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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 1, FEBRUARY 2015 3

Comparison of Reverberation Chamber ShapesInspired From Chaotic Cavities

K. Selemani, J.-B. Gros, E. Richalot, O. Legrand, O. Picon, and F. Mortessagne

Abstract—Using the knowledge gained from the wave chaostheory, we present simple shapes of resonant cavities obtainedby inserting metallic hemispheres or caps on the walls of aparallelepiped-shaped cavity. The presented simulation resultsshow a significant improvement of the field statistical propertieswhen the number of hemispheres or caps increases, and the com-parison with a classical reverberation chamber geometry shows animproved homogeneity and isotropy can be attained using thesenew proposed shapes.

Index Terms—Chaotic cavity, field homogeneity, field isotropy,reverberation chamber, statistical distribution, wave chaos.

I. INTRODUCTION

R EVERBERATION chambers (RCs) are used for electro-magnetic compatibility studies or antenna characteriza-

tions [1], [2]. Their operating range of frequency is above aminimum frequency, coined the lowest usable frequency (LUF).Above the LUF, the electromagnetic fields are supposed to fol-low ideal laws over a stirrer rotation, corresponding to statisti-cally isotropic, uniform, and depolarized fields [3]–[5]. Thesestatistical requirements are naturally fulfilled by most modes ofa chaotic cavity [6]–[8], without the help of any stirring process.Indeed, in a chaotic cavity, at high frequencies, generic modesare ergodic which, in a heuristic way, may be viewed as Gaus-sian random functions resulting from a random superpositionof plane waves (see, e.g., [9] for a definition of this term inthis context and also [10] for a recent review); each componentof such modes displays Gaussian statistics. However, properlydesigned cavities can display these ergodic properties of modeseven at low frequencies. Thus, chaotic reverberation chamberscan allow for an effective reduction of the LUF [11], [12].

Field isotropy and uniformity within a standardparallelepiped-shaped RC are usually obtained thanks toa mode stirrer (a rotating metallic object of complex shape [3]).In view of its supposedly indispensable nature to fulfill the

Manuscript received September 9, 2013; revised January 16, 2014; acceptedFebruary 20, 2014. Date of publication December 2, 2014; date of currentversion February 13, 2015. This work was supported by the French NationalResearch Agency (ANR) under the project CAOREV.

K. Selemani, E. Richalot, and O. Picon are with the Universite Paris-Est, ESY-COM (EA 2552), UPEMLV, ESIEE-Paris, CNAM, F-77454 Marne-la-Vallee,France (e-mail: [email protected]; [email protected]; [email protected]).

J.-B. Gros, O. Legrand, and F. Mortessagne are with the LPMC, CNRSUMR 7336, Universite de Nice-Sophia Antipolis, F-06108 Nice Cedex 2,France (e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEMC.2014.2313355

requirements of a well-operating RC, the stirring phenomenonin RCs has been widely studied [13]–[15]. In an ideal chaoticcavity, these statistical properties are a direct consequenceof the design without the help of any mechanical movement.Therefore, numerous studies on chaotic microwave cavitieshave focused on topological properties and several cavityshapes of intrinsic chaotic behavior [9], [16], [17]. A hybridapproach is adopted in this paper. Although each presentedcavity is provided with a mechanical stirring system, we aim atoptimizing the intrinsic properties of the cavity shape for a fixedposition of the stirrer. The basic idea is that the improvement ofthe field statistical properties for the static cavity might resultin a better operation over a stirrer rotation.

The impact of intrinsic (i.e., unstirred) reverberation cavityproperties on the field homogeneity with a stirrer in movementhas already been discussed [18] and cavity shape modificationshave been proposed, which consist for example in replacingflat walls by irregularly shaped ones [19] or placing corrugatedor curved wave diffractors on the chamber walls [11], [20]. Inthis paper, we take advantage of the knowledge gained fromthe study of chaotic cavities in various physical contexts, topropose new RC shapes leading to improvements of the fieldsproperties in comparison to regular RC cavities. By means ofthe finite element method, resonant modes of a classical RC andof three different chaotic chambers are obtained, and their sta-tistical properties are compared. Two chaotic RCs show remark-able features, namely almost perfect statistical homogeneity andisotropy of fields.

