced to a one-dimensional one. The resultant matrix equation, which is a tridiagonal, is then tenable to a recursive solution. 4. A Perfectly Matched Interface? A perfectly matched interface exists in the continuum space, as has been proved by previous works in this area. It will be prudent to investigate if this exists in the discretized space in accordance with the above analysis. The phase matching condition implies that K x = K x, and K y = K y. With the choice that =, =, S x = S x, and S y = S y, and from the dispersion relation, one arrives at The above implies that! = K iz = Ki S? K ix iz Six? K iy Siy ; i = ; : (3) Then, the reection coecient becomes K z =S z = K z =S z : (3) R T E = e?iz? e iz? e?iz + e iz e iz + e iz + e?iz? e iz : (33) Hence R T E is in general not zero when the conditions for perfectly matched interface from the continuum case is chosen. Due to the complexity of the expression for the reection coecient, it is unlikely that a condition can be found such that R T E = for all frequencies and all angles. When z!, all the z 's become very small, and the above becomes R T E i z? i( z + z )=: (34) In the above z is its value at p =. Since z = k z z =, the reection coecient is proportional to z in this limit. Also, s z S z z K? K S x x? K S y y + O( 3 z): 6

7 If we pick the S z at the interface to be the average of the material properties of the two media, viz., S z = :5(S z + S z ), then z :5( z + z ), and the leading order term in (34) vanishes. In this case, assuming that S x = S y =, then R T E?? z S 6 z? Sz K? Kx? K y + O( 3 z ): (35) Notice that the residual error is of second order, and is proportional to the contrast of the two media. The second-order error is consistent with the fact that the nite dierence method incurs second order error if done correctly. 5. Numerical Simulations Discrete-Space Theory In the following simulation, unless otherwise stipulated, a z is assumed to be throughout for simplicity. a z can be chosen to dier from to give the PML medium an added degree of freedom if desired, in order to absorb the evanescent wave well..45 Reflection Coeff. vs Alpha for Different Incident Angles.4 theta=.35 Reflection Coefficient.3.5. theta=3 Sz=, Sz=.5. theta= Alpha Figure. Residual reection coecient for a half space with dierent stretching variables in the z direction, S z and S z, and with dierent incident angle as a function of. The frequency is 3 GHz and the discretization size = : cm. The reection coecient does not vanish due to discretization error. In Figure, the reection coecient for a single interface is studied. Here, it is assumed that S z = :5(S z + S z ). According to the analysis, =. At an interface where both S z and S z are real, it is seen that this choice of is optimal for dierent angles of incidence. However, the reection coecient could not be made to vanish with dierent choice of. The residual reection coecient comes mainly from the discretization error of the continuum equation. As the angle of incidence increases, the residual error decreases, reducing the reection coecient. The reason being that as k z decreases, z increases increasing the eective discretization density per wavelength in the z direction. In Figure, S z is assumed complex with a large loss tangent of. In this case, a proper choice of could make the reection coecient almost zero. The reason is that the wrong choice of here leaves a residual rst-order error that cancels the higher-order errors. It fails to happen in the previous case, because then, the two errors are out of phase whereas with a large loss tangent, this error could be in phase and made to cancel each other. However, the optimal varies 7

8 .6 Reflection Coeff. vs Alpha for Different Incident Angles Sz=, Sz=.+i.5 Reflection Coefficient.4.3. theta= theta=3. theta= Alpha Figure. The same case as Figure, except that the stretching parameter in medium is complex with a large loss tangent, S z = + i. In this case, the wrong choice of introduces deliberate rst-order error to cancel the secondorder error so that the reection coecient becomes very small for certain. Reflection Coeff. vs kx for Different Discretization Sizes 3 GHz, Layers, LTMAX=, Linear Reflection Coefficient (db) Del=d=.5 cm Del=d=. cm Del=d/=.5 cm Del=d/3=.33 cm Analytic d=. cm AMAX= kx/k Figure 3. A PML medium with a linear prole. The reduction of discretization size (Del=) without reducing the PML thickness reduces the reection coef- cient. The solid line case corresponds to a simultaneous reduction of and PML thickness. An extra case for = d = : cm where a z is given the same linear prole as the loss tangent shows the absorption of the evanescent waves. as a function of angle. Therefore, no one universal exists for all angles and all frequencies. In the following simulations, =. Figure 3 shows the eect of discretization on the reection coecient when the PML is used as an absorber. In this example, the frequency is assumed to be 3 GHz, and PML with a linear prole is used. The maximum loss tangent (LTMAX= j x =(^ t a x )j) of the PML is. The continuum space theory (analytic) indicates that close to?4 db absorption is possible with such layers. Two numerical experiments are performed here: One whereby the layer thickness d is kept constant 8

