FDFD. The Finite-Difference Frequency-Domain Method. Hans-Dieter Lang

Transcription

1 FDFD The Finite-Difference Frequency-Domain Method Hans-Dieter Lang Friday, December 14, 212 ECE 1252 Computational Electrodynamics Course Project Presentation University of Toronto H.-D. Lang FDFD 1/18

2 The Finite-Difference Frequency-Domain Method Contents Derivation of the FDFD algorithm Eigenmode analysis H.-D. Lang FDFD 2/18

3 Starting position Maxwell s equations in phasor form E = jωμh H = jωεe + J Wave equations (frequency domain) ( 2 + k 2 )E = jωμ J Discretization of space H.-D. Lang FDFD 3/18

4 1D FDFD Maxwell s equations in phasor form E = jωμh H = jωεe + J ^k=^x E=^yE y, H=^zH z x E y = jωμh z x H z = jωεe y J y Finite differences in space E i+1 y E i y Δx = jωμh i+1/2 z H i+1/2 z Hz i 1/2 Δx = jωεe i y J i y E 1 = H 1 E 2 H 2 E 3 H 3 E 4 H 4 E 5 H 5 E 6 i = Δx H.-D. Lang FDFD 4/18

5 FDFD 1D FDFD Finite differences in space E i+1 y E i y Δx = jωμh i+1/2 z H i+1/2 z Hz i 1/2 Δx Matrix form /Δx jωμ 1/Δx... 1/Δx jωε 1/Δx. 1/Δx jωμ E 1 H 1 E 2 H 2.. = jωεe i y J i y = J 2. E 1 = H 1 E 2 H 2 E 3 H 3 E 4 H 4 E 5 H 5 E 6 i = H.-D. Lang FDFD 5/18

6 FDFD 1D FDFD Matrix form /Δx jωμ 1/Δx... 1/Δx jωε 1/Δx. 1/Δx jωμ E 1 H 1 E 2 H 2.. = J 2. Solve the linear system Ax = b x = A 1 b Direct inversion x=a\b Least-square, iterative methods etc. H.-D. Lang FDFD 6/18

7 FDFD PML for FDFD Similar to FDTD jωεe i+1 y jωμh i+1/2 z inside PML ( jω + σ ) 2i+1 ( ε jω + σ 2i ε εey i+1 ) μh i+1/2 y Gradual increase in conductivity σ 2i Empirical σ max, different from FDTD [2, 3] Anisotropic for > 1D H.-D. Lang FDFD 7/18

8 FDFD PML for FDFD No PML: shorted TL, VSWR.1.5 Field amplitude (a.u.) Field amplitude (a.u.) Re(E) Im(E) Cell number Abs(E) Re(H) With PML: VSWR 1.15 Im(H) Abs(H) Cell number H.-D. Lang FDFD 8/18

9 FDFD PML for FDFD Γ = VSWR 1 VSWR s 11 in db Nabs=5 Nabs=1 Nabs= Frequency (GHz) Used parameters: l = 3 mm, R = exp( 12), exp( 14), exp( 16) and p = 4, 6, 8 H.-D. Lang FDFD 9/18

10 Why FDFD? FDFD vs. FDTD Why frequency domain? Resonator characteristics (high Q long simulation time) Eigenmodes direct Dispersive media FDFD characteristics No stability issues Direct eigenmode analysis Solver less general Boundary conditions are more difficult to apply PML even more important Similar numerical dispersion issues FDTD: Broadband, FDFD: Narrow- (single) band H.-D. Lang FDFD 1/18

11 Dispersive media Time vs. frequency domain Different measurements Example: Lorentz media.4 Timestep n=971 Lorentz media 5 x 1 7 Lorentz media FDTD: 6 cells FDFD: 5 cells H.-D. Lang FDFD 11/18

12 Dispersive media Reflection coefficient Γ(ω) of Lorentz media interface 1.8 FDTD: s=.9 FDTD: s=1 analytic FDFD s11 (linear) Frequency (Hz) x 1 16 FDTD: 8192 values/1 s frequency band FDFD: 4 values/2.6 s specific frequencies H.-D. Lang FDFD 12/18

13 Eigenmode analysis E = jωμh H = jωεe [ 1 jε 1 jμ ] [ E ] [ ] E = ω H H Resonator-Q from resonance frequency ω Q = Re ω 2 Im ω = ω 2ω Propagation constant β(ω) (2.5D eigenmode analysis) [ ] E = E (x, y) e jβz β 2 Ex = ( x 2 + y 2 + ω 2 εμ ) [ ] E x E y E y H.-D. Lang FDFD 13/18

14 Eigenmode analysis in 1D Dipole resonances Problem size: 1 cells (l = 15 mm), t sim <.1 s Field amplitude (a.u.) Cell number f GHz = Field amplitude (a.u.) Cell number f GHz = H.-D. Lang FDFD 14/18

15 Eigenmode analysis in 2D Cavity resonator modes Problem size: cells, ( matrix), t sim 6.5 s H.-D. Lang FDFD 15/18

16 Eigenmode analysis in 2.5D Waveguide modes (dimensions a = 2b) Problem size: 16 8 cells, t sim.6 s 25 2 (rad/m), (Np/m) TE 1 TE 2 TE 1 TE3 analytic analytic FDFD FDFD TEM limit cutoffs Frequency (GHz) H.-D. Lang FDFD 16/18

17 Conclusions FDFD = FD in space of Maxwell s equations in phasor form Useful for: Simulations of dispersive media Eigenmode analysis Simulations of resonators with high Q Sparsity: Both matrix and literature on FDFD Steady-state simulation: Everything matters, everywhere! H.-D. Lang FDFD 17/18

18 References [1] Umran S. Inan, Robert A. Marshall Numerical Electromagnetics The FDTD Method Cambridge University Press 211 [2] C. M. Rappaport, B. J. McCartin FDFD Analysis of Electromagnetic Scattering in Anisotropic Media Using Unconstrained Triangular Meshes IEEE Transactions on Antennas and Propagation, Vol. 39, No. 3, March 1991 [3] C. M. Rappaport Perfectly Matched Absorbing Boundary Conditions Based on Anisotropic Lossy Mapping of Space IEEE Microwave and Guided Wave Letters, Vol. 5, No. 3, March 1995 [4] M.-L. Lui, Z. Cheng A direct computation of propagation constant using compact 2-D full-wave eigen-based finite-difference frequency-domain technique Proceedings of the 1999 International Conference on Computational Electromagnetics and Its Applications (ICCEA 99), p , 1999 [5] Y.-J. Zhao, K.-L. Wu, K.-K. M. Cheng A Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave Structures IEEE Transactions on Microwave Theory and Techniques, Vol. 5, No. 7, July 22 [6] L.-Y. Li, J.-F. Mao An Improved Compact 2-D Finite-Difference Frequency-Domain Method for Guided Wave structures IEEE Microwave and Wireless Components Letters, Vol. 13, No. 12, December 23 [7] Raymond C. Rumpf Design and Optimization of Nano-Optical Elements by Coupling Fabrication to Optical Behavior PhD Thesis, University of Central Florida, Orlando Florida, 26 [8] Aliaksandra Ivinskaya Finite-Difference Frequency-Domain Method in Nanophotonics PhD Thesis, Department of Photonics Engineering, Technical University of Denmark, Lyngby, 211 H.-D. Lang FDFD 18/18