II. CHAOTIC MICROWAVE CAVITIES

One of the most fruitful applications of the wave chaos the-ory to electromagnetic systems stems from the analogy be-tween the Schrodinger and Helmholtz equations in the caseof a flat 2-D cavity [6]. The Schrodinger equation being scalar,the small cavity height allows the problem reduction to a 2-Dsystem with a single field component, so that the analogy di-rectly appears. Within a flat cavity parallel to the xy plane, werecall that the Helmholtz equation for harmonic Ez componentreads

�2Ez + k2Ez = 0 (1)

where k is the wavenumber. The boundary conditions on thecavity vertical walls are Dirichlet boundary conditions Ez = 0.This is formally equivalent to the time-independent Schrodingerequation for a quantum particle of mass m and energy E =�

2k2/2m in a 2-D infinite potential well. Due to this for-mal correspondence between scalar Helmholtz and Schrodingerequations in 2-D [21], most studies of chaotic electromagnetic

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4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 57, NO. 1, FEBRUARY 2015

Fig. 1. Two modes of a Sinai-like 2-D cavity with Dirichlet boundary condi-tions: (a) an ergodic mode, (b) a bouncing-ball mode.

cavities were carried out using 2-D systems [16], [17], [22]. Al-though the extension of quantum techniques to a fully vectorialHelmholtz equation is not straightforward, a few studies of 3-Dcavities have shown similar properties [23]–[25].

In the Wave Chaos theory, it is assumed that deterministicwavefields in complex geometries can be represented and an-alyzed by statistical methods. This property relies on the factthat the spatial distributions of the eigenfunctions of a resonantergodic cavity appear to be uniform and isotropic. Accordingto Berry’s hypothesis [9], [26], originally proposed in the con-text of Quantum Chaos, a typical mode of an irregular cavityhas all the characteristics of a Gaussian random field in thesemiclassical limit (i.e., for high frequencies). This hypothesiscan be heuristically understood within the geometrical limit ofrays at high frequency [9]. Indeed, for ergodic enclosures thechaotic dynamics of rays leads to modeling eigenfields as a su-perposition of plane waves with fixed wavenumber but randomdirections and phases. An example of ergodic mode in a 2-Dchaotic cavity (half Sinai billiard [27]) is shown Fig. 1(a).

Such a semiclassical analysis also leads to statistical predic-tions about the sequence of eigenfrequencies. The salient featureof this approach lies in the fact that, whatever the specific detailsof a chaotic cavity, its spectral fluctuations exhibit a universalbehavior. This universality underlies the relevance of a globalapproach consisting of generating artificial spectra through theeigenvalues of large random matrices. This is precisely the con-cern of random matrix theory (RMT) [28]. The intimate connec-tion between semiclassics and RMT has been anticipated by thefamous Bohigas–Giannoni–Schmit conjecture [29] and verifiedmany times in various contexts [6]. Within the framework ofRMT, the Gaussian randomness of modes is directly obtainedfrom the statistical features of eigenvectors of random matrices.Deviations from universal behavior are an important issue andare equally manifested in the field distributions of modes and inthe fluctuations of the corresponding spectra [7].

As an illustration in a 2-D chaotic cavity, a mode showing aclear deviation with respect to the ergodic universal behavior ofthe field is depicted in Fig. 1(b). This mode is often referred as abouncing-ball mode since it is built on ray trajectories bouncingback and forth between parallel walls. The properties of sta-tistical homogeneity and isotropy are therefore not fulfilled forsuch modes. Due to their regular occurrence in the spectrum,bouncing-ball modes contribute to significant spectral fluctua-tions on a large frequency range (compared to the mean spac-ing between adjacent eigenfrequencies) as will be illustrated inSection V.

Fig. 2. Cavity shapes and 3-D grids for field values extraction. The red pointin (c) and (d) indicates the excitation position in Section VI. (a) Cavity_1: Withone hemisphere. (b) Cavity_2: With two hemispheres. (c) Cavity_3: With 2 capsand 1 hemisphere. (d) Cavity_4: With a mode stirrer.

In a 3-D parallelepiped-shaped cavity with defocusing partsof the boundary, chaotic ray motion is expected. However, reg-ularity of modes may subsist due to continuous families of raytrajectories which bounce in a plane orthogonal to a pair ofparallel walls. Indeed, such families of marginally stable (non-chaotic) orbits correspond to bouncing-balls orbits when onlytwo parallel walls are involved, or to trajectories lying withina plane perpendicular to the both pairs of walls interceptingthem [25]. A local quantization of the wave vector along theseorbits leads to modes we coin tangential modes (TgM) [30],which constitute the most important family of regular modes.However, it is to be noticed that, contrary to electromagnetictangential modes of a parallelepiped-shaped cavity, the vanish-ing of a wave vector components only occurs in a part of thecavity. In the following sections, we will show how to reducethe spatial and spectral influence of regular modes and thus ob-tain ideal homogeneity and isotropy for almost all modes of achaotic RC.

