FDFD - The Finite-Difference Frequency-Domain Method · A Compact 2-D Full-Wave Finite-Difference...
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Introduction Eigenmode Analysis FDFD The Finite-Difference Frequency-Domain Method Hans-Dieter Lang Friday, December 14, 2012 ECE 1252 – Computational Electrodynamics Course Project Presentation University of Toronto H.-D. Lang FDFD 1/18
Transcript of FDFD - The Finite-Difference Frequency-Domain Method · A Compact 2-D Full-Wave Finite-Difference...
IntroductionEigenmode Analysis
FDFDThe Finite-Difference Frequency-Domain Method
Hans-Dieter Lang
Friday, December 14, 2012ECE 1252 – Computational Electrodynamics
Course Project Presentation
University of Toronto
H.-D. Lang FDFD 1/18
IntroductionEigenmode Analysis
The Finite-Difference Frequency-Domain Method
Contents
Derivation of the FDFD algorithm
Eigenmode analysis
Examples
H.-D. Lang FDFD 2/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
Introduction
Starting position
Maxwell’s equations in phasor form
∇×E = −𝑗𝜔𝜇H
∇×H = 𝑗𝜔𝜀E+ J
Wave equations (frequency domain)
(∇2 + 𝑘2)E = 𝑗𝜔𝜇J
Discretization of space
H.-D. Lang FDFD 3/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
Introduction
1D FDFD
Maxwell’s equations in phasor form
∇×E = −𝑗𝜔𝜇H
∇×H = 𝑗𝜔𝜀E+ J
k=x−−−−−−−−−−→E=y𝐸𝑦 ,H=z𝐻𝑧
𝜕𝑥𝐸𝑦 = −𝑗𝜔𝜇𝐻𝑧
𝜕𝑥𝐻𝑧 = −𝑗𝜔𝜀𝐸𝑦 − 𝐽𝑦
Finite differences in space
𝐸𝑖+1𝑦 − 𝐸𝑖
𝑦
Δ𝑥= −𝑗𝜔𝜇𝐻𝑖+1/2
𝑧
𝐻𝑖+1/2𝑧 −𝐻
𝑖−1/2𝑧
Δ𝑥= −𝑗𝜔𝜀𝐸𝑖
𝑦 − 𝐽 𝑖𝑦
𝑖 =
𝐸1=0
1
𝐸2
2
𝐸3
3
𝐸4
4
𝐸5
5
𝐸6
6
𝐻1
1.5
𝐻2
2.5
𝐻3
3.5
𝐻4
4.5
𝐻5
5.5Δ𝑥
H.-D. Lang FDFD 4/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
FDFD
1D FDFDFinite differences in space
𝐸𝑖+1𝑦 − 𝐸𝑖
𝑦
Δ𝑥= −𝑗𝜔𝜇𝐻𝑖+1/2
𝑧
𝐻𝑖+1/2𝑧 −𝐻
𝑖−1/2𝑧
Δ𝑥= −𝑗𝜔𝜀𝐸𝑖
𝑦 − 𝐽 𝑖𝑦
Matrix form⎡⎢⎢⎢⎢⎢⎢⎣1 0 0 0 . . .
1/Δ𝑥 𝑗𝜔𝜇 −1/Δ𝑥 0 . . .0 1/Δ𝑥 𝑗𝜔𝜀 −1/Δ𝑥
0 0 1/Δ𝑥 𝑗𝜔𝜇. . .
......
. . . . . .
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣𝐸1
𝐻1
𝐸2
𝐻2...
⎤⎥⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎢⎣00
−𝐽20...
⎤⎥⎥⎥⎥⎥⎦
𝑖 =
𝐸1=0
1
𝐸2
2
𝐸3
3
𝐸4
4
𝐸5
5
𝐸6
6
𝐻1
1.5
𝐻2
2.5
𝐻3
3.5
𝐻4
4.5
𝐻5
5.5
H.-D. Lang FDFD 5/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
FDFD
1D FDFD
Matrix form⎡⎢⎢⎢⎢⎢⎢⎣1 0 0 0 . . .
