Editor’s Note: The paper on which this article is based was originally presented at the 2019 IEEE International Symposium on Electromagnetic Compatibility & Signal/Power Integrity (EMC, SI & PI), where it received recognition as the Best Symposium Paper. It is reprinted here with the gracious permission of the IEEE. Copyright 2020 IEEE.
Introduction
The directed energy (DE) high power radiofrequency (HPRF) weapons have become an increasing threat to various missioncritical and safetycritical electronic systems and facilities. The intentional electromagnetic interference (IEMI) from the HPRF sources may introduce noise or signals into circuits and electronics of the target system, thus degrade, disable, or damage the system. To rapidly assess the vulnerability and susceptibility of electronics due to the HPRF attack, it is essential to develop a deepdomain understanding of the physics of HPRFtotargettoelectronics coupling.
As recognized by many scientists and engineers in the IEMI research community, the physics of HPRFtotargettoelectronics interaction is extremely complex [1]–[4]. A generic problem statement is shown in Figure 1. Most of electronics are hosted inside protective metallic enclosures, metal castings, and computer boxes. The HPRF energy ﬁrst needs to couple into the target enclosure/casing through open apertures or seams, and then interact with the sensitive electronics.
In the highfrequency regime, the complex boundary of the enclosure can lead to high modal density and high modal overlap. Wave solutions inside these enclosures show strong ﬂuctuations that are extremely sensitive to the geometry of the enclosure, the location of internal sensors and electronics, and the operating frequency. Minor differences in the system conﬁguration can result in signiﬁcantly different EM ﬁeld distributions inside the enclosure. Research regarding HPRF effects on missiles has shown large variations between different serial numbers due to assembly methods, cable routing, and component variations [5].
Evidently, because of the increasing complexity of electronic systems and continually evolving DE HPRF waveforms, it is expensive and impractical to perform experimental tests for all possible HPRF effects. The objective of this work is to investigate physicsoriented mathematical and statistical models, which discover and replicate the fundamental physics of HPRF coupling on electronic systems. The main contributions are described as follows.
1) Wavechaotic coupling: To characterize the multipath, raychaotic propagation inside the enclosure, a stochastic Green’s function (SGF) representation formula is proposed. The SGF represents the fundamental solution of wave equations in the complex wavechaotic media (domains exhibiting raychaotic dynamics). The derivation of the scalar SGF is presented in the recent work [6]–[9], leveraging the physics of wavechaotic dynamics [10], [11] and the mathematics of random matrix theory (RMT) [12], [13]. In this paper, we propose the vector dyadic SGF and the statistical representation formula. The work rigorously characterizes both variations and correlations of EM ﬁelds inside complex enclosures.
2) Systemspeciﬁc aspects: Moreover, we recognize that the EM coupling inside complex systems may involve mixed chaotic and regular wave dynamics. The systemspeciﬁc coupling ranges from aperture/slot geometries, to sitespeciﬁc shortorbits, to the conﬁguration of electronic components. Those nonchaotic dynamical features result in deviations from universality and ergodicity in fully wave chaos. To answer this challenge, we propose a hybrid deterministic and stochastic formulation incorporating componentspeciﬁc characteristics and investigate an insitu SGF representation formula addressing the sitespeciﬁc shortorbits.
Comparing to existing literature [14]–[21], the work rigorously integrates the systemspeciﬁc coupling and the wavechaotic coupling, and seamlessly incorporating universal statistical properties and deterministic coupling characteristics within a comprehensive framework. The developed predictive models are validated and veriﬁed in various experimental settings, including complex 3D metallic cavities and modestirred reverberation chambers.
Methodology
Motivation for Stochastic Green’s Function
In the shortwavelength regime, the wave scattering process inside complex enclosures may exhibit chaotic ray dynamics [22]. EM ﬁelds show high variability and extreme sensitivity to small perturbations. Given the complexity of such environments, it is thus very desirable to have the statistical properties of EM ﬁelds within complex enclosures. Nevertheless, the standard statistical treatment of this problem depends on many parameters that are immensely complex [17].