Drawing inspiration from the 2-D chaotic cavity of Fig. 1 [27],we studied three parallelepiped-shaped cavities of dimensionsW = 0.785 m along x-axis, L = 0.985 m along y-axis, and H =0.995 m along z-axis, provided with metallic hemispheres orspherical caps on their walls (see Fig. 2). We denote as a hemi-sphere a spherical cap whose protrusion inside the cavity is equalto its radius. Hemispheres and caps positions are chosen to avoidany geometrical symmetry. Hemispheres have a radius of 15 cmand caps of 45 and 50 cm. The highest penetration depth of thecaps within the cavity is of 15 cm. The first cavity, Cavity_1[see Fig. 2(a)], is directly inspired from the 2-D cavity of Fig. 1.It comprises a hemisphere fixed on its wall at z = H. For the

SELEMANI et al.: COMPARISON OF REVERBERATION CHAMBER SHAPES INSPIRED FROM CHAOTIC CAVITIES 5

second cavity, Cavity_2 [see Fig. 2(b)], a second hemispherehas been added on the plane y = 0. The third cavity, Cavity_3[see Fig. 2(c)], comprises two caps on the planes z = H and y =L, as well as one hemisphere on the x = W plane. The choice ofthe geometric modifications from Cavity_1 to Cavity_3 will bediscussed later on while explaining how nonergodic modes canappear within these cavities. In these three cavities, the stirringprocess, beyond the scope of this paper, can be ensured by mov-ing one (or two) hemisphere(s) on the related wall, for example,by moving the hemisphere center along a circle.

To assess the respective performances of the chaotic cav-ities, the distributions of the three electric field componentsare examined for each of them and compared to the case of aparallelepiped-shaped cavity equipped with a mode stirrer [seeFig. 2(d)]. The shape and location of the latter conform to anindustrial RC, except a global scaling factor. As for the threeother cavities, the cavity with the mode stirrer is studied in afixed configuration, i.e., for a single position of the stirrer.

The three cavities are simulated, without considering any ex-citation antenna, by using the eigenmode solver of HFSS soft-ware. The first 450 resonant modes of each cavity are obtained.The corresponding resonant frequencies vary between 214 MHzand about 1.28 GHz. To study the field distributions, the val-ues of the three electric field components are recorded for eacheigenmode at 1001 points within the cavity volume. As shownin Fig. 2, these points are taken on a 3-D grid sampling a re-duced volume in the cavities. The distance between two adjacentlines as well as between the limits of the volume of use and thecavity is of 50 mm. This distance between two adjacent pointscorresponds to the recommended minimal quarter-wavelengthdistance [31] for the higher studied modes; however, we choseto keep constant the number of samples because of the use ofa goodness-of-fit test which is sensitive to it. For each eigen-mode, the mean of the square electric field amplitude on thewhole cavity volume is normalized to one. Simulations are per-formed without considering any losses in the cavity, so that theelectric field components are real.

In Section III, we focus on the electric field properties. Usingsimulation results, we first consider the distribution of eachCartesian field component and determine if a Gaussian lawis followed. In Section IV, dedicated to the field isotropy, wepropose to use the standard deviations of the electromagneticfield components to build an indicator of the field isotropy. InSection V, we show that the conclusions of the previous sections(related to spatial statistics) can also be established through thespectral statistics of the cavities: the more complex the geome-try, the more reduced the deviations from the universal behavior.In Section VI, the total response is rebuilt from the eigenmodesin order to apply standardized criteria to the studied cavities.

III. FIELD DISTRIBUTIONS OF EIGENMODES

We first examine the distribution of the three Cartesian electricfield components for each eigenmode. The orthonormal basis isdefined according to the cavity edges as reported in Fig. 2. In awell-operating reverberation chamber, as for the ergodic modes

Fig. 3. Electric field amplitude and associated distribution of Ex componentat 999 MHz (232th mode) for Cavity_1 (KS=0).

Fig. 4. Electric field amplitude and associated distribution of Ex componentat 994 MHz (230th mode) for Cavity_1 (KS = 1).

of a chaotic cavity, a centered normal distribution is expectedfor each real field component.

