Statistical Analysis of Cavity Quality Factor Due to Localized Losses With the Stochastic Green's Function Method

The statistical characterization of the cavity quality factor (<inline-formula><tex-math notation="LaTeX">$Q$</tex-math></inline-formula>-factor) holds significant practical importance, especially in the context of large and complex metallic enclosures. While there are existing methods for analyzing <inline-formula><tex-math notation="LaTeX">$Q$</tex-math></inline-formula>-factor statistics attributed to distributed and uniform losses, there is a noticeable gap in addressing the statistical cavity <inline-formula><tex-math notation="LaTeX">$Q$</tex-math></inline-formula>-factor arising from localized losses, such as aperture leakage, wall coating, and absorptive loading. This article introduces a physics-oriented, hybrid deterministic-stochastic approach to predict the statistical distribution of cavity <inline-formula><tex-math notation="LaTeX">$Q$</tex-math></inline-formula>-factor due to localized losses. The key ingredient of this method is the stochastic Green's function integral equation formulation, grounded in a statistical description of the cavity eigenmodes within an enclosed electromagnetic environment. The computational model is evaluated through both numerical and laboratory experiments, validating its reliability and applicability in real-world scenarios.

Abstract-The statistical characterization of the cavity quality factor (Q-factor) holds significant practical importance, especially in the context of large and complex metallic enclosures.While there are existing methods for analyzing Q-factor statistics attributed to distributed and uniform losses, there is a noticeable gap in addressing the statistical cavity Q-factor arising from localized losses, such as aperture leakage, wall coating, and absorptive loading.This article introduces a physics-oriented, hybrid deterministic-stochastic approach to predict the statistical distribution of cavity Q-factor due to localized losses.The key ingredient of this method is the stochastic Green's function integral equation formulation, grounded in a statistical description of the cavity eigenmodes within an enclosed electromagnetic environment.The computational model is evaluated through both numerical and laboratory experiments, validating its reliability and applicability in real-world scenarios.
Historically in the study of cavity resonators, the modal Qfactor was introduced as a measure to account for dielectric and wall losses inside the cavity for a single eigenmode.This parameter, defined by the ratio of stored energy to dissipated power, multiplied by the modal frequency [17], [18], laid the groundwork for understanding power and energy behavior within these resonators.However, the modal Q-factor becomes less effective in analyzing the high-frequency reverberation, as the complex boundary of the enclosure can lead to high modal density and high modal overlap.As a result, the literature introduced the composite, effective Q-factor as an average value (frequencyaveraged or stir-averaged) of the cavity Q-factor [19], [20].It is noted that the composite Q-factor typically varies smoothly with operating frequency.It does not characterize the quasi-random fluctuations between nearby frequencies or different cavity configurations.This limitation underscores the need for characterizing the cavity Q-factor in terms of a statistical measure, such as the probability density function (PDF).
Broadly speaking, the losses inside a realistic cavity can be characterized into two categories.One refers to homogeneous, distributed losses throughout the cavity surface and volume, e.g., wall loss due to the power dissipated in the metal walls and material loss from power absorbed in the dielectric materials uniformly filled in the cavity.For those distributed losses, it is predicted that the PDF of Q-factor exhibits a Fisher-Snedecor F -distribution [21], [22], [23], assuming statistically homogeneous and isotropic cavity fields.The theoretical prediction is verified experimentally in the MSRC setting.Another noticeable work utilizing the volumetric number of independent field samples and the central limit theorem is presented in [24], [25], and [26], which concludes that the Q-factor PDF is basically normal distribution in the overmoded regime of well-stirred reverberation chambers.
While existing research has primarily focused on the Q-factor statistics for cavities with homogeneous, distributed losses, there has been little discussion of the statistical cavity Q-factor due to localized losses (e.g., aperture leakage, absorptive loading).Unlike the case of homogeneous losses, localized losses depend on the specific lossy components involved.Taking the aperture leakage as an example, the power loss through the aperture depends on the aperture shape, surface area, and operating frequency of interest.Therefore, it becomes necessary to integrate these component-specific attributes into the statistical Q-factor prediction.
The contribution of this work is a hybrid deterministicstochastic model that is built upon a recently developed stochastic Green's function (SGF) method [27], [28], [29].The SGF can be considered as a physics-oriented, statistical surrogate model for the vector wave equation in large, complex enclosures.It provides a computationally efficient approximation of enclosed EM environments based on a statistical representation of the cavity eigenfunctions and eigenvalues.By leveraging the SGF method as a cavity surrogate model, we integrate the component-specific features of the localized losses through an SGF surface integral equation (IE) formulation.The statistical predictions of aperture leakage, absorptive loading, and wall coating are validated by numerical simulations and experimental results.To the best of authors' knowledge, the work represents an original contribution and the first result to analyze the statistical Q-factor for localized losses in the literature.
We remark that the initial study of this work has been reported in a conference paper [30].This article significantly extends and elaborates the approaches and numerical experiments presented in [30].

