Chapter 11. Perfectly Matched Layer Introduction

Transcription

1 Chapter 11 Perfectly Matched Layer 11.1 Introduction The perfectly matched layer (PML) is generally considered the state-of-the-art for the termination of FDTD grids. There are some situations where specially designed ABC s can outperform a PML, but this is ery much the exception rather than the rule. The theory behind a PML is typically pertinent to the continuous world. In the continuous world the PML should indeed work perfectly (as its name implies) for all incident angles and for all frequencies. Howeer, when a PML is implemented in the discretized world of FDTD, there are always some imperfections (i.e., reflections) present. There are seeral different PML formulations. Howeer, all PML s essentially act as a lossy material. The lossy material, or lossy layer, is used to absorb the fields traeling away from the interior of the grid. The PML is anisotropic and constructed in such a way that there is no loss in the direction tangential to the interface between the lossless region and the PML (actually there can be loss in the non-pml region too, but we will ignore that fact for the moment). Howeer, in the PML there is always loss in the direction normal to the interface. The PML was originally proposed by J.-P. Bérenger in In that original work he split each field component into two separate parts. The actual field components were the sum of these two parts but by splitting the field Bérenger could create an (non-physical) anisotropic medium with the necessary phase elocity and conductiity to eliminate reflections at an interface between a PML and non-pml region. Since Bérenger first paper, others hae described PML s using different approaches such as the complex coordinate-stretching technique put forward by Chew and Weedon, also in Arguably the best PML formulation today is the Conolutional-PML (CPML). CPML constructs the PML from an anisotropic, dispersie material. CPML does not require the fields to be split and can be implemented in a relatiely straightforward manner. Before considering CPML, it is instructie to first consider a simple lossy layer. Recall that a lossy layer proided an excellent ABC for 1D grids. Howeer, a traditional lossy layer does not work in higher dimensions where oblique incidence is possible. We will discuss this and show how the split-field PML fixes this problem. Lecture notes by John Schneider. fdtd-pml.tex 307

2 308 CHAPTER 11. PERFECTLY MATCHED LAYER 11.2 Lossy Layer, 1D A lossy layer was preiously introduced in Sec Here we will reisit lossy material but initially focus of the continuous world and time-harmonic fields. For continuous, time-harmonic fields, the goerning curl equations can be written ( ) H jωǫe+σe jω ǫ j σ ( ω E jωµh σ m H jω E jω ǫe, (11.1) ) H jω µh, (11.2) µ j σ m ω where the complex permittiity and permeability are gien by ǫ ǫ j σ ω, (11.3) µ µ j σ m ω. (11.4) For now, let us restrict consideration to a 1D field that isz-polarized so that the electric field is gien by E â z e γx â z E z (x) (11.5) where the propagation constant γ is yet to be determined. Gien the electric field, the magnetic field is gien by H 1 jω µ E â γ y jω µ E z(x). (11.6) Thus the magnetic field only has ay component, i.e.,h â y H y (x). The characteristic impedance of the medium η is the ratio of the electric field to the magnetic field (actually, in this case, the negatie of that ratio). Thus, η E z(x) H y (x) jω µ γ. (11.7) Since γ has not yet been determined, we hae not actually specified the characteristic impedance yet. To determineγ we use the other curl equation where we sole for the electric field in terms of the magnetic field we just obtained: E 1 jω ǫ H 1 jω ǫ γ 2 jω µ e γx â z. (11.8) Howeer, we already know the electric field since we started with that as a gien, i.e., E exp( γx)â z. Thus, in order for (11.8) to be true, we must hae or, soling forγ (and only keeping the positie root) γ 2 1, (11.9) (jω) 2 µ ǫ γ jω µ ǫ. (11.10)

