Polynomial Chaos Approach for Maxwell s Equations in Dispersive Media

Polynomial Chaos Approach for Maxwell s Equations in Dispersive Media Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 15, 2013 Prof. Gibson (OSU) PC-FDTD AMCS 2013 1 / 41

Acknowledgements Collaborators H. T. Banks (NCSU) V. A. Bokil (OSU) W. P. Winfree (NASA) Students Karen Barrese and Neel Chugh (REU 2008) Anne Marie Milne and Danielle Wedde (REU 2009) Erin Bela and Erik Hortsch (REU 2010) Megan Armentrout (MS 2011) Brian McKenzie (MS 2011) Prof. Gibson (OSU) PC-FDTD AMCS 2013 2 / 41

Outline 1 Maxwell s Equations Dispersive Media Polarization Models Distribution of Parameters 2 Dispersion Analysis Discrete Dispersion Prof. Gibson (OSU) PC-FDTD AMCS 2013 3 / 41

Maxwell s Equations Maxwell s Equations D t B t = H J in Ω (0, T] = E in Ω (0, T] D = B = 0 in Ω (0, T] E(0, x) = E 0 (x); H(0, x) = H 0 (x) in Ω E n = 0 on Ω, t (0, T] E = Electric field vector D = Electric flux density H = Magnetic field vector B = Magnetic flux density J = Current density n = Unit outward normal to Ω Prof. Gibson (OSU) PC-FDTD AMCS 2013 4 / 41

Maxwell s Equations Constitutive Laws Maxwell s equations are completed by constitutive laws that describe the response of the medium to the electromagnetic field. D = ɛe + P B = µh + M J = σe + J s P = M = J s = Polarization Magnetization Source Current ɛ = Electric permittivity µ = Magnetic permeability σ = Electric Conductivity Prof. Gibson (OSU) PC-FDTD AMCS 2013 5 / 41

Maxwell s Equations Dispersive Media Complex permittivity We can usually define P in terms of a convolution P(t, x) = g E(t, x) = t 0 g(t s, x; q)e(s, x)ds, where g is the dielectric response function (DRF). In the frequency domain ˆD = ɛê + ĝê = ɛ 0 ɛ(ω)ê, where ɛ(ω) is called the complex permittivity. ɛ(ω) described by the polarization model We are interested in ultra-wide bandwidth electromagnetic pulse interrogation of dispersive dielectrics, therefore we want an accurate representation of ɛ(ω) over a broad range of frequencies. Prof. Gibson (OSU) PC-FDTD AMCS 2013 6 / 41

Maxwell s Equations Dispersive Media Dry skin data 10 3 True Data Debye Model Cole Cole Model ε 10 2 10 1 10 2 10 4 10 6 10 8 10 10 f (Hz) Figure: Real part of ɛ(ω), ɛ, or the permittivity [GLG96]. Prof. Gibson (OSU) PC-FDTD AMCS 2013 7 / 41

Maxwell s Equations Dispersive Media Dry skin data 10 1 10 0 10 1 σ 10 2 10 3 10 4 True Data Debye Model Cole Cole Model 10 2 10 4 10 6 10 8 10 10 f (Hz) Figure: Imaginary part of ɛ(ω)/ω, σ, or the conductivity. Prof. Gibson (OSU) PC-FDTD AMCS 2013 8 / 41

Maxwell s Equations Polarization Models P(t, x) = g E(t, x) = t 0 g(t s, x; q)e(s, x)ds, Debye model [1929] q = [ɛ d, τ] or g(t, x) = ɛ 0 ɛ d /τ e t/τ τṗ + P = ɛ 0 ɛ d E or ɛ(ω) = ɛ + ɛ d 1 ωτ with ɛ d := ɛ s ɛ and τ a relaxation time. Cole-Cole model [1936] (heuristic generalization) q = [ɛ d, τ, α] ɛ d ɛ(ω) = ɛ + 1 + (iωτ) 1 α Prof. Gibson (OSU) PC-FDTD AMCS 2013 9 / 41

Maxwell s Equations Polarization Models Dispersive Dielectrics Debye Material Input is five cycles (periods) of a sine curve. Prof. Gibson (OSU) PC-FDTD AMCS 2013 10 / 41

Maxwell s Equations Polarization Models Dispersive Media 1000 Time=5.0ns 40 Time=10.0ns 500 20 E 0 500 E 0 20 40 60 1000 0 0.2 0.4 0.6 0.8 z 80 0 0.2 0.4 0.6 0.8 z Figure: Debye model simulations. Prof. Gibson (OSU) PC-FDTD AMCS 2013 11 / 41