Let us consider as an example the mode at 999 MHz (232thmode) for the cavity with one hemisphere, i.e., Cavity_1 (seeFig. 3). The mapping of the electric field amplitude shows,while comparing to the parallelepiped-shaped cavity, that thedisturbance associated with the hemisphere is global within thecavity, and not located in the vicinity of the hemisphere. Fromthe 1001 values of the electric field components extracted alongthe 3-D grid, we build the distribution of amplitude of each fieldcomponent of this mode. In Fig. 3, the numerically obtainedhistogram for Ex is successfully compared to the probabilitydistribution function (PDF) of the centered normal law havingthe same standard deviation as the simulation data. To quan-tify the agreement, the Kolmogorov–Smirnov (KS) test at 95%confidence is used. The result of the test is 0 if the histogramand the normal law match within the confidence interval and 1otherwise. For the mode studied in Fig. 3, the answer is 0. Letus now consider the mode at 994 MHz (230th mode) for thesame cavity (see Fig. 4). The regularity of the field mapping,with a periodic repetition along x- and y-axes of a nearly identi-cal sinusoidal pattern, indicates that this mode corresponds to atangential mode lying in xy plane. The associated histogram ofEx confirms this observation as it is not fitted by the centerednormal law of the same standard deviation: the answer of theKS test is 1 in this case.

The insertion of a second hemisphere [see Fig. 2(b)] aims toeliminate this kind of tangential mode because it breaks up thetwo facing planar surfaces. Tangential modes are however stillobserved, as with the 174th mode corresponding to a reflectionbetween the bottom and the top of the cavity (see Fig. 5). The


Fig. 5. Electric field amplitude and associated distribution of Ex componentat 913 MHz (174th mode) for Cavity_2 (KS = 1).

replacement of hemispheres by caps [see Fig. 2(c)] in the thirdcavity allows a reduction of facing plane surfaces. The suppres-sion of the reflection between the two vertical walls orthogonalto the x-axis is however ensured by inserting a hemisphere in-stead of a cap for stirring purpose; the stirring process, notconsidered here, will be presented in the conclusion. We willsee in the following that, according to these design principlesof a chaotic cavity, the field properties are progressively im-proved through the modifications from Cavity_1 to Cavity_3.However, even with Cavity_3 the statistical requirements arenot followed by all modes, and the stirring process could be ofsome help especially in the undermoded regime to improve thefield properties [18].

Before checking if the eigenmode field components are nor-mally distributed, we firstly search if modes having a vanishingcomponent appear because of the strong degradation of the fieldisotropy they would imply. As the numerical field values issuedfrom simulations are never strictly null, a criterion is necessaryto decide if a field component of a mode can be considered asvanishing (and thus only attributed to numerical noise) or not.To this aim, the empty parallelepiped-shaped cavity having thesame dimensions has been simulated in the same conditions;as the null components are analytically known in this case, therelated simulation results have been examined, and the criterionof (2) to consider a field component Ei of the electric field �E aspractically vanishing has been determined:

μ ( |Ei |)μ

(| �E |

) ≤ 0.006 (2)

where μ(x) indicates the mean value of x on the 1001 points.This test has been performed on all modes of the four cavities.Contrary to the empty parallelepiped-shaped cavity, no vanish-ing component has been found for the four presented cavities.It means that, even if the vanishing of one field component islocally possible, for example, in a plane where a bouncing ballphenomenon is observed, all the modes have three nonvanishingcomponents.

The KS test is performed with the first 450 eigenmodes ofall cavities. The results obtained for the Ex components of theeigenmodes are given in Fig. 6(a)–(d) for the four cavities. Theincrease of the number of zero-answers due to hemisphere andcap insertion clearly appears. Moreover, the Gaussian charac-ter of the field improves with increasing frequency in the four

Fig. 6. KS test for Ex component of each mode (mode order in abscisse)obtained with (a) Cavity_1, (b) Cavity_2, (c) Cavity_3, (d) Cavity_4.

TABLE ISUCCESS PERCENTAGE OF THE KS TEST PERFORMED ON THE DISTRIBUTION OF

THE THREE FIELD COMPONENTS FOR THE 450 RESONANT MODES OF CAVITIES,AND IN BRACKETS WHILE EXCLUDING THE FIRST 30 MODES

cavities: whereas the Ex components of a large majority of thefirst modes are not normally distributed, the zero answers aremajority after the 120th mode. The same tests performed on Ey

and Ez components of the modes confirm these findings.The results obtained for the three electric field components

are summarized in Table I. The normal law is largely followedin Cavity_3 with a success ratio of the KS test above 89% foreach electric field component when considering all modes. Thelowest ratios are related to the cavity with one single hemisphere.The results obtained for the two other cavities are similar, with amean success ratio over the three components of 84.4% with twohemispheres and 87.3% with a stirrer; however, the cavity witha stirrer presents a higher heterogeneity of the results associatedwith each component, the success ratio varying between 81%and 90% in this case against a variation between 84% and 88%with the two hemispheres.

As in all cavities, most of the first modes do not pass the testand the success ratio increases with the mode order, the successratio has been calculated by excluding the first 30 modes (inbrackets in Table I). All the success ratios are then increased.Thus, in Cavity_3, the success ratio of the KS test exceeds 92%after eliminating the first 30 modes.