3 11.2. LOSSY LAYER, 1D 309 Because µ and ǫ are complex,γ will be complex and we writeγ α+jβ whereαis the attenuation constant andβ is the phase constant (or wae number). Returning to the characteristic impedance as gien in (11.7), we can now write η jω µ jω µ ǫ µ ǫ. (11.11) Alternatiely, we can write ( ) j σm ωµ η ǫ ( ). (11.12) 1 j σ ωǫ Let us now consider a z-polarized plane wae normally incident from a lossless material to a lossy material. There is a planar interface between the two media at x 0. The (known) incident field is gien by E i z e j whereβ 1 ω ǫ 1. The reflected field is gien by H i y 1 η 1 e j (11.13) E r z Γe j H r y Γ η 1 e j (11.14) where the only unknown is the reflection coefficientγ. The transmitted field is gien by E t z Te γ 2x H t y T η 2 e γ 2x (11.15) where the only unknown is the transmission coefficientt. Both the electric field and the magnetic field are tangential to the interface at x 0. Thus, the boundary conditions dictate that the sum of the incident and reflected field must equal the transmitted field atx 0. Matching the electric fields at the interface yields Matching the magnetic fields yields 1+Γ T. (11.16) 1 η 1 + Γ η 1 T η 2 (11.17) or, rearranging slightly, 1 Γ η 1 T. (11.18) η 2 Adding (11.16) and (11.18) and rearranging yields Plugging this back into (11.16) yields T 2η 2 η 2 +η 1. (11.19) Γ η 2 η 1 η 2 +η 1. (11.20)

4 310 CHAPTER 11. PERFECTLY MATCHED LAYER Consider the case where the media are related by µ 2 ǫ 2 ǫ 1 and σ m µ 2 σ ǫ 2. (11.21) We will call these conditions the matching conditions. Under these conditions the impedances equal: ( ) η 2 µ ( ) 2 1 j σm ωµ 2 ) ǫ 2 (1 j 1 j σ ωǫ 2 µ1 ) η 1. (11.22) σ ωǫ 2 ǫ 1 (1 j σ ǫ 1 ωǫ 2 When the impedances are equal, from (11.20) we see that the reflection coefficient must be zero (and the transmission coefficient is unity). We further note that this is true independent of the frequency. As we hae seen preiously, this type of lossy layer can be implemented in an FDTD grid. To minimize numeric artifacts it is best to gradually increase the conductiity within the lossy region. Any field that makes it to the end of the grid will be reflected, but, because of the loss, this reflected field can be greatly attenuated. Furthermore, as the field propagates back through the lossy region toward the lossless region, it is further attenuated. Thus the reflected field from this lossy region (and the termination of the grid) can be made relatiely inconsequential Lossy Layer, 2D Since a lossy layer works so well in 1D and is so easy to implement, it is natural to ask if it can be used in 2D. The answer, we shall see, is that a simple lossy layer cannot be matched to the lossless region for obliquely traeling waes. Consider a TM z field where the incident electric field is gien by E i â z e jk 1 r, (11.23) â z e jβ 1cos(θ i )x jβ 1 sin(θ i )y, (11.24) â z e jx jβ 1y y. (11.25) Knowing that the angle of incidence equals the angle of reflection (owing to the required phase matching along the interface), the reflected field is gien by Combining the incident and reflected field yields The magnetic field in the first medium is gien by E r â z Γe jx jβ 1y y. (11.26) E 1 â z ( e j x +Γe jx ) e jβ 1yy â z E 1z. (11.27) H 1 1 jω E 1, (11.28) â x β 1y ω ( e jx +Γe x ) e jβ 1yy +â y ω ( e jx +Γe x ) e jβ 1yy. (11.29)

