Statistical Properties of Wave Chaotic Scattering and Impedance Matrices Collaborators: Xing Zheng, Ed Ott ExperimentsSameer 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: apertures resonant cavities transmission lines circuit 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)

Black Box Representation N - Port System connectors cables circuit boards Integrated circuits Schematic N- Port System Incoming RF Power Device Terminal

Z and S-Matrices What is S ij ? outgoing incoming S matrix Complicated function of frequency Details depend sensitively on unknown parameters Z(  ), S(  ) N- Port System N ports voltages and currents, incoming and outgoing waves voltagecurrent Z matrix

Frequency Dependence of Reactance for a Single Realization Lossless: Z cav =jX cav Z cav S - reflection coefficient Mean spacing  f ≈.016 GHz 

Two - Port Scattering Matrix S 1 2 S Reciprocal: Lossless: Transmission Reflection = 1 - Transmission

Statistical Model of Z Matrix Port 1 Port 2 Other ports Losses Port 1 Free-space radiation Resistance R R (  ) Z R (  ) = R R (  )+jX R (  ) R R1 (  ) R R2 (  ) Statistical Model Impedance Q -quality factor   n - mean spectral spacing Radiation Resistance R Ri (  ) w in - Guassian Random variables  n - random spectrum System parameters Statistical parameters

Two Dimensional Resonators Anlage Experiments Power plane of microcircuit Only transverse magnetic (TM) propagate for f < c/2h EzEz HyHy HxHx h Box with metallic walls ports Voltage on top plate

Wave Equation for 2D Cavity E z =-V T /h Voltage at j th port: Impedance matrix Z ij (k): Scattering matrix: k=  /c Cavity fields driven by currents at ports, (assume e j  t dependence) : Profile of excitation current

Preprint available: Five Different Methods of Solution 1. Computational EM - HFSS 2. Experiment - Anlage, Hemmady 3. Random Matrix Theory - replace wave equation with a matrix with random elements No losses 4. Random Coupling Model - expand in Chaotic Eigenfunctions 5. Geometric Optics - Superposition of contributions from different ray paths Not done yet Problem, find:

Expand V T in Eigenfunctions of Closed Cavity Where: 1.  n are eigenfunctions of closed cavity 2. k n 2 are corresponding eigenvalues Z ij - Formally exact

Replace  n by Chaotic Eigenfunctions Replace eigenfunction with superposition of random plane waves (Berry Hypothesis) Random amplitude Random direction Random phase Overlap integral - GRV Q: what is

Cavity Shapes Chaotic Integrable (not chaotic) Mixed Stable Unstable

Eigenfrequency Statistics Normalized Spacing: 2. Eigenvalues k n 2 are distributed according to appropriate statistics Mean Spacing : 2D 3D Distribution

Energy or frequency Wave Chaotic Spectra Spacing distributions are characteristic for many systems TRS = Time reversal symmetry TRSB = Time reversal symmetry broken Thorium Poisson TRS TRSB Harmonic Oscillator 2+O2+O

Overlap Integral and Radiation Impedance Radiation impedance: V = Z R I CAVITY LID Coaxial Cable Outgoing waves I, V Radiation Resistance: Cavity impedance:

Statistical Model for Impedance Matrix Q -quality factor  k  n =1/(4A) - mean spectral spacing -Radiation resistance for port i System parameters w in - Guassian Random variables k n - random spectrum Statistical parameters

HFSS - Solutions Bow-Tie Cavity Moveable conducting disk -.6 cm diameter “Proverbial soda can” Curved walls guarantee all ray trajectories are chaotic Cavity impedance calculated for 100 locations of disk 4000 frequencies 6.75 GHz to 8.75 GHz Expect: Z = j(X R +R R  )  - unit Lorentzian Krieger, Ann. Phys. 42, 375 (1967) Mello, “Mesoscopic Quantum Physics” 1995 Fyodorov and Sommers, J. Math Phys. 38, 1918 (1997)

Frequency Dependence of Reactance for a Single Realization  Mean spacing  f ≈.016 GHz Z cav =jX cav

Frequency Dependence of Median Cavity Reactance Effect of strong Reflections ? Radiation Reactance HFSS with perfectly absorbing Boundary conditions Median Impedance for 100 locations of disc  f =.3 GHz, L= 100 cm 

Distribution of Fluctuating Cavity Impedance GHz GHz GHz GHz R R ≈ 35  GHz Z = j(X R +R R  )

Large Contribution from Periodic Ray Paths ? 22 cm 11 cm Possible strong reflections L = 94.8 cm,  f =.3GHz Single moving perturbation not adequate 47.4 cm Bow-Tie with diamond scar

Impedance Transformation by Lossless Two-Port Port Free-space radiation Impedance Z R (  ) = R R (  )+jX R (  ) Port Cavity Impedance: Z cav (  ) Z cav (  ) = j(X R (  )+  R R (  )) Unit Lorenzian Lossless 2-Port Lossless 2-Port Z’ R (  ) = R’ R (  )+jX’ R (  ) Z’ cav (  ) = j(X’ R (  )+  R’ R (  )) Unit Lorenzian

Properties of Lossless Two-Port Impedance Eigenvalues of Z matrix Individually  1,2 are Lorenzian distributed 11 22 Distributions same as In Random Matrix theory

HFSS Solution for Lossless 2-Port 22 11 Joint Pdf for  1 and  2 Port #1: (14, 7) Port #2: (27, 13.5) Disc

Effect of Losses Port 1 Other ports Losses Z cav = jX R +(  +j  R R Distribution of reactance fluctuations P(  ) Distribution of resistance fluctuations P(  )

Predicted and Computed Impedance Histograms Z cav = jX R +(  +j  R R

Equivalence of Losses and Channels Z cav = jX R +(  +j  R R Distribution of reactance fluctuations P(  ) Distribution of resistance fluctuations P(  )    

EXPERIMENTAL SETUP Sameer Hemmady, Steve Anlage CSR Eigen mode Image at 12.57GHz 8.5” 17” 21.4” 0.310” DEEP Antenna Entry Point SCANNED PERTURBATION Circular ArcR=25” R=42”  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 0. 31” 1.05” 1.6” Perturbation

Importance of Normalization (2a) BottomPlate Top plate Diameter 2a=0.635mm 2a=1.27mm Raw Data X [ohms] Normalized  = (X-X R )/R R

  Testing the Effects of Increasing Loss High Loss Low Loss Intermediate Loss

Conclusions Random Coupling Model works Universal features - distributions of impedance fluctuations System specific features - role of radiation impedance Next step 1. Anomalies due to multipath effects - “Scars” 2. Time domain properties 3. Systems of circuits and resonators