A. Background and Problem Statement
The Q-factor of the enclosure is a key quality in calculating the statistical properties of internal EM fields in an electrically large cavity.Generally, the Q-factor Q is defined as where ω is the angular frequency, U s is the steady-state energy stored in the cavity, and P d is the dissipated power.It is noted that the macroscopic definition of Q-factor as in [21], [24], and [23] is considered here, instead of modal Q-factors for individual cavity eigenmodes.We are interested in studying the fluctuation of the Q-factor for an ensemble of statistically similar cavity environments, i.e., varying stir states in an MSRC, or changing the frequency within the coherence bandwidth.Broadly speaking, the statistical fluctuation of the Q-factor depends on the nature of the loss mechanisms contributing to the dissipated power P d .If the loss is distributed throughout the enclosure, the fluctuations in Q-factor are small.This was an assertion that was tested early in the development of what is known as "wave chaos" [31].However, if the loss is localized in the cavity, the Q-factor can fluctuate significantly, which is the primary interest of this article.
Consider an electrically large, complex cavity with both uniform distributed losses and localized losses, the Q-factor can be In the following, we will analyze the statistical Q-factor, denoted as Q loc , attributed to three distinct localized loss mechanisms: absorptive loading (discussed in this section), aperture leakage, and wall coating (covered in Section III).
Taking the case of dielectric loading as an example, the design of the numerical experiment is illustrated in Fig. 1.Here, a small electric dipole is placed inside the cavity as the transmitter (Tx).This Tx dipole is responsible for the cavity excitation.The other sets of receiving (Rx) electric dipoles are utilized as the probe devices to measure the received vectorial electric fields (E-fields), denoted as E x , E y , and E z .
Next, we introduce an artificial surface S d over the boundary of the dielectric object to facilitate the calculation of surface electric and magnetic currents.By leveraging the data collected from the internal E-fields and surface currents, we can obtain the stored energy U s within the cavity and the dissipated power P d,loc in the dielectric loading.The ratio between ωU s and P d,loc gives rise to the Q loc , according to (2).
To acquire a statistical ensemble of cavity Q-factor Q loc , one may rotate the mode stirrer in numerical experiment and collect the U s and P d,loc for each stirrer state.It is clear that when first-principles numerical modeling is used, we need many full-wave simulations to obtain a statistical ensemble of U s and P d,loc .The computation complexity is prohibitively expensive for electrically large cavities.In this work, we address this challenge by harnessing the capabilities of a recently developed statistical wave model known as the stochastic dyadic Green's function (S-DGF).This model serves as an efficient and effective statistical solution to the vector wave equation within large and complex enclosures.The utilization of S-DGF significantly enhances the computational efficiency of our numerical analysis.

B. Introduction of S-DGF
The S-DGF is based on a statistical description of the eigenmodes of an enclosed EM environment.Let us consider a 3-D metallic cavity of volume V featuring distributed losses.The cavity wall is modeled as a perfect electric conductor (PEC) boundary condition.Within the cavity, the volume is filled with a homogeneous medium characterized by permittivity 0 , permeability μ 0 , and conductivity σ.It is noted that the medium conductivity σ is introduced to account for the homogeneous, distributed losses occurring within the cavity.
Consider the second-order vector wave equation where k2 is the square of complex wave number, defined by in which k = ω √ μ 0 0 is the free-space wavenumber, and the cavity Q-factor Q uni = ω 0 /σ.The electric dyadic Green's function satisfies where I is the unit dyad (idemfactor) and superscript ee identifies the contribution from the electric current source to the E-field.The stochastic DGF in the dyadic form can be constructed through the expansion of eigenfunctions [29, Appendix A] where ⊗ indicates an outer product between two vectors.Ψ e i and k i are the ith electric eigenvector and eigenvalue of the cavity.In prior work, we proposed the substitution of approximate, statistically defined eigenvectors and eigenvalues.Specifically, the statistics of eigenvectors are derived from Berry's random wave model (RWM) [32], taking into account different orientations of polarization.The statistics for eigenvalues are generated using Wigner's random matrix theory (RMT) [33].
After incorporating the RWM and RMT approximation, the expression of S-DGF for a pair of source and field points is given by [29] where G ee 0 (r, r ) is the electric dyadic Green's function of free-space.D ee S (r, r ; k m ) is a dyadic product of correlated Gaussian random variables.We have shown that the mean value of D ee S (r, r ; k m ) is related to the homogeneous Green's dyadic as The eigenvalues λ are calculated from the Gaussian orthogonal ensemble (GOE) of random matrices, and α is a macroscopic dimensionless loss-parameter defined by