5 11.3. LOSSY LAYER, 2D 311 The transmitted electric field is E t â z Te γ 2 r (11.30) â z Te γ 2xx γ 2y y, (11.31) â z E t z. (11.32) Plugging this expression into Maxwell s equations (or, equialently, the wae equation) ultimately yields the constraint equation γ 2 2x +γ 2 2y ω 2 µ 2 ǫ 2. (11.33) Owing to the boundary condition that the fields must match at the interface, the propagation in the y direction (i.e., tangential to the boundary) must be the same in both media. Thus, γ 2y jβ 1y. Plugging this into (11.33) and soling forγ 2x yields γ 2x β1y 2 ω 2 µ 2 ǫ 2 α 2x +jβ 2x. (11.34) Note that when β 1y 0, i.e., there is no propagation in the y direction and the field is normally incident on the interface, this reduces to γ 2x jω µ 2 ǫ 2 which is what we had for the 1D case. On the other hand, when σ σ m 0 we obtain γ 2x j ( ω 2 µ 2 ǫ 2 β 2 1y) 1/2 where the term in parentheses is purely real (so that γ 2x will either be purely real or purely imaginary). The transmitted magnetic field is gien by H t 1 jω µ 2 E t, (11.35) â x β 1y ω µ 2 E t z â y γ 2x jω µ 2 E t z. (11.36) Enforcing the boundary condition on the electric field and the y-component of the magnetic field atx 0 yields or, rearranging the second equation, Adding (11.37) and (11.39) and rearranging yields 1+Γ T, (11.37) ( 1+Γ) γ 2x T, ω jω µ 2 (11.38) 1 Γ γ 2x j µ 2 T. (11.39) T j 2 µ 2 γ 2x Using this in (11.37) yields the reflection coefficient j µ 2 γ 2x +. (11.40) Γ j µ 2 γ 2x. (11.41) j µ 2 γ 2x +

6 312 CHAPTER 11. PERFECTLY MATCHED LAYER The reflection coefficient will be zero only if the terms in the numerator cancel. Let us consider these terms indiidually. Additionally, let us enforce a more restrictie form of the matching conditions where now µ 2, ǫ 2 ǫ 1 and, as before, σ/ǫ 2 σ m /µ 2. The first term in the numerator can be written j µ 2 γ 2x µ 2 ω 2 µ, (11.42) 2 ǫ 2 β1y 2 ) (1 j σ ωǫ 1 ), (11.43) 2 ω 2 ǫ 1 (1 j σ ωǫ 1 β 2 1y ω 2 ǫ 1 β1y 2 ( ) 2 1 j σ ωǫ 1. (11.44) The second term in the numerator of the reflection coefficient is. (11.45) ω 2 ǫ 1 β1y 2 Clearly (11.44) and (11.45) are not equal (unless we further require that σ 0, but then the lossy layer is not lossy). Thus, for oblique incidence, the numerator of the reflection coefficient cannot be zero and there will always be some reflection from this lossy medium Split-Field Perfectly Matched Layer To obtain a perfect match between the lossless and lossy regions, Bérenger proposed a non-physical anisotropic material known as a perfectly matched layer (PML). In a PML there is no loss in the direction tangential to the interface but there is loss normal to the interface. First, consider the goerning equations for the components of the magnetic fields for TM z polarization. We hae jωµ 2 H y +σ mx H y E z x jωµ 2 H x +σ my H x E z y (11.46) (11.47) whereσ mx andσ my are the magnetic conductiities associated not with thexandy components of the magnetic field, but rather with propagation in thexory direction. (Note that for 1D propagation in the x direction the non-zero fields are H y and E z while for 1D propagation in the y direction they areh x ande z.) For the electric field the goerning equation is jωǫ 2 E z +σe z H y x H x y (11.48)

7 11.4. SPLIT-FIELD PERFECTLY MATCHED LAYER 313 Here there is a single conductiity and no possibility to hae explicitly anisotropic behaior of the electrical conductiity. Thus, it would still be impossible to match the lossy region to the lossless one. Bérenger s fix was to split the electric field into two (non-physical) components. To get the actual field, we merely sum these components. Thus we write These components are goerned by E z E zx +E zy (11.49) jωǫ 2 E zx +σ x E zx H y x, (11.50) jωǫ 2 E zy +σ y E zy H x y. (11.51) Note that there are now two electrical conductiities: σ x and σ y. If we set σ x σ y σ and add these two equations together, we recoer the original equation (11.48). Further note that if σ y σ my 0 but σ x 0 and σ mx 0, then a wae with components H x and E zy would not attenuate while a wae with componentsh y ande zx would attenuate. Let us define the termss w ands mw as S w 1+ σ w jωǫ 2 (11.52) S mw 1+ σ mw jωµ 2 (11.53) wherew is eitherxor y. We can then write the goerning equations as jωǫ 2 S x E zx H y x, (11.54) jωǫ 2 S y E zy H x y, (11.55) jωµ 2 S mx H y x (E zx +E zy ), (11.56) jωµ 2 S my H x y (E zx +E zy ). (11.57) Taking the partial of (11.56) with respect toxand then substituting in the left-hand side of (11.54) yields ω 2 µ 2 ǫ 2 S x S mx E zx 2 x 2 (E zx +E zy ). (11.58) Taking the partial of (11.57) with respect toy and then substituting in the left-hand side of (11.55) yields ω 2 µ 2 ǫ 2 S y S my E zy 2 y (E 2 zx +E zy ). (11.59) Adding these two expressions together after diiding by thes terms yields ( 1 ω 2 2 µ 2 ǫ 2 (E zx +E zy ) S mx S x x + 1 ) 2 (E 2 S my S y y 2 zx +E zy ) (11.60)

