Efficient Statistical Model for Predicting Electromagnetic Wave Distribution in Coupled Enclosures
Shukai Ma, Sendy Phang, Zachary Drikas, Bisrat Addissie, Ronald Hong, Valon Blakaj, Gabriele Gradoni, Gregor Tanner, Thomas M. Antonsen, Edward Ott, Steven M. Anlage
EEfficient Statistical Model for Predicting Electromagnetic Wave Distribution inCoupled Enclosures
Shukai Ma, ∗ Sendy Phang,
2, 3
Zachary Drikas, Bisrat Addissie, Ronald Hong, Valon Blakaj, GabrieleGradoni,
2, 3
Gregor Tanner, Thomas M. Antonsen,
5, 6
Edward Ott,
5, 6 and Steven M. Anlage
1, 6 Quantum Materials Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA School of Mathematical Sciences, University of Nottingham, UK, NG7 2RD George Green Institute for Electromagnetics Research, University of Nottingham, UK, NG7 2RD U.S. Naval Research Laboratory, Washington, DC 20375, USA Department of Physics, University of Maryland, College Park, Maryland 20742, USA Department of Electrical and Computer Engineering,University of Maryland, College Park, Maryland 20742-3285, USA
The Random Coupling Model (RCM) has been successfully applied to predicting the statistics ofcurrents and voltages at ports in complex electromagnetic (EM) enclosures operating in the shortwavelength limit [1–4]. Recent studies have extended the RCM to systems of multi-mode aperture-coupled enclosures. However, as the size (as measured in wavelengths) of a coupling aperturegrows, the coupling matrix used in the RCM increases as well, and the computation becomes morecomplex and time consuming. A simple Power Balance Model (PWB) can provide fast predictionsfor the averaged power density of waves inside electrically-large systems for a wide range of cavityand coupling scenarios. However, the important interference induced fluctuations of the wave fieldretained in the RCM are absent in PWB. Here we aim to combine the best aspects of each modelto create a hybrid treatment and study the EM fields in coupled enclosure systems. The proposedhybrid approach provides both mean and fluctuation information of the EM fields without thefull computational complexity of coupled-cavity RCM. We compare the hybrid model predictionswith experiments on linear cascades of over-moded cavities. We find good agreement over a set ofdifferent loss parameters and for different coupling strengths between cavities. The range of validityand applicability of the hybrid method are tested and discussed.
I. INTRODUCTION
The ability to characterize and predict the natureof short-wavelength Electromagnetic (EM) waves insideinter-connected enclosures is of interest to various sci-entific fields. Applications include EM compatibilitystudies for electronic components under high-power mi-crowave exposure [5, 6], coupled quantum mechanicalsystems modeled with superconducting microwave bil-liards [7], cascades of quantum dots [8–11] by way ofanalogy, and
Smart Homes sensors in furnished indoorenvironments. The enclosures in these applications aregenerally electrically large with an operating wavelength λ (cid:28) V / , where V is the volume of the system. Theinterior geometry of these enclosures is often complexincluding wall features and internal objects acting asscatterers and geometrical details may not be preciselyspecified. These systems are then well-described as ray-chaotic enclosures, where the trajectories of rays withslightly different initial conditions diverge exponentiallywith increasing number of bounces off the irregular wallsand interior objects [12–14]. This ray-chaotic propertyhas inspired research in diverse contexts such as acoustic[15–17] and microwave cavities [18–24], the spectral prop-erties of atoms [25] and nuclei [26], quantum dot systems[27].Benefiting from the continuing advance in computa-tional capabilities, deterministic approaches utilizing nu- merical techniques are widely applied in simulating EMquantities inside chaotic systems with specified geome-tries [28]. The resolution required for deterministic meth-ods, such as the Finite Difference Time Domain (FDTD)or the Finite Element Method (FEM), scale with the in-verse of the wavelength and thus consume a large amountof computational resources in the short-wavelength limit(i.e., when typical wavelengths are small compared to thelinear scale of the enclosure). In addition, minute changesof the interior structure of a given system will drasticallyalter the solution of the EM field [4, 23]. Statistical meth-ods may thus be more appropriate when studying suchsystems. Many such approaches have been proposed, ex-amples include the Baum-Liu-Tesche technique [5, 29]which analyzes a complex system by studying the travel-ing waves between its sub-volumes, and the Power Bal-ance Model (PWB) [30–33] which predicts the averagedpower flow in systems. The PWB method makes predic-tions of the steady-state averaged energy density insideall system sub-volumes based on equating the incomingand out-going power in each connected sub-volume. Aversion of the PWB method estimating also the vari-ance of the fluctuations has been presented in [34]. ThePWB method is based on the assumption of a uniformfield distribution in each cavity which is often fulfilledin the weak inter-cavity coupling and low damping limit.Extensions of the PWB method that drop the uniformfield assumption are ray tracing (RT) methods [35, 36] a r X i v : . [ phy s i c s . c l a ss - ph ] M a y or the Dynamical Energy Analysis (DEA) method [37–39] which calculate local ray or energy densities. DEAand RT capture non-uniformity in the field distributionwithin a given sub-volume. Like PWB, they do not treatfluctuations in energy density due to wave interference.The Random Coupling Model (RCM) [1–4, 21, 40] al-lows for the calculation of the statistical properties of EMfields described in terms of scattering and impedance ma-trices that relate wave amplitudes, or voltages and cur-rents at ports. The RCM is based on Random MatrixTheory (RMT) originally introduced to describe complexnuclei [26]. It was later conjectured that any system withchaotic dynamics in the classical limit will also have waveproperties whose statistics are governed by random ma-trix theory [41, 42] Here the RCM is applied to wavechaotic systems in the short wavelength regime. Incontrast to the above mentioned methods, the RCM isable to describe the full probability distribution func-tions (PDFs) for voltages and currents at ports. Anexemplary application of RCM is the simulation of thefluctuating impedance matrix based on minimal systeminformation, namely a single-cavity loss parameter α (tobe defined) and several system-specific features [2, 3, 43–47]. The system-specific features include the radiationinformation of the ports (both emitting and absorbing),as well as short orbits inside the cavity. The parameter α reflects the loss level of the system, which can be derivedfrom the overall cavity dimension, Q-factor and operat-ing frequency. The RCM has been successfully appliedto single cavities and systems of coupled cavities withvarying losses, cavity dimensions, and in the presence ofnonlinear elements [48, 49].The computational complexity of the RCM grows,however, with the addition of large apertures connectingtogether multiple enclosures. In the RCM an apertureis treated as a set of M correlated ports, the number ofwhich scales with the area of the aperture as measuredin wavelengths squared. For example, a circular shapedaperture whose diameter corresponds to four operatingwavelengths allows ∼
100 propagating modes, leadingto M ∼
