Finite-Difference Time-Domain and Beam Propagation Methods for Maxwell s Equations
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1 Finite-Difference Time-Domain and Beam Propagation Methods for Maxwell s Equations Wolfgang Freude and Jan Brosi Institute of High-Frequency and Quantum Electronics (IHQ), University of Karlsruhe, Germany Universität Karlsruhe (TH) Institut für Hochfrequenztechnik und Quantenelektronik (IHQ) COST-P11 Training School: Modelling and Simulation Techniques June 19 22, 2006, University of Nottingham, UK
2 What s The Good Of This Course? COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 1
3 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 2
4 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 3
5 Maxwell s Fundamental Equations COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 4
6 Elementary Solution: Plane Waves and Spatial Frequencies COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 5
7 Elementary Solution: Plane Waves and Spatial Sampling COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 6
8 Sampling, Nyquist Frequency and Interpolation COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 7
9 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 8
10 Yee s Lattice with Cubic Unit Cells of Size Δ x Δ y Δ z H z E x, ε x E y, ε y E z, ε z H y H x z x y Yee, K. S.: Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media. IEEE Trans. Antennas Propag. AP-14 (1966) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 9
11 FDTD Names: Kane S. Yee and Allen Taflove IEEE AP February 1988 IEEE MTT May 2006 Yee, K. S.: Numerical solution of initial boundary value problems involving Maxwell s equations in isotropic media. IEEE Trans. Antennas Propag. AP-14 (1966) Taflove, A.; Brodwin, M. E.: Numerical solution of steady-stae electromagnetic scattering problems using the time-dependent Maxwell s equations. IEEE Trans. MTT-23 (1975) Tafove, A.: Application of the finite-difference time-domain method to sinusoidal steady-stae electromagnetic penetration problems. IEEE Trans. Electromagn. Compatibility 22 (1980) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 10
12 Finite Differences in Cartesian Coordinates H z E x, ε x E y, ε y E z, ε z (i, j, k) H y H x z x y COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 11
13 Central Differences and Discretization H y H z E x, ε x H z (i, j, k) H y z x y COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 12
14 Six Update Equations for Ampere s and Faraday s Laws H y H z E x, ε x H z E y, ε y (i, j, k) H y z x y COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 13
15 Why Leapfrog? Support at time n + ½ Arrive at time n + 1 Start at time n COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 14
16 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 15
17 Numerical Dispersion COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 16
18 Numerical Dispersion COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 17
19 Numerical Dispersion Differential Operators COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 18
20 1D Numerical Dispersion Artefact COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 19
21 1D Stability and Numerical Dispersion Artefact COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 20
22 1D Stability and Numerical Dispersion COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 21
23 Numerical Dispersion Instable Range COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 22
24 Numerical Dispersion Stable Range COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 23
25 1D Stability Intuitive Explanation COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 24
26 1D Stability and Numerical Dispersion COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 25
27 1D Plane Wave Dispersion and Accuracy Choice of Step Sizes COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 26
28 Numerical Dispersion Examples Taflove, A.; Hagness, S. C.: Computational electrodynamics: The finite-difference time-domain method, 2. Ed. Boston: Artech House Figs. 2.3,4 COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 27
29 Global Instability Examples Local Taflove, A.; Hagness, S. C.: Computational electrodynamics: The finite-difference time-domain method, 2. Ed. Boston: Artech House Figs. 2.6,7 COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 28
30 3D Stability and Numerical Dispersion COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 29
31 3D Plane Wave Dispersion and Accuracy Choice of Step Sizes COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 30
32 3D Stability Intuitive Explanation for COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 31
33 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 32
34 Perfectly Matched Layer (PML) Absorbing Boundary Cond. (ABC) Berenger, J. P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys. 114 (1994) Taflove, A.; Hagness, S. C.: Computational electrodynamics: The finite-difference time-domain method, 2. Ed. Boston: Artech House Chapter 7 COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 33