9 at. cm, and the discretization size is reduced. In this case, the smaller values of bridge the gap between the continuum-space theory and the discrete-space theory. In the solid-line case, the layer thickness is reduced the same time when the discretization size is reduced. For some k x 's, close to normal incidence, it outperforms the constant d case, and is inferior when close to grazing incidence. For the = d = : cm case, when a z is given a the same linear prole as the loss tangent with a z;max = AMAX =, it is seen that the evanescent spectrum is absorbed well also. This can be understood easily from Equation (9). Reflection Coeff. vs kx for Different Discretization Sizes 3 GHz, Layers, LTMAX=, Parabolic Reflection Coefficient (db) Del=d=.5 cm Del=d=. cm Del=d/=.5 cm Del=d/3=.33 cm AMAX= 9 Analytic d=. cm.5.5 kx/k Figure 4. The same case as Figure 3, except that a parabolic prole is used. The simultaneous reduction of PML thickness and does not improve the result. An extra case for = d = : cm shows the absorption of evanescent waves when a z is given the same parabolic prole as the loss tangent. Figure 4 shows the same experiment for a parabolic prole. In this case, the continuum space theory (analytic) predicts lower absorption compared to the linear prole. The reason is that the absorption of the PML in the continuum theory is proportional to the area under the prole via the equation e i R L Kz(z)dz ; where L is the total thickness of the PML and K z (z) is a complex number. Therefore, less area exists under a parabolic prole compared to a linear prole. However, the gap between the discrete-space theory and the continuum-space theory is smaller compared to the previous case. For the same discretization, the parabolic prole outperforms the linear prole slightly in this example. The explanation is that in a parabolic prole, the loss is changed more gradually from a lossless medium to a lossy medium, so that the contrast between adjacent layers is smaller. This tends to mitigate the discretization error. However, in the example whereby the PML thickness d is decreased simultaneously with the discretization size, the PML does not perform as well compared to the previous gure. The reason being that the eective area under the prole decreases more rapidly in this case as the PML thickness is decreased. Notice that when a z is given the same parabolic prole as the loss tangent with the maximum value of, the evanescent spectrum is well absorbed. In the continuum theory, it is well-known that the absorption will increase indenitely with the increase in the loss tangent of the PML since no reection occurs at any of the interfaces. However, this is not true in the discrete-space theory. In 9

10 Reflection Coeff. vs kx for Different LTMAX s 3 GHz, Layers, Linear, Del=d=. cm LTMAX= Reflection Coefficient (db) LTMAX=3 LTMAX=6 LTMAX= kx/k Figure 5. The reection coecient for dierent amount of loss in the PML medium. The increase of the loss in the PML by increasing the maximum loss tangent (LTMAX) of a linear prole does not improve the reection coecient indenitely. Figure 5, it is shown that increasing the loss tangent of the prole by increasing the maximum loss tangent (LTMAX) aids in the absorption initially, but eventually will saturate because increasing the loss prole will eventually increase the contrast between adjacent layers, accentuating the discretization error as indicated by Equation (35). 6. Optimization of PML for FEM Applications x PML Incident wave θ z Reflected wave L Figure 6. Plane wave reection by a metal backed PML. Much of the salient features of PML media observed in the nite dierence method are also observed in the nite element method. Since the PML contains

11 several parameters, it is necessary to nd a set of optimal parameters to achieve the best performance. For this purpose, we consider the problem where a plane wave is incident on a PEC-backed PML obliquely at an angle with the z-axis (Figure 6). For TM case, this incident wave can be represented by E inc y (x; z) = E e?ikz cos +ikx sin : (36) This plane wave will be reected by the PML and the reected eld can be represented by Ey ref (x; z) = R T M E e ikz cos +ikx sin ; (37) Therefore, for z L the total eld is E total y = E ref y + E inc y ; (38) from which we obtain dey dz? ik cos E y =?ik cos E e?ikl cos ; (39) at z=l+ or through the eld continuity condition de y S dz? ik cos E y =?ik cos E e?ikl cos : (4) at z=l? The eld in the PML region can be shown to satisfy the dierential equation d dz de y + k S z dz S z (? sin )E y = ; z L: (4) This dierential equation, together with the boundary condition at z = L given in (4) and E y = at z =, can be solved using the D FEM, from which we obtain the eld in the region z L. The reection coecient can then be calculated as R T M = E y(z = L)? E e?ikl cos E e ikl cos : (4) Similar procedure can be employed to calculate R T E. It is obvious that the reection coecients, R T M and R T E, are functions of k ; ; S z ; and L (S z is a function of z). For example, R T M can be written as R T M = f (k ; ; S z ; L): (43) If the number of PML is xed to be N and the form of S z is given such as m z? L S z = + i L t max; (44) L where m can be,,, or 3, then R T M can be written as R T M = f (k ; ; L t max ; ); (45)

