Univeral Statitic of the Scattering Coefficient of Chaotic Microwave Cavitie Sameer Hemmady 1,, Xing Zheng, Thoma M. Antonen Jr. 1,, Edward Ott 1, and Steven M. Anlage 1,3. Department of Phyic, Univerity of Maryland, College Park, MD-074. (Dated: March 6 th, 005 Abtract: We conider the tatitic of the cattering coefficient S of a chaotic microwave cavity coupled to a ingle port. We remove the non-univeral effect of the coupling from the experimental S data uing the iation impedance obtained directly from the experiment. We thu obtain the normalized, complex cattering coefficient whoe Probability Denity Function (PDF i predicted to be univeral in that it depend only on the lo (quality factor of the cavity. We compare experimental PDF of the normalized cattering coefficient with thoe obtained from Random Matrix Theory (RMT, and find excellent agreement. The reult apply to cattering meaurement on any wave chaotic ytem. PACS# : 05.45.Mt, 03.65.Nk,11.55.-M,03.50.De,84.40.-X,84.40.Az I. Introduction: The cattering of hort wavelength wave by chaotic ytem ha motivated intene reearch activity, both theoretically and experimentally [1 3]. Some example of wave chaotic ytem include quantum dot [4], atomic nuclei [5], microwave cavitie [, 6], and acoutic reonator [7]. Since thee ytem all have underlying chaotic ray dynamic, the wave pattern within the encloure, a well a the repone to external input, can be very enitive to mall change in frequency and to mall change in the configuration. Thi motivate a tatitical approach to the wave cattering problem. The univeral ditribution for chaotic cattering matrice can be decribed by Dyon circular enemble [9, 10]. However, the circular enemble cannot typically be directly compared with experimental data becaue it applie only in the cae of ideal coupling (which we define ubequently, while in experiment there are nonideal, ytem-pecific effect due to the particular mean of coupling between the cattering ytem (e.g, a microwave cavity and the outide world. Thi non-univerality of the raw experimental cattering data ha long been appreciated and addreed in theoretical work [11 13]. Of particular note i the work of Mello, Peveyra and Seligman (MPS which introduce the ditribution known a the Poion kernel, where a cattering matrix < S > i ued to parameterize the non-ideal coupling. To apply thi theory to an experiment it i typically neceary to pecify a procedure for determining a meaured etimate of < S >. The firt microwave experiment invetigating cattering tatitic by comparion with the theory of Ref. [11-13] were thoe of Ref. [15]. In analyzing their experimental data, the author of Ref.15 conidered the 1 Alo with the Department of Electrical and Computer Engineering. Alo with the Intitute for Reearch in Electronic and Applied Phyic. 3 Alo with the Center for Superconductivity Reearch. power reflection coefficient S and ued value of thi quantity averaged over a number of configuration and over a uitable frequency range ( f 0 f / to ( f 0 + f / to compare with theory. For a given number of averaging configuration, the choen frequency range mut be large enough that good tatitic are obtained, but at the ame time be mall enough that the variation of the coupling with frequency i not ignificant. Subequent work by the author of Ref.[15,16], focued on the complex cattering amplitude. Again average of meaured value were made over a range of frequencie and a number of different configuration. We denote thi experimentally obtained average a S. If the frequency range i mall enough and a ufficient number of independent meaurement are obtained S will yield a good approximation to the deired < S >. The quantity < S > in the theory decribe direct (or prompt procee [1,14] that depend only on the local geometry of the coupling port, a oppoed to complicated chaotic procee reulting from multiple reflection far removed from the coupling port. On the other hand, in obtaining S, a in Ref. [16], averaging of the cattering data of the full chaotic ytem i employed. Thu the data ued to obtain S i the ame meaured data whoe tatitic are being tudied. Alo note that < S > i preumed to characterize the coupling, which i independent of the chao of the ytem and i thu, in principle, a non-tatitical quantity. In thi paper we hall purue another approach [17,18]. Specifically we eek to characterize the coupling in a manner that i both independent of the chaotic ytem and obtainable in a non-tatitical manner (i.e. without employing average. A we explain in more detail ubequently, thi latter point i of practical importance becaue of the inherent inaccuracy and ample ize iue introduced by averaging a fluctuating quantity.