It has been recognized that the ergodic modes in cavity environments lead to certain universal statistical properties of EM ﬁelds [23]–[25]. For instance, in overmoded reverberation chambers, the probabilistic EM ﬁelds present uniform phase distribution and Rayleigh distributed amplitude. The variance of the Rayleigh distribution is related to the quality factor of the chamber, instead of the exact shape of the enclosure or the location of the measurement. The homogeneous and structureless statistical behavior motivates the investigation of the stochastic Green’s function, which describes the generic statistical properties of wave dynamics inside the cavity rather than detailed speciﬁcs.
Review of Green’s Function in Free Space
The Green’s function stands for the fundamental solution of a partial differential equation. In the electromagnetic theory, the Green’s function describes the physics of the wave propagation from the source point rʹ to the receiving point r. In the homogeneous, freespace scenario, we have the wellknown 3D dyadic Green’s function as:
(1)
where I is the identity tensor, and k is the wave number of the operating frequency.
The imaginary part of G_{0} can be expressed as the following plane wave integral:
(2)
By applying the SokhotskiPlemelj theorem, the real part, Re[G_{0} (r, rʹ; k)] can be expressed as:
(3)
Where denotes the Cauchy principal value. Equations (2) and (3) give rise to the KramersKronig relation.
Stochastic Green’s Function in Chaotic Media
The stochastic Green’s function represents the fundamental, probabilistic solution of the wave equation in chaotic environments. Consider the 2^{nd} order vector wave equation inside a metallic cavity, the Green’s function in the dyadic form can be constructed from the eigenfunction expansion:
(4)
Where ⊗ indicates a tensor product. Ψ_{i} and k_{i} are the ith eigenfunction and eigenvalue of the cavity.
1) Random wave model: Based on the Berry’s random plane wave hypothesis [26], eigenfunctions of the wavechaotic cavity can be locally modeled as an isotropic, random superposition of plane waves. Let’s start by assuming that ﬁeld and source points, r, rʹ are close to each other and away from the cavity boundary, the eigenfunction Ψ_{i} can thereby be expressed by the ensemble:
(5)
where the direction ê_{n}, amplitude α_{n}, and phase β_{n} are independent random variables. The r_{0} is the center of the plane wave expansion. Based on the central limit theorem, the summation over a large number N plane waves results in the Gaussiandistributed eigenfunction values. Equation (4) can be rewritten as:
(6)
where the ω_{i} is zero mean, unit width Gaussian random variable, and V is the enclosed volume of the cavity.
Comparing Eqs. (6) and (3), the sum over eigenmodes in G_{S} (r, rʹ; k) approximates, on average, the integral in Eq. 3 but with a statistically ﬂuctuating contribution coming from the denominator when eigenvalues k_{n} ≈ k. We can rewrite the expression of Eq. (6) as:
(7)
2) Random matrix theory: A key observation is that the statistical variation of SGF in Eq. (7) mainly depends on the sum of eigenmodes whose eigenvalues are close to k, and the distance between r and rʹ. It provides the inspiration for introducing a universal random quantity, g_{co}, deﬁned as:
(8)
where ∆ is the meanspacing between adjacent eigenvalues. For large metallic cavities, the meanspacing is given as ∆ = 4π^{2}/kV ) [20]. Moreover, to characterize the generic losses for realistic cavities (dielectric losses, ohmic losses, etc.), we introduce a dimensionless cavity lossparameter, α = k^{2}/ (∆Q), predicted from the Weyl law [27], where Q represents the average quality factor [19].
The eigenvalues k_{n} obey locally repulsion laws, which are distributed according to the random matrix hypotheses of Wigner. The statistics of the normalized eigenvalue distributions (k^{2} – k^{2}_{n}) /∆ is predicted by RMT with the Gaussian Orthogonal Ensemble of random matrices [10], [12].
3) Universality in stochastic Green’s function: We notice that the g_{co} is denoted as a universal statistical quantity, since its statistics do not depend on the explicit geometry of the enclosure or the location of the sources and receivers within the enclosure. Rather, they are determined by a few macroscopic parameters, including the operating frequency, the volume, and the lossparameter.
It can be easily shown that the stochastic Green’s function preserves the translation invariance. The G_{S} is a function of the difference r – rʹ and the statistical variable g_{co} only. Moreover, based on the stochastic Green’s function, we can derive the spatial ﬁeldﬁeld correlation function. The result gives a universal twopoint correlation function, which agrees with Berry’s random wave model conjecture.