12 where is the thickness of each PML, = L N : (46) In the above L t max is the maximum loss tangent. Similarly, we have R T E = f (k ; ; L t max ; ): (47) For a given frequency, we can dene an object function R = Z max [jf j + jf j] d = R(L t max ; ); (48) where max denes the range of the incident angle to be considered. To nd the best PML, we can employ one of optimization schemes to minimize R with respect to L t max and. In this work, we use Powell's method for this purpose (Press et al., 99; Jin et al., 99). 3 GHz, Layers, Del=.cm, LTMAX=8, Linear 3 GHz, Layers, Del=.cm, LTMAX=, Linear Reflection (db) E-pol, FEM H-pol, FEM Analytic Reflection (db) E-pol, FEM H-pol, FEM Analytic Theta (deg.) Theta (deg.) (a) (b) Figure 7. Reection coecient of a PML. (a) =. cm, LTMAX = 8. (b) =. cm, LTMAX =. First, to corroborate the results shown in Figures 3 and 4, the reection coef- cients are computed for a linear -layer PML with = : cm at 3 GHz. The results are given in Figure 7 for two dierent LTMAX, compared with the theoretical prediction based on the continuum theory. Similar phenomena are observed here for the nite element solution as in the nite dierence solution. Next, we show some results obtained using the optimization scheme. In all these results, the number of layers is xed at, the frequency is assumed to be at 3 GHz, and the loss tangent is chosen to be a parabolic function. The parameters to be optimized are and LTMAX for a chosen max. Figure 8 shows the results obtained for max = 4, 6, and 8 degrees. As can be seen, one is able to nd a set of optimal parameters, and LTMAX, to achieve the best performance within max. As is expected, the overall reection is smaller for a smaller max and for a large max the performance is slightly compromised. It is also interesting to note that contrary to the common belief that should be set at., it is found that the best performance is achieved with : :. In our optimization scheme described above, we have not specied the frequency range to be optimized since we expect that the PML will have a wide frequency band, as predicted theoretically. This is indeed true, as can be seen in Figure

13 3 GHz, Layers, Del=.94 cm, LTMAX=.37, Quadratic 3 GHz, Layers, Del=.364 cm, LTMAX=.86, Quadratic Reflection (db) E-pol H-pol Optimized for theta from to 4 degrees Reflection (db) E-pol H-pol Optimized for theta from to 6 degrees Theta (deg.) (a) Theta (deg.) (b) 3 GHz, Layers, Del=.76 cm, LTMAX=5.778, Quadratic Reflection (db) E-pol H-pol Optimized for theta from to 8 degrees Theta (deg.) (c) Figure 8. Reection coecient of an optimized PML. (a) Optimized for from to 4 degrees. (b) Optimized for from to 6 degrees. (c) Optimized for from to 8 degrees. 9, where the reection coecient is shown for two angles of incidence for a PML optimized at 3 GHz. The reection coecient is nearly a constant across the entire frequency range, even at very low frequencies. 7. Application of PML to FEM Solution of Waveguide Discontinuity Problems To illustrate the application of the PML, we consider the waveguide discontinuity problem shown in Figure (a), where a discontinuity (either geometrical, or material, or both) exists at the joint of two parallel plate waveguides. To solve this problem using the FEM, we truncate the solution region using the PML, at the two ends of the waveguide, as shown in Figure (b). For H-polarization, the eld in the entire region is y S + k S z r H y = ; (49) where S z = for non-pml region. To excite an incident wave, we place a magnetic current sheet M at the front of the PML on the left-hand side. This problem can be 3

14 Reflection (db) Layers, Del=.364 cm, LTMAX=.86, Quadratic Theta= deg. Theta=5 deg. Optimized for theta from to 6 degrees Frequency (GHz) Figure 9. Reection coecient of a PML optimized at 3 GHz. incident wave reflected wave 5 mm transmitted wave mm 5 mm (a) PML D (b) Figure. (a) A waveguide discontinuity problem. (b) PML truncation. analyzed easily using a D FEM. Once we have obtained the solution, the reection and transmission coecients for the dominant mode can be obtained as R = H e?ikz T = H e ikz Z a Z b Hy (z )? H inc y (z ) dx; (5) H y (z ) dx; (5) where z denotes a point between the discontinuity and the PML on the left-hand side, z denotes any point between the discontinuity and the PML on the right-hand side, a and b represent the width of the waveguide at z and z, respectively. Representative results are given in Figure where both the reection and transmission coecients associated with the dominant mode are shown for a dominant mode incidence. These results were obtained with the optimized PML given in Figure and the absorbing surface of the PML was placed mm and.5 mm 4