A dicued in Ref. 17 and 18, a direct mean of invetigating the tatitic of typical chaotic cattering ytem can be baed on determination of the iation impedance of the port Z or equivalently S, the complex iation cattering coefficient. Thee will be dicued in more detail ubequently. The iation cattering coefficient (repectively, iation impedance i the cattering coefficient (impedance that would be oberved if the ditant boundarie of the cavity were made perfectly aborbing or moved to infinity. Therefore, it decribe prompt procee at the port and can be hown to be equal to < S >. The perfect coupling cae correpond to = 0, in which all incident wave S energy enter the cavity. The iation cattering coefficient can be directly meaured in microwave experiment without reorting to averaging over a range of frequencie. Note, that S S depend on the microwave frequency and thu, for the purpoe of taking into account coupling, S can be a more ueful and robut mean of extracting univeral propertie from the data than S, which depend on the frequency range f over which the averaging i done. The impedance i another fundamental quantity characterizing coupling to a cavity i the impedance Z. It i related to S through the bilinear tranformation, where Z o S = ( Z + Z ( Z 1 o Z o, (1 i the characteritic impedance of the tranmiion line or waveguide feeding the port. The impedance linearly relate the voltage and the current phaor at the port and i determined olely by propertie of the cavity and it port. In what follow we only dicu the one-port cae, hence Z and S are calar (rather than matrice throughout thi paper. A more general dicuion involving multiple port can be found in [19]. Inverting the tranformation Eq.(1, we can relate the iation impedance Z = R + ix to the iation cattering coefficient Z S a, (1 + S = Z o. ( (1 S In Ref.[18-0] it wa hown that the cavity impedance Z can be expreed in term of a caled impedance z and the iation impedance a, Z Z = ix + R z, (3 where z i a complex random variable atifying < z >= 1. The random variable z ha univeral propertie and decribe the impedance fluctuation of a perfectly coupled cavity. The real part of z i well known in olid tate phyic a the local denity of tate (LDOS and it tatitic have been tudied [1,]. The imaginary part of z determine fluctuation in the cavity reactance. Dicuion of the probability ditribution of the normalized impedance z for microwave experiment ha been preented in a previou paper [18]. The purpoe of thi paper i to tudy the univeral tatitical propertie of the cavity cattering coefficient defined in term of the normalized impedance z, i = ( z 1 /( z + 1 = e φ, (4 which can be compared directly to theoretical prediction baed on ideal coupling. Combining Eq. (, (3 and (4, we obtain a formula relating to the cavity cattering coefficient S and the cavity iation cattering coefficient S, (1 + S (1 + S * ( S S (1 SS * =. (5 The invere of Eq.(5 giving the actual cattering amplitude S in term of the normalized cattering amplitude i a tatement of the Poion Kernel [11,1] for a ingle port cavity with internal lo. We note that the firt factor in Eq.(5 i imply a phae hift which depend only on the coupling geometry. Thu, the magnitude of atifie, S S 1 SS * = (6 According to the tatitical theory, the only parameter on which the tatitic of z and depend i the lo due to the cavity wall aborption, which can be controlled and quantified in our experiment. We will experimentally verify the key theoretical prediction baed on the circular enemble hypothei, uch a the tatitical independence of the magnitude and phae of the normalized cattering coefficient, and the uniform probability ditribution for the phae. Different degree of lo and different coupling tructure will be examined, and approximation to the probability denity function of and z will be compared with the theoretical prediction from Random Matrix Theory (RMT. An expreion imilar to Eq.(6 wa ued by Kuhl et al.[16] to generate ditribution of the cattering amplitude S baed on theoretical prediction for the normalized cattering amplitude. Thu, it wa aumed that the normalized cattering amplitude had the predicted propertie of independence of magnitude and phae, and uniform ditribution of phae. We will experimentally tet thee aumption by uing Eq.(5 directly to determine the propertie of.