1/Δ𝑥 𝑗𝜔𝜇 −1/Δ𝑥 0 . . .0 1/Δ𝑥 𝑗𝜔𝜀 −1/Δ𝑥
0 0 1/Δ𝑥 𝑗𝜔𝜇. . .
......
. . . . . .
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎣𝐸1
𝐻1
𝐸2
𝐻2...
⎤⎥⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎢⎣00
−𝐽20...
⎤⎥⎥⎥⎥⎥⎦Solve the linear system
Ax = b ⇒ x = A−1b
∙ Direct inversion x=A\b∙ Least-square, iterative methods etc.
H.-D. Lang FDFD 6/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
FDFD
PML for FDFD
Similar to FDTD
𝑗𝜔𝜀𝐸𝑖+1𝑦
𝑗𝜔𝜇𝐻 𝑖+1/2𝑧
inside PML−−−−−−−→
(𝑗𝜔 +
𝜎2𝑖+1
𝜀
)𝜀𝐸𝑖+1
𝑦(𝑗𝜔 +
𝜎2𝑖𝜀
)𝜇𝐻 𝑖+1/2
𝑦
Gradual increase in conductivity 𝜎2𝑖
Empirical 𝜎max, different from FDTD [2, 3]
Anisotropic for >1D
H.-D. Lang FDFD 7/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
FDFD
PML for FDFD
No PML: shorted TL, VSWR→ ∞
0 20 40 60 80 100 120 140 160 180 200
−0.1−0.05
00.05
0.1
Cell number
Fiel
d am
plitu
de (a
.u.)
0 20 40 60 80 100 120 140 160 180 200
−0.1−0.05
00.05
0.1
Cell number
Fiel
d am
plitu
de (a
.u.)
With PML: VSWR→1
0 20 40 60 80 100 120 140 160 180 200−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Re(E)Im(E)Abs(E)Re(H)Im(H)Abs(H)
H.-D. Lang FDFD 8/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
FDFD
PML for FDFD
|Γ| = VSWR− 1
VSWR+ 1
0 1 2 3 4 5−120
−110
−100
−90
−80
−70
−60
Frequency (GHz)
|s11
| in
dB
Nabs=5Nabs=10Nabs=16
Used parameters: 𝑙 = 300mm, 𝑅 = exp(−12), exp(−14), exp(−16) and 𝑝 = 4, 6, 8
H.-D. Lang FDFD 9/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
Why FDFD?
FDFD vs. FDTDWhy 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 10/18
IntroductionEigenmode Analysis
Frequency Domain ConsiderationsExamples
Examples
Dispersive mediaTime vs. frequency domainDifferent measurementsExample: Lorentz media
0 1000 2000 3000 4000 5000 6000
−0.1
0
0.1
0.2
0.3
0.4
Timestep n=9701
0 100 200 300 400 500−5
0
5x 10−7
Lorentz media Lorentz media
FDTD: 6000 cells FDFD: 500 cells
H.-D. Lang FDFD 11/18
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Frequency Domain ConsiderationsExamples
Examples
Dispersive media
Reflection coefficient |Γ(𝜔)| of Lorentz media interface
0 0.5 1 1.5 2 2.5 3x 1016
0
0.2
0.4
0.6
0.8
1s1
1 (li
near
)
Frequency (Hz)
FDTD: s=0.9FDTD: s=1analyticFDFD
∙ FDTD: 8192 values/10 s → frequency band∙ FDFD: 40 values/2.6 s → specific frequencies
H.-D. Lang FDFD 12/18
IntroductionEigenmode Analysis
IntroductionExamples
Eigenmode Analysis
Eigenmode analysis
∇×E = −𝑗𝜔𝜇H
∇×H = 𝑗𝜔𝜀E⇒
[0 1
𝑗𝜀 ∇×− 1
𝑗𝜇 ∇× 0
] [EH
]= 𝜔0
[EH
]
Resonator-𝑄 from resonance frequency 𝜔0
𝑄 =Re𝜔0
2 Im𝜔0=
𝜔′0
2𝜔′′0
Propagation constant 𝛽(𝜔) (2.5D eigenmode analysis)
E = E0(𝑥, 𝑦) e𝑗𝛽𝑧 ⇒ 𝛽2
[𝐸𝑥
𝐸𝑦
]=
(𝜕2𝑥 + 𝜕2
𝑦 + 𝜔2𝜀𝜇) [𝐸𝑥
𝐸𝑦
]
H.-D. Lang FDFD 13/18
IntroductionEigenmode Analysis
IntroductionExamples
Examples
Eigenmode analysis in 1DDipole resonancesProblem size: 100 cells (𝑙 = 150mm), 𝑡sim < 0.01 s
10 20 30 40 50 60 70 80 90 100−0.2
−0.1
0
0.1
0.2
Cell number
Fiel
d am
plitu
de (a
.u.)