100 ports in the RCM modelling of the inter-connected cavities. A cavity with M ports is describedby an ( M × M ) matrix [1, 3, 47, 50]. When large aper-tures are present, connecting multiple cavities, the RCMmodel can become cumbersome. First there is the needto calculate the matrix elements for each aperture thatdescribe the passage of waves through an aperture ra-diating into free space. Second these matrices mush becombined with random matrices that give the statisticalfluctuations. Third, the RCM is a Monte-Carlo methodin which the matrices simulating the cavities are con-structed for each realization, and many realizations maybe required to get accurate statistical results. Finally,the matrices must be connected together which involvesinverting the sub-matrices representing the sub-volumesinside a complex system for each realization [47]. There is thus a need to develop a simple statistical method thatapplies in cases where the apertures are larger than awavelength, but small enough so that the two enclosuresconnected by the aperture can be considered as two sep-arate volumes.Here, we introduce a hybrid approach that combinesthe PWB and RCM to generate statistical predictions ofthe EM field for multi-cavity systems without the com-putational complexity of a full RCM treatment. The hy-brid approach is valid in cases where the coupling be-tween adjacent cavities is carried by many channels dueto, for example, large apertures as described above. Us-ing the hybrid method, we apply the PWB method forcomputing the average EM field intensities in each cav-ity and use RCM to predict the fluctuations in the cavityof interest only. The modeling of multiple channels be-tween adjacent cavities is thus reduced to computing ascalar coupling coefficient in the PWB model, often giventhrough simple expressions involving the area of the aper-ture. A coupled RCM model still needs to be appliedwhere the number of connecting channels between enclo-sures is small. Such small apertures act as “bottlenecks”for the wave dynamics and a full RCM treatment is nec-essary to characterize the fluctuations correctly, see alsoSection V and Appendix C and D. From nested reverber-ation chamber modelling, it is known that weak coupling(bottlenecks) introduces statistical independence of thetwo cavity environments [51] in the sense that the cou-pled problem can be described in terms of a random mul-tiplicative process leading to products of random fields.This leads to strong deviations from a Gaussian randomfield hypothesis as discussed in more detail in SectionV and Appendix C. One may further reduce the com-putational cost of the hybrid model using a simplifiedtreatment as proposed in Appendix E. This simplifiedversion of the hybrid model does not require additionalknowledge of the frequency-dependent aperture admit-tance, which is usually obtained through full-wave simu-lations.We test the hybrid model using cavity-cascade systems,that is, linear arrays of coupled complex systems. RCMbased studies for a linear multi-cavity array with a sin-gle coupling channel and multiple channels were treatedin [5, 21, 50]. In the following, we introduce the exper-imental set-up of the cavity cascade system in SectionII. The formulation of the PWB-RCM hybrid model ispresented in Section III. In Section IV, we compare ourmodel simulations with experimental results. The limitsof the hybrid model are studied in Section V with conclu-sions presented in Section VI. We refer technical detailsto the Appendices A-E. These contain an in-depth anal-ysis of the limitations of the hybrid model depending onthe number of coupling channels in Appendix C, and ex-tension of the hybrid model including ”bottelnecks” inAppendix D and a simplified version of the original hy-brid model in Appendix E. FIG. 1. A schematic view of the experimental set-up. Wemeasure the 2 × → →
3. Rotatable mode stirrers are employed ineach cavity to generate different system configurations. The1- and 2-cavity system measurement are conducted by block-ing the apertures and moving the location of the RX port tocavity 1 and 2, respectively.
II. EXPERIMENTAL SET-UP
We study the transmission and reflection of EM wavesin chaotic multi-cavity systems. A series of individualcavities is connected into a linear cascade chain throughcircular shaped apertures as shown schematically in Fig.1. Each cavity is of the same size and shape, but containsa mode-stirrer that makes the wave scattering propertiesof each cavity uniquely different [52]. The total num-ber of connected cavities is varied from 1 to 3. Short-wavelength EM waves from 3.95 to 5.85 GHz are injectedinto cavities of dimension 0 . × . × . m throughWR187 single-mode waveguides, shown as T(R)X in Fig.1. The loss factor of the system is tuned by placingRF absorber cones inside each cavity. The cavities arelarge compared with typical wavelengths of the EM field(with λ = 7.49cm - 5.12cm and there are ∼ modesin the frequency range in operation) simulating realis-tic examples of wave chaotic enclosures. The diameterof the aperture is 0 . m which requires on the order of ∼
100 modes to represent the fields in the aperture atthe operating frequency. The thickness of the apertureis about 0 .
04 times the operating wavelength. We mea-sure the 2 × × S = Z / ( Z + Z ) − ( Z − Z ) Z − / , where Z is a diagonal matrix whose elements correspondto the characteristic impedances of the waveguide chan-nels leading to the ports. Independent mode stirrers areemployed inside each cavity to create a large ensembleof statistically distinct realizations of the system [53–56].All mode stirrers are rotated simultaneously to ensure alow correlation between each measurement. A total num-ber of 200 distinct realizations of the cavity cascade arecreated.In the RCM, the “lossyness” of a single cavity is char-acterized by the loss parameter α defined as the ratio ofthe 3-dB bandwidth of a mode resonance to the meanfrequency spacing between the modes [23, 45]. The lossparameter α has values larger than zero (with α = 0corresponds to no loss), where α corresponds roughly tothe number of overlapping modes at a given frequency.For the RCM to be valid, there is an upper limit on α determined by the following conditions: (i) the loss rateand mode density should be relatively uniform over therange of frequencies considered and (ii) the number ofoverlapping modes should be much less than the num-ber of modes used in the RMT construction of the RCMnormalized impedance matrix (see Section III).The magnitude of the induced voltage at a load at-tached to the last cavity in the chain, | U L | , can be cal-culated from the measured impedance matrix Z of thecavity cascade system [4]. In the experiment, the loadis the RX receiver in the VNA ( Z L = 1 /Y L = 50Ω).Our objective is to use the hybrid PWB-RCM model todescribe the statistics of the load voltage | U L | using aminimum amount of information about the cavities andminimal computational resources. III. THE HYBRID MODELIII.A PWB in the Hybrid Model
The PWB method can be used to determine mean val-ues of EM power flow and energy in systems of coupledcavities [30–33]. For a multi-enclosure problem, the PWBmethod solves for the mean power density S i in each en-closure ( i is the cavity index) by balancing the powersentering and leaving each cavity. These power transferrates are characterized in terms of area cross-sections ( σ ),such that the power transferred is σS i . Various loss chan-nels, such as aperture/port leakage, cavity wall absorp-tion and lossy objects inside the enclosure are charac-terised through the corresponding cross sections σ o , σ w and σ obj , respectively [32]. Constant power is injectedinto the coupled systems through sources in some or allof the enclosures. The method solves for a steady statesolution when the inputs and losses are made equal foreach individual cavity in the system reaching a power bal-anced state. The PWB method does not contain phaseinformation of the EM fields and thus does not describe FIG. 2. Schematic illustration of the hybrid model appliedto a 3-cavity cascade. In the hybrid model, we use the PWBmethod to characterize the power flow from the first cavityto the next to last cavity. The fluctuations in the final cavityare described using the RCM method using the mean powerflow values obtained from PWB as an input. fluctuations due to interference. This can lead to an in-complete prediction of enclosure power flow in the caseof small apertures as discussed in Section III.C, SectionV.B and Appendix C.