Our paper i organized a follow. Section II preent our experimental etup and data collection. Section III carrie out the normalization proce to recover univeral cattering characteritic and preent experimental hitogram approximation to the probability denity function of the magnitude and phae of the normalized cattering coefficient for different coupling tructure and loe. Section IV explore a predicted relationhip between the average cavity power reflection coefficient ( S and the magnitude of the iation cattering coefficient. Section V dicue the advantage of employing the iation cattering coefficient in uncovering univeral propertie, or of predicting raw cattering data. Section VI conclude the paper and give the ummary. far enough from the antenna o a not to alter it near-field characteritic. For each configuration, the cattering coefficient S i meaured in 8000 equally paced tep over a frequency range of 6 to 1 GHz. In total, one hundred different configuration are meaured, reulting in an enemble of 800,000 S value. We refer to thi tep a the cavity cae. II. Experimental Setup: Microwave cavitie with irregular hape (having chaotic ray dynamic have proven to be very fruitful for the tudy of wave chao, where not only the magnitude, but alo the phae of cattering coefficient, can be directly meaured from experiment. Our experimental etup conit of an air-filled quarter bow-tie chaotic cavity (Fig.1(a which act a a two dimenional reonator below about 19 GHz [3]. Ray trajectorie in a cloed billiard of thi hape are known to be chaotic. Thi cavity ha previouly been ued for the ucceful tudy of the eigenvalue pacing tatitic [6] and eigenfunction tatitic [4,5] for a wave chaotic ytem. In order to invetigate a cattering problem, we excite the cavity by mean of a ingle coaxial probe whoe expoed inner conductor, with a diameter (a extend from the top plate and make electrical contact with the bottom plate of the cavity (Fig.1(b. In thi paper we tudy the propertie of the cavity over a frequency range of 6-1 GHz, where the pacing between two adjacent reonance i on the order of 5 30 MHz. A in the numerical experiment in Ref.[18,19], our experiment involve a two-tep procedure. The firt tep i to collect an enemble of cavity cattering coefficient S over the frequency range of interet. Enemble averaging i realized by uing two rectangular metallic perturbation with dimenion 6.7 x 40.6 x 7.87 mm 3 (~1% of the cavity volume, which are ytematically canned and rotated throughout the volume of the cavity (Fig.1(a. Each configuration of the perturber within the cavity volume reult in a different value for the meaured value of S. Thi i equivalent to meaurement on cavitie having the ame volume, lo and coupling geometry for the port, but with different hape. The perturber are kept Fig.1: (a The phyical dimenion of the quarter bow-tie chaotic microwave reonator are hown along with the poition of the ingle coupling port. Two metallic perturbation are ytematically canned and rotated throughout the entire volume of the cavity to generate the cavity enemble. (b The detail of the coupling port (antenna and cavity height h are hown in cro ection. (c The implementation of the iation cae i hown, in which commercial microwave aborber i ued to line the inner wall of the cavity to minimize reflection. The econd tep, referred to a the iation cae, involve obtaining the cattering coefficient for the excitation port when wave enter the cavity but do not return to the port. In the experiment, thi condition i realized by removing the perturber and lining the ide wall of the cavity with commercial microwave aborber (ARC Tech DD10017D which provide about 5dB of reflection lo between 6 and 1 GHz (Fig.1.(c. Note that the ide wall of the cavity are outide the near field zone of the antenna. Uing the ame frequency tepping of 8000 equally paced point over 6 to 1 GHz, we meaure the iation cattering coefficient for the cavity. Such an approach approximate the ituation where the ide wall are moved to infinity; therefore S S doe not depend on the chaotic ray trajectorie of the cavity, and thu give a 3
characterization of the coupling independent of the chaotic ytem. Becaue the coupling propertie of the antenna depend on the wavelength and thu vary over S frequency, i uually frequency dependent. Having meaured the cavity S and S, we then tranform thee quantitie into the correponding cavity and iation impedance ( Z and repectively uing Z Eq. (1 and (. The normalized impedance z i obtained by Eq. (3. In order to obtain z, every value of the determined cavity impedance Z i normalized by the correponding value of at the ame frequency. The Z tranformation in Eq.(4 (or equivalently, Eq.