lecture1 cem sh final

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Computational Methods for Electromagnetics (and Photonics)

E-mail: sailing@kth.seTel: 08-7908465

Sailing HeDivision of Electromagnetic Engineering(also with Photonics Lab in Kista, KTH-ZJU Joint Research Centre of Photonics)

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Course introduction• Twelve LecturesMy part of lectures:Lecture 1: Introduction to CEM…. Lecture 9: Detailed application examples of CEM, from Stealth to

Cloaking, from RF antennas to optical nano-antennas, Computational project on waveguides.

Lecture 10: Application examples of FDTD for resonators, etc., Lecture 11: CEM for planar lightwave circuits, photonics, etc. • ….• Four Lab Works

• TA: Ning Zhu, ningz@kth.se , tel. 790 4266 (Kista)

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Lecture 1. Introduction to Computational Electromagnetics

I. Brief introduction to Computational Electromagnetics (CEM);

II. Classification of Computational Methods;

III. Some Applications

IV. Challenging Problems

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I. Brief introduction to CEM

• The Stage of Closed-Form Solutions(paper and pen)

• The Stage of Approximate Solutions(calculator, computer)

• The Stage of Numerical Solutions(computer, super computer)

Three Stages of EM Simulations

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The Stage of Closed-Form Solutions

• After the Maxwell’s equations were published in 1864, and in the beginning of 1900s, many closed-form solutions have been obtained:

Mie, ca. 1900: Mie series solution of scattering by a sphere (separation of variables)

Lord Rayleigh, 1897: Guided-wave solution in a hollow waveguide (separation of variables)

Lord Rayleigh: Rayleigh scattering by small particles

……

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The Stage of Approximate Methods

• High-Frequency MethodsGeometrical Optics (GO)Geometrical Theory of Diffraction (GTD)Uniform Theory of Diffraction (UTD)Physical Optics (PO)Physical Theory of Diffraction (PTD)Shooting and Bouncing Ray (SBR): XPATCH

combination of High-Frequency and Numerical Methods.

High-frequency methods are ray-based methods, which require information of shadow region and illuminated region.

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The Stage of Numerical MethodsAge of numerical methods: MOM, FDTD, FEM

Yee, 1966; Harrington, 1968; Silvester, 1972; Rao, Wilton & Glisson, 1982; Mittra, 1980+; Taflove, 1980+.

Differential Equation Solvers (FDTD, FEM)Integral Equation Solvers (MoM, BIM)

VIZ =⋅

Many numerical methods started in the electromagnetic community, and later spread to other communities and become popular…

The radiation condition at infinity is emulated by the use of absorbing boundary conditions (ABC), such as Perfect Matched Layer (PML)

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EMEMSIMULATIONSSIMULATIONSWireless

Comm. &Propagation

Physics BasedSignal Processing & Imaging

ComputerChip Design& Circuits

Lasers &Optoelectronics

MEMS &MicrowaveEngineering

RCS Analysis,Design, ATR& StealthTechnology

AntennaAnalysis &Design

EMC/EMIAnalysis

RemoteSensing &SubsurfaceSensing & NDE

BiomedicalEngineering& BioTech

Nano-phootnics

Education

New Materials (metamaterials)

Impact of EM simulations

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Computational electromagnetics (CEM) refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment, in which computationally efficient approximations to Maxwell's Equations are typically used.

What’s the CEM?

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Solving a Complex Problem Needs …• Electromagnetic Physics:

A correct and efficient problem definitionA good physical insight within calculationsA good physical model can reduce complexity

• MathematicsA correct and efficient mathematical descriptionMathematical analysis: convergence, stability, conditioning, error analysis, error control

• Computer ScienceEfficient algorithms for the math problemEfficient memory arrangement: shared memory and local memoryParallelization of computers and interprocessorcommunications

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Basic Theory in Electromagnetics

+t

0

0

t

t

ρ

ρ

με

∂∇ × = −

∂∂

∇ × =∂

∇ • =∇ • =

∂∇ ⋅ + =

∂==

BE

DH J

DB

J

B HD E

Faraday’s Law

Ampere’s Law

Gauss’s LawNo magnetic charge

James Clerk Maxwell (1831-1879)

Current Continuity

Constitute Relation

The most elegant equations in the universe.