4) The case of large separation: In the case of r and rʹ are well separated, we can use two random plane wave expansions to approximate locally Ψ_{i} (r, k_{i}) and Ψ_{i} (rʹ, k_{i}). The resulting Green’s function G_{S} (r, rʹ; k) is derived as:
(9)
where ω_{n}, ωʹ_{n} are two independent Gaussian random variables with zero mean and unit variance, and
The statistical variation can then be denoted as a universal random quantity, g_{uc}, deﬁned as:
(10)
It is straightforward to prove that the mean value of Eq. (9) is zero. The result can be understood via reverberation chambers. The perfect mixing (wellstirred) over a number of cavity realizations leads to a zero mean complex ﬁeld, when the measurement point is far away from the source. Due to page restrictions, the discussion of intermediate separation case will be deferred to the extended paper.
Statistical Representation Formula
The stochastic Green’s function characterizes the universal statistical behavior of the wavechaotic propagation. As a direct application, one can use it to obtain the probabilistic values of induced EM ﬁelds due to a distributed current source. It gives the following statistical representation formula (SRF):
(11)
The SRF also provides the impedance relation between the surface electric ﬁeld and electric current. By making use of the vectordyadic Green’s second identity, Equation 11 can be reduced a surface integral form:
(12)
which maps the tangential component of magnetic ﬁeld to the tangential component of electric ﬁeld.
Consider a source and a receiver (denoted as components) located within a complex metallic enclosure and far from boundaries. We introduce trial and testing functions on the surfaces of the components. The discrete matrix form of SRF can be decomposed as:
(13)
where Z_{0}^{1}^{,}^{2} are impedance matrices obtained by using freespace Green’s function, corresponding to the real part of the Green’s function in Eq. (7). The A_{0}^{1}^{,}^{2} are the outgoing plane wave representations of the source currents, where D_{0}^{1}^{,}^{2} = [A_{0}^{1}^{,}^{2}]^{†} denote the incoming representations. The g^{∗∗} represent the Gaussian diffuse energy propagator, whose statistical values are obtained from Eqs. (8) and (10) with random matrix theory.
Integration with SystemSpeciﬁc Aspects
1) Componentspeciﬁc features: Evidently, the HPRF coupling effects are not only determined by the cavity environment, but also by the electronics and antennas under test. To integrate the speciﬁc knowledge (types, geometries, materials, et al.) of the electronic components, we propose a hybrid deterministic and stochastic formulation.
In the proposed framework, electronic components (antennas, devices, apertures, etc.) in the computational domain are modeled by the volumebased ﬁnite element (FE) method. The statistical representation formula is placed on the exterior surfaces of those components, emulating the multipath, raychaotic propagation environments. The resulting system matrix equation can be written as:
(14)
In Eq. (14), A^{FE} represents the FE matrix, Z_{S}^{BE} is the boundary element (BE) discretization of the surface representation formula. The matrix C denotes the coupling between FE and BE matrices. The probabilistic solutions of EM and electrical quantities can be obtained by stochastic matrix collocation or Monte Carlo approaches. We notice that the matrix equation (14) only involves the unknowns on the surface and volume of the electronic components. Therefore, this is no need to discretize the cavity enclosure.
2) Sitespeciﬁc shortorbits: One important factor that causes the deviation from chaotic universality is due to shortorbits, whose phase accumulation along the trajectories may not be large enough to be considered as random. Consider an antenna mounted on the wall of the cavity enclosure, as illustrated in Figure 2. There exists direct, shortorbit couplings between the antenna and the neighboring wall. These short ray trajectories should be considered as classical coherent rays, rather than chaotic ray trajectories which ergodically sample the enclosure. Clearly, the existence of short orbits makes nonuniversal contributions to the statistical EM ﬁeld behavior.
