random coupling model: statistical predictions of

1

Random Coupling Model: Statistical Predictions of

Interference in Complex Enclosures

Steven M. Anlage

with Bo Xiao, Edward Ott, Thomas Antonsen

Research funded by AFOSR and ONR

COST Action IC1407 (ACCREDIT)

University of Nottingham

4 April, 2016

http://index.html/

2

The Maryland Wave Chaos Group

Tom Antonsen Steve AnlageEd Ott

Jen-Hao YehLPS

James HartLincoln Labs

Biniyam TaddeseFDA

Also:Undergraduate StudentsEliot BradshawJohn AbrahamsGemstone Team TESLA

Post-DocsGabriele GradoniMathew Frazier

NRL Collaborators: Tim Andreadis, Lou Pecora, Hai Tran, Sun Hong, Zach Drikas, Jesus Gil Gil

Funding: ONR, AFOSR, DURIP

John RodgersNRL, Naval Academy,

UMD

Ming-Jer LeeWorld Bank

Trystan Koch

Faculty

Graduate Students (current + former)

Mark HerreraHeron Systems

Bo Xiao Bisrat Addissie

3

Outline

• The Problem: Electromagnetic Interference

• Our Approach – A Wave Chaos Statistical Description

• The Random Coupling Model (RCM)

• Examples of the RCM in Practice

• Conclusions

4

Electromagnetic Interference and

High-Power Microwave Effects on ElectronicsHow to defend electronics from electromagnetic interference?

R.F.I. International

Electromagnetic Interference (EMI)

Electromagnetic Compatibility (EMC)

5

Electromagnetic Compatibility Issues in Automobiles

Germany Japan

How to defend electronics from electromagnetic interference?

6

What Happens to Electronics Housed in Electrically-Large Enclosures?

Examples: Aircraft cockpit, Rooms, Automobile engine electronics, Computer, etc. System size >> Wavelength

Empirical evidence suggests that some electronics are susceptible under some conditions …

Many failure modes have been identified:

Internal circuit signal disruption – spurious signal generation

Front-end diodes produce baseband + harmonic signal input

Burnout of traces, contacts, components

0

10

0 2000

FrequencyA

mp

litu

de

0

10

0 2000

Frequency

Am

pli

tud

e

HPMJohn Rodgers

F. Sonnemann, Diehl

UMD

Room Dim: 4 x 3.5 x 3 m Port 1: Hertzian Dipole

Port 2: PCB track

7

Electromagnetic Coupling to Enclosures and CircuitsA Complicated Problem

Coupling of external radiation to

computer chips is a complex process:

Apertures

Resonant cavities

Transmission Lines

Circuit Elements

System Size >> Wavelength

Chaotic Ray

Trajectories

What can we say about the nature of fields and induced

voltages inside such a cavity?

Statistical Description using Wave Chaos!!

Arbitrary Enclosure

(with losses “1/Q”)

Rather than make predictions for a specific configuration,

determine a probability distribution function (PDF) of the

relevant quantities

http://www.qksrv.net/click-1300521-10281960

8

Outline





• Conclusions

9

It makes no sense to talk about

“diverging trajectories” for waves

1) Waves do not have trajectories

Wave Chaos?

2) Linear wave systems can’t be chaotic

3) However in the semiclassical limit, you can think about rays

Wave Chaos concerns solutions of linear wave equations which,

in the semiclassical limit, can be described by chaotic ray trajectories

In the ray-limit

it is possible to define chaos

“ray chaos”

Maxwell’s equations, Schrödinger’s equation are linear

10

From Classical to Wave Chaos

wave

ray

trajectory

Quantum

Classical

(chaos)

Semiclassical limit

(quantum chaos)

11

H

Random Matrix Theory (RMT) and Wave ChaosWigner; Dyson; Mehta; Bohigas …

The RMT Approach:

Complicated Hamiltonian: e.g. Nucleus: Solve

Replace with a Hamiltonian with matrix elements chosen randomly

from a Gaussian distribution

Examine the statistical properties of the resulting Hamiltonians

This hypothesis has been tested in many systems:

Nuclei, atoms, molecules, quantum dots, acoustics (room, solid body, seismic),

optical resonators, random lasers,…

Some Questions:

Is this hypothesis supported by data in other systems?

