82 A.Klaedtke,J.Hamm,and O.Hess the computational cube it is terminated with a totally reflecting boundary. The workings of such an enclosing PM layer is shown in Fig.5.6.A wave coming from the left enters the PML.On the right side,the wave is totally reflected and crosses the PML again being further reduced in magnitude. To describe such a behaviour in terms of the Maxwell equations in fre- quency space,the dielectric constant and the permeability have to be tenso- rial quantities.It turns out that both can be separated into a scalar and a tensorial part,with the tensorial part s being the same for both. VXH=-wers·E (5.8) 7XE=w山rs·H The ansatz requiring a tensor s that absorbs an incoming wave without re- flecting it leads to 5.9. Oi 11=h- (5.9) w The parameters si-with i being one of x,y or z-can be separated into a real()and an imaginary part (oi/w).The real part degrades evanescent waves.The imaginary part absorbs energy. The conversion of the differential equation 5.8 from frequency space to time-domain requires the introduction of two new fields D and B as shown in 5.10 to avoid the calculation of a convolution integral. 0 0 E B=Hr (5.10) With this substitution,it is possible to separate the real and imaginary parts occurring on the right side of 5.8.The transformation to time-domain is then simply the substitution of -ww by the time derivative. 0 aD 0 VxH= Kz ·D Ot 0 Kx】 0 Ky 0 0 (5.11) 8B 7XE=- Kz ·B 0 8t82 A. Klaedtke, J. Hamm, and O. Hess the computational cube it is terminated with a totally reflecting boundary. The workings of such an enclosing PM layer is shown in Fig. 5.6. A wave coming from the left enters the PML. On the right side, the wave is totally reflected and crosses the PML again being further reduced in magnitude. To describe such a behaviour in terms of the Maxwell equations in fre￾quency space, the dielectric constant and the permeability have to be tenso￾rial quantities. It turns out that both can be separated into a scalar and a tensorial part, with the tensorial part s being the same for both. ∇ × H = − ıω rs · E ∇ × E = ıω µrs · H (5.8) The ansatz requiring a tensor s that absorbs an incoming wave without re- flecting it leads to 5.9. s =   sysz sx 0 sxsz sy 0 sxsy sz   , si = κi − σi ıω (5.9) The parameters si — with i being one of x, y or z — can be separated into a real (κi) and an imaginary part (σi/ω). The real part degrades evanescent waves. The imaginary part absorbs energy. The conversion of the differential equation 5.8 from frequency space to time-domain requires the introduction of two new fields D and B as shown in 5.10 to avoid the calculation of a convolution integral. D = r   sz sx 0 sx sy 0 sy sz   · E; B = µr   sz sx 0 sx sy 0 sy sz   · H (5.10) With this substitution, it is possible to separate the real and imaginary parts occurring on the right side of 5.8. The transformation to time-domain is then simply the substitution of −ıω by the time derivative. ∇ × H =   κy 0 κz 0 κx   · ∂ D ∂t +   σy 0 σz 0 σx   · D ∇ × E = −   κy 0 κz 0 κx   · ∂ B ∂t −   σy 0 σz 0 σx   · B (5.11)