(5 yield the normalized cattering coefficient = exp( i, which φ i the key quantity of interet in thi paper. Since the artifact of non-ideal coupling are uppoed to have been filtered out through thi normalization proce, the tatitic of the enemble of value hould depend only on the net cavity lo. In order to tet the validity of the theory for ytem with varying lo, we create different cavity cae with different degree of lo. Lo i controlled and parameterized by placing 15. cm-long trip of microwave aborber along the inner wall of the cavity. Thee trip cover the ide wall from the bottom to top lid of the cavity. We thu generate three different lo cenario (Lo Cae 0, Lo Cae 1, Lo Cae 3 (hown chematically in the inet to Fig 4(b. The number 0, 1, 3 correpond to the number of 15. cm -long trip placed along the inner cavity wall. The total perimeter of the cavity i 147.3 cm. The theory predict that a long a the lo i the ame, the normalized z or will have the ame tatitical behavior. Thi prediction will be teted in our experiment with two different coupling geometrie correponding to coaxial cable with two different inner diameter (a=1.7mm and a=0.65mm, chematically hown in Fig.1(b. III: Experimental Reult and Data Analyi: In thi ection, we preent our experimental finding for the tatitical propertie of the normalized cattering coefficient, for different coupling geometrie and degree of lo. Thi ection i divided into three part. In the firt part, we give an example for the PDF of at a pecific degree of quantified lo and a certain coupling geometry. In the econd part, we fix the degree of quantified lo, but vary the coupling by uing coaxial cable antenna having inner conductor of different diameter (a=1.67mm and a=0.635mm. The PDF hitogram for the magnitude and phae of in thee two cae will be compared. Finally, in the third part, we tet the trend of the PDF of for a given coupling geometry and for three different degree of quantified lo. Good agreement with random matrix theory i found in all cae. III (a:statitical Independence of and φ : Fig.: (a Polar contour denity plot for the real and imaginary component of the normalized cavity ( = exp( iφ for Lo cae 0 in the frequency range of 6 to 9.6 GHz. The angular lice with the ymbol (triangle, circle, quare indicate the region where the PDF of i calculated and hown in (b. Oberve that the PDF of the three region are eentially identical. (c The PDF of the phae φ of the normalized cattering coefficient for two annuli defined by 0. 3 0 (tar and 0.3 < 0.6 (hexagon. Oberve that thee phae PDF are nearly uniform in ditribution. The uniform ditribution i hown by the olid line ( P ( φ = 1/ π. Thi i conitent with the prediction that the i tatitically independent of the phae φ, of. The firt example we give i baed on lo cae 0 (i.e., no aborbing trip in the cavity and coupling through a coaxial cable with inner diameter a=1.7mm. Having obtained the normalized impedance z, we tranform z into the normalized cattering coefficient uing Eq. (4. Since the wall of the cavity are not perfect conductor, the normalized cattering 4
coefficient i a complex calar with modulu le than 1. (In Lo Cae 0, mot of the lo occur in the top and bottom cavity plate ince they have much larger area than the ide wall. Baed on Dyon circular enemble, one of the mot important propertie of i the tatitical independence of the cattering phae φ and the magnitude. Fig.(a how a contour denity plot of in the frequency range of 6 to 9.6 GHz for Lo Cae 0. The graycale level at a given point in Fig. (a indicate the number of point (for {Re(, Im(} that fall within a local rectangular region of ize 0.0 X 0.0. We next arbitrarily take angular lice of thi ditribution that ubtend an angle of π/4 ian at the center, and compute the hitogram approximation to the PDF of uing the point within thoe lice. The correponding PDF of for the three lice are hown in Fig.(b. We oberve that thee PDF are eentially identical, independent of the angular lice. Fig.(c how PDF of φ computed for all the point that lie within two annuli defined by 0 0.3 (tar and 0.3 < 0. 6 (hexagon. Thee plot upport the hypothei that the magnitude of i tatitically independent of the phae φ of and that φ i uniformly ditributed in 0 to π. To our knowledge, thi repreent the firt experimental tet of Dyon circular enemble hypothei for wave chaotic cattering. III(b : Detail Independence of : Fig.3: (a PDF for the un-normalized Lo Cae 0 cavity the frequency range of 6 to 11.85 GHz for two different coupling antenna diameter a = 0.635 mm (circle and a = 1.7 mm (olid tar. (b PDF for the normalized cavity in the frequency range S in of 6 to 11.85 GHz for two different coupling antenna diameter a = 0.635 mm (circle and a = 1.7 mm(olid tar. Note that the diparitie een in Fig. 3.