What new applications are enabled by wave chaos?Can losses / decoherence be included?

What causes deviations from RMT predictions?

Hypothesis: Complicated Quantum/Wave systems that have chaotic classical/ray

counterparts possess universal statistical properties described by

Random Matrix Theory (RMT) “BGS Conjecture”

Cassati, 1980

Bohigas, 1984

EH

Orthogonal (real matrix elements, b = 1)

Unitary (complex matrix elements, b = 2)

Symplectic (quaternion matrix elements, b = 4)Universality Classes of RMT:

12

Where is Wave Chaos Found?

Billiard

Incoming

Channel

Outgoing

Channel

|S|S1111||

|S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

|S|S1111||

|S|S2222||

|S|S2121||

Frequency (GHz)

|| xxS

Electromagnetic Cavities:

Complicated S11, S22, S21

versus frequency

B (T)

Transport in 2D quantum dots:

Universal Conductance Fluctuations

Res

ista

nce

(k

W) mm

Nuclear scattering:

Ericson fluctuations

d

d

Proton energyldeBroglie << nucleus size

Compound nuclear reaction

1

2

Incoming Channel

Outgoing Channel

ldeBroglie << billiard size

l << billiard size

Incoming Voltage waves

Outgoing Voltage waves

S matrix

×

+

+

+

NN V

V

V

S

V

V

V

2

1

2

1

][

Scattering matrix

+

+

+

NN V

V

V

S

V

V

V

2

1

2

1

][

13

The Difficulty in Making Predictions in Wave Chaotic Systems…

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Ab

s[Z

cav]

Frequency [GHz]

Perturbation Position 1

8.7 8.8 8.9 9.0 9.1 9.2 9.3

200

400

600

800

1000

Ab

s[Z

cav]

Frequency [GHz]

Perturbation Position 2

(W)

Ele

ctro

mag

net

ic

Wav

e Im

ped

ance

The extreme sensitivity of the properties of

wave chaotic systems to small perturbations

suggests a statistical approach (RMT)

The Random Coupling Model incorporates the

underlying chaos into a quantitative statistical

description of the wave properties of real systems

Ray-

Chaotic

Enclosure

14

Induced Voltage Statistical Distributions

for Objects in an Arbitrary Enclosure

Arbitrary Enclosure

(with losses “1/Q”)Port 1

Port 2Port 3Port 4

Port 5

Generalized Port

Concept

Port i

Port j

Our approach treats all objects of interest as “ports”

Incident rf energy enters the enclosure through one or more ports

The energy reverberates and is absorbed by one or more ports inside the enclosure

Formulate a quantitative statistical theory of absorbed energy

http://www.qksrv.net/click-1300521-10281960

15

N-Port Description of an Arbitrary Scattering System

N – Port

Scattering

System

N Ports

Voltages and Currents,

Incoming and Outgoing Waves

Z matrixS matrix

1V+

1V

V1 , I1

VN , IN

NV+

NV

Complicated

Functions of

frequency

Detail Specific (Non-Universal)

)(),( SZ

)()( 0

1

0 ZZZZS +

+

+

+

NN V

V

V

S

V

V

V

2

1

2

1

][

NN I

I

I

V

V

V

2

1

2

1

][

16

Traditional Approach to Describing Wave Chaotic

Scattering Systems – the Scattering Matrix

C. M. Marcus (1992)

2-D Electron Gas

electron mean free path >> system size

electron wavelength << system size

Ballistic Quantum Transport

Quantum interference Fluctuations in G ~ e2/h

“Universal Conductance Fluctuations”

Landauer-

Büttiker

1 2

1 1

222 N

n

N

m

nmSh

eG

Nuclear scattering:

Ericson fluctuations

Wd

d

Proton energy

Compound nuclear reaction

Incoming Channel/Port

Outgoing Channel/Port

In contrast, our approach uses the Impedance (Z)

description of wave scattering

17

Outline





• Conclusions

18

Statistical Model of Impedance (Z) MatrixS. Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005)

Port 1

Port 2

Other ports

Losses

Port 1

Free-space radiation

Resistance RR()

ZR() = RR() + jXR ()

RR1()

RR2()

Statistical Model Impedance

Q -quality factor

D2n - mean spectral spacing

Radiation Resistance RRi()

win- Guassian Random variables

n - random spectrum

System

parameters

Statistical

parameters

Zij ( ) j

RRi

1/2 (n )n

RRj1/2 (n )

Dn

2 winwin

2 (1+ jQ1) n

2

L. K. Warne, et al., IEEE Trans. on Anten. and Prop. 51, 978 (2003)

X. Zheng, et al., Electromagnetics 26, 3 (2006); Electromagnetics 26, 37 (2006)

modes n

win wjn

19

The Most Common Non-Universal Effects:

1) Non-Ideal Coupling between external scattering states and internal modes (i.e. Antenna/Port properties)

Universal Fluctuations are Usually Obscured by

Non-Universal System-Specific Details

Port

Ray-Chaotic

Cavity

Incoming wave

“Prompt” Reflection due to Z-Mismatch

between antenna and cavity

Z-mismatch at interface of port and cavity.

Transmitted wave

2) Short-Orbits between the antenna and fixed walls of the billiards

20

The Random Coupling ModelDivide and Conquer!

Enclosure Problem

Port 1

Port 2Port 3Port 4

Port 5

Port i

Port j

Coupling Problem

Mean

part

Fluctuating Part

(depends on a)

<Imx> = 1

<Rex > = 0

RadRad RiiXZZZ x++~

Solution: Random Matrix Theory;

Electromagnetic statistical properties are

governed by Loss Parameter a k2/(Dkn2 Q) df3dB/Dfspacing

Solution: Radiation Impedance Matrix Zrad

+ Short Orbits

Electromagnetics 26, 3 (2006)Electromagnetics 26, 37 (2006)Phys. Rev. Lett. 94, 014102 (2005)

Z matrix

NN I

I

I

V

V

V

2

1

2

1

][

IEEE Trans. EMC 54, 758 (2012)

21

xiRiXZ avgavgcavity +

Universally Fluctuating Complex

Quantity with Mean 1 (0) for the

Real (Imaginary) Part.

Predicted by RMT

Semiclassical

Expansion over

Short Orbits

Complex Radiation Impedance

(characterizes the non-universal coupling)Index of ‘Short Orbit’

of length l Stability of orbit

Action of orbit

Theory of Non-Universal Wave Scattering PropertiesIncluding Imperfect Coupling and Short Orbits

James Hart, T. Antonsen, E. Ott, Phys. Rev. E 80, 041109 (2009)

Port

Zcavity

Port

ZRad

The waves do

not return to the port

Perfectly absorbing

boundaryCavity

Orbit Stability Factor:

►Segment length

►Angle of incidence

►Radius of curvature of wall

Assumes foci and caustics are absent!

Orbit Action:

►Segment length

►Wavenumber

►Number of Wall Bounces

1-Port, Loss-less case:

+

+b(l)

iLlik

lblbRadRadavgPorteDpRZZ

4/)(

)()(

22

8

0 1 20

1

2

3

-1 0 10

1

2

3

]ˆRe[ zl

])ˆ(Re[ zP l ])ˆ(Im[ zP l

]ˆIm[ zl

16a

3.6a

9.1a

16a

3.6a

9.1a

Inclusion of loss: Pa(Z)

a = 3-dB bandwidth / mean-spacing

Fyodorov+Savin JETP Lett. 80, 725 (2004)

Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005)

