Notes on the UPML implementation in Meep

Steven G. Johnson

Posted August 17, 2009; updated March 10, 2010

1 OverviewIn this note, we go over the implementation of UPML absorbing boundary layersin the free Meep FDTD package. It is assumed that the reader is already familiarwith the basic stretched-coordinate derivation of PML (see http://math.mit.edu/~stevenj/18.369/pml.pdf ), as well as the general technique by which acoordinate transformation can be expressed as a transformation of ε and µ (seehttp://math.mit.edu/~stevenj/18.369/coordinate-transform.pdf), lead-ing to the so-called “UPML” formulation. The focus here is on how these trans-formed materials are expressed in Meep’s FDTD algorithm. The subtlety isthat the transformations/materials of PML are frequency-dependent, and soto express them in time domain involves the evolution of appropriate auxiliarydifferential equations. (Equivalently, multiplication by a frequency-dependentsusceptibility corresponds in the time domain to a convolution, leading to so-called “convolutional PML” formulations.) The trick is to keep the number ofauxiliary differential equations (and the resulting memory and computationalcosts) to a minimum, while not making the PML region too complicated (orrequire too much new code!) compared to the non-PML regions.

Because the treatment of ε and µ are identical except for interchange of ~Dwith ~B and ~E with ~H (and a sign flip from Ampere’s to Faraday’s law), we onlydescribe the ε and d ~D/dt equations (Ampere’s law) here.

We proceed slightly differently from the UPML as derived in e.g. the Tafloveand Hagness FDTD textbook. As reviewed in the appendix, this “standard”UPML performs a matrix factorization that relies on ε commuting with thePML Jacobian, and this is not the case for an arbitrary anisotropic ε. Ourfactorization, instead, works for anisotropic ε, and turns out to have a niceproperty: the PML just adds two auxiliary fields ~U and ~W with diagonal rela-tionships to ~D and ~E, and allow us to use the non-PML timestepping operationsunchanged except for the addition of these diagonal ODE updates. In particular,we get a true PML for anisotropic, dispersive, media “for free” in the sense thatthe code for those portions is unchanged (although the case of conducting medianecessitates an extra auxiliary field). This is nice because the timestepping foranisotropic media requires special care for stability, and dispersive media mayhave complicated polarization-update equations, and this way we don’t need to

1

modify any of that in the slightest for the PML. Our approach also makes aclean separation between the ~D update equation (from ∇× ~H) and terms thataffect the ~E update equation (from the constitutive equations), and correspond-ingly in the Meep code these are handled separately in step_db and update_eh,respectively.

2 Non-PML materials in MeepIn Meep, we support dispersive media ε(ω) of the form:

ε(ω) =(

1 +iσD

ω

)[ε∞ + χE(ω)] ,

where σD is a conductivity (which may be anisotropic, but must be diagonal inCartesian coordinates and hence commutes with J ), ε∞ is a non-dispersive partof the permittivity (which may be non-diagonal anisotropic) and χE(ω) is someadditional dispersive part (possibly anisotropic) implemented by auxiliary ODEsthat solve ~P = χ(ω) ~E (currently, a sum of Lorentzians). This corresponds, intime-domain, to Ampere’s law of the form:

~K = ∇× ~H =∂ ~D

∂t+ σD

~D, (1)

denoting ∇× ~H by ~K for later convenience, and a constitutive equation

~D = ε∞ ~E + ~P ,

where ~P is time-evolved via a system of ODEs derived from ~P = χ(ω) ~E. (Wedon’t include free currents ~J here, since they have no impact on the PMLequations, and indeed one rarely puts free currents inside the PML anyway.)

In FDTD, we discretize these in space and time. Let us denote the timediscretization by a superscript: ~Dn denotes ~D(n∆t), and similarly for ~Pn and~En. The magnetic fields are offset in time by half a time step, giving ~Hn+0.5 and~Kn+0.5. To time-step these fields in FDTD, we first compute ~Dn+1 from ~Dn and~Kn+0.5 by the curl equation, then compute ~En+1 from ~Dn+1 and ~Pn+1 by theconstitutive equation, and finally compute ~Pn+2. (Note that, with Lorentziandispersion, we can compute ~Pn+2 from ~Pn+1 and ~Pn given ~En+1.