This challenge is addressed by studying an insitu statistical representation formula, which integrates the sitespeciﬁc, deterministic interactions with the universal statistical behavior of the cavity. In the computational model, we shall include the sitespeciﬁc features, namely, the antenna and its neighboring cavity wall, as shown in Figure 2. The ﬁrst step is to evaluate the nonisotropic plane wave representation on the boundary surface of the antenna, denoted by Ã^{PA}, where the superscript indicates that it is a mapping from antenna surface current to plane wave spectrum. Taking the shortorbit coupling into consideration, Ã^{PA} is calculated as:
(15)
where Z_{0}^{CA} is the coupling impedance matrix from antenna to the surrounding cavity wall. The Z_{0}^{CC }is the freespace impedance matrix of the cavity wall. A_{0}^{PA} and A_{0}^{PC} are the freespace plane wave representations for surface currents on antenna and cavity wall, respectively.
Subsequently, the nonisotropic plane wave representation, Ã^{PA}, is substituted in Eq. (13) to obtain the insitu statistical representation formula. In practice, the size of the insitu shortorbit region is prequantiﬁed through the calculation of the Ehrenfest time [28], [29], which is the time scale of the transition from classical to chaotic ray trajectories.
Experiments
Experimental Validation with 3D Cavity
We consider a complicated 3D aluminum cavity to validate the proposed work. The geometry and photograph of the experimental setup are illustrated in Figure 3(a)(b). Two Xband waveguide (WG) antennas are mounted on the different locations of cavity walls as transmitter (Tx) and receiver (Rx). The cavity is signiﬁcantly overmoded and EM ﬁelds exhibit wave chaotic ﬂuctuations. A paddlewheel mode stirrer is used to generate an ensemble of measurements. At each frequency, the mode stirrer is rotated 200 positions over 360 degrees.
The volume of the cavity is 0.42m^{3} and the cavity lossparameter α = 9.9516. Based on Eqs. (8) and (10), we generate the universal statistical quantities, g_{co} and g_{uc}. The probability distribution function (PDF) of the real and imaginary parts of g_{co} and g_{uc} obtained by computational predictions and measurement results are compared in Figure 4.
In the computational setup, we only need to include the sitespeciﬁc features, namely, the WG antennas and their surrounding aluminum plates. The computational domains are illustrated in Figure 5(b) and Figure 6(a), where antennas are mounted on the adjacent walls and opposite walls in the experiments, respectively. The interaction with other parts of the 3D cavity is categorized statistically with the SGF on the exterior surface of the antennas. As a result, the solution of the reduced order, stochastic model can be obtained at the same cost as an antenna radiating in free space. The PDF of the antenna Sparameters is depicted in Figure 5(c) and Figure 6(b), where excellent agreements are observed.
Experimental Validation with Reverberation Chamber
Next, we present the experimental validation with a reverberation chamber (RC) measurement. RC has been used as a standard testing facility for electromagnetic compatibility (EMC) measurements. Recent works have also demonstrated the use of RC as an efﬁcient emulator for rich fading environments and overtheair (OTA) testing of wireless devices [30]– [33]. It is particularly useful to analyze the correlation and capacity of multipleinput multipleoutput (MIMO) systems in rich multipath environments.
Shown in Figure 7 is a reference setup presented in [30] for measuring a sixelement monopole circular MIMO array. The chamber is of size 0.8m × 1.05m × 1.5m, and equipped with two mechanical plateshaped stirrers. The sixelement monopole array is located on a rotatable platform (circular metal plate with radius 0.14m) and rotated inside the chamber (platform stirring). Three ﬁxed wall antennas are used for polarization stirring. The measurements were done at 900MHz. The spacing between monopole antennas is 0.24λ.
The computational model includes three wall antennas (Tx) and sixelement monopole array on the metal plate (Rx). The spatial multipath propagation between them is characterized by the stochastic Green’s function representation formula. The outcome provides a probabilistic ensemble of Sparameter matrices describing the transmission performance. The random transfer (channel) matrix, H, from the Tx antennas to the Rx antennas can then be calculated as:
(16)
where Z_{G} and Z_{L} are Tx and Rx load impedances.
The calculated ergodic channel capacity is compared to the measurement result in Figure 8, where a very good agreement is observed. Due to the close spacing between monopole antennas, the MIMO channel receives correlated wireless signals. In turn, it affects the channel capacity comparing to ideal Rayleigh fading channel. The results verify that the proposed work leads to a competent statistical model rigorously resolves the spatial correlation, propagating coherence, and antenna mutual coupling in the rich multipath environment.