8 314 CHAPTER 11. PERFECTLY MATCHED LAYER or, after rearranging and recalling thate zx +E zy E z, ( 1 S mx S x 2 x S my S y 2 y 2 +ω2 µ 2 ǫ 2 ) E z 0. (11.61) To satisfy this equation the transmitted field in the PML region would be gien by E t z Te j S mxs xβ 2x x j S mys yβ 2y y (11.62) where we must also hae β 2 2x +β 2 2y ω 2 µ 2 ǫ 2. (11.63) Using (11.56), they component of the magnetic field in the PML is gien by H t y 1 Ez t jωµ 2 S mx x (11.64) Smx S x β 2x E ωµ 2 S z, t (11.65) mx β 2x Sx E ωµ 2 S z. t (11.66) mx As always, the tangential fields must match at the interface. Matching the electric fields again yields (11.37). Matching they component of the magnetic fields yields 1 Γ µ 1β 2x Sx T. (11.67) µ 2 S mx Using this and (11.37) to sole for the transmission and reflection coefficients yields T Γ 2, (11.68) + β 2x S x µ 2 S mx β 2x S x µ 2 S mx. (11.69) + β 2x S x µ 2 S mx It is now possible to match the PML to the non-pml region so that Γ is zero. We begin by settingµ 2 and ǫ 2 ǫ 1. Thus we haeω 2 µ 2 ǫ 2 ω 2 ǫ 1 and β 2x ( ω 2 µ 2 ǫ 2 β 2 2y) 1/2, (11.70) ( ω 2 ǫ 1 β 2 2y) 1/2. (11.71) Recall that within the PML the propagation in the y direction is not gien by β 2y but rather by Sy S my β 2y. Phase matching along the interface requires that Sy S my β 2y β 1y. (11.72)

9 11.5. UN-SPLIT PML 315 If we let S y S my 1, which can be realized by setting σ y σ my 0, then the phase matching condition reduces to β 2y β 1y. (11.73) Then, from (11.71), we hae β 2x ( ω 2 ǫ 1 β 2 1y) 1/2, (11.74). (11.75) Returning to (11.69), we now hae Γ S x S mx + S x, (11.76) S mx S 1 x S mx. (11.77) S 1+ x S mx The last remaining requirement to achiee a perfect match is to hae S x S mx. This can be realized by haingσ x /ǫ 2 σ mx /µ 2. To summarize, the complete set of matching conditions for a constant-x interface are ǫ 2 ǫ 1 (11.78) µ 2 (11.79) σ y σ my 0 (11.80) σ x σ mx. (11.81) ǫ 2 µ 2 Under these conditions propagation in the PML is goerned by ( ( e jsxx jβ 1y y exp j 1+ σ ) ) x jβ 1y y, (11.82) jωǫ ( 1 exp β ) 1xσ x e jx jβ 1y y. (11.83) ωǫ 1 This shows that there is exponential decay in thexdirection but otherwise the phase propagates in exactly the same way as it does in the non-pml region Un-Split PML In the preious section we had S w and S mw where w {x,y}. Howeer, once the matching condition has been applied, i.e., σ w /ǫ 2 σ mw /µ 2, then we hae S w S mw. Hence we will drop the m from the subscript. Additionally, with the understanding that we are talking about the PML region, we will drop the subscript2from the material constants. We thus write S w 1+ σ w jωǫ. (11.84)