(a on account of the different coupling geometrie diappear after normalization. (c PDF for the unnormalized cavity phae ( φ S for Lo Cae 0 in the frequency range of 6 to 11.85 GHz for two different coupling antenna diameter a = 0.635 mm (circle and a = 1.7 mm (olid tar. (d PDF for the normalized cavity phae (φ in the frequency range of 6 to 11.85 GHz for two different coupling antenna diameter a = 0.635 mm (circle and a = 1.7 mm (olid tar. The normalized phae PDF for the tar and circle in Fig. 3(d are nearly uniformly ditributed (the gray line in Fig. 4(d how a perfectly uniform ditribution P ( φ = 1/ π. To further verify that the normalized doe not include any artifact of ytem-pecific, non-ideal coupling, we take two identical wave chaotic cavitie and change only the inner diameter of the coupling coaxial cable from a=1.7 mm to a=0.635 mm. Since the modification of the coaxial cable ize barely change the propertie of the cavity, we aume that the lo parameter in thee two cae are the ame. The difference in the coupling geometry manifet itelf a gro difference in the ditribution of the raw cavity cattering coefficient S. Thi i clearly obervable for the PDF of the cavity power reflection coefficient S a hown in Fig. 3(a and the PDF for the phae of S (denoted φ hown in Fig.3(c, for Lo Cae 0 over a frequency range of 6 to 11.85 GHz. However, after meaurement of the correponding iation impedance and the normalization procedure decribed above, we oberve that the PDF for the normalized power reflection coefficient are nearly identical, a hown in (Fig.3(b for and the phae φ in (Fig.3(d. Thi upport the theoretical ( prediction that the normalized cattering coefficient i a univeral quantity whoe tatitic doe not depend on the nature of the coupling antenna. Similarly, in Fig.3(c, the phae φ of the cavity cattering coefficient S how preference for S certain angle. Thi i expected becaue of the non-ideal coupling (impedance mimatch that exit between the antenna and the tranmiion line. After normalization, the effect of non-ideal coupling are removed and only the cattering phae of an enemble of ideally coupled chaotic ytem (in which all cattering phae are equally likely i een. Hence, conitent with theoretical expectation, the phae φ of normalized data how an approximately uniform ditribution (Fig.3(d. III(c : Variation of with Lo : Having etablihed that the coupling geometry i irrelevant for the ditribution of, we fix the coupling geometry (coaxial cable with inner diameter a=1.7 mm and vary the degree of quantified lo within the cavity. Three lo cae will be conidered, namely, lo cae 0, 1 and 3. In Ref.[18] the degree of lo i characterized by a ingle damping ~ ~ parameter k / Q. Here, k, where k=πf/c i the wave number for the incoming frequency f and S = k / k n k n i the 5
mean pacing of the adjacent eigenvalue k n. The quantity Q i the quality factor of the cavity. The ~ parameter k / Q repreent the ratio of the frequency width of the cavity reonance due to ditributed loe, and the average pacing between reonant frequencie. ~ for value of k / Q that are obtained by computing the variance, while the ymbol are obtained from hitogram approximation to the PDF of the normalized impedance z extracted from the experimental data over a frequency range of 6.5 to 7.8 GHz. Generally, a the lo of the cavity increae, the PDF of the normalized imaginary part of the impedance loe it long tail and begin to harpen up, developing a Gauian appearance (Fig.4(b. At the ame time, the PDF of the normalized real part moothly evolve from being peaked between Re( z = 0 and Re( z = 1, into a Gauian-like ditribution that peak at Re( z = 1 (Fig.4(a. Fig.4: PDF for the (a real and (b imaginary part of the normalized cavity impedance z for a wave chaotic microwave cavity between 6.5 and 7.8 GHz with h = 7.87 mm and a = 1.7 mm, for three value of lo in the cavity (open tar: 0, circle: 1, quare: 3 trip of aborber. Alo hown are ingle parameter, imultaneou fit for both ~ PDF (olid curve, where the lo parameter k /Q i obtained from the variance of the data in (a and (b. σ For ufficiently high lo, the variance ( of the PDF of the real and imaginary part of z can be related to ~ k / Q by [18], ~ σ = σ = Q /( π k. (7 Re( z Im( z Thi relation ha been experimentally validated for different cavitie and for different coupling geometrie [17]. Thu we can determine the damping parameter from meauring the variance of the PDF of the real or imaginary part of z (uch a thoe hown in Fig. 4. With thi parameter determined, we ue a Monte-Carlo imulation baed on random matrix theory (ee Eq.(9 of Ref.[17] and dicuion therein to calculate the theoretically predicted PDF of z and. (Approximate formula for the PDF of Re[z ] [1] and Im[z] [6] which agree well with the Monte Carlo reult are alo available. The olid curve in Fig. 4 are plot from RMT r = Fig.5: PDF for the normalized power reflection coefficient on a natural log cale for Lo Cae 0, 1, 3 (tar, circle and quare repectively in the frequency range of 6.5 to 7.8 GHz. Thee are from the ame data a ued to obtain Fig.4. Alo hown i the prediction of the model ~ in [17] (olid line for P(r uing the value of k / Q obtained from the variance of the ditribution in Fig.4. The ymbol in Fig.5 (preented on a emi-log cale how the PDF of the normalized power reflection coefficient ( r = in the frequency range 6.5 to 7.8 GHz for three different lo cae. The olid line are the prediction for the PDF of r, P(r for different value of the lo parameter / Q. ~ In Fig.5, the k / Q parameter are the ame a thoe for Fig. 4, and were obtained from the variance of the PDF of the real or imaginary part of z. We oberve that our data conform well to the prediction from random matrix theory for all degree of lo. ~ k 6