15 Discontinuity in a parallel-plate waveguide Discontinuity in a parallel-plate waveguide.8.8 Reflection Coef ABC PML Reflection Coef ABC PML Frequency (GHz) Frequency (GHz) Discontinuity in a parallel-plate waveguide Discontinuity in a parallel-plate waveguide.8.8 Transmission Coef ABC PML Transmission Coef ABC PML Frequency (GHz) Frequency (GHz) (a) (b) Figure. Reection and transmission coecients for the dominant mode with the PML placed (a) mm and (b).5 mm away from the discontinuity. away from the discontinuity for Figures and, respectively. These results were compared with the data obtained using the absorbing boundary condition (ABC) which was derived for the dominant mode (Jin, 993) and thus is applicable only to the frequencies below the cut-o of the second mode (5 GHz). The ABC was applied 5 mm away from the discontinuity and its result has an accuracy better than % from to GHz. The sampling rate for both ABC and PML solutions was points/mm in the non-pml region. It is clear that to obtain accurate result, the PML must be placed suciently far away from the discontinuity. The reason being that PML does not work as well for evanescent waves compared to plane waves. It is also obvious that the PML is not applicable at cut-o since it has a reection coecient close to one at the grazing incidence. 8. Conclusions The PML has been analyzed using nite-dierence equation. It is shown that a perfectly-matched interface does not exist in the discretized space due to discretization errors of the continuum equations. The errors are of second order, and are proportional to the contrast between two PML media. These errors decrease when the angle of incidence is close to grazing because the discretization density per wavelength is higher there. A stack of PML media can be used to design material absorbers for FDTD simulations. A PML tapered by parabolic prole is found to work better than one tapered by a linear prole. Since the discretization error is proportional to contrast, increasing the loss tangents of the PML does not reduce the reection coecient indenitely. The nite-element simulation of PML is shown to exhibit similar phenomena 5

16 as the nite dierence case. The Powell method can be used to design optimal PML for absorption. With layers of PML, close to?6 db absorption is possible from to 8 degrees. The absorption exhibits extremely broad bandwidth down to static. A simulation of PML for the absorption of waves is performed for D nite element method in a waveguide with a junction discontinuity. It is observed that the further the PML is placed away from the waveguide junction, the better is the simulation result. The reason being that PML does not work as well for evanescent waves. PML is observed to fail close to cut-o because then, the PML does not absorb well. References Berenger, J.-P., \A perfectly matched layer for the absorption of electromagnetic waves," J. Computational Physics, 4, 85{, 994. Chew, W. C., \Electromagnetic theory on a lattice," J. Applied Physics, 75,, pp , May 994. Chew, W. C., Waves and Fields in Inhomogeneous Media. Van Nostrand, New York, 99 (reprinted by IEEE Press, 995). Chew, W. C. and W. H. Weedon, \A 3D perfectly matched medium from modied Maxwell's equations with stretched coordinates," Micro. Opt. Tech. Lett., 7, , 994. Chew, W. C., W. H. Weedon, and A. Sezginer, \A 3-D perfectly matched medium by coordinate stretching and its absorption of static elds," ACES Digest, Monterey, CA, pp , March, 995. Gribbons, M., S. K. Lee, and A. C. Cangellaris, \Modication of Berenger's perfectly matched layer for the absorption of electromagnetic waves in layered media," ACES Digest, Monterey, CA, pp , March 995. Jin, J. M., The Finite Element Method in Electromagnetics. John Wiley & Sons, New York, 993. Jin, J. M., J. L. Volakis, and V. V. Liepa, \Fictitious absorber for truncating nite element meshes in scattering," IEE Proc. H, 39, 5, pp , 99. Katsz, D. S., E. T. Thiele, and A. Taove, \Validation and extension to three dimensions of the Berenger PML absorbing boundary condition for FD-TD meshes," IEEE Micro. Guided Wave Lett., 4, 8, pp. 68-7, 994. Mittra, R. and U. Pekel, \A new look at the perfectly matched layer (PML) concept for the reectionless absorption of electromagnetic waves," IEEE Micro. Guided Wave Lett., 5, 3, pp , 995. Navarro, E. A., C. Wu, P. Y. Chung, and J. Litva, \Application of PML superabsorbing boundary condition to non-orthogonal FDTD method," Electronics Lett., 3,, pp , 994. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, nd ed. Cambridge: Cambridge Univ. Press, 99. Sacks, Z. S., D. M. Kingsland, R. Lee, and J.-F. Lee, \A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag., vol. 43, no., pp , 995. Taove, A., \Review of the formulation and applications of the nite-dierence timedomain method for numerical modeling of electromagnetic wave interactions with arbitrary structures," Wave Motion,, 6, pp , 988. Yee, K. S., \Numerical solution of initial boundary value problems involving Maxwell's equation in isotropic media," IEEE Trans. Antennas Propag., AP-4, 3{37,