Lecture course "Integrated Nanophotonics" by Dmitry Fedyanin

Lecture 8: Finite-Difference Time-Domain Method (FDTD)

Transcript

1. Integrated Nanophotonics Dmitry Yu. Fedyanin Laboratory of Nanooptics and Plasmonics, MIPT Lecture 8: Finite-Difference Time-Domain Method
2. 2 WHAT IS FDTD? Light scattering at a dielectric wedge (the problem cannot be solved analytically)
3. 3 WHAT IS FDTD? A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 1995. In September 2012, this book was ranked as the 7th most- cited book in physics, according to Google Scholar.
4. 4 WHAT IS FDTD? S.C. Hagness et al., FDTD microcavity simulations: design and experimental realization of waveguide-coupled single- mode ring and whispering-gallery-mode disk resonators, Journal of Lightwave Technology 15, 2154 (1997).
5. 5 ORIGINAL YEE'S PAPER div D=4 πρ div B=0 rot E=− 1 c ∂ B ∂t Maxwell's equations rot H= 1 c ∂ D ∂t + 4 π c j D=ε E B=μ H j=σ E Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
6. 6 ORIGINAL YEE'S PAPER div D=4 πρ div B=0 rot E=− 1 c ∂ B ∂t Maxwell's equations rot H= 1 c ∂ D ∂t + 4 π c j D=ε E B=μ H j=σ E Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
7. 7 ORIGINAL YEE'S PAPER rot E=− 1 c ∂ B ∂t Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
8. 8 ORIGINAL YEE'S PAPER rot E=− 1 c ∂ B ∂t Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
9. 9 ORIGINAL YEE'S PAPER rot E=− 1 c ∂ B ∂t Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
10. 10 ORIGINAL YEE'S PAPER rot E=− 1 c ∂ B ∂t Kane Yee , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation 14, 302–307 (1966).
11. 11 1D FDTD 1D scalar wave function Two propagating-wave solutions forward and backward
12. 12 1D FDTD 1D scalar wave function Taylor's series expansion of f(x,t) around the point x
13. 13 1D FDTD 1D scalar wave function Taylor's series expansion of f(x,t) around the point x central difference approximation
14. 14 1D FDTD Discrete wave equation
15. 15 1D FDTD Discrete wave equation Explicit time-marching solution
16. 16 1D FDTD Figure: S. Gedney, Computational Electromagnetics: The Finite-Difference Time-Domain, EE699, Fall 2003
17. 17 1D FDTD Magic time step Figure: S. Gedney, Computational Electromagnetics: The Finite-Difference Time-Domain, EE699, Fall 2003
18. 18 1D FDTD Magic time step Stability requirement Δ t≤ Δ x min(c) A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 1995.
19. 19 1D MAXWELL'S EQUATIONS rot E=− 1 c ∂ H ∂t rot H= 1 c ∂ E ∂t − ∂ E x ∂ z = 1 c ∂ H y ∂t − ∂ H y ∂ z = 1 c ∂ E x ∂t
20. 20 1D MAXWELL'S EQUATIONS − ∂ E x ∂ z = 1 c ∂ H y ∂t − ∂ H y ∂ z = 1 c ∂ E x ∂t
21. 21 WHAT ABOUT MATERIALS? ε(ω)=ε r + 4 πσ ω i − ∂ E x ∂ z = 1 c ∂ H y ∂t − ∂ H y ∂ z = 1 c ∂ D x ∂t
22. 22 WHAT ABOUT MATERIALS? − ∂ E x ∂ z = 1 c ∂ H y ∂t − ∂ H y ∂ z = 1 c ∂ D x ∂t ε(ω)=ε r + 4 πσ i ω D(ω)=ε(ω)E(ω)=ε r E(ω)+ 4 π σ i ω E(ω) since exp(+i ωt−ik r)
23. 23 WHAT ABOUT MATERIALS? − ∂ E x ∂ z = 1 c ∂ H y ∂t − ∂ H y ∂ z = 1 c ∂ D x ∂t ε(ω)=ε r + 4 πσ i ω D(ω)=ε(ω)E(ω)=ε r E(ω)+ 4 πσ i ω E(ω) Fourier transform 1 iω →∫ 0 t D(t)=ε r E(t)+4 π σ∫ 0 t E(τ)d τ Dn=ε r En+4 πσ Δ t∑ i=0 n Ei=(ε r +4 πσ)En+4 πσ Δ t ∑ i=0 n−1 Ei Use the Z-transform for more complicated dependences ε(ω).
24. 24 PERFECTLY MATCHED LAYER (PML) J. P. Berenger was the first to introduce a technique that is matched for all angles of incidence and polarizations. This method is based on a mathematical step known as "Field splitting". R H = ζ 1 −ζ 2 ζ 1 +ζ 2 Normal incidence: ζ=√μ ε 1 2 TM wave For H field: R E = ζ 2 −ζ 1 ζ 2 +ζ 1 For E field: where R H =−R E =0 if μ 1 ε 1 = μ 2 ε 2