Maxwell s Equations Polarization Models Scalar Equations in Two Dimensions (cont) Letting H = H z, we have the 2D Maxwell-Debye TE scalar equations: H t = 1 ( Ex µ 0 y E ) y, x ɛ 0 ɛ E x t E y ɛ 0 ɛ t P x t P y t = H y P x t, = H x P y t, = ɛ 0ɛ d τ E x 1 τ P x, = ɛ 0ɛ d τ E y 1 τ P y. Prof. Gibson (OSU) PC-FDTD AMCS 2013 12 / 41

Maxwell s Equations Polarization Models Stability Estimates for Maxwell-Debye Theorem (Li2010) For Ω R 2, let E, P, and H be the solutions to the 2D Maxwell-Debye TE scalar equations with PEC boundary conditions. Then the Debye model satisfies the stability estimate where the energy is defined as E(t) E(0) E(t) = µ 0 H(t) 2 L + ɛ 2 0 ɛ E(t) 2 L + 1 2 P(t) ɛ0 ɛ d and the L 2 (Ω) norm is defined as U(t) 2 2 = Ω U(z, t) 2 dz. Prof. Gibson (OSU) PC-FDTD AMCS 2013 13 / 41 2 L 2.

Maxwell s Equations Distribution of Relaxation Times Motivation for Distributions The Cole-Cole model corresponds to a fractional order ODE in the time-domain and is difficult to simulate Debye is efficient to simulate, but does not represent permittivity well Better fits to data are obtained by taking linear combinations of Debye models (discrete distributions), idea comes from the known existence of multiple physical mechanisms: multi-pole debye (like stair-step approximation) An alternative approach is to consider the Debye model but with a (continuous) distribution of relaxation times [von Schweidler1907] Empirical measurements suggest a log-normal or Beta distribution [Wagner1913] (but uniform is easier) Using Mellin transforms, can show Cole-Cole corresponds to a continuous distribution Prof. Gibson (OSU) PC-FDTD AMCS 2013 14 / 41

Maxwell s Equations Fit to dry skin data with uniform distribution 10 3 True Data Debye (27.79) Cole Cole (10.4) Uniform (13.60) ε 10 2 10 2 10 4 10 6 10 8 10 10 f (Hz) Figure: Real part of ɛ(ω), ɛ, or the permittivity [REU2008]. Prof. Gibson (OSU) PC-FDTD AMCS 2013 15 / 41

Maxwell s Equations Fit to dry skin data with uniform distribution 10 1 10 0 True Data Debye (27.79) Cole Cole (10.4) Uniform (13.60) 10 1 σ 10 2 10 3 10 2 10 4 10 6 10 8 10 10 f (Hz) Figure: Imaginary part of ɛ(ω)/ω, σ, or the conductivity [REU2008]. Prof. Gibson (OSU) PC-FDTD AMCS 2013 16 / 41

Maxwell s Equations Distribution of Parameters Distributions of Parameters To account for the effect of possible multiple parameter sets q, consider h(t, x; F ) = g(t, x; q)df (q), where Q is some admissible set and F P(Q). Then the polarization becomes: P(t, x) = t 0 Q h(t s, x; F )E(s, x)ds. Prof. Gibson (OSU) PC-FDTD AMCS 2013 17 / 41

Maxwell s Equations Distribution of Parameters Random Polarization Alternatively we can define the random polarization P(t, x; τ) to be the solution to τṗ + P = ɛ 0 ɛ d E where τ is a random variable with PDF f (τ), for example, f (τ) = 1 τ b τ a for a uniform distribution. The electric field depends on the macroscopic polarization, which we take to be the expected value of the random polarization at each point (t, x) P(t, x) = τb τ a P(t, x; τ)f (τ)dτ. Prof. Gibson (OSU) PC-FDTD AMCS 2013 18 / 41

Maxwell-Random Debye system In a polydisperse Debye material, we have with E ɛ 0 ɛ t = H P t H t = 1 E µ 0 τ P t + P = ɛ 0ɛ d E P(t, x) = τb τ a P(t, x; τ)f (τ)dτ. (1a) (1b) (1c) Prof. Gibson (OSU) PC-FDTD AMCS 2013 19 / 41

Dispersion Relation Theorem (Gibson2013) The dispersion relation for the system (1) is given by ω 2 c 2 ɛ(ω) = k 2 where the complex permittivity is given by [ ] 1 ɛ(ω) = ɛ + ɛ d E 1 iωτ Here, k is the wave number and c = 1/ µ 0 ɛ 0 is the speed of light in freespace. Note: for a uniform distribution, this has an analytic form since [ ] 1 E 1 iωτ = 1 [arctan(ωτ) i 12 2τ r ω ln ( 1 + (ωτ) 2)] τ=τ m+τ r τ=τ m τ r Prof. Gibson (OSU) PC-FDTD AMCS 2013 20 / 41

