Statistical Properties of Wave Chaotic Scattering and Impedance Matrices

Collaborators: Xing Zheng, Ed Ott,

Experiments Sameer Hemmady, Steve Anlage,

Supported by AFOSR-MURI

Electromagnetic Coupling in Computer Circuits

connectors

cables

circuit boards

Integrated circuits

Schematic• Coupling of external radiation to computer circuits is a complex processes:

aperturesresonant cavitiestransmission linescircuit elements

• Intermediate frequency range involves many interacting resonances

• System size >>Wavelength

• Chaotic Ray Trajectories

• What can be said about coupling without solving in detail the complicated EM problem ?

• Statistical Description !(Statistical Electromagnetics, Holland and St. John)

Z and S-MatricesWhat is Sij ?

€

V1−

V2−

M

VN1−

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

= S

V1+

V2+

M

VN1+

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

outgoing incoming

S matrix

€

S = (Z + Z0 )−1(Z − Z0)

• Complicated function of frequency• Details depend sensitively on unknown parameters

Z() , S()

N- PortSystem

•••

€

V1+ ⇒

V1− ⇐

€

VN+ ⇒

VN− ⇐

N ports • voltages and currents, • incoming and outgoing waves

€

V1, I1

€

VN, IN

€

V1

V2

M

VN

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

= Z

I1

I2

M

I2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

voltage current

Z matrix

Frequency Dependence of Reactance for a Single Realization

Lossless:

Zcav=jXcav

Zcav

€

S =Zcav − Z0Zcav + Z0

= e jφ

S - reflection coefficient

Mean spacing f ≈ .016 GHz

Two - Port Scattering Matrix

0

0.2

0.4

0.6

0.8

1

1.2

55 60 65 70 75 80 85

S_matrix

mod-S_11mod-S_12

|Sij|

k2=2/c2

S1 2

€

=S11 S12

S21 S22

⎡

⎣ ⎢

⎤

⎦ ⎥S

Reciprocal:

Lossless:

€

S12 = S21 Transmission

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S112

= S222

=1− S122

Reflection = 1 - Transmission

Statistical Model of Z Matrix

Port 1

Port 2

Other ports

Losses

Port 1

Free-space radiationResistance RR()ZR() = RR()+jXR ()

RR1()

RR2()

€

Zij (ω) = −j

πRRi

1/2(ωn )RRj1/2(ωn)

Δωn2 winw jn

ω2(1+ jQ−1) −ωn2

n∑

Statistical Model Impedance

Q -quality factor

2n - mean spectral spacing

Radiation Resistance RRi()

win- Guassian Random variables

n - random spectrum

Systemparameters

Statisticalparameters

Two Dimensional Resonators

• Anlage Experiments• Power plane of microcircuit

Only transverse magnetic (TM) propagate forf < c/2h

Ez

HyHx

h

Box withmetallic walls

ports

€

Ez (x, y) = −VT (x, y) / h

Voltage on top plate

Wave Equation for 2D Cavity Ez =-VT/h

€

V j = dxdyu jVT∫• Voltage at jth port:

• Impedance matrix Zij(k):

€

Zij = − jkhη ui ∇⊥2 + k 2

( )∫−1

u j dxdy

• Scattering matrix:

€

S(k) = Z + Z0I( )−1

Z − Z0I( )

k=/c

€

∇⊥2 VT + k 2VT = − jkhη uiIi

ports∑

• Cavity fields driven by currents at ports, (assume ejt dependence) :

€

η = μ / ε

Profile of excitation current

Preprint available:

Five Different Methods of Solution

€

Zij = − jkhη ui ∇⊥2 + k 2

( )∫−1

u j dxdy

1. Computational EM - HFSS

2. Experiment - Anlage, Hemmady

3. Random Matrix Theory - replace wave equation with a matrix with random elementsNo losses

4. Random Coupling Model - expand in Chaotic Eigenfunctions

5. Geometric Optics - Superposition of contributions from different ray pathsNot done yet

Problem, find:

Expand VT in Eigenfunctions of Closed Cavity

€

Zij = − jkhη ui ∇⊥2 + k 2

( )∫−1

u j dxdy = − jkhηuiφn φnu j

k2 − kn2

modes−n∑

Where:

1. n are eigenfunctions of closed cavity

2. kn2 are corresponding eigenvalues

€

u jφn = dxdy∫ u jφn

€

kn = ωn / c

Zij - Formally exact

Random Coupling ModelReplace n by Chaotic Eigenfunctions

1. Replace eigenfunction with superposition of random plane waves

€

φn = limN →∞ Re2

ANak exp i knek ⋅x +θ k( )[ ]

k =1

N

∑ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

Random amplitude

€

Zij = − jkhηuiφn φn u j

k 2 − kn2

modes− n∑

Random direction Random phase

Chaotic Eigenfunctions

Rays ergodically fill phase space. Eigenfunctions appear to be a superposition of plane wave with random amplitudes and phases.