Fyodorov+Savin+Sommers J Phys. A 38, 10731 (2005)

Hemmady, et al., Phys. Rev. E 74 , 036213 (2006)

Mean of Pa(z)

Variance of Pa(z)

Universal Impedance (Z) Statistics in the Presence of Loss

a

1 1a

Comparison of data (symbols) and RMT (solid lines)

1ˆRe zE l 0ˆIm zE l Independent of a

23

8

0 1 20

1

2

3

-1 0 10

1

2

3

]ˆRe[ zl

])ˆ(Re[ zP l ])ˆ(Im[ zP l

]ˆIm[ zl

16a

3.6a

9.1a

16a

3.6a

9.1a

Inclusion of loss: Pa(Z)

PDF of the eigenvalues lz of the

universal impedance matrix (z)

Universal Impedance (z) Statistics in the Presence of Loss

a

1

Comparison of data (symbols) and RMT (solid lines)

1ˆRe zE l 0ˆIm zE l

a = 3-dB bandwidth / mean-spacing

Fyodorov+Savin JETP Lett. 80, 725 (2004)

Hemmady, et al., Phys. Rev. Lett. 94, 014102 (2005)

Fyodorov+Savin+Sommers J Phys. A 38, 10731 (2005)

Hemmady, et al., Phys. Rev. E 74 , 036213 (2006)

Frequency (GHz) Frequency (GHz)

|S21|2 |S21|2

a ~ 10-2 a ~ 1

24

Outline





• Conclusions

25

Microwave Cavity Analog of a

2D Quantum Infinite Square Well

Table-top experiment!

Ez

Bx By

( )

boundariesatwith

VEm

n

nnn

0

02

2

2

+ Schrödinger equation

boundariesatEwith

EkE

nz

nznnz

0

0

,

,

2

,

2

+Helmholtz equation

Stöckmann + Stein, 1990

Doron+Smilansky+Frenkel, 1990

Sridhar, 1991

Richter, 1992

d ≈ 8 mm

An empty “two-dimensional” electromagnetic resonator

Bow-Tie Billiard

A. Gokirmak, et al. Rev. Sci. Instrum. 69, 3410 (1998)

~ 50 cm

The only propagating

mode for f < c/d:Metal walls

26

The Experiment:

A simplified model of wave-chaotic scattering systems

A thin hollow metal box

ports

Side view

21.6 cm

43.2 cm

0.8 cm

λ

Ez

Coaxial

cable

27

Microwave-Cavity Analog of a 2D

Infinite Square Well with Coupling to Scattering States

Network Analyzer [measures Scattering (S)-matrix vs. frequency]

Thin Microwave Cavity PortsElectromagnet

We measure from 500 MHz – 19 GHz, covering about 750 modes in the semi-classical limit

28

Pro

bab

ilit

y D

ensi

ty

-2 -1 0 1 20.0

0.3

0.6

2a=0.635mm2a=1.27mm

)Im(z-500 -250 0 250 500

0.000

0.005

0.010

0.015

2a=1.27mm

2a=0.635mm

))(Im( WCavityZ

Testing Insensitivity to System Details

CAVITY BASE

Cross

Section

View of

Port

CAVITY LID

Diameter (2a)

Coaxial

Cable Freq. Range : 9 to 9.75 GHz

Cavity Height : h= 7.87mm

Statistics drawn from 100,125 pts.

Rad

RadCavity

Rad

Cavity

R

XXi

R

Rz

+

RAW Impedance PDF NORMALIZED Impedance PDF

Metallic Perturbations

Port 1

xiRiXZ RadRadCavity +

29

Data and Theory smoothed

with the same 125-cm

(240 MHz window)

low-pass filter

Nonuniversal Properties Captured by the Extended RCMEmpty Cavity Data

Theory includes all orbits to 200 cm length

6.0 6.5 7.0 7.5 8.0

0

20

40

60

80R

esis

tan

ce (

W)

Frequency (GHz)

Re[Z11

]

Re[Z(L)