Dispersion Relation Proof: (for 2D) Letting H = H z, we have the 2D Maxwell-Random Debye TE scalar equations: H t = 1 ( Ex µ 0 y E ) y, (2a) x E x ɛ 0 ɛ = H t y P x t, (2b) ɛ 0 ɛ E y t = H x P y t, τ P x t + P x = ɛ 0 ɛ d E x τ P y t + P y = ɛ 0 ɛ d E y. (2c) (2d) (2e) Prof. Gibson (OSU) PC-FDTD AMCS 2013 21 / 41

Dispersion Relation Proof: (cont.) Assume plane wave solutions of the form (kx x+ky y ωt) V = Ṽ e and let x = (x, y) T and k = (k x, k y ) T. Then (2) becomes iω H = 1 µ 0 ( ik y Ẽ x ik x Ẽ y ), (3a) ɛ 0 ɛ iωẽ x = ik y H ( iω P x ), ɛ 0 ɛ iωẽy = ik x H ( iω Py ), iωτ P x + P x = ɛ 0 ɛ d Ẽ x iωτ P y + P y = ɛ 0 ɛ d Ẽ y. (3b) (3c) (3d) (3e) Prof. Gibson (OSU) PC-FDTD AMCS 2013 22 / 41

Dispersion Relation Proof: (cont.) By (3d) we have [ ] 1 P x = E[ P x ] = ɛ 0 ɛ d Ẽ x E 1 iωτ (4) Substituting into (3b) we have ( [ 1 ɛ 0 ɛ iωẽ x = ik y H + iωɛ 0 ɛ d Ẽ x E 1 iωτ [ 1 iωɛ 0 (ɛ + ɛ d E 1 iωτ Similarly for combining (3e) and (3c) ]), ]) Ẽ x = ik y H, iωɛ 0 ɛ(ω)ẽx = ik y H. (5) iωɛ 0 ɛ(ω)ẽ y = ik x H. (6) Prof. Gibson (OSU) PC-FDTD AMCS 2013 23 / 41

Dispersion Relation Proof: (cont.) Substituting both (5) and (6) into (3a) yields iω H = 1 ( (iky ) 2 µ 0 iωɛ 0 ɛ(ω) + (ik x) 2 ) H, iωɛ 0 ɛ(ω) (iω) 2 µ 0 ɛ 0 ɛ(ω) = ky 2 + kx 2 ω 2 c 2 ɛ(ω) = k 2 The proof is similar in 1 and 3 dimensions. The exact dispersion relation will be compared with a discrete dispersion relation to determine the amount of dispersion error. Prof. Gibson (OSU) PC-FDTD AMCS 2013 24 / 41

Discrete Dispersion Finite Difference Methods The Yee Scheme In 1966 Kane Yee originated a set of finite-difference equations for the time dependent Maxwell s curl equations. The finite difference time domain (FDTD) or Yee algorithm solves for both the electric and magnetic fields in time and space using the coupled Maxwell s curl equations rather than solving for the electric field alone (or the magnetic field alone) with a wave equation. Approximates first order derivatives very accurately by evaluating on staggered grids. Prof. Gibson (OSU) PC-FDTD AMCS 2013 25 / 41

Discrete Dispersion Yee Scheme in One Space Dimension Staggered Grids: The electric field/flux is evaluated on the primary grid in both space and time and the magnetic field/flux is evaluated on the dual grid in space and time. The Yee scheme is H n+ 1 2 l+ 1 2 H n 1 2 l+ 1 2 t E n+1 l E n l t = 1 µ = 1 ɛ E n l+1 E n l z H n+ 1 2 l+ 1 2 H n+ 1 2 l 1 2 z t n+1... z 5 z 2 z 2 3 z 1 z 2 1 z 0 2 h z 1 z 1 z 3 z 2 z 5... 2 2 2 t n+ 1 2 Prof. Gibson (OSU) PC-FDTD AMCS 2013 26 / 41

Discrete Dispersion The FDTD or Yee grid in 1D ( E ( H ) n+ 3 2 ) n+2 k+ 1 2 k+2 n + 2 n + 3 2 n + 1 n + 1 2 n k k+ 1 2 k+1 k+ 3 2... Prof. Gibson (OSU) PC-FDTD AMCS 2013 27 / 41