III.B RCM in the Hybrid Model
As introduced in Section I, the Random CouplingModel provides an alternative method to describe thestatistics of the EM fields in a wide variety of complexsystems. In contrast with PWB, the RCM deals withboth the mean and fluctuations of the cavity fields.For coupled-cavity systems, the RCM multi-cavitytreatment begins with the modeling of the fluctuatingimpedance matrix of each individual cavity [47]. Thesematrices relate the voltages and currents at the ports ofa cavity. When cavities are connected the voltages atthe connecting ports are made equal and the connectingcurrents sum to zero. The input port on the first cavityis excited with the known signal. This leads to a linearsystem of equations that can be solved for all the voltagesand currents. This system is resolved for each realizationof the cavity impedance matrices.Model realizations of the fluctuating cavity impedancematrix of an individual cavity are created via a normal-ized impedance matrix ξ RCM derived from a randommatrix ensemble [1–3, 55]. In terms of the normalizedimpedance matrix, the fluctuating cavity impedance ma-trix is written as Z cav = iIm [ Z avg ] + Re [ Z avg ] / · ξ RCM · Re [ Z avg ] / , where the quantity Z avg is discussed below. Here, ξ RCM is defined as ξ RCM = − iπ (cid:88) n w n w Tn ( k − k n ) / ∆ k n + iα , where the sum over n represents a sum over the modes in-side the cavity. The vector w n , whose number of elementsequals the number of ports, consists of independent, zeromean, unit variance random Gaussian variables whichrepresent the coupling between each port and the n th cavity mode. This random choice of mode-port couplingoriginates from the so-called Berry hypothesis, where thecavity modes can be modeled as a superposition of ran-domly distributed plane waves [57]. The quantities k and k n are wavenumbers corresponding to the operat-ing frequency ω = k c and the resonant frequencies ofthe cavity modes, ω n = k n c . Rather than use the trueresonant frequency of the cavity, a representative set offrequencies is generated from a set of eigenvalues of a ran-dom matrix selected from the relevant Random MatrixTheory ensemble [1, 19, 58, 59]. These RMT eigenvaluesare appropriately normalized to give the correct spec-tral density via the parameter ∆ k n . The loss parameter α = k / ( Q ∆ k n ) where k is the wave vector of interestand Q is the quality factor [45].System-specific information about the enclosure is cap-tured in the averaged impedance matrix for each cavity, Z avg [43]. Here, the average impedance matrix can bethought of in two ways. First it can be considered asa window average of the exact fluctuating cavity matrixover a frequency range, ω , centered at frequency ω . Inthe case in which the windowing function is Lorenzianthis average is equivalent to evaluating the exact cav-ity matrix at complex frequency ω + i ω . This in turnis the response matrix for exponentially growing signalswith real frequency ω and growth rate ω . This leads tothe second way of understanding the average impedancematrix. It is the early time ( ω · t <
1) response of theports of the cavity. Thus, the average impedance matrixcan be calculated by assuming the walls of the cavity havebeen moved far from each port, and each port respondsas if there were only outgoing waves from the port. Inthe case of apertures as ports, the transverse electric andmagnetic fields in the aperture opening are expanded ina set of basis modes with amplitudes that are treated asport voltages in the case of electric field, and port cur-rents in the case of magnetic field. The linear relationbetween these amplitudes is calculated for the case ofradiation into free space, and this becomes the averageaperture admittance/impedance matrix. Each mode inthe aperture field representation is treated as a separateport in the cavity matrix. Thus, the dimensions of thematrix grows rapidly with the addition of a large aper-ture [50].In the cavity cascade system, the above mentionedapertures (with M propagating modes) are adopted asthe connecting channel between neighbouring cavities.With M connecting channels between cavities, the di-mension of the above defined cavity impedance matrixbecomes M × M . The matrix multiplications and inver-sions in the calculation of RCM multi-cavity formulations[47] thus have complexity which grows as O ( M . ) usingcommon algorithms [60]. Thus for large M , the compu-tational cost of the RCM scales roughly as N × M . ,where N represents the number of cavities in the system.Here we propose a hybrid method for multi-cavityproblems that combines both PWB and RCM as shownschematically in Fig. 2. In an N − cavity cascade systemwith multiple channel connections between adjacent cav-ities, we utilize the PWB to characterize the mean flow ofEM waves from the input port of the first cavity to the in-put aperture of the last ( N th ) cavity. The RCM methodis now applied to the last cavity and the connected loadat the single-mode output port of that cavity. Thus thehybrid method combines the strengths of both methods:the fluctuations of the EM field will be captured withRCM, and the computational cost is greatly reduced us-ing the PWB method. A quantitative comparison of thecomputational costs between full RCM method and thehybrid method is discussed later in Section III.D.In the following, we discuss the hybrid model formu-lation in detail based on the 3-cavity system exampleshown in Fig. 2. We will first introduce the PWB treat-ment to the first two cavities in the chain, followed bythe modeling of the last cavity using RCM in subsectionIII.C. We will then discuss how to connect the PWB andRCM models at the aperture plane between the last twocavities in subsection III.D. With the model formulationintroduced, we will look into the validity of the hybridmodel in subsection III.E. A step-by-step protocol to ap-ply the PWB-RCM fusion to generic cavity systems isdetailed in Appendix D. III.C Detailed Cavity Treatments
PWB characterizes the flow of high-frequency EMwaves inside a complex inter-connected system based onthe physical dimensions of the cavities, the cavity qualityfactors Q , and the coupling cross sections σ o as well asthe incident power P in driving the system [30, 32]. PWBthen calculates the power densities of each individual cav-ity in steady state. For the 3-cavity cascade system inFig. 2, the PWB equations are σ w + σ o + σ o − σ o − σ o σ w + σ o + σ o − σ o − σ o σ w + σ o + σ o S S S = P in (1)where the σ wi ’s refer to the wall loss cross section and S i is the power density of the i th cavity, σ o and σ o arethe cross section of the input and output ports, while σ o and σ o represent the aperture cross sections, see Fig. 2.The cross sections can be expressed explicitly from knownphysical dimensions and cavity wall properties [32]. P in is the (assumed steady) incident power flow into the firstcavity. In this example, we assume that only the first cavity receives EM power from external sources. Thebalance between the input and loss is achieved by solv-ing Eq. (1) for the steady state power densities S i . Forexample, the power balance condition of the first enclo-sure is expressed as ( σ w + σ o + σ o ) · S = P in + σ o · S .The LHS of this equation represent the loss channels ofthe cavity, including the cavity wall loss and the leak-age through the input port and the aperture. The RHSdescribes the power fed into the cavity, consisting of theexternal incident power and the power flow from the sec-ond cavity. The net power that flows into the last cavityin the cascade is expressed as P → = σ o ( S − S ).The last cavity is characterized by the RCM method.With the knowledge of the cavity loss parameter α andthe port coupling details, the full cavity admittance ma-trix of the last cavity can be expressed as [50] Y cav = i · Im (cid:16) Y rad (cid:17) + Re (cid:16) Y rad (cid:17) . · ξ · Re (cid:16) Y rad (cid:17) . . The quantity Y rad is a frequency-dependent block-diagonal