As the three field components are normally distributed foran ergodic mode, we then search the modes whose three com-ponents follow a normal distribution. Table II summarizes theresults of this global test for all the modes in all cavities. Itclearly indicates that the insertions of a second hemisphere andthen caps significantly increase the number of modes with threenormally distributed components. Once again, the best homo-geneity is obtained for the cavity with two caps and a hemi-sphere: 79% of the modes have their three components followinga normal distribution, and the success ratio reaches 84% after


TABLE IISUCCESS PERCENTAGE OF THE GLOBAL HOMOGENEITY TEST PERFORMED ON

ALL MODES, OR BY EXCLUDING THE FIRST 30, 50, AND

100 MODES, RESPECTIVELY

Fig. 7. Standard deviation of Ez for (a) Cavity_1 and (b) Cavity_3.

eliminating the first 30 modes. According to this test, the homo-geneity obtained with two hemispheres is better than with thestirrer (with a success ratio of 76% against 72%). As expected,the cavity with one single hemisphere leads to the lowest successratio, with a value of 64%. The success ratios obtained whileexcluding the first 30, 50 or 100 modes are also indicated inTable II. All the success ratios increase with the number of ex-cluded modes, but the conclusions on the comparison betweenthe four cavities remain unchanged.

Besides homogeneity, the field is also required to be isotropicin a well-operating reverberation chamber. We recall that anergodic field is also isotropic [9]. The field isotropy will beexamined in Section IV.

IV. STANDARD DEVIATION AND ISOTROPY OF EIGENMODES

To investigate the field isotropy, the standard deviations ofthe three Cartesian field components of each mode will now beexamined.

The field isotropy implies an equality of the standard devia-tions of each field component. Therefore, we examine here thestandard deviations of the three electric field components foreach eigenmode, and use the difference between them as anindicator of the field isotropy.

The standard deviations σi are calculated for each eigenmodefrom the component values at the 1001 sampling points, for eachcavity. The mode dependence of the standard deviation of Ez

component is depicted in Fig. 7 for Cavity_1 and Cavity_3; sim-ilar fluctuations are obtained for the other field components. Wenotice a decrease of the standard deviations fluctuation when fre-quency increases for both cavities. Whatever the frequency, thestandard deviation fluctuations are much smaller in Cavity_3.The value of 1/

√3 ≈ 0.577 indicated in Fig. 7 corresponds to

TABLE IIIMEANS OF THE STANDARD DEVIATIONS OF EACH COMPONENT AND OF Δσ

FROM THE 31TH MODE

Fig. 8. Δσ of each mode (mode order in abscisse) for (a) Cavity_1,(b) Cavity_2, (c) Cavity_3, (d) Cavity_4.

an ideal Gaussian mode of unit mean amplitude and whose threeorthogonal components have a zero mean and identical standarddeviations. It is expected in a chaotic cavity that the standarddeviations fluctuate around this value. This behavior is observedwith a smaller fluctuation in Cavity_3 than in Cavity_1.

The mean values of the standard variations of each com-ponents are given in Table III; the first 30 modes have beenexcluded to the mean calculation as we already showed that thefirst modes are not homogeneous. The obtained mean valuesare compared (between brackets) to the ideal 0.577 value. Itappears that the smallest differences are obtained with Cavity_3then Cavity_2, whereas Cavity_1 presents the largest ones.

Thus, in the ideal case of an isotropic field, the standarddeviations of the three electric field components are equal. Toevaluate the isotropy of the modes, we propose the indicator Δσdefined in (3). Its value, constrained between 0 and 1, decreaseswhen the three standard deviations become similar. Δσ is equalto 1 when one field component vanishes whereas, in the idealisotropic case, it is vanishing.

Δσ =max(σx, σy , σz ) − min(σx, σy , σz )max(σx, σy , σz ) + min(σx, σy , σz )

(3)

Fig. 8(a)– (d) indicates Δσ varies highly with the mode order.However, this indicator globally decreases with the frequencyin all cavities, with a stabilization of its mean and standarddeviation around the 150th mode. In the whole frequency band,we notice that the lowest mean and maximum values of Δσare obtained with Cavity_3. The mean values of Δσ, calculatedfor each cavity from its 31th mode, are given in Table III. The


Fig. 9. Evolution of the spectral fluctuations. Fluctuating part of the countingfunctions, N − Nav , for (a) Cavity_1, (b) Cavity_2, (c) Cavity_3 and (d) Cav-ity_4 (continuous lines); NTgM is the contribution of the tangential modes (reddotted line). The value of the standard deviation σ of N − Nav is indicated inthe inset.

lowest mean values are obtained with Cavity_3 then Cavity_2,indicating an improvement of the field isotropy compared to theclassical RC equipped with a mode stirrer.