C. Extension of S-DGF
In order to conduct surface IE-based modeling of the dielectric object, it is imperative to extend the S-DGF of electric type to general forms of Green's dyadic.These dyadics characterize the contributions from the magnetic current source to the magnetic field (G mm S ), from the electric current source to the magnetic field (G me S ), and from the magnetic current source to the E-field (G em S ).Consider the second-order vector wave equation The magnetic dyadic Green's function satisfies It is easy to show that a similar eigenfunction expansion exists for G mm , denoted by where Ψ m i is the ith magnetic eigenvector of the cavity.In a cavity resonator, these magnetic eigenvectors can be constructed from the electric eigenvectors by the relation We recall that after incorporating the RWM, the vector components of Ψ e i are expressed as a superposition of many plane waves with uniformly distributed orientation and polarization.In the Cartesian coordinate system, we have where the vector components are Ψ e,y i (r) lim Ψ e,z i (r) lim The polarization angle ψ n , direction ên and phase β n are independent, uniform random variables.The amplitude a n satisfies a m a n = 2 (NV ) δ mn , in which V is the volume of the cavity.Correspondingly, the vector components of magnetic eigenvector Ψ m i can be approximated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Ψ m,y i (r) lim By leveraging the central limit theorem, we note that all three vector components Ψ m,x i , Ψ m,y i , and Ψ m,z i are zero mean Gaussian random variables.The statistics of the ) can be elucidated through the outer product of correlated Gaussian random variables, denoted as D mm S (r, r ; k i ).This is achieved using the discrete Karhunen-Loeve expansion, a method well-documented in the literature [35], [36].Specifically, considering the source point r and the observation point r are located along z-axis, as illustrated in Fig. 2, the covariance function between vectorial components can be derived and the proofs for these derivations are provided in Appendix A where the distance R = |r − r |, and f ⊥ (k i R) and f // (k i R) represent transversal and longitudinal correlation described by The results are essentially the same as the case of electric eigenvectors in [29].The mean value of the dyadic expression in the numerator of ( 11) is where G mm 0 (r, r ) is the magnetic dyadic Green's function in free space, defined by By following the derivation in Section II.B [29], we can derive the magnetic S-DGF as Next, we will derive Green's dyadic, denoted as G me S , representing the contribution from the electric current source to the magnetic field.It satisfies To reveal the statistical property of G me S , we first derive the covariance function between vectorial components of Ψ m i and Ψ e i .As is shown in Appendix B, the results are obtained as We recall the expression of the free-space dyadic Green's function G me 0 (r, r ), given by The explicit dyadic form of (32) can be written as Regarding the two points given in Fig. 2, we have the real and imaginary parts By comparing ( 28)-( 31) with (35), we have the mean value of the dyadic expression in ( 27) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The mean of stochastic Green's dyadic G me S (r, r ) can be obtained as follows: In the abovementioned derivation, we have utilized the expression of mean space between two adjacent squared eigenvalues The Green's dyadic G em S (r, r ) has the same expression as G me S (r, r ) based on the duality principle.

D. Analysis of Electric Dipoles Inside Enclosures
The electric S-DGF can be directly utilized to analyze electric dipole problems shown in Fig. 1.Specifically, we consider the Tx dipole characterized by a thin, short conducting wire of length l placed inside a 3-D metallic enclosure.In this scenario, both the electrical length and the radius of the wire are significantly smaller than the wavelength, allowing us to apply the thin-wire approximation.We assume that the electric current on the dipole follows a vector triangle function, denoted as v t (r ), with the peak current at the center represented by J t .The radiated E-field can be expressed using the electric field integral equation (EFIE) The Tx dipole acts as the source to generate the stochastic EM field within the cavity.We may introduce another small electric dipole as an Rx probe device.Utilizing the Galerkin method and vector triangle basis and testing functions, the EFIE can be formulated as where J r is the electric current at the center of the Rx dipole.The self-term and couplingterm impedances are determined by the following expressions: Substituting ( 7) into ( 40) and ( 41), we can obtain the explicit form of Z S and C S as (see Appendix C for detailed derivation) where R r 0 and X r 0 are the real and imaginary parts of the dipole impedance, Z r 0 = jX r 0 + R r 0 , using free-space dyadic Green's function as the kernel.w r m and w t m represent independent, zero mean, unit variance Gaussian random variables.
In order to measure received vectorial E-fields, we can place three independent Rx dipoles along with the x-, ŷ-, and ẑ-axis, respectively.The resulting matrix equation is given by ⎡ Equation ( 42) can be straightforwardly extended to include many Rx dipoles inside the enclosure.