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1 1 2 2

1 1 2 2

1 1 2 2

1 1 2 2

s

s

s

ms

ρρ

× + × =⎧⎪ × + × =⎪⎨ ⋅ + ⋅ =⎪⎪ ⋅ + ⋅ =⎩

n H n H JE n E n Mn D n Dn B n B

1ε1μ

2ε2μ

1n̂

2n̂

…with Boundary Conditions

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Solving Maxwell’s Equations

Time Domain Frequency Domain

Fourier transform

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Time vs. Frequency Domain

Broadband simulation/frequency sweepNonlinear material modelingTransient phenomena

Time domain:

Frequency domain:Multiple excitation/angular sweepDispersive material modelingSteady state phenomena

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II. The Classification of CEM• Analytical methods• Numerical Methods

– Finite Difference Time Domain (FDTD);– Finite Element Method (FEM); – Method of Moments (MoM); – Beam Propagation Method (BPM) -- Lecture 11;– Transmission-line modeling;– Monte Carlo method;

– …• Hybrid Methods

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CEM Overview

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Numerical Methods(Labs 2 and 4 with commercial softwares)

• Method of moments (Harrington, 1960s)– Integral equation based– Versatile geometry handling– Small number of unknowns

• Finite Difference Time Domain Method (Yee, 1960s)– Differential equation based– Simplicity– Large number of unknowns– Sparse matrix system

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(1). Method of Moments (MoM)MoM for EM Problems

Derive integral equation (IE)

Convert the IE into a matrix equation using basis functions and weighting functions

Evaluate matrix elements

Solve the matrix equation and obtain the poarameters of interest

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Conducting Cylinders, TM caseConsider a perfectly conducting cylinder excited by an impressed electric field Ez

i

PP ′

PP ′−nld

φφ ′

C

The impressed field induces surface currents Jz on the conducting cylinder, which produce a scattered field Ez

s.

x

y

o

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Conducting Cylinders, TM caseThe field due to Jz is given by

∫ ′′−′−=C

zsz ldkHJkE )()(

4)( )2(

0 ρρρηρ

On the cylinder surface C, the boundary condition is

C on ,0=+= sz

izz EEE

C on ,)()(4

)(

C on 0)()(4

)(

)2(0

)2(0

ρρρρηρ

ρρρρηρ

∫

∫

′′−′=⇒

=′′−′−⇒

Cz

iz

Cz

iz

ldkHJkE

ldkHJkE

where )(ρizE is known and zJ is the unknown to be determined.

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Method of Moments

gLf =Consider inhomogeneous equation

L: operator g: excitation (source) (known function) f: field or response (unknown function to be determined)

n

N

nnuaf ∑

=

=1

~functions (basis) expansion:

constant:

n

n

ua

For exact solutions the summation is usually infinite and un form a complete set of basis functions

For approximate solutions the summation is usually finite.

(*)

(**)

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Method of Moments

gLua nn

n =∑Substitude (**) in (*) and use the linearity of L

Now define a set of weighting functions or testing functions, w1,w2, w3 ,... İn the range of L, take the inner product with each wm.

,...3,2,1,,, ==∑ mgwLuwa mnmn

n

This set of equations can be written in matrix form:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

.........

...,,

...,,

2212

2111

LuwLuwLuwLuw

lmn [ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= 2

1

aa

an [ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡= gw

gwgm ,

,

2

1

[ ][ ] [ ]mnmn gal =

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More About MoM• MOM is an old topic, which was proposed by Harrington in 1966.