matrix whose components are the radiation ad-mittance matrices of all the ports and apertures of thatcavity. We assume no direct couplings between aperturesbecause the direct line-of-sight effect is small in the ex-perimental set-up. Consider a cavity with two M − modeaperture connections, the dimension of the correspondingmatrix Y rad is 2 M × M . The matrix elements are com-plex functions of frequency in general and can be calcu-lated using numerical simulation tools. We use here thesoftware package CST Studio to calculate the apertureradiation admittance (see Ref. [61] and the AppendixB.2 in Ref. [47]). The RCM normalized impedance ξ is a detail-independent fluctuating “kernel” of the totalcavity admittance Y cav . With known α , an ensemble ofthe normalized admittance ξ can be generated throughrandom matrix Monte Carlo approaches [4]. Combin-ing the fluctuating ξ and Y rad , an ensemble of “dressed”single cavity admittance matrices for the final cavity canbe generated. It is later shown in Appendix E that asubstantial reduction of the hybrid model computationalcost is made possible using an aperture-admittance-freetreatment, at the price of reduced accuracy for longercascade chains. III.D The Hybrid Model
We next connect the PWB and RCM treatments atthe interface between the second and the third cavity.As discussed in the previous section, the power flow intothe third cavity, P → , is calculated from the 3-cavityPWB calculation. Identical system set-ups are utilizedin the PWB and RCM treatments, including the operat-ing frequency range, the dimensions of the cavities, portsand apertures, and the loss of the cavity (achieved by asimple analytical relationship between the RCM α pa-rameter and σ w , see Appendix A for more details). Totransfer the scalar power values P → generated by PWBinto an aperture voltage vector required for RCM, we as-sign random voltages U o drawn from a zero mean, unitvariance Gaussian distribution for the M -mode apertureand calculate the random aperture power using P o = 12 Re ( U ∗ o · Y rad · U o ) . These randomly assigned aperture voltages U o are thennormalized by the ratio P → /P o to match with thevalue calculated from PWB. Combined with the RCMgenerated cavity admittance matrix Y cav , an ensembleof induced voltage values U L at the load on the last cav-ity is computed utilizing Eq. (A.9) in Ref. [47]. Thepower delivered to the load is obtained using P L = 12 Re ( U L ∗ · Y L · U L ) , where Y L is the load admittance, taken to be 1 / (50Ω)here.With the formulation of the hybrid model now ex-plained, here we discuss the improvement in compu-tational time and memory usage by replacing the fullRCM multi-cavity method with the hybrid model. Foran N − cavity system connected through M − mode aper-tures, the hybrid model requires only a fraction of1 / [( N − M ] the memory consumption as comparedto the full RCM method, enabled by the reduced cavityimpedance matrix storage for the first N − N andlarger apertures (having M modes). III.E Limits of the Hybrid Model
The hybrid model is based on the assumption that thefluctuations in a given cavity are independent of the fluc-tuations in adjacent cavities and thus of the fluctuationsin the power flowing between cavities (as a function offrequency, for example). We assess the validity of theseassumptions by analysing a multi-channel cascaded cav-ity system in Appendix C. We study in particular theeffect of the total number of effective cavity-cavity cou- pling channels M n between the n th and ( n + 1)st cavityon the fluctuation levels of the load-induced power P L connected to the final cavity. Since P L ∝ | U L | , the con-clusions drawn from the power flow studies can also beapplied to voltage-related results as presented below inSection IV. Defining the load power fluctuation levels asthe ratio κ = (cid:104) P L (cid:105) / (cid:10) P L (cid:11) , where (cid:104)· · · (cid:105) represents aver-aging over an ensemble, we treat κ as a measure char-acterising the level of fluctuations of the power. Here, κ ∼ κ (cid:29) κ ∝ N (cid:89) n =1 (1 + M − n ) , (2)where the product is over all the cavities in the cascade.If cavities n = 1 to n = N − M n (cid:29) M − n → (cid:81) N − n =1 (1 + M − n ) → M n ≈
100 at the frequencies considered. The quan-tity M n is small when a single-mode waveguide connectsthe last cavity ( N ) to the load (the experiment in SectionIV). At the last cavity ( N ) there is a single mode outputport, M n = 1 and the 1 + M − n = 2 which induces higherfluctuations at the load compared to the case where allapertures are large. Similar small M n situations appearwhen a “bottleneck” is introduces between cavities (theexperiments in Section V.B). It is therefore sensible toadopt RCM for just the last cavity to capture the powerfluctuations at the load, while it is sufficient to includethe influence of the intervening cavities with PWB onlygiving the required information about the mean powerflow. This case is discussed in Section IV. In Section V,we will also consider the effect of having small apertureconnections – referred to as “bottlenecks” – at interme-diate locations in the cavity cascade where we see devia-tions of the hybrid model from a full multi-cavity RCMtreatment. IV. COMPARISON OF HYBRID MODEL WITHEXPERIMENTAL RESULTS AND DISCUSSION
We now conduct the PWB-RCM hybrid analysis forthe multi-cavity experiment and compare with measure-ments. We consider 2- and 3-cavity cascades with large(on the scale of the wavelength) apertures between thecavities and single-mode connections to the load in thelast cavity. The induced voltages at the load | U L | arecalculated from data using the methods reported in Refs.[4, 62], and these experimental results are shown as solidlines in Fig. 3. An ensemble of induced load voltages for FIG. 3. The PDFs of load induced voltage | U L | of 2- and3-cavity experiments (solid) and hybrid model calculations(dashed). The single cavity loss parameter is varied from 9.7,7.5, 5.7 and 1.7 from (a-d), respectively. The inset in (b)shows the 2- and 3-cavity experiment averaged induced volt-age values (cid:104)| U L |(cid:105) with respect to different loss parameters α .Multi-mode ( ∼
100 modes) circular apertures are employedbetween the cavities. the multi-cavity system is created by moving the modestirrers in all cavities between each measurement. Thelosses in the single cavities is altered by inserting equalamounts of RF absorbers in each cavity. In addition,the hybrid PWB-RCM method is used to calculate | U L | and the resulting distributions are shown as dotted linesin Fig. 3. Good agreement between the measured andmodel generated results are observed over a range of dif-ferent total cavity numbers and single cavity loss values.Under varying cavity loss conditions, the probability den-sity function (PDF) of the induced load voltage | U L | ofthe 3-cavity system has a lower mean value and smallerfluctuations compared to the 2-cavity system results. Go-ing from two to three cavities will decrease the energydensity in the last cavity and thus the power deliveredto the load. This difference between the 2- and 3-cavity | U L | becomes smaller when the single cavity loss decreaseas can be seen following Fig. 3 (a) to Fig. 3 (d), see alsothe inset in Fig. 3 (b).The induced load voltage PDFs of the multi-cavitysystem can be generated solely with the RCM formu-lation [47]. A comparison between the | U L | PDFs gener-ated with full RCM method, the hybrid method, and theexperiments are shown in Fig. 4. Both theoretical ap-proaches are able to generate statistical ensembles whichagree well with the experimental results. We find that theRCM results (dashed lines) slightly outperform the hy-brid method (dotted lines) for the two-cavity case. How-
FIG. 4. Comparison of induced load voltage statistics for thehybrid model (dotted), RCM predictions (dashed) and exper-imental results (solid) for the case of 2 and 3 cavity cascadeswith single-cavity loss parameter α = 9 .