E. Analysis of Cavity Dielectric Loading
Considering a homogeneous dielectric object illuminated by an electric dipole inside the enclosure, as shown in Fig. 3.The boundary of the dielectric object is denoted as S d .The boundary value problem can be decomposed into the interior subproblem, i.e., the solution of the wave equation within the dielectric object, and the exterior subproblem with the wave solution exterior to the dielectric object.The entire domain solution is obtained by the enforcement of electric and magnetic field continuities across the bounding surface of the dielectric object.
In this work, we employ a surface IE method known as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [37] to analyze the dielectric object.The homogeneous (DGF) is utilized for the interior dielectric subregion, while the stochastic DGF is employed for the cavity subregion outside of the dielectric object.
Regarding the interior dielectric subregion, we have EFIE and magnetic field integral equation (MFIE) using Green's function in the homogeneous dielectric medium Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
For the cavity subregion, we have EFIE and MFIE using the SGF To derive the discrete matrix equation, the dielectric surface S d is discretized using a collection of triangular meshes denoted by K h .Both trial and test functions employ the surface divconforming vector Rao-Wilton-Glisson (RWG) functions [38], v h , which are defined over K h .
By applying the interface conditions, i.e., tangential continuities of EM fields across the dielectric boundary, and Galerkin testing method [39], the surface IE in ( 43), ( 44), (45), and ( 46) can be represented in a matrix equation as The impedance submatrix blocks labeled with subscript S utilize the SGF as the integral kernel, while those marked with H utilize the homogeneous Green's function of the dielectric medium.The J d and M d represent the solution vectors for surface electric and magnetic currents on the dielectric surface S d .The right-hand side characterizes the cavity electric and magnetic fields radiated from the electric dipole.The explicit expression of ( 47) is provided in Appendix D.

F. Calculation of Statistical Cavity Q-factor
We are now ready to analyze the statistical cavity Q-factor for the problem statement illustrated in Fig. 1.The wave physics involving the Tx dipole, Rx dipoles, and the dielectric object is modeled by an IE matrix equation of the following compact form: In (48), J x/y/z r are the solution vectors of electric current for a finite number N sets of x-, ŷ-, and ẑdirected Rx dipoles.In addition to the matrix blocks described in previous sections, we have introduced matrices C e/h,x/y/z S accounting for the coupling between the dielectric surface currents and Rx dipole sets.It is noted that there are no coupling terms between Rx dipoles, as they are used as probing devices.
We remark that the matrix equation ( 48) only involves the wire currents on the Tx/Rx dipoles and the surface currents on the dielectric surface.The coupling between these currents and their interaction with the cavity interior are statistically characterized through the S-DGF model.Therefore, this is no need for the discretization of the cavity enclosure and mode stirrer.Compared to the original problem statement in Fig. 1, the reduced-order model is illustrated in Fig. 4.
After obtaining the solution of (48), the dissipated power P d,abl is then calculated by integrating the Poynting vector over the surface of the dielectric object, that is where M d i and J d i represent the surface electric and magnetic currents on each triangle.n i is the corresponding normal direction.
Given the use of linear triangular basis functions for the Rx dipole currents, retrieving the vectorial E-field from the electric current solution is straightforward.For example, the x-directed E-fields at Rx probe locations can be obtained by Subsequently, the cavity energy density can be approximated by The energy stored in the cavity is calculated as Finally, the cavity Q-factor due to the absorptive dielectric loading is obtained by using (2) By repeating the abovementioned procedure with different S-DGF IE matrices, we can predict the statistics of cavity Q-factor due to the dielectric loading.Based on the equivalence principle, an equivalent interior problem can be derived by considering the aperture covered by a PEC [40].At the aperture surface, the magnetic field relation is expressed as Here, H − t represents the tangential components of the total magnetic field.The H Tx t denotes the tangential components of the magnetic field radiated by the Tx dipole.The H R,− t represents the tangential components of the radiated magnetic field by the surface magnetic current where the η = μ is the wave impedance, π τ (•) is a tangential trace operator defined by: π τ (u Another equivalent problem can be established for the exterior region by filling the interior cavity region with a PEC.The aperture surface IE for the exterior region in terms of the magnetic field is given by Finally, by applying the boundary condition at the aperture, i.e., the continuity of magnetic fields for exterior and interior regions, To derive the discrete matrix equation, the aperture surface S a is discretized using triangular elements denoted by K h .Both trial and test functions make use of the RWG functions v h defined over K h .Through the application of interface condition and Galerkin testing method [39], the SIEs in (54), (55), and (56) can be expressed in a matrix equation as Here, M a is the solution vector for the magnetic current at the aperture.Y a 0 is the aperture admittance matrix for the exterior region in terms of free-space dyadic Green's function, which is calculated by The matrix Y a 0 consists of real and imaginary parts, expressed as Y a 0 = R a 0 + jX a 0 .Following the steps outlined in Appendix F [29], we can construct the aperture admittance matrix Y a S for the interior cavity region The coupling matrix between the Tx dipole and aperture currents is given as The solution of the aperture magnetic current can then be used to analyze the power leakage from the aperture.

