FDFD - hade.ch

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
Introduction
Eigenmode Analysis
The Finite-Difference Frequency-Domain Method
Contents
Derivation of the FDFD algorithm
Eigenmode analysis
Examples
H.-D. Lang
FDFD
2/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
Introduction
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
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
Introduction
1D FDFD
Maxwell’s equations in phasor form
∇ × E = −𝑗𝜔𝜇H
𝜕𝑥 𝐸𝑦 = −𝑗𝜔𝜇𝐻𝑧
^ x
k=^
∇ × H = 𝑗𝜔𝜀E + J
−−−−−−−−−−→
𝜕𝑥 𝐻𝑧 = −𝑗𝜔𝜀𝐸𝑦 − 𝐽𝑦
E=^
y𝐸𝑦 , H=^
z𝐻 𝑧
Finite differences in space
𝐸𝑦𝑖+1 − 𝐸𝑦𝑖
= −𝑗𝜔𝜇𝐻𝑧𝑖+1/2
Δ𝑥
𝑖+1/2
𝑖−1/2
− 𝐻𝑧
Δ𝑥
𝐻𝑧
= −𝑗𝜔𝜀𝐸𝑦𝑖 − 𝐽𝑦𝑖
𝐸1= 0 𝐻1 𝐸2 𝐻2 𝐸3 𝐻3 𝐸4 𝐻4 𝐸5 𝐻5 𝐸6
𝑖=
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Δ𝑥
H.-D. Lang
FDFD
4/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
FDFD
1D FDFD
Finite differences in space
𝑖+1/2
𝐸𝑦𝑖+1 − 𝐸𝑦𝑖
= −𝑗𝜔𝜇𝐻𝑧𝑖+1/2
Δ𝑥
𝑖−1/2
− 𝐻𝑧
Δ𝑥
𝐻𝑧
= −𝑗𝜔𝜀𝐸𝑦𝑖 − 𝐽𝑦𝑖
Matrix form
⎡
⎤ ⎡ ⎤ ⎡
⎤
1
0
0
0
...
𝐸1
0
⎢1/Δ𝑥 𝑗𝜔𝜇 −1/Δ𝑥
⎢
⎥
0
. . .⎥
𝐻1 ⎥
⎢
⎥ ⎢
⎥ ⎢ 0 ⎥
⎢ 0
⎥ ⎢
⎢
⎥
⎢
1/Δ𝑥
𝑗𝜔𝜀
−1/Δ𝑥
⎢
⎥ ⎢ 𝐸2 ⎥ = ⎢−𝐽2 ⎥
⎥
⎢
⎥
.
⎢ 0 ⎥
..⎥ ⎢
⎢ 0
𝐻2 ⎥
0
1/Δ𝑥
𝑗𝜔𝜇
⎣
⎦
⎣
⎦
⎣
⎦
.
.
..
.
..
.
..
.
..
..
.
..
𝐸1= 0 𝐻1 𝐸2 𝐻2 𝐸3 𝐻3 𝐸4 𝐻4 𝐸5 𝐻5 𝐸6
𝑖=
1
1.5
2
2.5
3
H.-D. Lang
3.5
4
FDFD
4.5
5
5.5
6
5/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
FDFD
1D FDFD
Matrix form
⎤ ⎡ ⎤ ⎡
⎤
1
0
0
0
...
0
𝐸1
⎢1/Δ𝑥 𝑗𝜔𝜇 −1/Δ𝑥
⎥
⎢
0
. . .⎥
𝐻1 ⎥
⎢
⎥ ⎢
⎥ ⎢ 0 ⎥
⎢ 0
⎥ ⎢
⎢
⎥
⎢
1/Δ𝑥
𝑗𝜔𝜀
−1/Δ𝑥
⎢
⎥ ⎢ 𝐸2 ⎥ = ⎢−𝐽2 ⎥
⎥
⎢
⎥ ⎢ ⎥ ⎢
.
⎥
.
⎢ 0
⎥
0
𝐻
2
.
0
1/Δ𝑥
𝑗𝜔𝜇
⎣
⎦ ⎣ . ⎦ ⎣ . ⎦
⎡
..
.
..
.
..
..
.
.
..
..
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
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
FDFD
PML for FDFD
Similar to FDTD
𝑗𝜔𝜀𝐸𝑦𝑖+1
𝑗𝜔𝜇𝐻𝑧𝑖+1/2
(︁
inside PML
−−−−−−−→
𝜎2𝑖+1 )︁ 𝑖+1
𝑗𝜔 +
𝜀𝐸
𝜀 )︁ 𝑦
(︁
𝜎2𝑖
𝑗𝜔 +
𝜇𝐻𝑦𝑖+1/2
𝜀
Gradual increase in conductivity 𝜎2𝑖
Empirical 𝜎max , different from FDTD [2, 3]
Anisotropic for > 1D
H.-D. Lang
FDFD
7/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
0.1Examples
FDFD
0.05
PML for FDFD
No PML: shorted TL, VSWR→ ∞
Field amplitude (a.u.)
0
0.1
0.05
−0.05
0
−0.05
−0.1
0
20
40
60
80
Field amplitude (a.u.)