Conclusion
The proposed work aims to establish a physicsoriented, reducedorder modeling capability, which predicts the statistical EM coupling rapidly while retaining the underlying ﬁrstprinciples analysis. The statistical model only needs the knowledge of sitespeciﬁc features and deterministic coupling channels, e.g. the electronics, apertures, antennas, and coaxial cables. The interaction with other parts of the complex cavity is characterized statistically with a novel stochastic Green’s function representation formula. We can rapidly evaluate the susceptibility and/or vulnerability of the system to various HPRF waveforms and engagement angles. The advancements reduce the need for costly and timeconsuming measurements.
Authors’ Note: The work summarized in this paper was supported by U.S. NSF CAREER award, #1750839, and U.S. AFOSR/AFRL Center of Excellence on the Science of Electronics in Extreme Electromagnetic Environments, Grant FA95501510171.
References

 S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal impedance ﬂuctuations in wave chaotic systems,” Phys. Rev. Lett., vol. 94, no. 1, p. 014102, 2004.
 S. Hemmady, X. Zheng, T. M. Antonsen, E. Ott, and S. M. Anlage, “Universal statistics of the scattering coefﬁcient of chaotic microwave cavities,” Phys. Rev. E, vol. 71, no. 5, p. 056215, 2005.
 G. Gradoni, T. M. Antonsen, S. M. Anlage, and E. Ott, “A statistical model for the excitation of cavities through apertures,” IEEE Transactions on Electromagnetic Compatibility, vol. 57, pp. 1049–1061, October 2015.
 J. Fan, “A new EMC challenge: Intrasystem EMI and RF interference,” Safety & EMC, pp. 3–5, 2015.
 S. Moran, “The basics of electric weapons and pulsedpower technologies,” The Leading Edge: Directed Energy Technology, Modeling, and Assessment, vol. 7, no. 4, pp. 50–57, 2012.
 S. Lin and Z. Peng, “A novel stochastic wave model statistically replicating reverberation chambers,” in 2017 IEEE 20th Conference on Electrical Performance of Electronic Packaging and System, October 2017.
 S. Lin, Z. Peng, and T. Antonsen, “A hybrid method for quantitative statistical analysis of insitu ic and electronics in complex and wavechaotic enclosures,” in 2016 Progress in Electromagnetic Research Symposium (PIERS), pp. 2503–2510, August 2016.
 S. Lin, Z. Peng, and T. Antonsen, “Quantitative statistical analysis with physicsbased surrogate modeling for wave chaotic systems,” in 2017 IEEE International Symposium on Antennas and Propagation (APSURSI), July 2017.
 Z. Peng and S. Lin, “EMCAD for complex electronics systems: A journey from order to chaos,” in 2017 IEEE MTTS International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization for RF, Microwave, and Terahertz Applications (NEMO), pp. 61–63, May 2017.
 E. Wigner, “Random matrices in physics,” SIAM Review, vol. 9, no. 1, pp. 1–23, 1967.
 P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, and G. Vattay, Chaos: Classical and Quantum. ChaosBook.org, 2017. version 15.9.
 V. Kravtsov, “Random matrix theory: Wignerdyson statistics and beyond.(lecture notes of a course given at sissa (trieste, italy)),” arXiv preprint arXiv:0911.0639, 2009.
 T. Tao., Topics in random matrix theory. American Mathematical Society, 2012. Graduate Studies in Mathematics, vol. 132.
 D. A. Hill, “Plane wave integral representation for ﬁelds in reverberation chambers,” IEEE Transactions on Electromagnetic Compatibility, vol. 40, pp. 209–217, August 1998.
 J. Lefèbvre, J. Hagen, D. Lecointe, and W. Tabbara, “Coupling of electromagnetic energy to a scatterer inside a cavity: A combined statistical and integral representation approach,” Radio Science, vol. 35, no. 2, pp. 427–434, 2000.
 L. R. Arnaut, “Statistical distributions of dissipated power in electronic circuits immersed in a random electromagnetic ﬁeld,” Radio Science, vol. 40, pp. 1–10, December 2005.
 H. W. L. Naus, “Statistical electromagnetics: Complex cavities,” IEEE Transactions on Electromagnetic Compatibility, vol. 50, pp. 316–324, May 2008.