However, new bloods have been put in MOM in the past decades.• For Basis Function (for PEC, for example):

Rao-Wilton-Glisson (RWG) basis , triangular patch (1982)High-order basis, triangular patch or quadr. patchHigh-order basis, curvelinear triangular patchWire-surface junctionSurface-surface junctionNew physics-based basis functions

Fewer Unknowns,ComplicatedEvaluation

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Download software Feko-lite: http://www.feko.info/sales

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CADFeko

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Basic steps• Establish the model in CADFEKO

– Create CAD geometry using canonical structures and perform boolean operations on these.

– Add the properties, e.g. dielectric constant, coating, conductivity

• Add the excitation– Set solution parameters (e.g. frequency, loads).– Set excitations (e.g. frequency, loads).

• Add the request for results – e.g. far-fields, near-fields, S-parameters, SAR analysis

• Mesh and Calculation– Create mesh (surface and volume meshes) from CAD geometries.

Once the model preparation is complete (geometry, mesh, excitations and calculation requests), the Solution is obtained by running the solver FEKO.

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Lab 2. Design a rectangle patch antenna of a given size (10cmx7.8cm) working at 915MHz on a substrate withThe patch is located at the middle of the substrate (20cmx15cm). Find the best feeding point (i.e., |S11| is minimal and at least less than -10dB when the input port impedance is 50Ω ). Show the far field radiation and S-parameters of the antenna

4.4, 1.55r h mmε = =

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(2). FDTD Method

• The acronym of FDTD stands for Finite Difference Time Domain.

• first developed by Kane S. Yee in 1966.

• A method to simulate electromagnetic wave propagation in any kind of materials (including metals with dispersion).

• Very useful tool for simulating waves in subwavelength scale object (e.g. near-field optics).

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Electromagnetic waveH Et

E Ht

μ

ε

∂= −∇×

∂∂

= ∇×∂ finite-

difference approximation

( ) ( )x

xxFxxFxF

ΔΔ−−Δ+

=∂∂ 2/2/

Finite difference scheme (discretization)

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electric field points are spatially half-grid offset from the magnetic field points.

Yee’s mesh1 cell

( , , , ) ( , , , )

( , , )n

F x y z t F i x j y k z n t

F i j k

= Δ Δ Δ Δ

=

In 1966, K. S. Yee in the first time presented the finite difference approximation of Maxwell’s equations. These formulas are called Yee’s formulas. This method is called Finite Difference Time Domain now.

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FDTD method

( ) ( )⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ +−= −−−

21

21

21

211 kHkHCkEkE n

ynyEXLZ

nx

nx

( ) ( ){ }1121

21

21

21

−−+−⎟⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ + −+ kEkECkHkH n

xnxHYLZ

ny

ny

εztCEXLZ Δ

Δ=

μztCHYLZ Δ

Δ=

• Maxwell equation for plane wave traveling in the positive z direction (in one dimension):

• time-stepping in the discretized form:zE

tH

zH

tE

∂∂

=∂∂

∂∂

=∂∂ με ,

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This flow chart often called ‘Leap-flog Algorithm’

Algorithm

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FDTD Resultswhat will you see with FDTD?

a “movie” of the field propagating in or being scattered by the object.

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Cell phone interaction with the human head.Digested from: Recom Inc. Website:

http://www.recomic.com/html/index.html.

FDTD simulation: Cell phone interaction with a human head

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A commercial software for FDTD simulation: OptiFDTD

OptiFDTD is a powerful, highly integrated, and user friendly CADenvironment that enables the design and simulation of advanced passive and non-linear photonic components.

Some applications:

• Waveguide-based planar light circuits such as

splitters, couplers and resonators

• Photonic band gap materials and devices

• Surface plasmon devices

• Nonlinear materials and dispersive materials

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Main components of OptiFDTD

• DesignerCreates a layout of devices on a wafer.

• SimulatorProcesses data in designer files,

monitors the progress and stores the results.

• AnalyzerLoads and analyzes the result files by simulator.

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Designer

• Very user-friendly CAD interface, in which you can easily draw your structures.

• Conveniently switching between different views.

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Simulator and Analyzer

• You can edit any simulation parameters in simulator.