2) Calculation of Statistical Cavity Q-factor:
To analyze the statistical cavity Q-factor for aperture leakage, the design of the numerical experiment is illustrated in Fig. 6.By leveraging the proposed statistical surrogate model using S-DGF, the wave physics associated with the Tx dipole, Rx dipoles, and the aperture is modeled by an IE matrix equation of the following compact form: After obtaining the solution of (61), the power leakage through the aperture can be calculated as Following the similar procedure given in Section II-F, the probabilistic Q-factor (Q lkg ) caused by aperture leakage can be generated.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. Extension to Wall Coating
We extend our methodology to predict the statistical properties of the Q-factor, denoted as Q wcl , attributed to the presence of a localized wall coating.In Fig. 7, we observe the application of a lossy, thin coating on the cavity wall.This coating consists of a single layer of dense dielectric material with a high dielectric constant.The thickness of the coating layer is denoted as τ .
In the subsequent analysis, particularly when dealing with a dielectric material characterized by a high relative permittivity, we utilize the equivalent surface impedance of the coating, denoted as Z c .To provide a visual representation of this concept, refer to the transmission line analogy depicted in Fig. 8.For the details of the impedance calculation, readers are directed to [41,Table I].
In comparison to the aperture leakage problem elaborated in the previous section, the wall coating problem exhibits a similar reduced-order model and the corresponding surface IE matrix equation.The distinction arises from the admittance matrix governing the relationship between the electric current J and the magnetic current M on the coating surface.By using the equivalent impedance Z c , we have the following impedance boundary condition at the coating surface Through the Galerkin testing method, the admittance matrix for the coating region can be described as Here, G m is a Gram matrix, with matrix entries determined by the inner product of the testing and basis RWG functions.We can then replace the matrix Y a 0 by Y c 0 in (61) to obtain the surface IE matrix equation for the system model.Following the same procedure described in the previous section, we can generate samples of cavity Q-factor, Q wcl , due to the wall coating.

A. Impact of the Choice of α
Initially, it may seem counterintuitive that the construction of the S-DGF in ( 7) also requires an input parameter known as the cavity loss-parameter α.It is crucial to note, however, that the choice of α does not affect the statistical distribution of the calculated cavity Q-factor due to localized losses.This can be explained as follows.The parameter α is related to the Q-factor, Q uni , which is related to the uniform loss of the cavity.In this article, the stochastic Q-factor of the localized losses is directly calculated by Q loc (ω) = ωU s /P d,loc .As we are solving a coupled stochastic system of equations involving the Tx dipole, Rx dipoles, and lossy components, different α values give rise to different cavity E-field solutions.But both the stored energy (numerator) and dissipated power (denominator) in ( 2) are proportional to the E-field square.Thereby, the calculated statistical cavity Q-factor due to localized losses remains unaffected by the choice of α.
The methodology can be conceptually compared to the absorbing cross-section (ACS) measurement of dielectric loading in reverberation chambers.For each mode stirrer configuration, the cavity stored energy is expressed as 2 is the spatial averaged squared magnitude of the cavity E-field, also known as the chamber constant.The dissipated power due to dielectric loss is denoted by in which σ a is the ACS of the dielectric loading.As both U and P d are contingent on the chamber constant, the ACS value calculated by the ratio of dissipated power to stored energy is related to the dielectric object itself, independent of the specific reverberation chambers under consideration.

B. Applicability of the Proposed Work
The research results are particularly relevant in the context of electrically large cavities exhibiting high modal density and modal overlap.In such scenarios, where a macroscopic cavity Q-factor is more relevant than the modal Q-factor, the proposed methodology is shown to be useful and valuable.It is essential to emphasize that, owing to the stochastic nature of the S-DGF, the results do not provide precise Q-factor values for a specific, welldefined cavity geometry or locations of the lossy components.Instead, they provide predictions for the probability distribution of the cavity Q-factor across an ensemble of statistically similar cavity environments.