Discrete Dispersion This gives an explicit second order accurate scheme in both time and space. It is conditionally stable with the CFL condition ν = c t h 1 d where ν is called the Courant number and c = 1/ ɛµ and d is the spatial dimension. The initial value problem is well-posed and the scheme is consistent and stable. The method is convergent by the Lax-Richtmyer Equivalence Theorem. The Yee scheme can exhibit numerical dispersion. Dispersion error can be reduced by decreasing the mesh size or resorting to higher order accurate finite difference approximations. Prof. Gibson (OSU) PC-FDTD AMCS 2013 28 / 41

Discrete Dispersion Extensions of the Yee Scheme to Dispersive Media The ordinary differential equation for the polarization is discretized using an averaging of zero order terms. The resulting scheme remains second-order accurate in both time and space with the same CFL condition. However, the Yee scheme for the Maxwell-Debye system is now dissipative in addition to being dispersive. Prof. Gibson (OSU) PC-FDTD AMCS 2013 29 / 41

Discretization Yee Scheme for Maxwell-Debye System (in 1D) become E n+ 1 2 E n 1 2 ɛ 0 ɛ t H n+1 H n + µ 1 + 1 2 2 0 t 1 τ Pn+ 2 P n 1 2 t E ɛ 0 ɛ t = H z P t H µ 0 t = E z τ P t = ɛ 0ɛ d E P = H n H n + 1 1 2 2 z = E n+ 1 2 +1 E n+ 1 2 z E n+ 1 2 + E n 1 2 = ɛ 0 ɛ d 2 1 Pn+ 2 P n 1 2 t 1 Pn+ 2 + P n 1 2. 2 Prof. Gibson (OSU) PC-FDTD AMCS 2013 30 / 41

Discretization Discrete Debye Dispersion Relation (Petropolous1994) showed that for the Yee scheme applied to the (deterministic) Maxwell-Debye, the discrete dispersion relation can be written ω 2 c 2 ɛ (ω) = K 2 where the discrete complex permittivity is given by ( ) 1 ɛ (ω) = ɛ + ɛ d 1 iω τ with discrete representations of ω and τ given by ω = sin (ω t/2), τ = sec(ω t/2)τ t/2 Prof. Gibson (OSU) PC-FDTD AMCS 2013 31 / 41

Discretization Discrete Debye Dispersion Relation (cont.) The quantity K is given by K = sin (k z/2) z/2 in 1D and is related to the symbol of the discrete first order spatial difference operator by ik = F(D 1, z ). In this way, we see that the left hand side of the discrete dispersion relation ω 2 c 2 ɛ (ω) = K 2 is unchanged when one moves to higher order spatial derivative approximations (Bokil-Gibson2011) or even higher spatial dimension. Prof. Gibson (OSU) PC-FDTD AMCS 2013 32 / 41

Discretization Polynomial Chaos Apply Polynomial Chaos (PC) method to approximate the random polarization τṗ + P = ɛ 0 ɛ d E, τ = τ(ξ) = τ r ξ + τ m resulting in or (τ r M + τ m I ) α + α = ɛ 0 ɛ d Eê 1 A α + α = f. The macroscopic polarization, the expected value of the random polarization at each point (t, x), is simply P(t, x; F ) = E[P] α 0 (t, x). Prof. Gibson (OSU) PC-FDTD AMCS 2013 33 / 41

The discretization of the PC system Discretization A α + α = f is performed similarly to the deterministic system in order to preserve second order accuracy. Applying second order central differences to α n+ 1 2 = α(t n, z ): 1 A αn+ 2 α n 1 2 t 1 + αn+ 2 + α n 1 2 2 Couple this with the equations from above: E n+ 1 2 E n 1 2 ɛ 0 ɛ t H n+1 H n + µ 1 + 1 2 2 0 t = n+ f 1 2 + f n 1 2. (7) 2 H n H n 1 + = 1 2 2 αn+ 2 0, α n 1 2 0, z t (8a) = E n+ 1 2 +1 E n+ 1 2. z (8b) Prof. Gibson (OSU) PC-FDTD AMCS 2013 34 / 41

Discretization Energy Decay and Stability Theorem (Gibson2013) For n 0, let U n = [H n, E n x, E n y, α n 0,x,..., αn 0,y,...]T be the solutions of the 2D TE mode Maxwell-PC Debye FDTD scheme with PEC boundary conditions. If the usual CFL condition for Yee scheme is satisfied c t z/ 2, then there exists the energy decay property E n+1 h E n h where Eh n = µ0 H n 2 + ɛ 0 ɛ E n 2 E + 1 2 H α n. ɛ0 ɛ d α Note: E[ P 2 ] = E[P] 2 + Var(P) 2 α 2 α Energy decay implies that the method is (conditionally) stable and hence convergenent. Prof. Gibson (OSU) PC-FDTD AMCS 2013 35 / 41

