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500 chapter 18xE xtH yzyH yE xFigure 18.11 A single electromagnetic pulse traveling along the z axis. The two pulsescorrespond to two different times, and the coupled electric and magnetic fields areindicated by solid and dashed curves, respectively.∂E∂t =+1 ɛ 0⃗ ∇×H ⇒∂E x∂t= − 1 ∂H y (z,t), (18.59)ɛ 0 ∂z∂H∂t = − 1 µ 0⃗ ∇×E ⇒∂H y∂t= − 1 ∂E x (z,t). (18.60)µ 0 ∂zAs indicated in Figure 18.11, we have chosen the electric field E(z,t) to oscillate (bepolarized) in the x direction and the magnetic field H(z,t) to be polarized in they direction. As indicated by the bold arrow in Figure 18.11, the direction of powerflow for the assumed transverse electromagnetic (TEM) wave is given by the righthandrule for E × H. Note that although we have set the initial conditions such thatthe EM wave is traveling in only one dimension (z), its electric field oscillates in aperpendicular direction (x) and its magnetic field oscillates in yet a third direction(y); so while some may call this a 1-D wave, the vector nature of the fields meansthat the wave occupies all three dimensions.18.11 FDTD AlgorithmWe need to solve the two coupled PDEs (18.59) and (18.60) appropriate for ourproblem. As is usual for PDEs, we approximate the derivatives via the centraldifferenceapproximation, here in both time and space. For example,∂E(z,t)∂t≃∆t∆tE(z,t+2) − E(z,t−∆t2 ), (18.61)−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 500