IV: Relationhip between Cavity and Radiation Power Reflection Coefficient : Fig.6: Dependence of the average of the cavity power reflection coefficient coefficient on the magnitude of the iation cattering, for different lo cae (Lo Cae 0: tar; Lo Cae 1: circle; Lo Cae 3: quare. The data i hown for the frequency range of 6.5 to 7.8 GHz. Alo hown are the numerical imulation from ~ the RMT baed upon the k / Q value 0.81,.4 and 6.5 for lo cae 0, 1 and 3 repectively (olid line. A a final experimental tet, we would like to examine how the meaured cavity power reflection coefficient depend only on the iation cattering coefficient and loe in the cavity. Ref.[18] predict that the average value of the cavity power reflection coefficient S depend only on the magnitude of the iation cattering coefficient ( S and the lo parameter ~ k / Q S, and i independent of the phae of. The quantity S i related to the iation impedance ( Z = R + ix through the tranformation, S ( R Z o + ( X =. (8 ( R + Z + ( X o We conider a cavity having quantified lo (Lo Cae 0, 1 and 3, with a coupling port of diameter a=1.7 mm and over the frequency range of 6.5 to 7.8 GHz. Having experimentally generated the normalized z a decribed above, we then imulate an enemble of imilar cavitie but with different coupling geometrie. Thi i done by mean of a lole two-port impedance tranformation [18] of our z data, a decribed by the relation, z 1 (1/ z + iβ =. (9 which correpond to adding a reactive impedance i / β in parallel with the impedance z. The quantity z thu imulate the impedance of a hypothetical cavity that i non-ideally coupled to the excitation port, and the coupling geometry i characterized by the real factor β, which can be varied in a controlled manner. We alo define a tranformed iation impedance ( For the generation of 1 z (1 + iβ z given by, =. (10 z, the factor β i varied over z. Having the ame range of value a ued to generate determined z and it correponding, we determine the cattering coefficient and through the tranformation, z = ( z 1 /( z +1 (11 ( z 1 /( z +1 (1 = A range of β value are choen to cover all poible coupling cenario. We then plot the average of (i.e., a a function of. Thi approach i followed for all three lo cae (Lo cae 0, 1 and 3 reulting in the data et with tar, circle and quare, repectively, in Fig.6. Firt note that all curve originate from the point = 1, = which may be thought of a the perfectly mimatched cae. Next conider < 1, and oberve that a the loe increae, the curve hift downward for a fixed coupling (characterized by. Thi i intuitively reaonable becaue, a the aborption (loe within the cavity increae, we expect le ignal to return to the antenna (i.e. maller for a given coupling. From the variance of the PDF of ~ Re[z] for the above lo cae, we determine k / Q to be 0.81,.4 and 6.5 for Lo Cae 0,1 and 3 repectively. The olid line in Fig.6 are obtained from the RMT theoretical prediction for the perfectly coupled cattering coefficient with the appropriate value for ~ k / Q. Next, thee are tranformed uing Eq. (9 and (10 with the ame range of coupling factor ( β a ued for the experimentally determined z. We oberve good agreement between the numerical imulation from RMT (olid line in Fig. 6 and our experimentally derived point. 7
For a given loy cavity one can alo conider it lole N -port equivalent. By the lole N -port equivalent we mean that the effect of the loe ditributed in the wall of our cavity can be approximated by a lole cavity with N 1 extra perfectly-coupled (pc port through which power coupled into the cavity can leave. The point in Fig.6 correpond to perfect coupling. In thi cae, Ref. [18] predict that the vertical axi intercept of thee curve correpond to the lole N -port equivalent of the ditributed loe within the cavity; i.e., at S = = 0 = 0 we have = /( N + 1 ( for time-reveral ymmetric wave chaotic ytem. Thu, in our experiment the quantified lo in Lo Cae 0, 1 and 3 i equivalent to ~ 11, 4 and 45 perfectly-coupled port, repectively. In other word, for all intent and purpoe, the cavity can be conidered lole but perfectly coupled to thi number of port. pc produce the black trace denoted a < S > 100. Note that even with one hundred cavity rendition, the fluctuation in < S are till preent and are een a the meander in the > 100 black trace. The red trace, which