Statistical Properties of Wave Chaotic Scattering and Impedance Matrices

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Statistical Properties of Wave Chaotic Scattering
and Impedance Matrices

• MURI Faculty Tom Antonsen, Ed Ott, Steve Anlage,
  
• MURI Students Xing Zheng, Sameer Hemmady, James
  Hart

AFOSR-MURI Program Review
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Electromagnetic Coupling in Computer Circuits
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Z and S-MatricesWhat is Sij ?
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Random Coupling Model
1. Formally expand fields in modes of closed
cavity eigenvalues kn wn/c
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Statistical Model of Z MatrixFrequency Domain
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Model ValidationSummary
Single Port Case Cavity Impedance Zcav RR
z jXR Radiation Impedance ZR RR
jXR Universal normalized random impedance z
r jx Statistics of z depend only on damping
parameter k2/(QDk2) (Q-width/frequency
spacing) Validation HFSS simulations Experi
ment (Hemmady and Anlage)
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Normalized Cavity Impedance with Losses
Theory predictions for Pdfs of z rjx
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Two Dimensional Resonators
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HFSS - SolutionsBow-Tie Cavity
Cavity impedance calculated for 100 locations of
disk 4000 frequencies 6.75 GHz to 8.75 GHz
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Comparison of HFSS Results and Modelfor Pdfs of
Normalized Impedance
Normalized Reactance
Normalized Resistance
x
r
Zcav jXR(rjx) RR
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EXPERIMENTAL SETUP Sameer Hemmady, Steve Anlage
CSR
Circular Arc R42
Antenna Entry Point
0.310 DEEP
8.5
Circular Arc R25
SCANNED PERTURBATION
21.4
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• ? 2 Dimensional Quarter Bow Tie Wave Chaotic
  cavity
• Classical ray trajectories are chaotic - short
  wavelength - Quantum Chaos
• 1-port S and Z measurements in the 6 12 GHz
  range.
• Ensemble average through 100 locations of the
  perturbation

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Comparison of Experimental Results and Modelfor
Pdfs of Normalized Impedance
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Normalized Scattering AmplitudeTheory and HFSS
Simulation
Actual Cavity Impedance Zcav RR z
jXR Normalized impedance z r jx Universal
normalized scattering coefficient s (z -1)/(z
1) s exp if Statistics of s depend only
on damping parameter k2/(QDk2)
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Experimental Distribution of Normalized
Scattering Coefficient
ssexpif
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Frequency Correlations in Normalized
ImpedanceTheory and HFSS Simulations
(f1-f2)
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Properties of Lossless Two-Port Impedance(Monte
Carlo Simulation of Theory Model)
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HFSS Solution for Lossless 2-Port
Joint Pdf for q1 and q2
Disc
q2
Port 1
Port 2
q1
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Comparison of Distributed Lossand Lossless
Cavity with Ports(Monte Carlo Simulation)
Distribution of resistance fluctuations P(r)
Distribution of reactance fluctuations P(x)
r
x
Zcav jXR(rjx) RR
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Time Domain Model for Impedance Matrix
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Incident and Reflected Pulsesfor One Realization
Incident Pulse
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Decay of MomentsAveraged Over 1000 Realizations
Prompt reflection eliminated
Log Scale
Linear Scale
ltV2(t)gt1/2
ltV3(t)gt1/3
ltV(t)gt
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Quasi-Stationary Process
2-time Correlation Function (Matches initial
pulse shape)
Normalized Voltage u(t)V(t) /ltV2(t)gt1/2
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Histogram of Maximum Voltage
Vinc(t)max 1V
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Progress
Direct comparison of random coupling model with
-random matrix theory P -HFSS solutions
P -Experiment P Exploration of increasing
number of coupling ports P Study losses in
HFSS P Time Domain analysis of Pulsed
Signals -Pulse duration -Shape (chirp?)
Generalize to systems consisting of circuits and
fields
Current
Future
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Role of Scars?
Eigenfunctions that do not satisfy random
plane wave assumption
Scars are not treated by either random
matrix or chaotic eigenfunction theory
Semi-classical methods
Bow-Tie with diamond scar
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Future Directions
Can be addressed -theoretically
-numerically -experimentally
Features Ray splitting Losses
Additional complications to be added later
HFSS simulation courtesy J. Rodgers