• In simulation, an animation of what is happening makes you intuitively understand the phenomenon.

• Analyzer offers you convenient tools for dealing with the data provided by the simulator.

Edit parameters

Simulating Data analyzing

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Lab 4. 2D ring resonator

• You are required to design a ring resonator in 2D case based on the provided materials, which acts like a filter for specific resonant wavelengths.

• You need to optimize the structure parameters, such as the radius of the ring, distance between the waveguides, etc.

• You can use the commercial software for the simulation and test your own design. schematic diagram

for a ring resonator

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Commercial FDTD Codes are found on the Web:• APLAC http://www.aplac.hut.fi/aplac/general.html• Apollo Photonics http://www.apollophoton.com/• Applied Simulation Technology http://www.apsimtech.com/• CFD Research http://www.cfdrc.com/datab/software/maxwell/maxwell.html• Cray http://lc.cray.com/• Empire http://www.empire.de/• EMS Plus http://www.ems-plus.com/ezfdtd.html• ETH

http://www.iis.ee.ethz.ch/research/bioemc/em_simulation_platform.en.html• Optima Research http://www.optima-

research.com/Software/Waveguide/fullwave.htm• Optiwave http://www.optiwave.com/• Quick Wave http://www.ire.pw.edu.pl/ztm/pmpwtm/qw3d/• Remcom http://www.remcominc.com/html/index.html• RSoft http://www.rsoftinc.com/fullwave_info.htm• Schmid http://www.semcad.com/solver_performance.html• Vector Fields http://www.vectorfields.com/concerto.htm• Virtual Science http://www.virtual-

science.co.uk/celia/Celia_code/celia_home.htm• Zeland Software http://www.zeland.com/fidelity.html

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References:• Yee, K. S., “Numerical solution of initial

boundary value problems involving Maxwell’s equations in isotropic media,”IEEE Trans. Antennas Propagat., vol. 14, 1966, pp. 302-307.

• A. Taflove, Computational Electrodynamics-The Finite Difference Time-Domain Method, Norwood: ArtechHouse, 2005.

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III. Some Applications

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1. Applications in Microwave AntennaDipoles--the Simplest Antennas

base stationReal-time evolution of the electric

field of an oscillating electric dipole

Pictures form http://en.wikipedia.org/wiki/

III. Some Applications

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CST Microwave Studio™

HFSS Finite Element Method (FEM)

FEKO Method of Moments (MoM)

XFDTD Finite Difference time-domain Methods (FDTD)

Common Commercial Softwares for Antenna Simulation

Finite-Volume Time-Domain (FVTD) method

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Reflector Antennas

National Radio Astronomy Observatory in U.S

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Patch Antenna

FR4 substratemicrostrip fed

air substratecoaxial line fed

PIFA (Planar Inverted-F Antenna)

feeding probe

shorted patch

ground plane

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VSWR

Antenna Structure Animated Current Distribution

Example I : UWB (Ultra-Wide Band) Antenna (using HFSS software)

Bandwidth: VSWR < 1.5

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0

30

60

90

120

150

180

210

240

270

300

330

-100

-80

-60

-40

-20

0

-100

-80

-60

-40

-20

0

3GHz 10GHz

0

30

60

90

120

150

180

210

240

270

300

330

-100

-80

-60

-40

-20

0

-100

-80

-60

-40

-20

0

Radiation Patterns (H plane)

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Example II: E Shaped Patch Antenna (using CST)

Antenna Structure Model in CST

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Broad band formed by 3 resonances

f1=2.252GHz

f2=2.422GHz

f3=2.553GHz

Return Loss

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f1=2.252GHz

Animate Current Distribution

E field and surface current

3D pattern

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f1=2.422GHz

Animate Current Distribution

E field and surface current

3D pattern

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f1=2.553GHz

Animate Current Distribution

E field and surface current

3D pattern

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Array Applicationsarray antenna

on F22 combat aircraft

reflector antenna arraymicrostrip fed planar phase array

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2. Radar Applications

RCS (Radar cross section) is the unit of measure of how detectable an object is with a radar. For example, a stealth aircraft (which is designed to be undetectable) will have design features that give it a low RCS, as opposed to a passenger airliner that will have a high RCS.