correpond to the iation cattering coefficient for thi antenna geometry, i devoid of uch fluctuation (becaue there are no reflected wave from the far wall back to the port and i eaily obtainable in practice without reorting to generating large enemble et of cavity configuration. Moreover, ince the iation impedance of the port i alo a function of frequency, there i no contraint on the frequency pan where the analyi for obtaining the univeral tatitic of (or z can be carried out. V : Validating the Ue of Radiation Impedance to Characterize Non-ideal Coupling. In ection III we ued the iation impedance ( Z, or the iation cattering coefficient ( S, a a tool to characterize the non-ideal coupling (direct procee between the antenna and the cavity. Thi quantity i meaurable and i only dependent on the local geometry around the port. A previouly noted, Ref.[15,16] ue configuration and frequency averaged cattering data to obtain an approximation to < S >. For a given center frequency, thi procedure relie on the f 0 atifaction of two requirement: firt the range of f mut be large enough to include a large number of mode; econd, S mut vary little over the range of f. The nature of the variation of S with frequency i illutrated in Fig. 7, where a plot (Re( S, Im( S of the complex cattering coefficient for a cavity in the frequency range of 6 to 1 GHz i hown. The blue trace how reult for S for a ingle configuration of the cavity correponding to a given poition and orientation of the perturber (Fig.1(a. Iolated reonance are een a circular loop in the polar plot. The degree of coupling i indicated by the diameter of the loop. Frequency range where the coupling i good would manifet themelve a large loop, while thoe frequency range with poor coupling reult in maller loop. Enemble averaging one hundred uch different configuration of thi cavity for different poition and orientation of the perturber Fig.7: (color online: Polar plot for the cavity cattering coefficient S = Re( S + i Im( S i hown for a frequency range of 6 to 1 GHz for Lo Cae 0 and with a coupling port of diameter a = 1.7 mm. The blue trace repreent one ingle rendition of the cavity for a elected poition and orientation of the perturber. Each circular loop repreent an iolated reonance. The black trace i the enemble average < S over one hundred different location and orientation of the perturber within the cavity. The meandering nature of the black trace how that remnant of the cavity reonance are till preent becaue of the finite number of enemble average. The red trace how the iation cattering coefficient for the ame port. Thi trace i mooth becaue the iation cattering coefficient approximate the cavity boundarie being extended to infinity. To quantitatively illutrate thi point, we imulate the nonuniveral cattering tatitic of a given cavity for a given type of coupling uing only the meaured iation impedance of the coupling port and the numerically generated normalized impedance z from RMT, which depend only upon the net loe within the cavity. We conider a Lo Cae 0 cavity, over a frequency range of 6 to 7.5GHz, which i excited by mean of a coaxial cable of inner diameter (a=1.7mm. The variation in < S > (inet of Fig.8(b indicate that the coupling 100 characteritic for thi etup fluctuate over the given frequency range, undergoing roughly four or five ocillation over a > 100 8
range in < S > 100 of order 0.. Thu the frequency averaged < S would be expected to be an unreliable > 100 etimate to parameterize the coupling over thi frequency range. Eq.(5 to obtain an etimate of the non-univeral ytempecific cattering coefficient, which we denote a S ~. ~ In Fig.8(a, the PDF of S i hown a the olid trace, while the experimentally meaured PDF of S i hown a the tar. While in Fig. 8(b, the PDF of φ ~ i hown a the olid S trace with experimentally meaured PDF of φ S hown a the tar. We oberve good agreement between the numerically generated etimate and the actual data. Thi reult validate the ue of the iation impedance (cattering coefficient to accurately parameterize the ytem-pecific, non-ideal coupling of the port and alo provide u with a way to predict beforehand the tatitical propertie of other complicated encloure non-ideally coupled to external port. VI : Concluion : Fig.8: (a The experimental PDF for the Lo Cae 0 cavity power reflection coefficient (S (tar over a frequency range of 6 to 7.5GHz. Alo hown i the numerical etimate P( S (olid trace determined from RMT and the experimentally meaured iation impedance of the port ( Z. (b The experimental PDF for the Lo Cae 0 cavity cattering phae (φ (tar over a frequency range of 6 to 7.5GHz. Alo hown i the numerical etimate P φ (olid trace determined from RMT and the experimentally meaured iation impedance of the port. We oberve good agreement between the meaured data and the numerically etimated PDF. The inet how the fluctuation in < S > (tar over the frequency range of 6 to 100 7.5GHz, while the olid trace how the magnitude of the experimentally meaured iation cattering coefficient ( S. We can etimate the parameter frequency ( k = 141.3m 1 eigenmode for our cavity ( ~ k / Q ( ~S ~ uing the center, the average pacing between k n = 109.m [], and typical Q value of ( Q ~ 5, yielding an etimated ~ / k Q = 0.8. We ue thi parameter to generate an enemble of (ω following Ref.