Using the standard second-order center-difference approximations, the equa-tion for ~D becomes:

~Kn+0.5 =~Dn+1 − ~Dn

∆t+ σD

~Dn+1 + ~Dn

2,

giving

~Dn+1 =(

1 +σD∆t

2

)−1 [(1− σD∆t

2

)~Dn + ~Kn+0.5

]. (2)

2

The constitutive equation is simply:

~En+1 = ε−1∞

(~Dn+1 − ~Pn+1

). (3)

(This is nontrivial to implement correctly for anisotropic media [Werner andCary, 2009], but the details need not concern us here.)

3 PML formulation in frequency domainPML is simplest to derive in frequency domain, where the fields all have time-dependence e−iωt. An ordinary PML in Cartesian coordinates is derived by acomplex coordinate stretching, where each coordinate is stretched by a factor

sx,y,z = 1 +iσx,y,z

ω,

where ω is the frequency and σ is the PML “conductivity.” For example, toterminate the cell in the x direction, only σx is nonzero. These coordinatestretchings can be absorbed into Maxwell’s equations as a change in ε and µ.The original permittivity ε in the PML region is replaced by an effective tensorε̃ given by

ε̃ =J εJT

detJ,

where J = diag(s−1x , s−1

y , s−1z ) is the Jacobian matrix of the coordinate stretch-

ing.In frequency domain, replacint ε(ω) with ε̃, eq. (1) becomes:

~K = ∇× ~H = −iωsxsysz

(1 +

iσD

ω

) s−1x

s−1y

s−1z

[ε∞ + χE(ω)]

s−1x

s−1y

s−1z

~E.

(4)Although this is straightforward to implement in frequency domain, where oneε(ω) is as good as another, in the time domain a frequency-dependent termrequires care. In time-domain, a frequency-dependent term can be thought ofas a convolution with a filter (hence the viewpoint of a “convolutional PML”adopted by some authors), where in the language of signal-processing we wishto find a stable recursive filter to implement this convolution with as few tapsas possible (to minimize the memory and computational burden). Equivalently,the frequency dependence may be implemented by auxiliary ordinary differentialequations (discretized ODE = recursive filter). The most convenient ODEs todiscretize are first-order ODEs. For example, a = sb = (1 + iσ/ω)b gives−iωa = −iωb+ σb, which corresponds to the first-order ODE da

dt = dbdt + σb.

So, before we proceed to time domain, we want to factorize eq. (16) intoterms with only one factor of s (or σ/ω) each (or ratios of single s factors). Indoing so, we are free to change the definition of ~D and introduce new auxiliary

3

fields as desired, since the fields in the PML region are not physical. A key trickis the factorization:

sxsysz

s−1x

s−1y

s−1z

=

sy

sz

sx

sz

sx

sy

.

Using this factorization, and defining new auxiliary fields, ~C, ~U , and ~W , we canfactorize eq. (16) into the following equivalent form (giving the same relationshipbetween ~H and ~E):

~K = ∇× ~H = −iω(

1 +iσD

ω

)~C (5)

~U =

s−1y

s−1z

s−1x

~C (6)

~D =

s−1z

s−1x

s−1y

~U (7)

~W = ε−1∞

(~D − ~P

)(8)

~P = χE(ω) ~W (9)

~E =

sx

sy

sz

~W (10)

4 PML formulation in time domainGiven eqs. (5–10), a time-domain formulation is now easy because each equationonly includes first-order factors in ω (corresponding to first-order time deriva-tives). Moreover, all of the nontrivial equations, namely (5), (8), and (9), areexactly the same as in the non-PML case, meaning that we can use exactly thesame code except passing new fields ~C and ~W instead of ~D and ~E. The otherequations are diagonal ODEs that are trivial to implement. It may seem waste-ful to have three new auxiliary fields, but in many cases they can be omitted:except in corners of the computational cell, one has PML only in one directionthat only one component of sx,y,z is 6= 1, and in these cases one need only storeone component of ~U and one component of ~W (with the other components re-placed by ~D and ~E, respectively); ~C can be omitted in non-conducting materials(σD = 0).

