The only source of electrical currents within dielectric media are the oscillating dipoles, their(bound )current density J in a dispersionless medium being given by J s=dp/dt= Eo(E- l)dE/dr n this and the following equations, the underline denotes time dependence, symbols without the underline are used to represent the complex amplitudes of time-harmonic fields. Assuming time-harmonic fields with the time-dependence factor exp(i@n), one can rewrite q (2)as Jb=-iaE(E-1). The B-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely, F=/Real(Jbx B ) where F is the (time-averaged) force per unit volume In FDTD simulations involving dispersive media, specific models (i.e, Debye, Drude, or Lorentz models) are used to represent the frequency dependence of the dielectric function @). Maxwell's equations are integrated in time over a discrete mesh, with the(model- dependent) dispersive behavior of the medium cast onto a time-dependent polarization vector We assume E(@)=Eo+AE(o), where the real-valued aoo denotes the relative permittivity of the medium in the absence of dispersion, the effects of dispersion enter through the complex valued function AE(@). The fraction of the polarization current density that embodies the contribution of Ae(@)is denoted by p. As the FDTD simulation progresses, determining the E-and H-fields as functions of time, Ip is computed concurrently by solving the relevant differential equations of the chosen model To derive an expression for the total (i.e, free carrier bound) current density i in a generally absorbing and dispersive medium, we begin with the following Maxwell equation Vx丑=aE+oDot Here the real-valued o denotes the material's conductivity for the free carriers. In our simulations, o is assumed to be a constant, independent of the frequency @; that is, the free carriers' conductivity is assumed to be dispersionless. Note that setting o=0 does not necessarily guarantee a transparent medium, as absorption could enter through the imaginary component of the complex permittivity, AE(o) Substituting for D in Eq ( 3)from the constitutive relation D=EE+P, we arrive at Eor=V×丑-(σE+Po)=V×丑-J, where, by definition, the total current density _ is given by D computations, Eq ( 3)is often rearranged as follows EnEo=V×旦-σE-Lp, where, for dispersive media(e.g, Debye, Drude, and Lorentz models), Eoo represents the frequency-independent part of the relative permittivity, while !p models dispersion (p=0 for non-dispersive dielectrics). When computing it is convenient to eliminate the time- derivative of E from Eq (5), as only one time-slice of E is normally stored during FDTD computations. From Eq (6), dE/Ot=(V XH-OE-Lp)Eoo, which, when substituted in Eq (5), yields J=(OE+Lp)lE+(1-llEoo)VXH Computing the total current density 2(i.e, the sum of the conduction and polarization current densities) at a given instant of time thus requires only the contemporary values of E, H, and Lp, which are readily available during the normal progression of an FDTD simulation #6863·$1500US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005 (C)2005OSA 4 April 2005/VoL 13, No. 7/OPTICS EXPRESS 2323The only source of electrical currents within dielectric media are the oscillating dipoles, their (bound) current density Jb in a dispersionless medium being given by Jb = ∂P/∂t = εo(ε – 1)∂E/∂t. (2) (In this and the following equations, the underline denotes time dependence; symbols without the underline are used to represent the complex amplitudes of time-harmonic fields.) Assuming time-harmonic fields with the time-dependence factor exp(−iω t), one can rewrite Eq. (2) as Jb = −iω εo(ε – 1)E. The B-field of the electromagnetic wave exerts a force on the bound current according to the Lorentz law, namely, F = ½Real (Jb × B*), where F is the (time-averaged) force per unit volume. In FDTD simulations involving dispersive media, specific models (i.e., Debye, Drude, or Lorentz models) are used to represent the frequency dependence of the dielectric function ε(ω). Maxwell’s equations are integrated in time over a discrete mesh, with the (model￾dependent) dispersive behavior of the medium cast onto a time-dependent polarization vector. We assume ε (ω) = ε ∞ +∆ε (ω), where the real-valued ε ∞ denotes the relative permittivity of the medium in the absence of dispersion; the effects of dispersion enter through the complex￾valued function ∆ε (ω). The fraction of the polarization current density that embodies the contribution of ∆ε (ω) is denoted by J p. As the FDTD simulation progresses, determining the E- and H-fields as functions of time, J p is computed concurrently by solving the relevant differential equations of the chosen model. To derive an expression for the total (i.e., free carrier + bound) current density Ĵ in a generally absorbing and dispersive medium, we begin with the following Maxwell equation: ∇ × H = σ E + ∂D/∂t. (3) Here the real-valued σ denotes the material’s conductivity for the free carriers. In our simulations, σ is assumed to be a constant, independent of the frequency ω ; that is, the free carriers’ conductivity is assumed to be dispersionless. Note that setting σ = 0 does not necessarily guarantee a transparent medium, as absorption could enter through the imaginary component of the complex permittivity, ∆ε (ω). Substituting for D in Eq.(3) from the constitutive relation D = εoE + P, we arrive at εo ∂E/∂t = ∇ × H − (σ E + ∂P/∂t ) = ∇ × H − Ĵ , (4) where, by definition, the total current density Ĵ is given by Ĵ = ∇ × H − εo ∂E/∂ t. (5) In FDTD computations, Eq.(3) is often rearranged as follows: εoε ∞ ∂E/∂t = ∇ × H − σ E − J p , (6) where, for dispersive media (e.g., Debye, Drude, and Lorentz models), ε ∞ represents the frequency-independent part of the relative permittivity, while J p models dispersion (J p = 0 for non-dispersive dielectrics). When computing Ĵ , it is convenient to eliminate the time￾derivative of E from Eq.(5), as only one time-slice of E is normally stored during FDTD computations. From Eq.(6), εo ∂E/∂t = (∇ × H − σ E − J p) /ε ∞ , which, when substituted in Eq.(5), yields Ĵ = (σ E + J p)/ε ∞ + (1 − 1/ε ∞ )∇ ×H. (7) Computing the total current density Ĵ (i.e., the sum of the conduction and polarization current densities) at a given instant of time thus requires only the contemporary values of E, H, and J p , which are readily available during the normal progression of an FDTD simulation. (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2323 #6863 - $15.00 US Received 14 January 2005; revised 15 March 2005; accepted 15 March 2005