35 2D Plane Waves in Berenger s PML Medium Berenger, J. P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comp. Phys. 114 (1994) Taflove, A.; Hagness, S. C.: Computational electrodynamics: The finite-difference time-domain method, 2. Ed. Boston: Artech House Chapter 7 IEEE AP January 1996 COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 34
36 2D Plane Waves and Berenger s PML Medium COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 35
37 2D Plane Waves and Berenger s PML Medium COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 36
38 2D Plane Waves and Berenger s PML Medium COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 37
39 Symmetric and Anti-Symmetric Boundary Conditions COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 38
40 Periodic Boundary Conditions (PBC) Kittel, C.: Introduction to solid state physics, 3rd ed. New York: John Wiley & Sons 1966 (Bloch s theorem: Chapter 9 p. 259 ff.) Myers, H. P.: Introductory solid state physics. New Delhi: Viva Books Private Ltd (Bloch s theorem: Sect. 7.6 p. 190 ff.) Morse, P. M.; Feshbach, H.: Methods of theoretical physics, Band 1 und 2. New York: McGraw-Hill 1953 (Floquet theorem: Vol. 1 Sect. 5.2 p. 557) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 39
41 Periodic Boundary Conditions Formulation Ko, W. L.; Mittra, R.: Implementation of Floquet boundary condition in FDTD for FSS analysis. IEEE APS Int. Symp. Dig., June 28-July 2 (1993), vol. 1, pp COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 40
42 Periodic Boundary Conditions Procedure x z y Ko, W. L.; Mittra, R.: Implementation of Floquet boundary condition in FDTD for FSS analysis. IEEE APS Int. Symp. Dig., June 28 July 2 (1993), vol. 1, pp Harms, P.; Mittra, R.; Ko, W.: Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures. IEEE Trans. Antennas Prop. 42 (1994) Turner, G.; Christodoulou, C.: Broadband periodic boundary condition for FDTD analysis of phased array antennas. IEEE APS Int. Symp. Dig., June (1998), vol. 2, pp COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 41
43 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 42
44 Dispersive and Nonlinear Media Fujii, M.; Tahara, M; Sakagami, I; Freude, W.; Russer, P.: High-order FDTD and auxiliary differential equation formulation of optical pulse propagation in 2D Kerr and Raman nonlinear dispersive media. IEEE J. Quantum Electron. 40 (2004) Fujii, M.; Omaki, N.; Tahara, M.; Sakagami, I.; Poulton, C.; Freude, W.; Russer, P.: Optimization of nonlinear dispersive APML ABC for the FDTD analysis of optical solitons. IEEE J. Quantum Electron. 41 (2005) Fujii, M,; Koos, C.; Poulton, C.; Sakagami, I.; Leuthold, J.; Freude, W.: A simple and rigorous verification technique for nonlinear FDTD algorithm by optical parametric four-wave mixing. Microwave and Optical Technol. Lett. 48 (2006) Fujii, M; Koos, C.; Poulton, C.; Leuthold, J.; Freude, W.: Nonlinear FDTD analysis and experimental validation of four-wave mixing in InGaAsP/InP racetrack microresonators. IEEE Photon. Technol. Lett. 18 (2006) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 43
45 Nonlinear and Dispersive Media Examples ADE Method Optical Kerr effect polarization: P t E t (3) 3 K() = ε0 χk () finite difference eq. Lorentz dispersion ~ polarization: PL ( ω) ~ = ~ χ ( ω) E( ω) = F -1 L const. nonlinear susceptibility 2 ε Δε L ωl 2 ω + 2 jδ ω ω 2 L ~ E( 0 ω) L ω P 2 L L ( t) + 2δ L dpl ( t) dt + d P dt 2 L 2 ( t) = ε Δε 0 L 2 ω E( t) L finite difference eq. solved with Yee's leapfrog algorithm Fujii, M.; Koos, C.; Poulton, C.; Leuthold, J.; Freude, W.: FDTD modelling and experimental verification of FWM in semiconductor micro-resonators. Proc. 15th International Workshop on Optical Waveguide Theory and Numerical Modelling (OWTNM'06), Varese, Italy, April 20 21, 2006, p. 79 COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 44
46 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 45
47 Scalar Solution of Maxwell s Equations COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 46
48 Scalar Paraxial Wave Equation COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 47
49 Scalar Paraxial Beam Propagation Method COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 48
50 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 49
51 Formal Operator Solution Error in hand-out COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 50
52 Split-Step Fourier Method COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 51
53 Complexity of SSFM and FD-BPM = diagonal matrix FFT diagonal matrix IFFT diagonal matrix Press, W. H.; Flannery, B. P.; Teuklsky, S. A.; Vetterling, W. T.: Numerical Recipes: The Art of Scientific Computing. Cambridge Univ., New York 1986 Scarmozzino, R.; Osgood, Jr., R. M.: Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications. J. Opt. Soc. Amer. A 8 (1991) Kremp, T.; Freude, W.: Fast split-step wavelet collocation method for WDM system parameter optimization. J. Lightwave Technol. 23 (2005) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 52
