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502 chapter 18Maxwell’s equations (18.59) and (18.60) now become the discrete equationsE k,n+1/2xH k+1/2,n+1y− Exk,n−1/2∆t∆t− H k+1/2,ny= − Hk+1/2,n y− H k−1/2,nyɛ 0 ∆z= − Ek+1,n+1/2xµ 0 ∆z,− E k,n+1/2x.To repeat, this formulation solves for the electric field at integer space steps (k)but half-integer time steps (n), while the magnetic field is solved for at half-integerspace steps but integer time steps.We convert these equations into two simultaneous algorithms by solving for E xat time n + 1 2 , and H y at time n:Exk,n+1/2 = Ex k,n−1/2 − ∆tɛ 0 ∆zHyk+1/2,n+1 = Hy k+1/2,n − ∆tµ 0 ∆z(Hk+1/2,ny(Ek+1,n+1/2x− Hyk−1/2,n ), (18.64)− Exk,n+1/2 ). (18.65)The algorithms must be applied simultaneously because the space variation of H ydetermines the time derivative of E x , while the space variation of E x determinesthe time derivative of H y (Figure 18.12). This algorithm is more involved than ourusual time-stepping ones in that the electric fields (filled circles in Figure 18.12) atfuture times t = n + 1 2are determined from the electric fields at one time step earliert = n − 1 2, and the magnetic fields at half a time step earlier t = n. Likewise, themagnetic fields (open circles in Figure 18.12) at future times t = n +1are determinedfrom the magnetic fields at one time step earlier t = n, and the electric field at halfa time step earlier t = n + 1 2. In other words, it is as if we have two interleavedlattices, with the electric fields determined for half-integer times on lattice 1 andthe magnetic fields at integer times on lattice 2.Although these half-integer times appear to be the norm for FDTD methods[Taf 89, Sull 00], it may be easier for some readers to understand the algorithm bydoubling the index values and referring to even and odd times:Exk,n = Ex k,n−2 − ∆tɛ 0 ∆zH k,ny= H k,n−2y − ∆tµ 0 ∆z(Hk+1,n−1y(Ek+1,n−1x− Hyk−1,n−1 ), k even, n odd, (18.66)− Exk−1,n−1 ), k odd, n even. (18.67)This makes it clear that E is determined for even space indices and odd times, whileH is determined for odd space indices and even times.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 502