V. SPECTRAL FLUCTUATIONS

RMT has introduced various statistical tools to analyze spec-tral fluctuations of chaotic cavities [6]. We use here the simplestone: the fluctuations of the counting function N(f). The lattergives the actual number of resonant modes up to the frequencyf. Its smooth part Nav is related to the geometry of the cav-ity through Weyl’s law that takes the presence of the caps andhemispheres into account [32]. In our case, Nav is given by apolynomial expression whose leading term at high frequenciesis 8π V (f/c)3 , with V the cavity volume and c the speed oflight. According to RMT, the fluctuations of the modal densityaround the mean modal density characterize the cavity behaviorfrom regular to chaotic [25]. In Fig. 9(a)–(c), the fluctuatingpart of the counting functions N − Nav respectively associatedwith the three chaotic cavities is shown within the frequencyrange [0.4, 1.28 GHz]. The first 19 modes are not shown sincethe large scale fluctuations computed from the tangential modes(TgM) are not pertinent in the corresponding frequency range[0.214, 0.4 GHz]. For Cavity_1, with only one hemisphere, thedeviations from RMT predictions concerning these fluctuationsare important, thus indicating the significant presence of noner-godic modes in the spectrum. Indeed, in spite of the defocusingpart of the boundary, the cavity still possesses parallel walls andremains close to a parallelepiped-shaped room. In such a cavity,many modes are reminiscent of the modes of a regular one forwhich ergodic features are not expected. As already mentioned,the most important family of regular modes consists of TgMwhich, in the parallelepiped-shaped cavity, have a wave vectorquantized in the planes parallel to the walls of the cavity. The

counting function for TgM perpendicular to each axis can beobtained through a procedure introduced in [25] and has re-cently been extended by Gros et al. [33] to the case of RCs. InFig. 9, the red dotted curve corresponds to the contribution ofTgM to N − Nav . Note that the amplitudes of the fluctuationsare progressively reduced from Cavity_1 to Cavity_3 due to thecorresponding suppression of TgM. In Cavity_3, the remain-ing fluctuations are much closer to the amplitude predicted byRMT. As in the previous sections, the comparison is extended toCavity_4 [see Fig. 9(d)] which, here again, exhibits a behaviorintermediate between Cavity_1 and Cavity_2. And this is par-ticularly well illustrated by evaluating the standard deviation σof N − Nav as shown in the inset for each cavity.

VI. FIELD HOMOGENEITY AND ISOTROPY

Uniformity and anisotropy coefficients defined in the norm[31] apply to the field excited within the RC and not to thecavity eigenmodes. In this part, we will rebuild the field at eachfrequency from the previously determined eigenmodes to allowa comparison of the studied cavities regarding the standardizedcriteria. For an enclosure excited at the angular frequency ω, thefield at ω can be expanded on the cavity eigenmodes as [34]

−→E (ω ) = − jωμ

V

∞∑m=1

1k2

m − k2

(∫

V

�J · �em dV

)�em (4)

where k2 = k20 [1 − (−1 + j) · ωm / (Qm ω)], with Qm the

quality factor of the mth eigenmode. �em is the mth eigenmodefield whose mean of the square electric field amplitude on thewhole cavity volume V is normalized to one. km is related to themth resonant angular frequency ωm by km = ωm /c, with c thespeed of light in vacuum. The current �J represents the source.

In the following, a pointlike current source of amplitude 1/√

3along each axis of the Cartesian coordinate system is placed atthe position indicated in Fig. 2(c) and (d). The quality factorsQm are calculated by considering the effects of the Joule losseson the cavity walls and of the cavity loading by an antenna,according to (Qm )−1 = (QJ oule

m )−1 + (Qantm )−1 . The quality

factors QJ oulem and Qant

m , respectively, associated with the Jouleand antenna effects, are calculated in the high frequency ap-proximation [35], [36]:

QJ oulem =

3V

2S δand Qant

m = 16π2 V

(f

c

)3

(5)

with S the total surface area of the cavity walls and δ the wallskin depth.

The modal expansion (4) shows that the spectral width ofeach resonance is nonzero. Thus, the modes are not only ex-cited at their resonance frequencies, but their weighting rapidlydecreases outside the vicinity of the resonant frequency. Thisrebuilt field will be used to apply uniformity and anisotropystandardized criteria to Cavity_3 and Cavity_4. The frequencystep, chosen equal to the third of the smallest frequential modebandwidth, is of 54 kHz for Cavity_3 and 52 kHz for Cavity_4.