A. Verification of Cavity Q-Factor for Absorptive Loading 1) Problem Description:
The first numerical study concerns the cavity Q-factor resulting from power dissipation under lossy objects within the enclosure.The goal is to validate the predicted statistics of Q-factor with full-wave numerical simulations.2) Numerical Approaches: We have considered the following three numerical approaches to characterize the cavity Qfactor.In the first approach, we conduct full-wave simulations using commercial software.The mode stirrer is rotated through 12 positions over 360 • for a frequency range of 1.99 to 2.01 GHz with 2 MHz frequency stepping.At each stirrer position and frequency, we uniformly collect E-fields within the cavity to compute the stored energy stored U s , and the S-parameter S 11 of the radiating dipole.The dissipated power is then approximated by the radiated power as P d ≈ P t = 1 − |S 11 | 2 .The total number of cavity Q-factor samples is 132, obtained using (1).
As an additional comparison, we explore the well-known random plane wave (RPW) [42] approach, which has been applied to determine the absorption cross section of lossy materials under a random field excitation [43].The mean-value ACS, σ a , is obtained by averaging M = 5000 RPW simulations.Each full-wave simulation uses N p = 100 RPWs as excitation and is solved by the PMCHWT IE matrix equation.Consequently, a deterministic Q-factor, denoted as Qabl , is obtained using the formula The third approach utilizes the proposed work for the statistical prediction of cavity Q-factor for absorptive loading.We employ electrically small electric dipoles of length l = 5 mm as both the cavity excitation and E-field probes.These dipoles are modeled by the EFIE (42).The dielectric loading is modeled by the PMCHWT (47), incorporating both the homogeneous DGF and stochastic DGF as integral kernels.The overall system of matrix equation is provided in (48).To verify the assertion  that the choice of α in the S-DGF does not affect the PDF of calculated cavity Q-factor, we have considered three different choices, α = 0.1, 1, and 10, corresponding to a cavity factor Q uni = 37311.11,3731.11,373.11 for homogeneous, uniform loss.A statistical ensemble of S-DGF IE matrices in (48) was generated using 5000 GOE random matrices with dimensions of 2000 × 2000.This results in a total number of 5000 Q abl samples.
3) Comparison: The PDF of cavity Q-factor obtained from full-wave simulation and the proposed work are plotted in Fig. 10.The mean value of Q abl predicted by the RPW approach is also included.As is seen in Fig. 10, the results show generally good agreement between the first-principles simulation and the statistical prediction.Next, we replace the dielectric cube with a dielectric sphere loading of diameter 0.1 m.The same constitutive parameters are used.The comparison of full-wave simulation, RPW approach, and the proposed work is shown in Fig. 11.The results again agree well.
Table I presents quantitative measures for comparing the mean values and probability distributions obtained through both full-wave simulation and the proposed methodology.The PDF obtained from the full-wave simulation serves as the reference distribution for comparison.Statistical distance is evaluated using the Kullback-Leibler Divergence metric [44].The results indicate a generally good agreement between the results of the first-principles simulation and the statistical predictions.The Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.small discrepancy observed may be attributed to the limited size of our cavity ensemble (12 stirrer configurations) in the full-wave simulation.
Both the stochastic and full-wave simulations were conducted on a workstation with Intel Xeon silver 4114 CPU and 208 GB of memory.Taking the dielectric cube case as an example, the construction of S-DGF IE matrices took 18.4 min, and the solution of the IE matrices was obtained in 12 min.Regarding the full-wave simulations of 12 cavity configurations, each simulation took 7 h and 31 min on average.Thereby a total time of 5412 min is required to obtain the results of the configuration ensemble.

B. Verification of Cavity Q-Factor for Wall Coating
The next numerical experiment is to evaluate the statistics of Q-factor due to the power dissipated in the wall coating.Specifically, we consider an Eccosorb LS-24 absorber with a thickness of τ = 5 mm serving as a wall coating.This thin absorber is applied to a confined area on the PEC wall of a 3-D closed cavity with a volume of V = 1 m 3 .The relative permittivity and permeability of the absorber are r = 13.7666− 14.9681j and μ r = 1 at 1 GHz, respectively.The composite Q-factor (Q ref ) obtained using the formulation (7.116) in [17] are considered as reference results.
In our proposed approach, we employ simple small electric dipoles with a length of l = 5 mm as the cavity excitation and E-field probes.Following the procedure outlined in Section II-I-B, we use 5k GOE random matrices with dimensions of 2000 × 2000 to generate a statistical ensemble of S-DGF IE matrices.Consequently, a total of 5k Q-factor samples are computed.The average Q-factor ( Q wcl ) is obtained by averaging these samples.The results for varying surface coating areas (A) are plotted in Fig. 12, demonstrating a good agreement between the proposed approach and the reference results.
Moreover, Fig. 13 plots the PDFs of the calculated Q-factor concerning the wall coating areas.For each of the probability distributions, we calculate the mean value μ and the standard derivation σ.It is noted that as the coating area increases, the loss mechanism tends toward the distributed case.The statistics of the cavity Q-factor closely resemble a Gaussian distribution N (μ, σ), as discussed in [24].In addition, the standard  derivation in the Q-factor decreases as the surface coating area increases.