 A. Gifuni, “On the expression of the average power received by an antenna in a reverberation chamber,” IEEE Transactions on Electromagnetic Compatibility, vol. 50, pp. 1021–1022, November 2008.
 S. Hemmady, T. Antonsen, E. Ott, and S. Anlage, “Statistical prediction and measurement of induced voltages on components within complicated enclosures: A wavechaotic approach,” Electromagnetic Compatibility, IEEE Transactions on, vol. 54, pp. 758–771, August 2012.
 L. K. Warne and R. E. Jorgenson, “Highfrequency electromagnetic scarring in threedimensional axisymmetric convex cavities,” Electromagnetics, vol. 36, no. 3, pp. 186–213, 2016.
 M. Mehri and N. Masoumi, “Statistical prediction and quantiﬁcation of radiated susceptibility for electronic systems PCB in electromagnetic polluted environments,” IEEE Transactions on Electromagnetic Compatibility, vol. 59, pp. 498–508, April 2017.
 G. Gradoni, J.H. Yeh, B. Xiao, T. M. Antonsen, S. M. Anlage, and E. Ott, “Predicting the statistics of wave transport through chaotic cavities by the random coupling model: A review and recent progress,” Wave Motion, vol. 51, no. 4, pp. 606–621, 2014.
 J. H. Yeh, J. A. Hart, E. Bradshaw, T. Antonsen, E. Ott, and S. Anlage, “Determination of universal and nonuniversal properties of wave chaotic scattering systems,” Phys. Rev. EStatist., Nonlinear Soft Matter Phys., vol. 81, p. 025201, 2010.
 J. H. Yeh, J. A. Hart, E. Bradshaw, T. Antonsen, E. Ott, and S. Anlage, “Experimental examination of the effect of short ray trajectories in twoport wavechaotic scattering systems,” Phys. Rev. EStatist., Nonlinear Soft Matter Phys., vol. 82, p. 041114, 2010.
 J. B. Gros, O. Legrand, F. Mortessagne, E. Richalot, and K. Selemani, “Universal behavior of a wave chaos based electromagnetic reverberation chamber,” Wave Motion, vol. 51, no. 4, pp. 664–672, 2014.
 M. V. Berry, “Regular and irregular semiclassical wavefunctions,” Journal of Physics A: Mathematical and General, vol. 10, no. 12, p. 2083, 1977.
 E. Ott, Chaos in Dynamical System. Cambridge, U.K.: Cambridge University Press, 1993.
 L. Aleiner and A. I. Larkin, “Divergence of classical trajectories and weak localization,” Physical Review B, vol. 54, pp. 14423–14444, 1996.
 H. Schomerus and P. Jacquod, “Quantumtoclassical correspondence in open chaotic systems,” Journal of Physics A: Mathematical and General, vol. 38, no. 49, p. 10663, 2005.
 P. Kildal and K. Rosengren, “Correlation and capacity of mimo systems and mutual coupling, radiation efﬁciency, and diversity gain of their antennas: simulations and measurements in a reverberation chamber,” IEEE Communications Magazine, vol. 42, pp. 104–112, December 2004.
 M. A. GarciaFernandez, J. D. SanchezHeredia, A. M. MartinezGonzalez, D. A. SanchezHernandez, and J. F. ValenzuelaValdes, “Advances in modestirred reverberation chambers for wireless communication performance evaluation,” IEEE Communications Magazine, vol. 49, pp. 140–147, July 2011.
 D. Micheli, M. Barazzetta, F. Moglie, and V. M. Primiani, “Power boosting and compensation during OTA testing of a real 4G LTE base station in reverberation chamber,” IEEE Transactions on Electromagnetic Compatibility, vol. 57, pp. 623–634, August 2015.
 K.A. Remley, J. Dortmans, C. Weldon, R. D. Horansky, T. B. Meurs, C. M. Wang, D. F. Williams, C. L. Holloway, and P. F. Wilson, “Conﬁguring and verifying reverberation chambers for testing cellular wireless devices,” IEEE Transactions on Electromagnetic Compatibility, vol. 58, pp. 661–672, June 2016.
Dear sir
Experimental statical formula ,integration of systems analysis graphs ,em radiation in pcb.
Very good artical in em radiation of electronic components.