In particular, an article on stealth provides an overview of various methods used in designing aircraft so that they are more difficult to detect.

A USAF B-2 Spirit in flight

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Concept of pulse radar

http://www.aerospaceweb.org/question/electronics/q0168.shtml

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1. The power transmitted in the direction of the target 2. The amount of power that impacts the target and is reflected back in

the direction of the radar 3. The amount of reflected power that is intercepted by the radar antenna 4. The length of time in which the radar is pointed at the target

Factors that determine the energy returned by a target

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T-33 jet trainer

T-33 radar cross section

T-33 medianized radar cross section

Typical RCS diagram

Website: http://www.cst.com/

5959

Invisible (transparent) cloaking with metamaterials (n=0)

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Obtain macroscopic (effective) material parameters from an

artificial structure of microscopicelements through CEM

6161

Our world (refractive index n>0)

Positive refraction

slab: defocusing

n>0

>0.5 λ

6262

3 decades ago，Veselagopredicted theoretically：

Metamaterials with negative refractive index

refractive

index n<0permeabilityμ<0

permittivityε<0

slab: focusing

n<0

V.G. Veselago, Sov. Phys. Usp. 10, 509, 1968

6363SCIENCE VOL 292 6 APRIL 2001

lattice of split ring resonators

First experiment for metamaterial with n<0

---- at microwave frequencies

6464

Negative permeability at 100 THz(metal’s permittivity is already negative at optical

frequencies)

Retrieved real part of Effective Permeability

M. Wegener et al., Science 306, 1351 (2004)

Au

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•Can be applied to both simple and complicated structures

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Cloaking, realized with SRRs

First experimental demonstration of cloaking at 8.5GHz

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Cloak OFF Cloak ON

Non-magnetic cloak @ 632.8nm with silver wires in silica

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3. Circuit design(1). Microwave circuits.Waveguide based on Split Ring Resonators (SRRs) (with CST STUDIO SUITE)

Dual band-rejection filterOverall size: 15mm*10mmRejection bands: 2.47~2.7GHz, 4.7~5.3GHz

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Current distribution

f = 2.5GHz

f = 5GHz

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(2). Electronic integrated circuits(with CST STUDIO SUITE)

System-in-Package model with material definitions

Applications in EMC Transient Simulation of a System-in-Package (SiP), shown in figure 1 and imported from the CDS Cad design System, consists of copper (lossy metal), polyimide and Silicon with bond wires and through vias.

72Definition of the discrete ports in the SiP model

Figure 2 shows the discrete port assignment for the power suppy pin (1) and the signal pins (2,3,4,5).

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Surface currents in the SiP at 10 GHz for Port 1 excitation - some materials have been hidden for clarity

Figure 4 shows an animated field plot of the surface currents at 10GHz as a function of phase.

Pictures from http://www.cst.com

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a bad design may result in:

Breakdown

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(3). Photonic Integrated CircuitsOptical Coupler. Waveguide ports have been defined at the waveguide feeds. The ring and feed lines are defined with a dielectric constant of 9. The frequency range simulated was between 170 THz and 250 THz.

Geometry of the Optical Coupler with Waveguide Ports

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frequency dependent behaviour of the field. Ports 3 and 4 are isolated at 211.6 THz whereas ports 1 and 2 are isolated at 250 THz.