€

φn = limN →∞ Re2

ANa j exp i k j ⋅ x + θ j( )[ ]

j =1

N

∑ ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

€

φn = limN →∞

1

2ANa j exp i k j ⋅ x + α j( )[ ]

j =1

N

∑

Time reversal symmetry

kj uniformly distributed on a circle |kj|=kn

Time reversal symmetry broken

is a Gaussian random variable

€

φn

Cavity Shapes

Integrable(not chaotic)

Chaotic

Chaotic Mixed

Eigenfrequency Statistics

€

Zij = − jkhηuiφn φn u j

k 2 − kn2

modes− n∑

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

P(s)

integrablechaotic A

s

chaotic B

2. Eigenvalues kn2 are distributed according

to appropriate statistics

€

sn = (kn+12 − kn

2) / Δk2

Normalized Spacing:

Mean Spacing:

€

kn+12 − kn

2 ≡ Δk 2

€

k2 = 4πA( )−1

€

k2 = 2π 2kV( )−1

2D

3D

ener

gy

Wave Chaotic Spectra

Spacing distributions are characteristic for many systems

TRS = Time reversal symmetryTRSB = Time reversal symmetry broken

Thorium

Poisson

TRS TRSB

HarmonicOscillator

2+O

Statistical Model for Impedance Matrix

€

Zij (k) = −j

πRRi

1/2(kn)RRj1/2(kn)

Δkn2 winw jn

k2(1− jQ−1) − kn2

n∑

Q -quality factor

k2n=1/(4A) - mean spectral spacing

€

RRi(k) =kη4

dθk2π∫ u i (k)

2= Re ZRi{ }

-Radiation resistance for port i

System parameters

win- Guassian Random variables

kn - random spectrum

Statistical parameters

Predicted Properties of Zij

€

Zij = Zij + δZij• Mean and fluctuating parts:

• Fluctuating part: Lorenzian distribution-width radiation resistance RRi

€

P(X ii) =RRi

π X ii2 + RRi

2( )

• Mean part:(no losses)

€

Zij = 0, i ≠ j

€

Zii

= jXR, i

(k) Radiation reactance

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XR,i

(k) = Im ZR

{ }

HFSS - SolutionsBow-Tie Cavity

Moveable conductingdisk - .6 cm diameter“Proverbial soda can”

Curved walls guarantee all ray trajectories are chaotic

Cavity impedance calculated for 100 locations of disk4000 frequencies6.75 GHz to 8.75 GHz

Frequency Dependence of Reactance for a Single Realization

Mean spacing f ≈ .016 GHz

Zcav=jXcav

Frequency Dependence of Median Cavity Reactance

Effect of strong Reflections ?

Radiation ReactanceHFSS with perfectly absorbingBoundary conditions

Median Impedance for 100 locations of disc

f = .3 GHz, L= 100 cm

Distribution of Fluctuating Cavity Impedance

€

ξ =(X − XR(ω)) / R

R(ω)

6.75-7.25 GHz 7.25-7.75 GHz

7.75-8.25 GHz 8.25-8.75 GHz

ξ ξ

ξ ξ

Rfs ≈ 35

Impedance Transformation by Lossless Two-Port

Port

Free-space radiationImpedanceZR() = RR()+jXR()

Port

Cavity Impedance: Zcav()

Zcav() = j(XR()+ξ RR())

Unit Lorenzian

Lossless2-Port

Lossless2-Port

Z’R() = R’R()+jX’R()

Z’cav() = j(X’R()+ξ R’R())

Unit Lorenzian

Properties of Lossless Two-Port Impedance

Eigenvalues of Z matrix

€

det Z − jX1 = 0

€

X1,2

= XR

+ ξ1, 2

RR

€

ξ1,2 = tanθ1,2

2 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Individually ξ1,2 are Lorenzian distributed

1

2

Distributions same asIn Random Matrix theory

HFSS Solution for Lossless 2-Port

2

1

Joint Pdf for 1 and 2

Port #1: (14, 7)

Port #2: (27, 13.5)

Disc

Effect of Losses

Port 1Other ports

Losses

Zcav = jXR+(+jξ RRDistribution of reactance fluctuationsP(ξ)

Distribution of resistance fluctuationsP()

Predicted and Computed Impedance Histograms

Zcav = jXR+(+jξ RR

Equivalence of Losses and Channels Zcav = jXR+(+jξ RR

Distribution of reactance fluctuationsP(ξ)

Distribution of resistance fluctuationsP()

ξ

ξ

Role of Scars?

• Eigenfunctions that do not satisfy random plane wave assumption

Bow-Tie with diamond scar

• Scars are not treated by either random matrixor chaotic eigenfunction theory

• Semi-classical methods

Large Contribution from Periodic Ray Paths ?

22 cm

11 cmPossible strong reflectionsL = 94.8 cm, f =.3GHz

47.4 cm