[18] and Eq.(4, combine it with the meaured S (ω of the antenna, and employ The reult dicued in thi paper erve to etablih a olid ground that can be ued to extract univeral tatitical propertie from data on wave chaotic ytem, or to engineer wave chaotic cavitie with pecific tatitical tranport propertie. In addition, given the frequency, volume and amount of loe (parameterized by Q within the encloure, and the iation impedance of the port, Ref. [17] and [18] provide u with a tool to predict the tatitic of the cavity repone ( Z and S a priori. We have hown that a imple normalization baed on the iation impedance can be ued to remove non-univeral, ytem-pecific coupling detail and bring out the univerality in the meaured impedance and cattering tatitic of wave chaotic ytem. Thi normalization procedure ha allowed u to experimentally verify theoretical prediction for the univeral propertie of a one-port wave chaotic ytem. We have alo teted everal apect of the theory in the realm of intermediate to high lo and for different coupling geometrie and find good agreement with theoretical prediction. We have hown that the average of the cavity power reflection coefficient S depend only on the magnitude of the iation cattering coefficient S and the degree of lo, and have obtained good agreement between theory and experiment. Finally, we alo demontrate the ability of thi normalization procedure to faithfully reproduce the nonuniveral tatitic of the cattering coefficient phae and magnitude of chaotic cavitie when < S > i not contant over the frequency range examined. Thee reult hould not be regarded a limited to microwave cavitie or any pecific 9
coupling tructure, but a applying to any wave chaotic ytem coupled to the outide world. Reference: [1] F. Haake, Quantum Signature of Chao (Springer- Verlag, 1991. [] H. -J Stockmann, Quantum Chao (Cambridge Univerity Pre, 1999, and reference therein. [3] R. Holland and R. St. John, Statitical Electromagnetic (Taylor and Franci, 1999, and reference therein. [4] Y. Alhaid, Rev. Mod. Phy. 7, 895 (000. [5] R. U. Haq, A. Pandey, and O. Bohiga, Phy. Rev. Lett. 48, 1086 (198. [6] P. So, S. M. Anlage, E. Ott and R. N. Oerter, Phy. Rev. Lett. 74 66 (1994. [7] O.I.Lobki, I. Rozhkov and R. Weaver,Phy. Rev. Lett. 91 194101-1 (003. [8] E. P. Wigner, Ann. Math. 53, 36 (1951; 6, 548 (1955; 65, 03 (1957; 67, 35 (1958. [9] F. J. Dyon, J. Math. Phy. 3,140 (196. [10] M. L. Mehta, Random Matrice (Academic Pre, SanDiego, CA, 1991. [11] P.A. Mello, P. Peveyra, and T.H. Selgiman, Ann. Of Phy. 161, 54 (1985. [1] P. W. Brouwer, Phy. Rev. B 51, 16878 (1995. [13] D.V. Savin, Y. V. Fyodorov and H. J Sommer, Phy. Rev. E 63, 0350 (001. [14] E. Kogan, P. A. Mello, and H. Liqun, Phy. Rev. E. 61, R17 (000. [15] R. A. Mendez-Sanchez, U. Kuhl, M. Barth, C. H. Lewenkopf, and H. J Stockmann, Phy. Rev. Lett. 91, 17410 (003. Acknowledgement: We acknowledge ueful dicuion with R. Prange and S. Fihman, a well a comment from Y. Fyodorov, D.V. Savin and P. Brouwer. Thi work wa upported by the DOD MURI for the tudy of microwave effect under AFOSR Grant F49600110374 and an AFOSR DURIP Grant FA9550041095. [16] U. Kuhl, M. Martinez-Mare, R.A. Mendez-Sanchez and H. J Stockmann, http://arxiv.org/ab/cond-mat/0407197. [17] X. Zheng, T. M. Antonen, and E. Ott, Statitic of Impedance and Scattering Matrice in Chaotic Microwave Cavitie: Single Channel Cae, J. Electromag., in pre (005. Alo available at http://arxiv.org/ab/cond-mat/040837. [18] S. Hemmady, X. Zheng, E. Ott, T. Antonen, and S. M. Anlage, Phy. Rev. Lett. 94, 01410 (005. [19] X. Zheng, T. M. Antonen, and E. Ott, Statitic of Impedance and Scattering Matrice in Chaotic Microwave Cavitie with Multiple Port, ubmitted to J. Electromag. Alo available at http://arxiv.org/ab/cond-mat/0408317. [0] L. K. Warne, K. S. H. Lee, H. G. Hudon, W. A. Johnon, R. E. Jorgenon and S. L. Stronach, IEEE Tran. Ant. Prop. 51, 978 (003. [1] K.B. Efetov and V.N. Prigodin, Phy. Rev. Lett. 70, 1315 (1993. [] C. W. J. Beenakker, Phy. Rev. B 50, 15170 (1994. [3] Ali Gokirmak, Dong-Ho Wu, J. S. A. Bridgewater and Steven M. Anlage, Rev. Sci. Intrum. 69, 3410 (1998. [4] D.-H. Wu, J. S. A. Bridgewater, A. Gokirmak, and S. M. Anlage, Phy. Rev. Lett., 81, 890 (1998. [5] S.-H. Chung, A. Gokirmak, D.-H. Wu, J. S. A. Bridgewater, E. Ott, T. M. Antonen, and S. M. Anlage, Phy. Rev. Lett. 85, 48 (000. [6] Y. Fyodorov and D.V. Savin, JETP Letter 80, 75 (004 [Pima v ZhETP 80, 855 (004]. 10