With PML: VSWR→1
−0.1
100
Cell number
Re(E)
180
Im(E)
Abs(E)
Re(H)
Im(H)
Abs(H)
120
140
160
0
20
40
60
80
100
Cell number
120
140
160
180
−0.15
200
0.1
0.05
−0.2
0
100
120
−0.05
−0.1
0
20
40
60
80
H.-D. Lang
FDFD
200
8/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
FDFD
PML for FDFD
|Γ| =
VSWR − 1
VSWR + 1
−60
|s11| in dB
−70
Nabs=5
Nabs=10
Nabs=16
−80
−90
−100
−110
−120
0
1
2
3
Frequency (GHz)
4
5
Used parameters: 𝑙 = 300 mm, 𝑅 = exp(−12), exp(−14), exp(−16) and 𝑝 = 4, 6, 8
H.-D. Lang
FDFD
9/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
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
10/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
Examples
Dispersive media
Time vs. frequency domain
Different measurements
Example: Lorentz media
−7
Timestep n=9701
5
0.4
x 10
Lorentz media
Lorentz media
0.3
0.2
0
0.1
0
−0.1
0
1000
2000
3000
4000
5000
6000
−5
FDTD: 6000 cells
0
100
200
300
400
500
FDFD: 500 cells
H.-D. Lang
FDFD
11/18
Introduction
Eigenmode Analysis
Frequency Domain Considerations
Examples
Examples
Dispersive media
Reflection coefficient |Γ(𝜔)| of Lorentz media interface
1
FDTD: s=0.9
FDTD: s=1
analytic
FDFD
s11 (linear)
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Frequency (Hz)
2.5
3
16
x 10
∙ FDTD: 8192 values/10 s → frequency band
∙ FDFD: 40 values/2.6 s → specific frequencies
H.-D. Lang
FDFD
12/18
Introduction
Eigenmode Analysis
Introduction
Examples
Eigenmode Analysis
Eigenmode analysis
∇ × E = −𝑗𝜔𝜇H
∇ × H = 𝑗𝜔𝜀E
[︃
⇒
0
1
− 𝑗𝜇 ∇×
1
𝑗𝜀
]︃ [︂ ]︂
[︂ ]︂
∇×
E
E
= 𝜔0
0
H
H
Resonator-𝑄 from resonance frequency 𝜔0
𝑄=
Re 𝜔0
𝜔′
= 0′′
2 Im 𝜔0
2𝜔0
Propagation constant 𝛽(𝜔) (2.5D eigenmode analysis)
E = E0 (𝑥, 𝑦) e 𝑗𝛽𝑧
⇒
𝛽2
H.-D. Lang
[︂
]︂
[︂ ]︂
(︀
)︀ 𝐸𝑥
𝐸𝑥
= 𝜕𝑥2 + 𝜕𝑦2 + 𝜔 2 𝜀𝜇
𝐸𝑦
𝐸𝑦
FDFD
13/18
Introduction
Eigenmode Analysis
Introduction
Examples
Examples
Field amplitude (a.u.)
Field amplitude (a.u.)
Eigenmode analysis in 1D
Dipole resonances
Problem size: 100 cells (𝑙 = 150 mm), 𝑡sim < 0.01 s
0.2
𝑓GHz =
0.1
0.9799
1.9556
2.9388
3.9173
4.8949
0
−0.1
−0.2
10
20
30
40
50
60
Cell number
70
80
90
100
0.2
𝑓GHz =
0.1
0
0.9896
1.973
2.9079
3.956
0
−0.1
−0.2
10
20
30
40
50
60
Cell number
H.-D. Lang
FDFD
70
80
90
100
14/18
Introduction
Eigenmode Analysis
Introduction
Examples
Examples
Eigenmode analysis in 2D
Cavity resonator modes
Problem size: 36 × 36 cells, (1369 × 1369 matrix), 𝑡sim ≈ 6.5 s
30
0.04
20
0.05
30
20
0
0.05
30
20
0
0.02
10
10
10
20
30
10
−0.05
0
10
20
0.05
30
0
10
10
0.05
30
20
20
0
10
20
30
20
30
0.06
0.04
0.02
0
−0.02
−0.04
30
20
10
−0.05
−0.05
10
−0.05
30
10
20
H.-D. Lang
30
FDFD
10
20
30
15/18
Introduction
Eigenmode Analysis
Introduction
Examples
Examples
Eigenmode analysis in 2.5D
Waveguide modes (dimensions 𝑎 = 2𝑏)
Problem size: 16 × 8 cells, 𝑡sim ≈ 0.6 s
250
(rad/m),
(Np/m)
200
150
TE10
100
analytic
analytic
FDFD
FDFD
TEM limit
cutoffs
TE20
TE01
TE30
50
0
2
4
6
8
10
Frequency (GHz)
H.-D. Lang
FDFD
12
14
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Introduction
Eigenmode Analysis
Introduction
Examples
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
Introduction
Eigenmode Analysis
Introduction
Examples
References
[1]
Umran S. Inan, Robert A. Marshall
Numerical Electromagnetics – The FDTD Method
Cambridge University Press 2011
[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.
78-81, 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. 50, No. 7, July 2002
[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 2003
[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, 2006
[8]
Aliaksandra Ivinskaya
Finite-Difference Frequency-Domain Method in Nanophotonics
PhD Thesis, Department of Photonics Engineering, Technical University of Denmark, Lyngby, 2011
H.-D. Lang
FDFD
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