10 316 CHAPTER 11. PERFECTLY MATCHED LAYER The conductiity in the PML is not dictated by underlying parameters in the physical space being modeled. Rather, this conductiity is set so as to minimize the reflections from the termination of the grid. In that senseσ w is somewhat arbitrary. Therefore let us normalize the conductiity by the relatie permittiity that pertains at that particular location, i.e., S w 1+ σ w jωǫ 0 (11.85) whereσ w σ w /ǫ r. Howeer, since the conductiity has not yet been specified, we drop the prime and merely write S w 1+ σ w jωǫ 0 (11.86) Note that there could potentially be a problem with S w when the frequency goes to zero. In practice, in the curl equations this term is also multiplied by ω and in that sense this is not a major problem. Howeer, if we want to moe these S w terms around, low frequencies may cause problems. To fix this, we add an additional factor to ensure thats w remains finite as the frequency goes to zero. We can further generalize S w by allowing the leading term to take on alues other than unity. This is effectiely equialent to allowing the relatie permittiity in the PML to change. The general expression fors w we will use is S w κ w + σ w a w +jωǫ 0. (11.87) For the sake of considering 3D problems we also assumew {x,y,z}. Diiding by thes terms, the goerning equations for TM z polarization are Adding the first two equations together we obtain jωǫe zx 1 S x H y x, (11.88) jωǫe zy 1 S y H x y, (11.89) jωµh y 1 S x E z x, (11.90) jωµh x 1 S y E z y. (11.91) jωǫe z 1 S x H y x 1 S y H x y. (11.92) Note that in all these equations each S w term is always paired with the deriatie in the w direction. Let us define a new del operator that incorporates this pairing 1 â x S x x +â 1 y S y y +â 1 z S z z. (11.93)

11 11.5. UN-SPLIT PML 317 Using this operator Maxwell s curl equations become jωǫe H, (11.94) jωµh E. (11.95) Note that these equations pertain to the general 3D case. This is known as the stretch-coordinate PML formulation since, as shown in (11.93), the complex S terms scale the arious coordinate directions. Additionally, note there is no explicit mention of split fields in these equations. If we can find a way to implement these equations directly in the FDTD algorithm, we can aoid splitting the fields. From these curl equations we obtain scalar equations such as (using the x-component of (11.94) and (11.95) as examples) Conerting these to the time-domain yields jωǫe x 1 S y H z y 1 S z H y z, (11.96) jωµh x 1 S y E z y + 1 S z E y z. (11.97) ǫ E x t µ H x t S y H z y S z H y z, (11.98) S y E z y + S z E y z, (11.99) where indicates conolution and S w is the inerse Fourier transform of1/s w, i.e., [ ] 1 S w F 1. (11.100) S w The reciprocal ofs w is gien by 1 S w 1 κ w + σw a w+jωǫ 0 a w +jωǫ 0 a w κ w +σ w +jωκ w ǫ 0. (11.101) This is of the form (a + jωb)/(c + jωd) and we cannot do a partial fraction expansion since the order of the numerator and denominator are the same. Instead, we can diide the denominator into the numerator to obtain Recall the following Fourier transform pairs: a+jωb c+jωd b d + a cb d c+jωd b a d + b c d 1+jω d c. (11.102) 1 δ(t), (11.103) jωτ τ e t/τ u(t), (11.104)