E-Field Plot at 211.6 THz

E-Field Plot at 250 THz

Website: http://www.cst.com/

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Ultra-compact resonator filterHere is an ultra-compact thermally-tunable microring resonator filter with a submicron heater on silicon nanowires.

pad

pad

metal SiO2 Si

Input

Through

Output

metal

Si core SiO2 up-cladding

SiO2 insulator layer

The 3D view of schematic configuration of the present tunable MRR filter

J. Lightwave Technol. 26(6): 704-709, 2008.

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the designed mask and the details of the structure

Heater Pad

Heater Pad Lmid

wT

Ltp Ltp

wtp

R wg

100μm

100μ

m

Inpu

t Th

roug

h

Dro

p

T-junction

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TE

Ltp=1μm

Lmid=0.4μm wT=0.2μm

(a)

TM

Ltp=1μm

Lmid=0.4μm wT=0.2μm

(b)

The 2D-FDTD simulated light propagation in an optimally designed T-junction for (a) TE polarization; (b) TM polarization.

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Thermal characteristics

T(x,y) @ P=5mW:Below heater: 350ºC Core: 152ºC~155ºC

x (μm)

y (μ

m)

SiO2 insulator

Si core

cladding Unit:ºC

2.44

2.46

2.48

2.5

2.52

2.54

0 1 2 3 4 5 6Power(mW)

Loss

(dB

/cm

)

TETM

P (mW)

n eff

numerically solve the Laplacian Equation with appropriate boundary conditions.

Δneff/ΔP: ~0.00644 mW-1

P=5mW Δneff=0.0322,Δλ ~ 20nm from Δλ=(Δneff/neff)λ

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The spectral response (TE) as power increases

• Power: 0mW 5mW, 1551.7nm 1571.8nm, Δλ: ~20nm;

• IL: ~0.25dB; Extinction ratio: >20dB;

• Q(=λ/Δλ): ~1000;

1550 1560 1570 1580-30

-20

-10

0

1550 1560 1570 1580-30

-20

-10

0

1550 1560 1570 1580-30

-20

-10

0

1550 1560 1570 1580-30

-20

-10

0

1550 1560 1570 1580-30

-20

-10

0

1550 1560 1570 1580-30

-20

-10

0

wavelength (nm)

spec

trum

(dB

)

λres:1551.7nm λres:1555.3nm

λres:1559.1nm λres:1563.1nm

λres:1567.35nm λres:1571.8nm

(a) 0mW (b) 1mW

(c) 2mW (d) 3mW

(e) 4mW (f) 5mW

• Higher Q: decrease the coupling by increasing the gap width. E.g., Q~104 when Gap=200nm.

1

3

2

4

2’

4’

1’

3’

l44’

l22’

Gap

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IV. Challenging Problems

Scattering by an airplane

Current distribution

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Faster solver for a very large object

• Currents are induced on a PEC scatterer illuminated by a source.

• The induced currents adjust themselves to cancel the incident field.

• Hence, every current element needs to talk to the other elements.

Js Einc

PECPEC

Example: Fast Multipole Method

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Fast Multipole Method

• all current elements talk directly to each other. The number of “links” is proportional to N 2 , where N is the number of current elements. => large matrix

OneOne--Level AlgorithmLevel Algorithm

• “hubs” are established to reduce the number of direct “links” between the current elements.

TwoTwo--Level AlgorithmLevel Algorithm

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Fast Multipole Method (cont’d)

• A tree structure showing the procedure to form a multilevel algorithm. (N log N)

MultiMulti--Level AlgorithmLevel Algorithm

"Efficient MLFMA, RPFMA and FAFFA Algorithms for EM Scattering by Very Large Structures,“IEEE Transactions on Antennas Propagation, vol. 52, no. 3, pp. 759-770, March 2004.

86

an airplane hit by lightning:(with CST STUDIO SUITE)

The lightning strike, modelled by the shown double exponential waveform, is applied to the nose of the aircraft using a discrete current port. A 300 Ohm load from the tail to the electric boundary forms the discharge channel.

Lightning strikes most commonly occur in clouds: either inter- or intra-cloud or cloud-to-ground. The simulation of indirect lightning effects on structures with metallic shells

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The surface current magnitude on the aircraft due to the lightning strike is shown as it varies in time.

Pictures from http://www.cst.com

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CEM for a tank

On the ground

89

Car communications

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Homework:

Derive the boundary conditions on page 12 from Maxwell’s equations(on page 11).