11]

Re[ZR,11

]

6.0 6.5 7.0 7.5 8.0

-40

-20

0

20

40

Rea

ctan

ce (

W)

Frequency (GHz)

Im[Z11

]

Im[Z(L)

11]

Im[ZR,11

]

6.0 6.5 7.0 7.5 8.0

-40

-20

0

20

40

-40

-20

0

20

40

R

eact

an

ce (

W)

Smoothed Im[Z11

]

Smoothed Im[Z(L)

11]

Im[ZR,11

]

Res

ista

nce

(W

)

Frequency (GHz)

Smoothed Re[Z11

]

Smoothed Re[Z(L)

11]

Re[ZR,11

]

J.-H. Yeh, et al., Phys. Rev. E 81, 025201(R) (2010); J.-H. Yeh, et al., Phys. Rev. E 82, 041114 (2010).

43 cm

Random Matrix Theory

describes the fluctuations

away from Z(L)

1 2

a ~ 1

30

The Random Coupling Model Applied to 3-Dimensional Enclosures

Uncovering Universal

Impedance Statistics

Conductor

(Copper tube) at port 2

Monopole antenna at

port 1

tMeasuremen RadZ Induced Voltage Statistics

Z. B. Drikas, et al., IEEE Trans. Electromag. Compat. 56, 1480 (2014)

US Naval Research Laboratory collaboration

1. Inject microwaves at port 1 and measure induced voltage at port 2

2. Rotate mode-stirrer and repeat

3. Plot the PDF of the induced voltage and compare with RCM prediction

31

RCM Predictions for Electrically-Large AperturesG. Gradoni, et al.

Enclosure

Incident Wave

( ) 2/12/1Im

radradradcav GiGYiY x+RCM for

admittance

Port 2

Port 1

Aperture Modes

US Naval Research Laboratory collaboration

32

Induced Voltage Statistics for Enclosures with Mixed

Regular and Chaotic Behavior

Chaotic EigenmodeRegular Eigenmode

Im(Z12)

Old RCM

New RCM

Numerical (Monte Carlo)

Relevant variables:

• 3D enclosures with parallel walls

• Illuminate through regular and irregular apertures

….Ming Jer Lee, et al., Phys. Rev. E 87, 062906 (2013)

Random Coupling Model still works!

𝒁𝒊𝒋 = 𝒁𝒊𝒋,𝑹𝒆𝒈𝒖𝒍𝒂𝒓 + 𝒁𝒊𝒋,𝑪𝒉𝒂𝒐𝒕𝒊𝒄

33

Outline





• Related Work

• Conclusions

34

Conclusions

Many thanks to: P. Brouwer, M. Fink, S. Fishman,

Y. Fyodorov, T. Guhr, U. Kuhl, P. Mello, R. Prange, A. Richter,

D. Savin, F. Schafer, T. Seligman, L. Sirko, H.-J. Stöckmann,

J.-P. Parmantier

The Random Coupling Model constitutes a comprehensive (statistical) description of the

wave properties of wave-chaotic systems in the short wavelength limit

We believe the RCM is of value to the EMC / EMI community for predicting the statistics

of induced voltages on objects in complex enclosures, for example.

This description should apply to any wave system in the

‘mesoscopic’, ‘mid-frequency’, … limit

Acoustics

Mechanical vibrations

Quantum mechanical

Electromagnetic

…

RCM Review articles:

G. Gradoni, et al., Wave Motion 51, 606 (2014)

Z. Drikas, et al., IEEE Trans. EMC 56, 1480 (2014)

Research funded by ONR,

ONR/DURIP, and AFOSR

http://index.html/

35

Students and Post-docs

• Present Students Wave Chaos

• Recent Post-docs

Gabriele Gradoni,

U. Nottingham, UK

Trystan Koch Bisrat AddissieBo Xiao

Faranstul Adnan Ke Ma Ziyuan Fu

Min Zhou Alan Liu

random coupling model: statistical predictions of

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