C. Validation of Cavity Q-Factor for Aperture Leakage
Finally, we experimentally validate the proposed work for predicting the statistics of Q-factor attributed to the power loss through aperture leakage.The experimental setup is illustrated in Fig. 14.A SMART 800 MSRC is placed inside an anechoic chamber.The dimension of the MSRC is 0.784 m × 1.494 m × 1 m.Two square apertures of dimension 0.3048 m × 0.3048 m are located on the opposite walls of MSRC.We can open one aperture at a time to examine the effects of power loss, as depicted in Fig. 15(a).The Tx and receiver are implemented using two monocone antennas, as shown in Fig. 15(b).The configuration ensemble is generated by rotating a paddle wheel acting as the mechanical mode stirrer.
We first perform the S-parameter measurement with both apertures closed.The stirrer is rotated 100 positions over 360 • .At each stir state, S-parameters of Tx and Rx antennas, S 11 and S 21 , are measured and recorded from 1 to 1.25 GHz with 0.025 MHz frequency stepping.To avoid measurements at multiple locations to calculate cavity energy density, we utilize the ensemble average over the stirrer states.The frequencydependent Q-factor is obtained by [45], [46] where V is the volume of MSRC, η TX and η RX are the Tx and Rx antenna efficiencies.A total number of 10 001 Q-factor Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.samples are obtained from 1 to 1.25 GHz.We then perform the measurement with the same experimental setting while opening the aperture at the front panel of the MSRC.The resulting cavity Q-factor is denoted by Q t .Therefore, the Q-factor due to the aperture leakage Q lkg can be recovered by Finally, we repeat the experiment by opening the aperture on the back panel of the MSRC.Thereby, two sets of measurement data (denoted by front panel and back panel) for Q lkg are obtained.
As depicted in Fig. 16, quasi-random fluctuations between frequencies are observed due to the stochastic nature of the cavity field.
The objective of this study is to predict the probability distribution of Q lkg within the frequency range 1-1.25 GHz.To achieve this, we divide the range into five frequency bands, each with bandwidth 50 MHz.Our proposed procedure, detailed in Section III-A, is then applied at frequencies ranging from 1 to 1.25 GHz with increments of 0.05 GHz.At each frequency point, we employ 10 k GOE random matrices to generate an ensemble of S-DGF IE matrices in (61).Therefore, a total of 60 k Q-factor samples are computed with the proposed work.Moreover, we have used the RPW method at these frequency points and computed the mean value of the cavity Q-factor.The results are illustrated in Fig. 17.We notice that the prediction from the proposed work shows a good agreement with the measurement results.The research findings hold significant practical applications in the power balance method [20], [47] for characterizing the response of a complex EM system to a high-frequency external excitation, in which the cavity Q-factor is one of the most important input parameters.Given that the proposed work provides a statistical cavity Q-factor, a natural extension is to develop a statistical power balance model to analyze the stochastic power flow in complex systems.Moreover, leveraging the calculated cavity Q-factor, the applications extend to the computation of the energy decay factor, the analysis of the power delay profile, and the control of the coherence bandwidth in the wireless OTA testing in loaded reverberation chambers [48].

A. Covariance of Eigenvectors in G mm S
As illustrated in Fig. 2, the covariance function of transverse vector components can be calculated by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Similarly, we could also derive the other two covariance functions

B. Covariance of Eigenvectors in G
me S Substituting ( 16) and ( 14) into C me xy (R), we can obtain Similarly, we could also have and all the other elements in the covariance matrix are zero.

C. Finite-Dimensional Discretization of SIE Matrices for Electric Dipole Case
Let us start with the general expression of the impedance matrix Z S described as Obviously, Z S = Z r,r S and C S = −Z r,t S will satisfy ( 40) and ( 41), respectively.Substituting ( 7) into (68), we have For the second part, we first consider the outer product in the numerator Based on the property of RWM approximation, it is easy to show that both Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In the abovementioned derivation, we have used the result of ( 8) and the fact of k m ≈ k.Then, the correlated Gaussian random variables associated with individual triangle functions can be constructed using the discrete Karhunen-Loeve expansion Therefore, the impedance matrix Z S is written as

D. Finite-Dimensional Discretization of SIE Matrices for Dielectric Loading Case
In (47), we first have the impedance matrices for the dielectric medium His research interests include computational electromagnetics and stochastic methods for wavechaotic problems.
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Fig. 1 .
Fig. 1.Illustration of the design of the numerical experiment for dielectric loading.
the Sokhotski-Plemelj theorem for the real line, and the Kramer-Kronig relation between the real part and imaginary part of G me 0 (r, r ).Finally, the Green's dyadic G me S (r, r ) can be constructed as

Fig. 4 .
Fig. 4. Illustration of the reduced-order model for the dielectric loading problem.

