1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Finite-Difference Time-Domain Simulation of Electromagnetic waves Yoon Tiem Leong School of Physics, USM, Penang May 17, 2007 Presented at “Mathematica Expository Workshop” Physics Department, UPM Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications Table of contents 1 1D scalar wave 2 (Sourceless) Maxwell Equation 2D sourceless Maxwell Equations 1D sourceless Maxwell Equations 1D Discretization 3 2D EM wave propagation from point source 4 Absorbing Boundary Condition in 1D Perfectly Matching Layers (PML) for 1D 2D Perfectly Matching Layers (PML) 5 2D EM plane waves 6 Total field/scattered field method 7 Putting everything together 8 Further applications Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave ∂ t 2 = c 2 ∂ 2 u ∂ 2 u One-dimension scalar wave equation: ∂ x 2 . u = u ( x , t ) one-dimension scalar wave. Use finite difference method to discretize the scala wave equation Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave Taylor expand u ( x , t n ) about x i , keeping time t n fixed, ∂ x | x i , t n + (∆ x ) 2 · ∂ 2 u u | x i , t n + ∆ x · ∂ u u ( x i + ∆ x ) | t n = ∂ x 2 | x i , t n + 2 (∆ x ) 3 · ∂ 3 u ∂ x 3 | x i , t n + (∆ x ) 4 · ∂ 4 u ∂ x 4 | ξ 1 , t n (1) 6 24 ∂ x | x i , t n + (∆ x ) 2 · ∂ 2 u u | x i , t n − ∆ x · ∂ u u ( x i − ∆ x ) | t n = ∂ x 2 | x i , t n + 2 (∆ x ) 3 · ∂ 3 u ∂ x 3 | x i , t n − (∆ x ) 4 · ∂ 4 u ∂ x 4 | ξ 2 , t n (2) 6 24 Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave Combining both Eqs, 2 u | x i , t n + (∆ x ) 2 · ∂ 2 u u ( x i + ∆ x ) | t n + u ( x i − ∆ x ) | t n = ∂ x 2 | x i , t n + (∆ x ) 4 · ∂ 4 u ∂ x 4 | ξ 3 , t n 12 where x i − ∆ x ≤ ξ 3 ≤ x i + ∆ x . Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave Short-hand notation: u n i ≡ u ( x i , t n ) ≡ u ( i ∆ x , n ∆ t ) u n ± 1 i ± 1 ≡ u ( x i ± ∆ x , t n ± ∆ t ) ≡ u [( i ± 1)∆ x , ( n ± 1)∆ t ] Second-order accurate, central difference approximation to ∂ 2 u ∂ x 2 | x i , t n : ∂ 2 u ∂ 2 x | x i , t n = u n i +1 − 2 u n i + u n � (∆ x ) 2 � i − 1 + O (3) (∆ x ) 2 Likewise, ∂ 2 t | x i , t n = u n +1 i + u n − 1 ∂ 2 u − 2 u n � (∆ t ) 2 � i i + O (4) (∆ t ) 2 Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave Plug the difference approximation of ∂ 2 u ∂ 2 x | x i , t n and ∂ 2 u ∂ 2 t | x i , t n into the one-dimension scalar wave equation: ∂ 2 u ∂ t 2 = c 2 ∂ 2 u ∂ x 2 , we then obtain the iterative difference equation for the scalar wave ( c ∆ t ) 2 [ u n i +1 − 2 u n i + u n i − 1 u n +1 ] + 2 u n i − u n − 1 = + i (∆ x ) 2 i O [(∆ x ) 2 + (∆ t ) 2 ] (5) Eq. (5) allows us to implement numerical iteration to simulate propagation of scalar wave. Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source Absorbing Boundary Condition in 1D 2D EM plane waves Total field/scattered field method Putting everything together Further applications 1D scalar wave ∆ x = x i − x i − 1 = x i +1 − x i is the spatial interval (usually expressed in unit of wavelength, λ , e.g. ∆ x = λ 10 . ∆ t time step 1D grid points : x ilast x 0 x 1 x 2 x i − ∆ x x i • x i +∆ x • • • · · · • • · · · • i is known, u n +1 Given u n can be calculated by iteration i Say u n i = i 0 = sin( n ω ∆ t ) hyperlink to 1Dscalar.nb Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source 2D sourceless Maxwell Equations Absorbing Boundary Condition in 1D 1D sourceless Maxwell Equations 2D EM plane waves 1D Discretization Total field/scattered field method Putting everything together Further applications (Sourceless) Maxwell Equation ∇ · � � Gauss’s Law for electric field D = 0. ∇ · � � Gauss’s Law for magnetic field B = 0. � D = ǫ� D :electric flux density, � � E E : electrin field σ ∗ : equivalent magnetic loss σ : electric conductivity, ǫ : electrical permitivity, µ : magnetic permeability � B = µ� B : magnetic flux density, � � H H : magnetic field ∂� ∂ t = − 1 H E − 1 ∇ × � � µσ ∗ � (6) H µ ∂� ∂ t = 1 E H − 1 ∇ × � � ǫ σ� H (7) ǫ Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source 2D sourceless Maxwell Equations Absorbing Boundary Condition in 1D 1D sourceless Maxwell Equations 2D EM plane waves 1D Discretization Total field/scattered field method Putting everything together Further applications (Sourceless) Maxwell Equation The system of six coupled partial differential equations of the curl operator in Eq. (6) and Eq. (7) forms the basis of the FDTD numerical algorithm for electromagnetic wave interactions with general 3-D objects. ∂ H x 1 µ [ ∂ E y ∂ z − ∂ E z ∂ y − σ ∗ H x ] = ∂ t ∂ H y µ [ ∂ E z 1 ∂ x − ∂ E x ∂ z − σ ∗ H y ] = ∂ t ∂ H z 1 µ [ ∂ E x ∂ y − ∂ E y ∂ x − σ ∗ H z ] = (8) ∂ t Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source 2D sourceless Maxwell Equations Absorbing Boundary Condition in 1D 1D sourceless Maxwell Equations 2D EM plane waves 1D Discretization Total field/scattered field method Putting everything together Further applications (Sourceless) Maxwell Equation ∂ E x 1 ǫ [ ∂ H z ∂ y − ∂ E y = ∂ z − σ E x ] ∂ t ∂ E y 1 ǫ [ ∂ H x ∂ z − ∂ H z = ∂ z − σ E y ] ∂ t ∂ E z 1 ǫ [ ∂ H y ∂ x − ∂ H x = ∂ y − σ E z ] (9) ∂ t A general medium is characterized by ǫ ( i , j , k ), µ ( i , j , k ), σ ( i , j , k ), σ ∗ ( i , j , k ) A total of 6 fields present in the most general 3D case: E x , E y , E z , H x , H y , H z Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
1D scalar wave (Sourceless) Maxwell Equation 2D EM wave propagation from point source 2D sourceless Maxwell Equations Absorbing Boundary Condition in 1D 1D sourceless Maxwell Equations 2D EM plane waves 1D Discretization Total field/scattered field method Putting everything together Further applications (Sourceless) Maxwell Equation The main purpose of FDTD: Given a set of fields E x , E y , E z , H x , H y , H z are known for the entire domain of known spatial structure in terms of ǫ ( i , j , k ), µ ( i , j , k ), σ ( i , j , k ) at time t n times march the fields E x , E y , E z , H x , H y , H z to the next time step, t n +1 Yoon Tiem Leong School of Physics, USM, Penang Finite-Difference Time-Domain Simulation of Electromagnetic w
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