Discretization Energy Decay and Stability (cont.) Proof. Involves recognizing that (E n h )2 = µ 0 H n 2 H + ɛ 0ɛ (E n, A h E n ) E + 1 ɛ 0 ɛ d ( α n Eê 1, A 1 ( α n Eê 1 )) 2 α with A h positive definite when the CFL condition is satisfied, and A 1 is always positive definite (eigenvalues between τ m τ r and τ m + τ r ). Prof. Gibson (OSU) PC-FDTD AMCS 2013 36 / 41

Discretization Theorem (Gibson2013) The discrete dispersion relation for the Maxwell-PC Debye FDTD scheme in (8) and (7) is given by ω 2 c 2 ɛ (ω) = K 2 where the discrete complex permittivity is given by ɛ (ω) := ɛ + ɛ d ê T 1 (I iω A ) 1 ê 1 and the discrete PC matrix is given by A := sec(ω t/2)a. The definitions of the parameters ω and K are the same as before. Recall the exact complex permittivity is given by [ ] 1 ɛ(ω) = ɛ + ɛ d E 1 iωτ Prof. Gibson (OSU) PC-FDTD AMCS 2013 37 / 41

Discretization Proof: (for 1D) Assume plane wave solutions of the form and Substitute into (8b) V n = Ṽ e i(k z ωn t) α n l, = α le i(k z ωn t) µ 0 He i(k(+ 1 2 ) z ω(n+ 1 ) t) ( ) 2 e iω t 2 e iω t 2 / t = Ẽe i(k(+ 1 2 ) z ω(n+ 1 ) t) ( ) 2 e ik z 2 e ik z 2 / z ( ) ( ) 2i 2i µ 0 H t sin(ω t/2) = Ẽ z sin(k z/2) ( ) µ0 z H sin(ω t/2) = Ẽ sin(k z/2) (9) t Prof. Gibson (OSU) PC-FDTD AMCS 2013 38 / 41

Discretization Proof: (cont.) Similarly (8a) yields ( z ɛ 0 ɛ t Ẽ + z ) t α 0 sin(ω t/2) = H sin(k z/2) (10) and (7) yields ( ) 2i A α t sin(ω t/2) + cos(ω t/2) α = ɛ 0 ɛ d cos(ω t/2)ẽê 1 (11) which implies The rest of the proof follows as before. α 0 = ê T 1 (I iω A ) 1 ê 1 ɛ o ɛ d Ẽ. (12) Note that the same relation holds in 2 and 3D as well as with higher order accurate spatial difference operators. Prof. Gibson (OSU) PC-FDTD AMCS 2013 39 / 41

Discretization Dispersion Error We define the phase error Φ for any method applied to a particular model to be Φ = k EX k k EX, (13) where the numerical wave number k is implicitly determined by the corresponding dispersion relation and k EX is the exact wave number for the given model. We wish to examine the phase error as a function of ω t in the range [0, π]. We note that ω t = 2π/N ppp, where N ppp is the number of points per period, and is related to the number of points per wavelength as, N ppw = ɛ νn ppp. We assume the following parameters which are appropriate constants for modeling water Debye type materials: ɛ = 1, ɛ s = 78.2, τ m = 8.1 10 12 sec, τ r = 0.5τ m. Prof. Gibson (OSU) PC-FDTD AMCS 2013 40 / 41

0.7 Debye dispersion for FD with h τ =0.1 Dispersion Analysis Discretization 0.25 Debye dispersion for FD with h τ =0.1 0.6 0.2 0.5 0.4 0.15 Φ Φ 0.3 0.1 0.2 0.1 degree=0, r=0.5τ degree=1, r=0.5τ degree=2, r=0.5τ 0.05 degree=0, r=0.5τ degree=1, r=0.5τ degree=2, r=0.5τ 0 0.5 1 1.5 2 2.5 3 ω t Debye dispersion for FD with h =0.01 τ 0.14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ω t Debye dispersion for FD with h =0.01 τ 0.05 0.12 0.1 degree=0, r=0.5τ degree=1, r=0.5τ degree=2, r=0.5τ 0.045 0.04 0.035 Φ 0.08 0.06 0.04 0.02 Φ 0.03 0.025 0.02 0.015 0.01 0.005 degree=0, r=0.5τ degree=1, r=0.5τ degree=2, r=0.5τ 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ω t ω t Prof. Gibson (OSU) PC-FDTD AMCS 2013 41 / 41