7, and circular multi-mode apertures between the cavities. The frequency range isfrom 3.95 to 5.85 GHz. ever, the computation time and storage cost of the fullRCM method is N times larger than the hybrid method,where N refers to the total number of connected cavitiesin the cascade.It is well-established that the distribution of fieldsinside sufficiently lossy single/nested cavity systems isRayleigh/double-Rayleigh distributed [48, 51, 63, 64]. Itwas shown by the authors of Ref. [48] that the cavityfield distribution deviates from a Rayleigh distributionin the low-loss limit ( α < V. TESTING THE LIMITS OF THE HYBRIDMODEL
As already discussed in Section III C, we expect thatthe hybrid model will break down in the limit where ei-ther the RCM and/or PWB methods are no longer valid.We consider two generic types of limitations, namely sys-tems with high loss and systems having “bottlenecks” inthe middle of the cavity cascade. These two conditionsare experimentally studied using a ×
20 scaled down ver-sion of the cavity system [23, 47]. The dimension of thesingle miniature cavity is 0 . × . × . m . EMwaves from 75-110GHz are fed into the cavity ( ∼ modes at and below the operating frequency range)through single-mode WR10 waveguides from VirginiaDiodes VNA Extenders and the S-parameters are mea-sured by a VNA. We have previously demonstrated that FIG. 5. (a) The load induced voltage | U L | statistics P ( | U L | )of the 2-cavity experiment and models in the high loss limit( α ∼ ∼
100 modes at 110GHz. The insetshows the experimental set-up schematically. (b) 3-cavity ex-perimental and model generated load induced voltage statis-tics. The cavities ( α ∼ .
1) are connected by rectangularshaped apertures with just 5 propagating modes. The insetis the schematic of the experimental set-up. identical statistical electromagnetic properties are foundin this and the full-scale configuration described in Sec-tion II [47].
V.A High and Inhomogeneous Cavity Losses
The proposed hybrid method is not expected to gener-ate accurate predictions for extreme high and inhomoge-neous lossy systems [62]. Both PWB and RCM modelsrequire uniform power distribution inside the studied sys-tem. Such a presumption no longer holds when the loss ofthe cavity wall becomes so high such that the energy dis-tribution near the system boundaries and from the inputto the output aperture drop considerably. In this case,PWB needs to be replaced with other methods such asray tracing [36] or the DEA analysis [39] or, in the case ofmultiple scatterers in each cavity, using an approximateflow solver based on a 3D diffusion model [65]; all thesemethods have a larger computational overhead comparedto PWB. Strong damping also violates the random planewave hypothesis crucial to the RCM [57, 62, 66]. Weexperimentally examine the applicability of the hybridmodel in the extremely high loss limit. A 2-cavity cas-cade system is designed where electrically-large (manywavelength in size) ARC RF absorber panels are placedon a wall inside each cavity. The detailed experimentalset-up of the extreme high-loss cases can be found in Ap-pendix B. The inclusion of absorber walls creates effec-tively “open-wall” high loss cavities ( α ∼
25 for a singlecavity) [62]. The measured and the model-generated loadinduced voltage statistics are shown in Fig. 5 (a) for thiscase. Neither the hybrid model nor the full RCM modelis expected to work in this high and inhomogenous losslimit. As seen in Fig. 5 (a), both models show strong disagreement with the measured data. These inhomo-geneities in each cavity are well captured using eitherray-tracing or DEA methods as demonstrated in [67].The hybrid model can be applied to the lower loss sys-tems ( α ∼
1) as demonstrated in Section IV. We pointout, however, that the stronger impedance fluctuations oflow-loss systems poses greater challenges for the acquisi-tion of good statistical ensemble data for both numericaland experimental methods [64, 68].
V.B Weak Inter-Cavity Coupling
Another assumption of the hybrid model is that in-termediate cavities have multiple connecting channels,see Section III C. We expect that the hybrid model asintroduced in Section III fails when some or all of theapertures are small, effectively acting as “bottlenecks”.The transmission rates then become strongly frequencydependent thus adding to the overall fluctuations in thesystem and deviating strongly from the aperture crosssections assumed for the PWB in Section III.E. In theexperiments, we create a “bottleneck” carrying only 5channels between the intermediate cavities of the chainand study how this brings out the limitations of the hy-brid model. As discussed in Section III.E and AppendixC, the fluctuations of the wave flow are then correlated asthe energy propagates through the cavity cascade chain;these increased fluctuations of the input power at the lastcavity are not captured in a PWB treatment. It is impor-tant to point out that the fluctuations of input currentat a cavity beyond a bottleneck can no longer be consid-ered Gaussian. The latter aspect is known from cavitysystems studied in electromagnetic compatibility, see forexample the deviation of the received power in a weaklycoupled nested reverberation chamber [51]. In AppendixC, arguments are developed to quantify these deviationsfor cavity cascade systems.We construct a 3-cavity experiment to test this ef-fect; for more details about the experiment, see AppendixB. The cavities are connected through small rectangularshaped apertures which allow only 5 propagating modesas opposed to 100 modes utilized in experiments dis-cussed earlier; we have α = 9 . | U L | , while the full RCM cavity cascade for-mulation retains the ability to correctly characterize thestatistics of the system.To summarize the section, we study the applicabilityof the PWB-RCM hybrid model to cases where it is ex-pected to fail. It is found that enclosures with large andinhomogeneous losses violate the conditions for the hy-brid method. The case of intermediate “bottlenecks”can be handled with RCM, but not the hybrid modelas formulated. A generalization of the PWB-RCM hy-brid that can handle arbitrary “bottlenecks” is outlinedin Appendix D. The proposed hybrid model is expectedto generate statistical mean and fluctuations of the EMfields for systems with moderate cavity losses and largeinter-cavity couplings strengths. Expanding the hybridmodel to systems with more complex topology and ex-ploring new types of hybrid models by combining RCMwith other statistical or deterministic methods has beenconsidered in [69]. VI. CONCLUSION
In this manuscript, we propose a PWB-RCM hybridmodel for the statistical analysis of EM-fields in complex,coupled cavity systems based on minimal information ofthe system. The method is tested and found to be in goodagreement with cavity cascade experiments under variousconditions, such as varying single cavity loss and the to-tal number of connected cavities. The limitations of thehybrid model are also discussed and demonstrated exper-imentally. The hybrid model is computationally low-costand able to describe the statistical fluctuations of the EMfields under appropriate conditions. We believe that thehybrid method may find broad applications in the analy-sis of coupled electrically large systems with sophisticatedconnection scenarios.We thank B. Xiao who designed the scaled cavity sys-tems and M. Zhou for the help in conducting numeri-cal computer simulations. We are thankful for R. Gun-narsson’s warmhearted help with computer simulationtechniques. This work was supported by the Office ofNaval Research under ONR Grant No. N000141512134,ONR Grant No. N000141912481, ONR DURIP GrantN000141410772 and ONR Grant No. N62909-16-1-2115as well as by the the Air Force Office of Scientific Re-search AFOSR COE Grant FA9550-15-1-0171.