Fig. 6 .
Fig. 6.Illustration of the design of the numerical experiment for the aperture leakage problem.

Fig. 7 .Fig. 8 .
Fig. 7. Illustration of the design of the numerical experiment for Q wcl .

Fig. 10 .
Fig. 10.PDF of the cavity Q-factor for dielectric cube loading.

Fig. 11 .
Fig. 11.PDF of the cavity Q-factor for dielectric sphere loading.

Fig. 12 .
Fig. 12.Comparison of the cavity Q-factor for wall coating case.

Fig. 13 .
Fig. 13.PDFs of Q-factor as the wall coating area increases.

Fig. 17 .
Fig. 17.PDF of the cavity Q-factor for aperture leakage.
VI. CONCLUSION This article aims to address an open question about the statistical characterization of cavity Q-factor caused by localized losses, such as aperture leakage and absorptive loading.A novel hybrid deterministic-stochastic computing model is proposed that integrates component-specific (loading) and site-specific (aperture) features with the statistical representation of complex cavity environments.Experimental results are supplied to validate the proposed work.
) Accordingly, the impedance matrix Z S consists of two part.The first part gives us jωμ v(r) • Re G ee 0 (r, r ) • v(r )dr dr = jX 0 .
r )] follow Gaussian distribution.After multiplying the remaining coefficient jωμ kV 2π 2 in (70), the matrix entries in (71) are the results of the product of Gaussian random variables, whose covariance between trial and testing triangle function is expressed byjωμ v(r) • D ee S (r, r ; k m ) kV 2π 2 • v(r ) dr dr = jωμ v(r) • D ee S (r, r ; k m ) kV 2π 2 • v(r )dr dr ≈ jωμ v(r) • − 1 π Im G 0 (r, r ; k) • v(r ) dr dr = j π R 0 .

.
Thus, we have the explicit form of Z S and C S as

Z.
e,j H = jωμ v(r)• π τ G ee H (r, r ) •v(r ) dS dS (72) Z e,m H = v(r)• π τ G em H (r, r ) •v(r ) dS dS (73)Z h,j H = v(r)• π τ G me H (r, r ) •v(r ) dS dS (74) Z h,m H = jω v(r)• π τ G mm H (r, r ) •v(r ) dS dS.(75)For the impedance matrices using the stochastic DGF, we haveZ e,j S = jωμ v(r)• π τ G ee S (r, r ) •v(r ) dS dS (76) Z e,m S = v(r)• π τ G em S (r, r ) •v(r ) dS dS (77) Z h,j S = v(r)• π τ G me S (r, r ) •v(r ) dS dS (78) Z h,m S = jω v(r)• π τ G mm S (r, r ) •v(r ) dS dS.(79)Following the similar procedure of the derivation in Appendix C, we can derive Z e,j S = jX e,j 0 (80)-(83), we have the final expression of the impedance matrix for dielectric loading as Z wm are correlated Gaussian random variables associated with individual RWG functions, which can be constructed using the discrete Karhunen-Loeve expansion [ And w e m and w m m are zero mean, unit variance independent Gaussian random variables.Shen Lin (Member, IEEE) received the B.E. degree in optical and electronic information engineering and the M.S. degree in instrument and meter engineering from the Harbin Institute of Technology, Harbin, China, in 2012 and 2014, respectively, and the M.S. degree in optical science and engineering from the University of New Mexico, Albuquerque, NM, USA, in 2019 and the Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign (ECE ILLINOIS), Champaign, IL, USA, in 2022.He is currently a Postdoctoral Fellow with ECE ILLINOIS.His research interests are in the areas of computational and statistical electromagnetics, including stochastic electromagnetics, quantitative statistical analysis of IC and electronics in complex environments/systems, and nonparametric stochastic algorithms.His research work has an impact on electromagnetic interference and compatibility, signal and power integrity, and wireless communication.The goal is to create a strong bond between classical and quantum electrodynamic physics.Sangrui Luo (Student Member, IEEE) received the B.S. degree in applied physics from the University of Science and Technology of China, Hefei, China, in 2020.He is currently working toward the Ph.D. degree in electrical engineering with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Champaign, IL, USA.
Statistical Analysis of Cavity Quality Factor Due to Localized Losses With the Stochastic Green's Function Method Shen Lin , Member, IEEE, Sangrui Luo , Student Member, IEEE, Yang Shao, Member, IEEE, Bisrat D. Addissie , Zachary B. Drikas , Member, IEEE, Gabriele Gradoni , Member, IEEE, and Zhen Peng , Senior Member, IEEE