The field uniformity is characterized by four standard devia-tions. The standard deviation associated with a field component


Fig. 10. Standard deviations of the three Cartesian field components (blue forEx , red for Ey , green for Ez ,) for (a) Cavity_3 and (b) Cavity_4.

Fig. 11 Total standard deviations for (blue) Cavity_3 and (red) Cavity_4.

Ei (i = x, y or z) is calculated as [31]

σi,dB = 20 · log(

1 +σi

〈Ei max〉

)(6)

where the standard deviation σi and the mean value 〈Ei max〉are calculated from the maximal field amplitudes attained overa stirrer rotation at each of the eight vertices of a cube. A globalindicator σtot is also calculated using the maximal field ampli-tudes obtained by combining the three field components.

As in our study the stirrer (or hemisphere) is modeled at afixed location, we replace the stirrer motion by the translationof the eight sampling points. For that purpose, we consider acube of size 0.25 m whose edges are parallel to the Cartesiancoordinate axes, and translate it by steps of 0.1 m along the axeswithin the cavity central volume of dimensions 0.45, 0.55, and0.55 m, respectively, along x-, y-, and z-axes. The three Cartesiantotal field components are sampled at the eight vertices of thiscube for each of the 48 positions of the cube, and the eightmaximal values over the cube positions are determined. Thefour resulting standard deviations are plotted with respect to thefrequency in Figs. 10–11 for Cavity_3 and Cavity_4.

According to the norm [31], a good approximation of the LUFis given by the 60th mode resonance frequency. This correspondsto about 665 MHz for the studied cavities. We can notice thatthe 3 dB limit for the total standard deviation (see Fig. 11) is notrespected for the conventional cavity in the studied frequencyband. For Cavity_3, the 3 dB limit is respected by the totalstandard deviation from 546 MHz except for three frequencies(associated with peaks between 3 and 3.31 dB); because of thetolerance of the norm for a maximum of three peaks above 3 dBper octave, the criterion relative to the standard deviation can be

TABLE IVMEANS (IN DB) OF THE STANDARD DEVIATIONS (6) OF EACH CARTESIAN FIELD

COMPONENT AND OF THE TOTAL STANDARD DEVIATION, FOR CAVITY_3AND CAVITY_4

Fig. 12. Field anisotropy coefficients versus frequency for (a) Cavity_3 and(b) Cavity_4. The blue curve corresponds to Axy , the red one to Ay z , the greenone to Az x , and the black one to Atot .

TABLE VMEANS (IN DB) OF THE ANISOTROPY COEFFICIENTS (8) FOR CAVITY_3

AND CAVITY_4

considered as respected from this frequency. The mean valuesof the four standard deviations obtained for both cavities (seeTable IV) clearly indicate that a better homogeneity is attainedwith Cavity_3.

Using the same procedure, we also determined the fieldanisotropy coefficients given by [11]

〈Aαβ 〉 =⟨

|Eα |2 − |Eβ |2|Eα |2 + |Eβ |2

⟩(7)

where α and β indicate directions of mutually orthogonal Carte-sian components, and the average is performed first on the cubepositions then on the eight cube vertices. A global anisotropyindicator can also be calculated using

〈Atot〉 =√[

A2xy + A2

yz + A2zx

]/3. (8)

At each frequency, a volumetric average of these four coeffi-cients is then performed by calculating their arithmetic meansover the eight cube vertices. The obtained anisotropy coefficientsare given in Fig. 12. The comparison of the curves obtained forboth cavities (see Fig. 12), as well as the mean values of the fouranisotropy coefficients (see Table V) clearly show that a betterfield isotropy is attained with Cavity_3, with anisotropy coeffi-cients below the expected −3 dB limit from 622 MHz exceptfor a few isolated frequencies.


VII. CONCLUSION

As most resonant modes in chaotic cavities present spatiallyhomogeneous and isotropic fields, we drew inspiration fromstudies developed in the field of wave chaos to propose threesimple geometric modification of a parallelepiped-shaped cavitywith the aim of following the requirements for well-operatingRCs. Using simulation results, comparisons between these threechaotic cavities and a classical RC equipped with a mode stirrerhave been performed. In a first stage, we demonstrated that theratio of field components of modes following a normal distribu-tion is, for two chaotic cavities, higher than for the classical RC,and that this ratio grows with increasing frequency. The fieldisotropy was also discussed from the analysis of the standarddeviations of the three components and, for each mode, of theirdispersions; the good performances of the chaotic cavities wereconfirmed. In complete accordance with these findings in thespatial domain, we also performed a comparison in the spectraldomain. We showed how the modifications of the chaotic cav-ities allow a significant reduction of the role of regular modeson the spectral fluctuations: by reducing the amount of facingparallel surfaces in the chaotic cavities, the spectral fluctuationsbecome closer to those expected from RMT predictions. By re-building the total electric field from the simulated eigenmodes,we then applied standard homogeneity and isotropy criteria,which also permitted to conclude that better performances areobtained with the chaotic cavity than with the conventional RC.