Appendix A: Relating Cavity Internal LossParameters in RCM and PWB
The parameters used to model the cavity loss in theRCM ( α ) and PWB ( σ w ) are different. We propose a sim-ple analytical relationship between these internal loss pa-rameters, namely α = γ · σ w . In the RCM, the single cav-ity loss parameter α can be written as α = k V / (2 π Q )for three-dimensional enclosures, where V , Q and k arethe cavity volume, closed cavity quality factor and theoperating wavenumber [23, 45]. In the PWB, the wallloss cross-section σ w is defined as σ w = P w /S where P w is the steady state power loss caused by wall absorption,and S represents the unidirectional power density at any FIG. A.1. The single-cavity radiated power through theaperture opening P aper vs the cavity loss calculated withPWB ( γ · σ w ) and RCM methods ( α ). Input power is setas 1 Watt. Inset: schematic diagram of the single cavity ra-diation set-up. point inside the cavity. To arrive at an analytical expres-sion for S , we first write the Poynting vector S p whichflows uniformly in all directions as S p = cW/ (4 π · V ),where W is the stored energy inside the cavity volume V . Thus S can be calculated through the integral S = (cid:90) π dφ (cid:90) π cosθ S p sinθ dθ = cW V .
Combined with Q = ωW/P w (which assumes that wallloss dominates the closed-cavity Q), we have σ w = P w S = ωWQ · VcW = 4 kVQ and finally arrive at α = γ · σ w where γ = 1 / (2 λ ).We next conduct a sanity check of the α to σ w re-lationship by creating a practical scenario. Consider asingle lossy cavity with a single input port and a radiat-ing aperture. We choose this particular scenario becauseit reflects the range of situations where we believe theRCM/PWB hybrid model will be relevant and valid. Wecalculate the radiated power from the cavity as a functionof the internal loss parameter in each model, for fixed in-put power. In the PWB treatment, a single-mode portand an aperture are opened on the cavity with their PWBcross sections σ o and σ o , respectively. Waves are inci-dent into the cavity through the port and exit the cavitythrough either the aperture or the port into vacuum. Inthe RCM treatment of the same system, the port and theaperture are modelled with their corresponding radiationimpedance (for the port), and radiation admittance (for0the aperture).For the study shown in Fig. A.1, the dimension of thesingle cavity is set to 0 . × . × . m , which isthe exact cavity dimension used in the experimental set-up. The aperture radiated power P aper is calculated withboth treatments at 5GHz under various cavity loss val-ues. The aperture is set to be a circular shaped aperturewhich allows ∼
75 propagating modes at 5GHz. Theresults for RCM are shown as red circles in Fig. A.1.The PWB results are obtained for a series of σ w val-ues and shown as a blue line in Fig. A.1. The factor γ = 1 / (2 λ ) = 139 m − is applied for the PWB calcu-lation to scale the σ w value to α on the horizontal axis.A good agreement of aperture radiated power is foundbetween the two models. We note that a finite discrep-ancy is observed in the range α ∈ [0 . , Appendix B: The Experimental Set-up of HybridModel Limitation Studies
As discussed in section III and IV, both the PWB andRCM methods assume that the wave energy inside thesub-volumes are homogeneously distributed. We wouldnot expect the hybrid model to be effective when this as-sumption is violated. We experimentally test this limitof the hybrid method by creating a system with non-uniform energy distributions (Fig. A.2 (a)). In the 2-cavity cascade set-up, we first cover one of the cavitywalls with RF absorbing materials to create an effective‘open-window’ cavity. The results for the load inducedvoltage statistics in this cavity can be found in Fig. 5 (a).The single cavity loss parameter is α = 25, estimatedby calculating the Q-factor of the S-parameter measure-ments [23]. It is shown that neither the hybrid methodnor the RCM method can correctly reproduce the exper-imental results for the load induced voltage statistics.We also test the case where the apertures are changedfrom a large circular shaped aperture with approximately100 propagating modes into a small rectangular shapedaperture with only 5 propagating modes (Fig. A.2 (b)).As shown in Fig. 5 (b), the hybrid model prediction de-viates from the experimental results, while the full RCMprediction retains the ability to accurately predict theinduced-voltage PDF. FIG. A.2. (a) The open-wall view of the single cavity underextreme high-loss conditions. A piece of the RF absorbermaterial (black rectangular shaped) is attached at one cavitywall. (b) The schematic picture of how the rectangular shapedaperture is created. Copper tape with a small rectangularopening (shown in blue) is placed over the original circularaperture. The inset shows a picture of the rectangular shapedaperture.
Appendix C: Multi-channel analysis for the cavitycascade system
Within the RCM description, the n th cavity in the cav-ity cascade chain can be modeled by the following equa-tion: (cid:20) V in V on (cid:21) = (cid:34) Z iin Z ion Z oin Z oon (cid:35) (cid:20) I in I on (cid:21) . (A.3)The superscripts i and o refer to the input and outputside of the n th cavity where we adopt the convention that“input” is on the side closest to the input TX port of thecavity chain and “output” is on the side with the RX port(see Fig. 1). The voltage vectors V and current vectors I at the input and output sides of a cavity are connectedthrough the cavity impedance matrix as shown in Eq.((A.3)). The Z-matrix of the n th cavity is written as a2 × n th cavity has m and m coupling channels at its input and output sides, theinput vectors ( V in , I in ) are m ×
1, and the output vectors( V on , I on ) are m ×
1. Correspondingly, the dimension ofthe Z-matrix elements are m × m , m × m , m × m and m × m for Z iin , Z oon , Z ion and Z oin , respectively.For simplicity, we assume the RCM description of theoff-diagonal impedance matrices as Z oin = ( R on ) / ξ n ( R in ) / , (A.4)and the matrices Z oo,iin are diagonal. The R i,on are theinput and output radiation resistance matrices, and ξ n is the RCM normalized impedance matrix. Since theoutput side of the n th cavity is connected to the inputside of the ( n + 1) st cavity, the continuity relationshipsof voltages and currents between the two cavities are: V on = V in +1 , I on = − I in +1 . (A.5)1In the high-loss limit, the output current at the ( n + 1) st cavity I on +1 is much smaller than the input current atthe n th cavity I in . By solving Eqs. (A.3) and (A.5) andneglecting terms Z ion +1 I on +1 (small compared to the Z ii terms), the relationship between the input currents at the n th and ( n + 1) st cavity in the high-loss limit is writtenas ( Z iin +1 + Z oon ) − Z oin I in = I in +1 . We then rewrite this equation as: ψ n +1 = τ n → n +1 ξ n ψ n with τ n → n +1 = ( R in +1 ) / ( Z iin +1 + Z oon ) − ( R on ) / , (A.6)a transition matrix which describes the coupling from the n th cavity to the ( n + 