12 318 CHAPTER 11. PERFECTLY MATCHED LAYER whereδ(t) is the Dirac delta function andu(t) is the unit step function. Thus, we hae [ a F 1 b d + ] b c d b ad bc δ(t)+ e ct/d u(t). (11.105) 1+jω d d d 2 c For the problem at hand, we hae a a w, b ǫ 0, c a w κ w + σ w, and d κ w ǫ 0. Using these alues in (11.105) yields S w 1 δ(t) σ ( [ w aw exp t + σ ]) w u(t). (11.106) κ w κ 2 wǫ 0 ǫ 0 κ w ǫ 0 Let us defineζ w (t) as ζ w (t) σ ( [ w aw exp t + σ ]) w u(t) (11.107) κ 2 wǫ 0 ǫ 0 κ w ǫ 0 so that S w 1 κ w δ(t)+ζ w (t). (11.108) Recall that the conolution of a Dirac delta function with another function yields the original function, i.e., δ(t) f(t) f(t). (11.109) Incorporating this fact into (11.98) yields ǫ E x t 1 H z κ y y 1 H y κ z z +ζ y (t) H z y ζ z(t) H y z. (11.110) Note that the first line of this equation is almost the usual goerning equation. The only differences are the κ s. Howeer, these are merely real constants. In the FDTD algorithm it is triial to incorporate these terms in the update-equation coefficients. The second line again inoles conolutions. Fortunately, these conolution are rather benign and, as we shall see, can be calculated efficiently using recursie conolution FDTD Implementation of Un-Split PML We now wish to deelop an FDTD implementation of the PML as formulated in the preious section. We start by defining the functionψ q E as uw Ψ q E ζ uw w(t) H w (11.111) tq t q t τ0 ζ w (τ) H (q t τ) dτ (11.112) w

13 11.6. FDTD IMPLEMENTATION OF UN-SPLIT PML 319 where E u w in the subscript indicates this function will appear in the update of the E u component of the electric field and it is concerned with the spatial deriatie in thew direction. In (11.112) the deriatie is of theh component of the magnetic field wherew,u, and are such that{w,u,} {x,y,z} and w u. In (11.112) note that ζ w (τ) is zero for τ < 0 hence the integrand would be zero for τ < 0. This fixes the lower limit of integration to zero. On the other hand, we assume the fields are zero prior to t 0, i.e., H (t) is zero for t < 0. In the conolution the argument of the magnetic field is q t τ. This argument will be negatie when τ > q t. Thus the integrand will be zero for τ > q t and this fixes the upper limit of integration toq t. Let us assume the integration ariable τ in (11.112) aries continuously but, since we are considering fields in the FDTD method, H aries discretely. We can still express H in terms of a continuously arying argument t, but it takes on discrete alues. Specifically H (t)/ w can be represented by Imax H (t) w wherep i (t) is the unit pulse function gien by { 1 if i t t < (i+1) p i (t) t, 0 otherwise. i0 H (i t ) p i (t) (11.113) w (11.114) To illustrate this further, for notational conenience let us write f(t) H (t)/ w. The stepwise representation of this function is shown in Fig. 11.1(a). Although not necessary, as is typical with FDTD simulations, this function is assumed to be zero for the first time-step. At t t the function isf 1 and it remains constant untilt 2 t when it changes tof 2. Att 3 t the function isf 3, and so on. The conolution contains the functionf(q t τ). At time-step zero (i.e.,q 0), this is merely f( τ) which is illustrated in Fig. 11.1(b). Here all the sample pointsf n are flipped symmetrically about the origin. We assume that the function is constant to the right of these sample points so that the function is f 1 for t τ < 0, it is f 2 for 2 t τ <, f 3 for 3 t τ < 2 t, and so on. Fig. 11.1(c) shows an example of f(q t τ) when q 0, specifically for q 4. Recall that in (11.112) the limits of integration are from zero to q t we do not need to concern ourseles with τ less than zero nor greater than q t. As shown in Fig. 11.1(c), the first pulse extending to the right of τ 0 has a alue of f q, the pulse extending to the right of τ t has a alue of f q 1, the one to the right of τ 2 t has a alue of f q 2, and so on. Thus, this shifted function can be written as q 1 f(q t τ) f q i p i (τ). (11.115) Returning to the deriatie of the magnetic field, we write H (q t τ) w i0 q 1 i0 H q i w p i(τ) (11.116) whereh q i is the magnetic field at time-stepq i and, when implemented in the FDTD algorithm, the spatial deriatie will be realized as a spatial finite difference.