The major conclusion from the results presented in this paperis that facing parts of parallel walls should be eliminated fromany RC where ergodicity of modes is wanted. The very simplecavity modifications we propose, consisting of inserting metal-lic hemispheres or caps on the cavity walls, while also avoid-ing placement that results in geometrical symmetries, permit asignificant improvement of the field statistical properties. Thestirring process is not addressed in this paper but the adaptationof these new cavity shapes to mechanically stirred reverberationchambers could be performed in two ways. The first one wouldconsist in improving a classical RC equipped with a stirrer byinserting three metallic caps on its walls, so that no parallel flatfacing surfaces subsist; the advantage of this approach residingin the very simple and low cost required modification. In thesecond one, the chaotic cavity itself could be used as an RC andone hemisphere could be used as a mode-stirrer by moving iton the cavity wall. In both approaches, it is expected that thespectral overlap of homogeneous and isotropic modes will leadto better statistical field properties than when the modes do notindividually meet the required statistical properties. It would re-sult in the improvement of the reverberation chambers operationespecially in the weak overlap regime, with a potential decreaseof their LUF.

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Kamardine Selemani was born in the Comoros in1986. He received the M.Sc. degree in High frequencycommunication systems and the Ph.D. degree intelecommunications and electronics from UniversiteParis-Est Marne-la-Vallee, Marne-la-Vallee, France,in 2010 and 2014, respectively.

He is a Graduate Teaching/Research Assistantwith Universite Paris-Est-Marne-la-Vallee. His re-search interests include reverberation chamber, com-putational electromagnetics, and EMC.

Jean-Baptiste Gros was born in Cannes, France, in1987. He received the Master’s degree in fundamentalphysics from University of Nice-Sophia Antipolis,Nice, France, in 2011. He is currently working towardthe Ph.D degree.

His research interests include wave chaos, chaoticscattering in complex systems.

Elodie Richalot received the Diploma and Ph.D.degrees in electronics engineering from theEcole Nationale Superieure d’Electronique,d’Electrotechnique, d’Informatique, etd’Hydraulique de Toulouse, Toulouse, France,in 1995 and 1998, respectively.

Since 1998, she has been with the Universityof Marne-la-Vallee, Champs-sur-Marne, France,where she became a Professor in electronics in 2010.Her current research interests include modelingtechniques, electromagnetic compatibility, urban

propagation, and millimeter wave transmission lines.

Olivier Legrand was born in France in 1961. He re-ceived the Ph.D. degree in theoretical physics fromthe Universite d’Aix-Marseille II, Marseille, France,in 1987, and the Habilitation degree from the Univer-site de Nice-Sophia Antipolis, Nice, France, in 1998.

In 1990, he joined the Universite de Nice-SophiaAntipolis, where, since 2001, he has been a Profes-sor of physics. His current research interests includelinear and nonlinear waves, wave chaos, and wavepropagation in complex media.

Odile Picon received the Agregation de Physique de-gree from the Ecole Normale Superieure de Fontenayaux Roses, Lyon, France, in 1976, the Doctor degreein external geophysics from the University of Orsay,Orsay, France, in 1980, and the Doctor in Physicsdegree in microwave CAD from the University ofRennes, Rennes, France, in 1988.

She was a Teacher from 1976 to 1982. Then shewas a Research Engineer with the Space and Ra-dioelectric Transmission Division, Centre Nationald’Etudes des Telecommunications (CNET), Paris,

France, from 1982 to 1991. Since 1991, she has been a Professor of electricalengineering, first at the Paris7-University, Paris, France, then at the Universityof Paris-Est Marne-la-Vallee, Marne-la-Vallee, France, where she is headingthe ESYCOM Laboratory. She has published more than 150 technical papers inbooks, journals, and conferences. Her research work deals with electromagnetictheory, numerical methods for solving field problems, and design of millimeter-wave passive devices.

Fabrice Mortessagne is currently a Physicist, a Pro-fessor at the University Nice Sophia Antipolis (UNS),and the Director of “Laboratoire de Physique de laMatiere Condensee” (UNS and CNRS).

His research activities include field of wave prop-agation in complex media and mesoscopic physics.His current works include chaotic scattering experi-ments in optics and microwaves, and microwave re-alizations of artificial graphene and other relativisticmeta-materials.

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