1) st with R i,on as in (A.4), and the ψ n = ( R in ) / I in is a current-like vector that describesthe input signal of the cavity n . The power entering the n th cavity is given by, P n = 12 Re (cid:104) I in † Z iin I in (cid:105) = 12 ψ † n ψ n (A.7)We first assume the transition matrix τ to be diagonal.The input power at the ( n + 1) st cavity is written as: P n +1 = 12 ψ † n +1 ψ n +1 = 12 (cid:16) τ n → n +1 ξ n ψ n (cid:17) † (cid:16) τ n → n +1 ξ n ψ n (cid:17) = 12 (cid:88) i,j,k (cid:16) ψ ∗ n,i ξ ∗ ji | τ j | ξ jk ψ n,k (cid:17) (A.8)The more general cases, where the constraint of τ be-ing diagonal is lifted, will be discussed later in this sec-tion. With (cid:104) P n (cid:105) = (cid:68) ψ † n ψ n (cid:69) being the average inputpower at the n th cavity, and the fact that (cid:10) ξ ∗ ji ξ j (cid:48) i (cid:48) (cid:11) = (cid:10) ξ ∗ ji ξ ji (cid:11) δ ii (cid:48) δ jj (cid:48) , we can further simplify Eq. (A.8) as: (cid:104) P n +1 (cid:105) = (cid:104) P n (cid:105) · (cid:68) | ξ | (cid:69) · (cid:88) j | τ j | = (cid:104) P n (cid:105) (cid:68) | ξ | (cid:69) (cid:88) j T j (A.9)where T j = | τ j | and the sum on j runs from alltransmitting channels. Based on the relationship inEq. (A.9), we next analyze the fluctuation level of thepower delivered to the load ( P L ) by studying the ratio κ = (cid:10) P L (cid:11) / (cid:104) P L (cid:105) . To simplify the following expression, X j = ψ ∗ n,j ψ n,j and Y j = ψ ∗ n +1 ,j ψ n +1 ,j are employed inthe following calculations. Equation (A.8) is now rewrit-ten as Y j = T j (cid:80) i,i (cid:48) ψ ∗ n,i ψ n,i (cid:48) ξ ∗ ji ξ ji (cid:48) , and further we have (cid:104) Y j (cid:105) = T j (cid:68) | ξ | (cid:69) (cid:80) i (cid:104) X i (cid:105) . Utilizing (cid:68) | ξ | (cid:69) = 2 (cid:68) | ξ | (cid:69) and (cid:10) ξ ∗ ji ξ ji (cid:48) ξ ∗ ji ξ ji (cid:48) (cid:11) = (cid:10) ξ ∗ ji ξ ∗ ji (cid:11) (cid:104) ξ ji (cid:48) ξ ji (cid:48) (cid:105) = 0 when i (cid:54) = i (cid:48) , we have: (cid:88) jj (cid:48) (cid:104) Y j Y j (cid:48) (cid:105) = (cid:68) | ξ | (cid:69) (cid:88) i,i (cid:48)(cid:48) (cid:104) X i X i (cid:48)(cid:48) (cid:105) (cid:88) jj (cid:48) T j T j (cid:48) + (cid:88) j T j (A.10)We then introduce a mean power transmission coefficientfor the n th cavity ¯ T n and an effective total number ofchannels M n , defined as:¯ T n = (cid:68) | ξ | (cid:69) (cid:88) j T j , M − n = (cid:88) j T j / (cid:88) j T j , (A.11)respectively. Thus we have the power transmitted intothe ( n + 1) st cavity (Eq. (A.9)) written as: (cid:104) P n +1 (cid:105) = ¯ T n (cid:104) P n (cid:105) , (A.12)and (cid:10) P n +1 (cid:11) = ¯ T n (cid:0) M − n (cid:1) (cid:10) P n (cid:11) . (A.13)The fluctuation level of the power after N cavities is: κ = (cid:10) P L (cid:11) (cid:104) P L (cid:105) = (cid:10) P (cid:11) (cid:104) P (cid:105) N (cid:89) n =1 (cid:0) M − n (cid:1) = N (cid:89) n =1 (cid:0) M − n (cid:1) . (A.14)Here we utilized the fact that the power injected intothe first cavity ( P ) is a fixed scalar in the case of asteady input. Since the last cavity is connected to theload with a single-mode port in experiments, we have (cid:0) M − N (cid:1) = 2 at the last cavity. The overall fluctuatinglevel of load power is κ = (cid:10) P L (cid:11) (cid:104) P L (cid:105) = 2 × (cid:34) N − (cid:89) n =1 (cid:0) M − n (cid:1)(cid:35) . (A.15)For a system with a large number of connecting channels( M n (cid:29)
1) between the neighbouring cavities, such as thecircular aperture connection which allows ∼
100 propa-gating modes, the factor (1+ M − n ) →
1. In this limit, themean field methods are sufficient to describe the powerflow for cavities with strong coupling. On the contrary,the hybrid model is not expected to work for system withfew coupling channels inside a cavity cascade. The studyexplains the experimental observations that hybrid mod-els are successfully applied to systems with multi-modeapertures connections (Fig. 3), while fail to generate pre-dictions for systems with “bottlenecks” (Fig. 5 (b)).We also note that the proposed hybrid model assumeseach input current throughout the chain is Gaussian dis-tributed while the RCM predicts non-Gaussian statistics[21]. In the presence of a “bottleneck” connection, a de-viation from Gaussian statistics of the input current dis-tribution will be present caused by the fluctuating nature2of the narrow aperture coupling.We next expand the analysis to the more general caseswhere the transition matrix τ is no longer a diagonalmatrix. With Eqs. (A.8) and (A.10), the expression for Y j becomes: Y j = (cid:88) j (cid:48) j (cid:48)(cid:48) τ ∗ jj (cid:48) τ jj (cid:48)(cid:48) (cid:88) i (cid:48) ,i (cid:48)(cid:48) ψ ∗ n,i (cid:48) ψ n,i (cid:48)(cid:48) ξ ∗ j (cid:48) i (cid:48) ξ j (cid:48)(cid:48) i (cid:48)(cid:48) . (A.16)And correspondingly the average of Y j is (cid:104) Y j (cid:105) = (cid:80) j (cid:48) τ ∗ jj (cid:48) τ jj (cid:48) (cid:68) | ξ | (cid:69) (cid:80) i (cid:104) X i (cid:105) . With the updated ¯ T n = (cid:68) | ξ | (cid:69) (cid:80) jj (cid:48) τ ∗ jj (cid:48) τ jj (cid:48) , we sum on j and have (cid:88) j (cid:104) Y j (cid:105) = (cid:68) | ξ | (cid:69) (cid:88) jj (cid:48) τ ∗ jj (cid:48) τ jj (cid:48)(cid:48) (cid:88) i (cid:104) X i (cid:105) = ¯ T n (cid:88) i (cid:104) X i (cid:105) , (A.17)and the average of the product of Y ’s is (cid:104) Y k Y k (cid:48) (cid:105) = (cid:88) j,j (cid:48) ,j (cid:48)(cid:48) ,j (cid:48)(cid:48)(cid:48) τ ∗ kj τ kj (cid:48) τ ∗ k (cid:48) j (cid:48)(cid:48) τ k (cid:48) j (cid:48)(cid:48)(cid:48) · (cid:88) i,i (cid:48) ,i (cid:48)(cid:48) ,i (cid:48)(cid:48)(cid:48) (cid:10) ψ ∗ n,i ψ n,i (cid:48) ψ ∗ n,i (cid:48)(cid:48) ψ n,i (cid:48)(cid:48)(cid:48) (cid:11) (cid:10) ξ ∗ ji ξ j (cid:48) i (cid:48) ξ ∗ j (cid:48)(cid:48) i (cid:48)(cid:48) ξ j (cid:48)(cid:48)(cid:48) i (cid:48)(cid:48)(cid:48) (cid:11) . (A.18)With careful calculation, the double summation of Eq.(A.18) over k and k (cid:48) gives: (cid:88) k,k (cid:48) (cid:104) Y k Y k (cid:48) (cid:105) / (cid:68) | ξ | (cid:69) = (cid:88) ii (cid:48) (cid:104) X i X i (cid:48) (cid:105) (cid:88) jk τ ∗ kj τ kj + (cid:88) ii (cid:48) (cid:104) X i X i (cid:48) (cid:105) (cid:88) jj (cid:48) (cid:34)(cid:88) k τ ∗ kj τ kj (cid:48) (cid:35) . (A.19)With the updated ¯ T n , we reach a similar expression of thepower flow as in Eq. (A.13) with the new M − n writtenas M − n = (cid:88) jj (cid:48) (cid:34)(cid:88) k τ ∗ kj τ kj (cid:48) (cid:35) / (cid:88) jj (cid:48) τ ∗ jj (cid:48) τ jj (cid:48) . (A.20)Thus the conclusions of the multi-channel analysis canbe applied to the more general cases with the refined τ ,¯ T n and M − n . Appendix D: Applying the Hybrid Model to GenericCavity Systems
Here we propose a scheme of applying the hybrid modelto more general cases, where there exists several limited-channel connections between the intermediate cavities.Due to the large power flow fluctuations induced bythese “bottlenecks”, we apply RCM to better character-ize these connections. A general case is shown in Fig.