54 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 53
55 Boundary Conditions Hadley, G. R.: Transparent boundary condition for the beam propagation method. Opt. Lett. 16 (1991) Hadley, G. R.: Transparent boundary condition for the beam propagation method. IEEE J. Quantum Electron 28 (1992) 363 Yevick, D.: A guide to electric field propagation techniques for guided-wave optics. Opt. Quant. Electron. 26 (1994) S185 S197 Vassalo, C.; Collino, F.: Highly efficient absorbing boundary condition for the beam propagation method. J. Lightwave Technol.14 (1996) Huang, W. P.; Xu, C. L.; Lui, W.; Yokoyama, K.: The perfectly matched layer (PML) boundary condition for the beam propagation method. IEEE Photon. Technol. Lett. 8 (1996) Chiou, Y. P.; Chang, H. C.: Complementary operators method as the absorbing boundary condition for the beam propagation method. IEEE Photon. Technol. Lett. 10 (1998) Agrawal, Arti; Sharma, Anurag: Perfectly matched layer in numerical wave propagation: factors that affect its performance. Appl. Opt. 43 (2004) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 54
56 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 55
57 Mode Solving with Imaginary Distance BPM Feit, M. D.; Fleck, J. A.: Computation of mode properties in optical fiber waveguides by a propagating beam method. Appl. Opt. 19 (1980) Yevick, D.; Hermansson, B.: New formulations of the matrix beam propagation method: Application to rib waveguides, IEEE J. Quantum Electron. 25 (1989) Jüngling, S.; Chen, J. C.: A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method. IEEE J. Quantum Electron. 30, (1994) Yevick, D.; Bardyszewski, W.: Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis. Opt. Lett. 17 (1992) Hadley, G. R.; Smith, R. E.: Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions. J. Lightwave Technol. (1995) Chen, J. C.; Jüngling, S.: Computation of higher-order waveguide modes by the imaginary-distance beam propagation method. Opt. Quantum Electron. 26 (1994) S199 S205 Poulton, C. G.; Koos, C.; Fujii, M.; Pfrang, A.; Schimmel, Th.; Leuthold, J.; Freude, W.: Radiation modes and roughness loss in high index-contrast waveguides. IEEE J. Sel. Topics Quantum Electron. Nov. (2006), in press COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 56
58 Mode Solving with BPM and Correlation Method Feit, M. D.; Fleck, J. A.: Computation of mode properties in optical fiber waveguides by a propagating beam method. Appl. Opt. 19 (1980) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 57
59 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 58
60 Polarization Effects with Full-Vectorial BPM Huang, W. P.; Xu, C. L.: Simulation of three-dimensional optical waveguides by a full-vector beam propagation method. IEEE J. Quantum Electron. 29 (1993) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 59
61 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 60
62 Wide-Angle and Bi-Directional BPM Sharma, Anurag; Agrawal, Arti: Perfectly matched layer in numerical wave propagation: factors that affect its performance. Appl. Opt. 43 (2004) Sharma, Anurag; Agrawal, Arti: New method for nonparaxial beam propagation. J. Opt. Soc. Am. A 21 (2004) Sharma, Anurag; Agrawal, Arti: A new finite-difference based method for wide-angle beam propagation. IEEE Photon. Technol. Lett. 18 (2006) Sharma, Anurag; Bhattaharya, Debjani; Agrawal, Arti: Analytical computation of the propagation matrix for the finite-difference split-step non-paraxial method. Proc. 5th International Workshop on Optical Waveguide Theory and Numerical Modelling (OWTNM'06), Varese, Italy, April 20 21, 2006, p. 38 Sharma, Anurag; Singh Shishodia, Manmohan: A non-iterative bi-directional wave propagation method based on the split-step non-paraxial (SSNP) method. Proc. 5th International Workshop on Optical Waveguide Theory and Numerical Modelling (OWTNM'06), Varese, Italy, April 20 21, 2006, p. 69 Sharma, Anurag; Agrawal, Arti: Non-paraxial split-step finite-difference method for beam propagation. Opt. Quantum Electron. (2006), in press COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 61
63 Finite-difference time-domain (FDTD) method Maxwell s equations, PDE classes, plane waves, sampling Yee s leapfrog algorithm Numerical dispersion, stability, accuracy and examples Boundary conditions: PML, symmetries, periodicity Dispersive and nonlinear media Beam propagation method (BPM) Scalar Helmholtz equation Split-step algorithm Boundary conditions: Dirichlet, TBC, ABC, PML BPM mode solving: Imaginary distance, correlation Full-vectorial and semi-vectorial BPM Wide-angle and bi-directional BPM Further reading Outline COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 62
64 Further Reading COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 63
65 Further Reading (2) COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 64
66 What Should You Have Learned? COST-P11, June 06 Institut für Hochfrequenztechnik und Quantenelektronik (IHQ), Universität Karlsruhe 65
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