25. 25 PERFECTLY MATCHED LAYER (PML) J. P. Berenger was the first to introduce a technique that is matched for all angles of incidence and polarizations. This method is based on a mathematical step known as "Field splitting". R H = ζ 1 −ζ 2 ζ 1 +ζ 2 Normal incidence: ζ=√μ ε R H = ζ 1 γ 1 −ζ 2 γ 2 ζ 1 γ 1 −ζ 2 γ 2 1 2 TM wave For H field: R E = ζ 2 −ζ 1 ζ 2 +ζ 1 For E field: The general case: where where γ 1 =cosθ γ 2 = √1− ε 1 μ 1 ε 2 μ 2 sin2θ
26. 26 PERFECTLY MATCHED LAYER (PML) J. P. Berenger was the first to introduce a technique that is matched for all angles of incidence and polarizations. This method is based on a mathematical step known as "Field splitting". R H = ζ 1 −ζ 2 ζ 1 +ζ 2 Normal incidence: ζ=√μ ε R H = ζ 1 γ 1 −ζ 2 γ 2 ζ 1 γ 1 −ζ 2 γ 2 1 2 TM wave For H field: R E = ζ 2 −ζ 1 ζ 2 +ζ 1 For E field: The general case: where where γ 1 =cosθ γ 2 = √1− ε 1 μ 1 ε 2 μ 2 sin2θ For R = 0, ε 2 and μ 2 should be functions of θ.
27. 27 PERFECTLY MATCHED LAYER (PML) H z =H zx +H zy Field splitting: 1 2 TM wave J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114, 185 (1994). ε r c ∂ E x ∂t +0= ∂(H zx +H zy ) ∂ y ε r c ∂ E y ∂t + 4 πσ E c E y =− ∂(H zx +H zy ) ∂ x μ r c ∂ H zx ∂t + 4 πσ H c H zx =− ∂ E y ∂ x μr c ∂ H yz ∂ t +0= ∂ E x ∂ y
28. 28 PERFECTLY MATCHED LAYER (PML) H z =H zx +H zy ε r c ∂ E x ∂t = ∂(H zx +H zy ) ∂ y μ r c ∂ H yz ∂ t = ∂ E x ∂ y ε r c ∂ E y ∂t + 4 πσ E c E y =− ∂(H zx +H zy ) ∂ x μr c ∂ H zx ∂t + 4 πσH c H zx =− ∂ E y ∂ x Field splitting: 1 2 TM wave J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114, 185 (1994). The (E y ,H zx ) wave (wave X) is absorbed, while the (E x ,H zy ) wave (wave Y) is not absorbed.
29. 29 PERFECTLY MATCHED LAYER (PML) 1 2 TM wave The wave X should not be reflected at the interfaces ε r c ∂ E y ∂t + 4 πσ E c E y =− ∂(H zx +H zy ) ∂ x μ r c ∂ H zx ∂t + 4 πσ H c H zx =− ∂ E y ∂ x R H = ζ 1 −ζ 2 ζ 1 +ζ 2 ζ 1 = √μ 1 ε 1 = √μ r + 4πσ H i ω ε r + 4πσ E i ω =ζ 2
30. 30 PERFECTLY MATCHED LAYER (PML) 1 2 TM wave The wave Y in the PML should travel along the y-axis as the incident/reflected plane wave in medium 1. ε 1 =ε r ; μ 1 =μ r
31. 31 PERFECTLY MATCHED LAYER (PML) 1 2 TM wave The wave Y in the PML should travel along the y-axis as the incident/reflected plane wave in medium 1. ε 1 =ε r ; μ 1 =μ r ε PML =ε 1 + 4πσ E iω =ε 1 (1+ 4 πσ E iωε 1 )=ε 1 S μ PML =μ 1 + 4πσH iω =μ 1 (1+ 4 πσH iωμ 1 )=μ 1 S*
32. 32 PERFECTLY MATCHED LAYER (PML) 1 2 TM wave J.-P. Berenger, Perfectly Matched Layer (PML) for Computational Electromagnetics, Morgan & Claypool Publishers, 2007. ε PML =ε 1 + 4πσ E iω =ε 1 (1+ 4 πσ E iωε 1 )=ε 1 S μ PML =μ 1 + 4 πσH iω =μ 1 (1+ 4 πσH iωμ 1 )=μ 1 S* μ 1 ε 1 = μ PML ε PML = μ 1 S* ε 1 S S=S*; σE ε 1 = σH μ 1
33. 33 UNIAXIAL PML (UPML) S. D. Gedney, An anisotropic perfectly matched layer absorbing media for the truncation of FDTD lattices, IEEE Trans. Antennas and Propagation 44, 1630 (1996). 1 2 TM wave The original split-field PML can be equally represented by an anisotropic perfectly matched medium εPML =ε1 S=ε1 [S x −1 0 0 0 S x 0 0 0 S x ] μPML =μ1 S* If one needs absorption in all three directions, then ε PML =ε 1 [S x −1 0 0 0 S x 0 0 0 S x ]× [S y 0 0 0 S y −1 0 0 0 S y ]× [S z 0 0 0 S z 0 0 0 S z −1 ]=ε 1 [S y S z S x 0 0 0 S x S z S y 0 0 0 S x S y S z ] S x = (1+ 4 πσ E iωε1 ) where S x *= (1+ 4 πσ H iωμ1 )
34. 34 NEXT CLASS Electro-optic modulators. Homework: A. Taflove, S.C. Hagness, Computational Electrodynamics:The Finite-Difference Time- Domain Method, Artech House, 2000. Chapter 2 Literature: D.M. Sullivan, Electromagnetic Simulations Using the FDTD Method, Wiley-IEEE Press, 2013. A. Taflove, S.C. Hagness, Computational Electrodynamics:The Finite-Difference Time- Domain Method, Artech House, 2000. J.-P. Berenger, Perfectly Matched Layer (PML) for Computational Electromagnetics, Morgan & Claypool Publishers, 2007.