FIG. A.3. Schematic of a cavity cascade array with many-channel connecting apertures between all cavities except thatbetween cavity i and cavity i + 1. Note the definition of inputand output powers at each cavity aperture. A.3. We study the treatment of limited channel connec-tions between the i th and ( i + 1) st cavities in a N-cavitycascade chain. The treatment involves four steps:1. Use the PWB N-cavity model to calculate theinput, output power of the i th cavity P iin | P W B , P iout | P W B , and the input power at the last ( N th )cavity P Nin | P W B .2. For the case of a bottleneck between cavity i andcavity i + 1 (Fig. A.3), use the calculated P iin | P W B and RCM treatment to calculate an ensemble of theoutput power of the i th cavity P iout | RCM .3. Update the input power at the last cavity by P Nin | RCM = P iout | RCM · P Nin P iout | P W B . Note that we therefore obtain an ensemble of theinput power to the last cavity P Nin | RCM .4. Use RCM to calculate an ensemble of the outputpower of the N th cavity P Nout | RCM with the ensem-ble of P Nin | RCM obtained from (3).For cases where there exist multiple “bottlenecks” in-side the chain (beyond cavity i ), one may repeat the pro-cedure (2)-(4) by actively changing the cavity index N to the next cavity connected through a bottleneck. Appendix E: Hybrid Model Simplification
Here we present a simplified treatment of the PWB-RCM hybrid model where the full-wave simulated aper-ture radiation admittance matrix is no longer required.We assume that all inter-cavity apertures are large on thescale of a wavelength, ie. the original PWB-RCM treat-ment is applicable. We also will assume that all cavitiesare in the high-loss ( α >
1) limit. This Y aper -free treat-ment will further decrease the computational cost of thehybrid model. We utilize Eqs. (A.6-A.8) from AppendixC to introduce the new treatment.Consider an N-cavity cascade chain, one may treat theload connected to the output port of the N th cavity as3 FIG. A.4. The PDFs of load induced voltage | U L | of 2-and 3-cavity experiments (solid), hybrid model (dashed), andthe simplified hybrid model (dash-dotted). The single cavityloss parameter is varied from 9.7, 7.5, 5.7 and 1.7 from (a-d),respectively. the ( N +1) st effective cavity. The induced power is equalto the input power of the ( N + 1) st cavity: P load = P N +1 = 12 ψ N +1 † ψ N +1 = 12 (cid:16) τ N → N +1 ξ N ψ n (cid:17) † (cid:16) τ N → N +1 ξ N ψ n (cid:17) (A.21)where τ N → N +1 = ( R iN +1 ) / ( Z iiN +1 + Z ooN ) − ( R oN ) / isa scalar transition factor due to the fact that a single-mode port acts as a single-mode ‘aperture’ connectingthe N th and ( N + 1) st cavity (the load). As introducedin the main text, the radiation impedance of the port,which is used to calculate τ N → N +1 , is obtained from thescattering matrix measurements of the waveguide ports.Equation (A.21) is further generalized as: P load = 12 ψ N +1 † ψ N +1 = 12 | τ N → N +1 | ψ † n ξ † N ξ N ψ n . (A.22)The ensemble average of P load is (cid:104) P load (cid:105) = 12 | τ N → N +1 | (cid:68) ψ † n ξ † N ξ N ψ n (cid:69) = 12 | τ N → N +1 | ψ † n ψ n (cid:68) ξ † N ξ N (cid:69) . (A.23)In the above equation, the quantity ψ n is moved out ofthe ensemble averaging because only ξ N is varying underensemble average while ψ n and τ N → N +1 are not. Asdefined in Appendix C, we write ψ n = R iiN / · I in under the high-loss assumption. Here R iiN is set as the radiationimpedance of the aperture which does not change valueunder ensemble averaging. The input current vector I in is also considered to be not changing since the cavity off-diagonal impedance element Z ion (Eq. (A.3)) is small athigh-loss limit, and the input power of the N th cavity isfixed from the prior PWB calculation. Since the inputpower of the N th cavity P N = ψ † N ψ N is given by thePWB calculation, we may further simplify Eq. (A.23) as (cid:104) P load (cid:105) = | τ N → N +1 | ψ † n ψ n (cid:68) ξ † N ξ N (cid:69) = P N | τ N → N +1 | · (cid:68) ξ † N ξ N (cid:69) . The vector element of ξ N , ξ i , is generated usingthe same method as in Section III. The vector product ξ † N ξ N is the summation of a large number of ξ ∗ i ξ i s. Thus ξ † N ξ N and ξ ∗ i ξ i have the same expected value accordingto the law of large numbers. We finally write (cid:104) P load (cid:105) = P N | τ N → N +1 | (cid:104) ξ ∗ i ξ i (cid:105) and Eq. (A.22) is reduced to scalarcomponents P load = P N | τ N → N +1 | ξ ∗ i ξ i .The comparison of the original hybrid model and the Y aper -free hybrid model is presented in Fig. A.4. Goodagreement between the simplified hybrid model predic-tions and the experimental cases for the PDF of loadinduced voltages are found. The original hybrid modelslightly outperforms its simplified version for the 3-cavitycases (red curves in Fig. A.4 (a-d)), and it is expectedthat the accuracy of predictions will decrease for longerchains. Such an effect may be caused by the absenceof an aperture admittance matrix in the simplified hy-brid model. The Y aper -free treatment would broaden theapplicable range of the PWB-RCM hybrid model to situ-ations where the exact shape of the aperture is unknownand the numerical simulation of the aperture is unavail-able. However, this shortcut shares the same high-loss( α >
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