For completeness, we write out the equations here, in the order that theywould be evaluated. The ~C update from eq. (5) is identical to the ~D timestep

4

from eq. (11)

~Cn+1 =(

1 +σD∆t

2

)−1 [(1− σD∆t

2

)~Cn + ~Kn+0.5

]. (11)

Each component Uk of ~U is updated from the corresponding component of ~Cby the ODE dUk/dt+ σk+1Uk = dCk/dt (where σk±1 is interpreted as a cyclicshift, e.g. σy+1 = σz, σz+1 = σx), giving:

Un+1k =

(1 +

σk+1∆t2

)−1 [(1− σk+1∆t

2

)Un

k + Cnk − Cn−1

k

]. (12)

Each component of ~D is then updated from the corresponding component of ~Uby the ODE dDk/dt+ σk−1Dk = dUk/dt, giving:

Dn+1k =

(1 +

σk−1∆t2

)−1 [(1− σk−1∆t

2

)Dn

k + Unk − Un−1

k

]. (13)

~W is then updated from ~D − ~P exactly as in eq. (14):

~Wn+1 = ε−1∞

(~Dn+1 − ~Pn+1

). (14)

Then, each component of ~E is updated from each component of ~W by the ODEdEk/dt = dWk/dt+ σkWk, giving:

En+1k = En

k +(

1 +σk∆t

2

)Wn+1

k −(

1− σk∆t2

)Wn

k . (15)

Finally, ~Pn+2 is computed using ~P = χE(ω) ~W , exactly as for the non-PMLcase (but with ~W replacing ~E).

Note that, in order to avoid having to save the fields from the previoustimestep in yet more auxiliary arrays, the ~C, ~U , and ~D updates [eqs. (11–13)]have to be performed in a single loop body, while the ~W and ~E updates [eqs. (14–15)] must be performed in another single loop body. [One cannot easily mergethe two loops because the offdiagonal anisotropic terms in ε−1

∞ combined with thestaggered Yee grid mean that eq. (14) is effectively nonlocal in space, requiring~Dn+1 components at several spatial points to determine ~Wn+1. Separatingthe two loops has the additional advantage that it reduces the combinatorialexplosion of the number of material cases that must be handled.]

5 Appendix: Textbook isotropic, nondispersivePML

In this appendix, we review the standard “textbook” UPML for isotropic nondis-persive media as found in, for example, Taflove and Hagness. It is important

5

to distinguish mere differences in notation (the derivation in T&H is far lesscompact than the one here) from substantive differences, and in particular theusual UPML corresponds to a different factorization of eq. (16) than ours. Thistextbook formulation only works for isotropic (or at least diagonal) ε, becauseit assumes that an arbitrary diagonal matrix commutes with ε.

In particular, dropping the σD and χE terms (assuming nondispersive me-dia), the textbook PML formulation corresponds to the following refactorizationof eq. (16):

~K = ∇× ~H = −iω

sy

sz

sx

ε∞

sz

sxsx

sysy

sz

~E. (16)

This then factorizes into two equations:

~K = ∇× ~H = −iω

sy

sz

sx

~D,

~E = ε−1∞

sx

sz sy

sxsz

sy

~D.

The former discretizes similarly to eq. (2), with the PML σy,z,x taking the placeof σD, and the latter turns into the ODE dEk/dt+σk−1Ek = ε−1

∞ [dDk/dt+σkDk]which is easily discretized.

Only two fields need be stored: ~D and ~E. If a conductivity σD (or similar)is included, however, one needs an additional auxiliary field ~C (or similar). Itturns out that the case of a dispersive ~P = χE(ω) ~E can also be handled with noadditional storage compared to the non-PML case, at least for a Lorentzian χE

that requires both ~Pn+1 and ~Pn to be stored anyway. So, the main cost of theMeep formulation is that two additional auxiliary fields ~U and ~W must be storedin the PML; however, this is actually only a slight additional burden in storagesince for most PML regions only one component of ~U and ~W actually mustbe stored (see above). The main problem with the textbook implementationis that it fails for anisotropic media (even if one naively plugs a tensor ε−1

∞into the ~E update, the reflection will not go to zero with increasing resolution).This is especially important given the fact that, even for nominally isotropicmedia, accurate subpixel averaging requires the discretization to use effectiveanisotropic media at material boundaries.

6