10 20 30 40 50 60 70 80 90 100−0.2
−0.1
0
0.1
0.2
Cell number
Fiel
d am
plitu
de (a
.u.)
𝑓GHz =0.97991.95562.93883.91734.8949
𝑓GHz =00.98961.9732.90793.956
H.-D. Lang FDFD 14/18
IntroductionEigenmode Analysis
IntroductionExamples
Examples
Eigenmode analysis in 2D
Cavity resonator modes
Problem size: 36× 36 cells, (1369× 1369 matrix), 𝑡sim ≈ 6.5 s
10 20 30
10
20
30
0
0.02
0.04
10 20 30
10
20
30
−0.05
0
0.05
10 20 30
10
20
30
−0.05
0
0.05
10 20 30
10
20
30
−0.05
0
0.05
10 20 30
10
20
30
−0.05
0
0.05
10 20 30
10
20
30
−0.04−0.0200.020.040.06
H.-D. Lang FDFD 15/18
IntroductionEigenmode Analysis
IntroductionExamples
Examples
Eigenmode analysis in 2.5DWaveguide modes (dimensions 𝑎 = 2𝑏)Problem size: 16× 8 cells, 𝑡sim ≈ 0.6 s
2 4 6 8 10 12 140
50
100
150
200
250
Frequency (GHz)
` (ra
d/m
), _
(Np/
m)
` analytic_ analytic` FDFD_ FDFDTEM limitcutoffs
TE10TE20
TE01
TE30
H.-D. Lang FDFD 16/18
IntroductionEigenmode Analysis
IntroductionExamples
Conclusions
FDFD = FD in space of Maxwell’s equations in phasor formUseful 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
IntroductionEigenmode Analysis
IntroductionExamples
References
[1] Umran S. Inan, Robert A. MarshallNumerical Electromagnetics – The FDTD MethodCambridge University Press 2011
[2] C. M. Rappaport, B. J. McCartinFDFD Analysis of Electromagnetic Scattering in Anisotropic Media Using Unconstrained Triangular MeshesIEEE Transactions on Antennas and Propagation, Vol. 39, No. 3, March 1991
[3] C. M. RappaportPerfectly Matched Absorbing Boundary Conditions Based on Anisotropic Lossy Mapping of SpaceIEEE Microwave and Guided Wave Letters, Vol. 5, No. 3, March 1995
[4] M.-L. Lui, Z. ChengA direct computation of propagation constant using compact 2-D full-wave eigen-based finite-difference frequency-domaintechniqueProceedings of the 1999 International Conference on Computational Electromagnetics and Its Applications (ICCEA ’99), p.78-81, 1999
[5] Y.-J. Zhao, K.-L. Wu, K.-K. M. ChengA Compact 2-D Full-Wave Finite-Difference Frequency-Domain Method for General Guided Wave StructuresIEEE Transactions on Microwave Theory and Techniques, Vol. 50, No. 7, July 2002
[6] L.-Y. Li, J.-F. MaoAn Improved Compact 2-D Finite-Difference Frequency-Domain Method for Guided Wave structuresIEEE Microwave and Wireless Components Letters, Vol. 13, No. 12, December 2003
[7] Raymond C. RumpfDesign and Optimization of Nano-Optical Elements by Coupling Fabrication to Optical BehaviorPhD Thesis, University of Central Florida, Orlando Florida, 2006
[8] Aliaksandra IvinskayaFinite-Difference Frequency-Domain Method in NanophotonicsPhD Thesis, Department of Photonics Engineering, Technical University of Denmark, Lyngby, 2011
H.-D. Lang FDFD 18/18