14 320 CHAPTER 11. PERFECTLY MATCHED LAYER f(t) f 1 f 2 f 4 f 5 f f t, t (a) f( τ) f 5 f 4 f 3 f 2 f f τ, t (b) f(4 t τ) f 5 f 4 f 3 f 2 f f τ, t (c) Figure 11.1: (a) Stepwise representation of a function f(t). The function is a constant f 0 for 0 t < t, f 1 for t t < 2 t, f 2 for 2 t t < 3 t, and so on. (b) Stepwise representation of a function f( τ). Here the constants f n are flipped about the origin but the pulses still extend for one time-step to the right of the corresponding point. Hence the function is a constant f 1 for t τ < 0, f 2 for 2 t τ < t, etc. (c) Stepwise representation of the function f(q t τ) when q 4.

15 11.6. FDTD IMPLEMENTATION OF UN-SPLIT PML 321 At time-stepq, Ψ Euw is gien by q t q 1 Ψ q E uw ζ w (τ) τ0 Interchanging the order of summation and integration yields q 1 Ψ q E uw i0 q 1 i0 i0 H q i w H q i w H q i w p i(τ)dτ (11.117) q t τ0 ζ w (τ)p i (τ)dτ, (11.118) (i+1) t τi t ζ w (τ)dτ, (11.119) where, in going from (11.118) to (11.119), the pulse function was used to establish the limits of integration. Consider the following integral (i+1) i e at dt 1 (i+1) a e at (11.120) i 1 ( e a(i+1) e ai ) (11.121) a 1 a ( e a 1 ) e ai (11.122) 1 a ( e a 1 )( e a ) i (11.123) Keeping this in mind, the integration in (11.119) can be written as where (i+1) t σ w ζ w (τ)dτ κ 2 wǫ 0 τi t (i+1) t τi t exp ( [ aw τ ǫ 0 + σ w κ w ǫ 0 ]) dτ (11.124) C w (b w ) i (11.125) ( [ aw b w exp C w ǫ 0 + σ w κ w ǫ 0 ] t ), (11.126) σ w σ w κ w +κ 2 wa w (b w 1). (11.127) Note that in (11.125) b w is raised to the power i which is an integer index. It is now possible to expressψ q E as uw q 1 Ψ q E H q i uw w C w(b w ) i. (11.128) i0

16 322 CHAPTER 11. PERFECTLY MATCHED LAYER Let us explicitly separate thei 0 term from the rest of the summation: Ψ q E C H q q 1 uw w w + Replacing the indexiwithi i 1 (so thati i +1), this becomes Ψ q E C H q q 2 uw w w + i 0 Dropping the prime from the index and rearranging slightly yields i1 Ψ q E C H q [q 1] 1 uw w w +b w i0 H q i w C w(b w ) i. (11.129) H q i 1 w C w(b w ) i +1. (11.130) H [q 1] i C w (b w ) i. (11.131) w Comparing the summation in this expression to the one in (11.128) one sees that this expression can be written as Ψ q E C H q uw w w +b wψ q 1 E. (11.132) uw Note thatψ Euw at time-stepq is a function ofψ Euw at time-stepq 1. ThusΨ Euw can be updated recursiely there is no need to store the entire history of Ψ Euw to obtain the next alue. As is typical with FDTD, one merely needs to knowψ Euw at the preious time step. We now hae all the pieces in place to implement a PML in the FDTD method. The algorithm to update the electric fields is 1. Update the Ψ Euw terms employing the recursie update equation gien by (11.132). Recall that theseψ Euw functions represent the conolutions gien in the second line of (11.110). 2. Update the electric fields in the standard way. Howeer, incorporate theκ s where appropriate. Essentially one employs the update equation implied by the top line of (11.110) (where that equation applies toe x and similar equations apply toe y and E z ). 3. Apply, i.e., add or subtract, Ψ Euw to the electric field as indicated by the second line of (11.110). This completes the update of the electric field. The magnetic fields are updated in a completely analogous manner. First the Ψ functions that pertain to the magnetic fields are updated (in this case there are Ψ Huw functions that inole the spatial deriaties of the electric fields), then the magnetic fields are updated in the usual way (accounting for any κ s), and finally the Ψ functions are applied to the magnetic fields (i.e., added or subtracted).