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)

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

Random Coupling Model Replace  n by Chaotic Eigenfunctions 1. Replace eigenfunction with superposition of random plane waves Random amplitude Random direction Random phase

Chaotic Eigenfunctions Time reversal symmetry k j uniformly distributed on a circle |k j |=k n Time reversal symmetry broken is a Gaussian random variable

Cavity Shapes Integrable (not chaotic) Chaotic Mixed

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

energy 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

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

Predicted Properties of Z ij Mean and fluctuating parts: Fluctuating part: Lorenzian distribution -width radiation resistance R Ri Mean part: (no losses) Radiation reactance

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

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 fs ≈ 35 

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(  )    

Role of Scars? Eigenfunctions that do not satisfy random plane wave assumption Bow-Tie with diamond scar Scars are not treated by either random matrix or chaotic eigenfunction theory Semi-classical methods

Large Contribution from Periodic Ray Paths ? 22 cm 11 cm Possible strong reflections L = 94.8 cm,  f =.3GHz 47.4 cm