Aspects of the Scattering and Impedance Properties of Chaotic Microwave Cavities

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1 Aspects of the Scatterig ad Impedace Properties of Chaotic Microwave Cavities Sameer Hemmady 1,,3, Xig Zheg 3, Thomas M. Atose Jr. 1,3, Edward Ott 1,3 ad Steve M. Alage 1, Departmet of Physics, Uiversity of Marylad, College Park, MD USA Abstract: We cosider the statistics of the impedace Z of a chaotic microwave cavity coupled to a sigle port. We remove the o-uiversal effects of the couplig from the experimetal Z data usig the radiatio impedace obtaied directly from the experimets. We thus obtai the ormalized impedace whose Probability Desity Fuctio (PDF) is predicted to be uiversal i that it depeds oly o the loss (quality factor) of the cavity. We fid that impedace fluctuatios decrease with icreasig loss. The results apply to scatterig measuremets o ay wave chaotic system. PACS# : Mt, Nk, M,03.50.De, X,84.40.Az 1. Itroductio: Quatum chaotic scatterig is a subject that first arose i the cotext of uclear scatterig [1]. It is ow fidig icreasig applicatios i codesed matter ad atomic physics to uderstad the properties of larger scale complicated quatum systems [, 3, 4]. Quatum scatterig is also of iterest for uderstadig the uiversal statistical properties of complicated electromagetic eclosures [5]. Here we are cocered with systems that display ray chaos i the limit of high quatum umber, or wave eergy. The quatities of iterest i the correspodig wave/quatum system are the impedace ( Z ) ad scatterig ( S ) matrices. The system has a fiite umber of ports or scatterig chaels, ad the S-matrix relates outgoig waves i terms of a liear combiatio of icomig waves o the system. The Z matrix relates the total voltage at oe port to a liear combiatio of the currets eterig all of the ports. They are related by a simple bi-liear 1 trasformatio S = ( Z + Z ) ( Z ), where o Z o Z 0 is the characteristic impedace of the ports. Oe successful approach to quatifyig the statistical properties of quatum scatterig systems is the Poisso Kerel (PK) [6-8]. The PK is a global approach to uderstadig the statistical properties of scatterig systems. It describes the statistical properties of the scatterig matrix S i the presece of imperfect couplig, i terms of the mea value of S. The mea value S characterizes the imperfect couplig betwee the system ad the exterior scatterig chaels, ad ca be approximately evaluated from data by takig the mea of a large amout of data o a esemble of similar systems, or by eergy averagig, or both. This approach has prove to be quite useful for describig microwave scatterig data, for istace [9,10]. We have itroduced a complemetary ew approach to quatum scatterig that makes use of differet coceptual pieces to achieve a similar outcome. Our approach is to directly characterize the o-ideal couplig betwee the outside world ad the system through a determiistic quatity kow i electromagetism as the radiatio impedace, Z. Oe ca the defie ormalized impedace (z) ad scatterig (s) matrices that directly reveal the uiversal fluctuatig properties of the scatterig system [11-13]. More explicitly, the radiatio impedace Z = R + ix of each chael is determied i a separate measuremet ad combied with the cavity impedace Z = R + ix to create a ormalized impedace matrix z as; R X X z = + i. This ormalized impedace R R matrix has statistical properties that are itrisic to the scatterig system ad idepedet of the couplig. A umber of experimets have bee performed to test the uiversality of the ormalized z [14, 15]. The 1 Also with the Departmet of Electrical ad Computer Egieerig. Also with the Ceter for Supercoductivity Research. 3 Also with the Istitute for Research i Electroics ad Applied Physics. 1

2 ormalized impedace approach has also bee employed by the Warszawa group to examie their data o quatum graph microwave aalogs [16].. Experimet: Our experimetal setup cosists of a airfilled quarter bow-tie chaotic cavity (Fig.1(a)) which acts as a two dimesioal resoator below about 19 GHz [17]. Ray trajectories i a closed billiard of this shape are kow to be chaotic. This cavity has previously bee used for the study of the eigevalue spacig statistics [18] ad eigefuctio statistics [19, 0] for a wave chaotic system. I order to ivestigate a scatterig problem, we excite the cavity by meas of a sigle coaxial probe whose exposed ier coductor, with a diameter (a) exteds from the top plate ad makes electrical cotact with the bottom plate of the cavity (Fig.1(b)). I this paper we study the properties of the cavity over a frequecy rage of 6-1 GHz, where the spacig betwee two adjacet resoaces is o the order of 5 30 MHz. As i the umerical experimets i Refs. [1, 13], our experimet ivolves a two-step procedure. The first step is to collect a esemble of cavity scatterig coefficiets S over the frequecy rage of iterest. Esemble averagig is realized by usig two rectagular metallic perturbatios which are systematically scaed ad rotated throughout the volume of the cavity (Fig.1(a)). Each cofiguratio of the perturbers withi the cavity volume results i a differet measured value of S. The perturbers are kept far eough from the atea so as ot to alter its ear-field characteristics. I total, oe hudred differet cofiguratios are measured, resultig i a esemble of 800,000 S values. We refer to this step as the cavity case. Fig. : The magitude of the cavity impedace with o absorbig strips is show as a fuctio of frequecy. The dots idicate a sigle reditio of the cavity impedace ad perturbatios. The dashed lie is the magitude of the complex cavity impedace obtaied after esemble averagig over 100 differet perturbatio positios withi the cavity. The solid lie is the magitude of the measured radiatio impedace for the same atea ad couplig detail as show i Fig.1 (b). Note that eve after 100 reditios of the perturbers withi the cavity, < Z > 100 is still a poor approximatio to. Z Fig.1: (a) The physical dimesios of the quarter bow-tie chaotic microwave resoator are show alog with the positio of the sigle couplig port. Two metallic perturbatios are systematically scaed ad rotated throughout the etire volume of the cavity to geerate the cavity esemble. (b) The details of the couplig port (atea) ad cavity height h are show i cross sectio. (c) The implemetatio of the radiatio case is show, i which commercial microwave absorber is used to lie the ier walls of the cavity to miimize reflectios. The secod step, referred to as the radiatio case, ivolves obtaiig the scatterig coefficiet for the excitatio port whe waves eter the cavity but do ot retur to the port. I the experimet, this coditio is realized by removig the perturbers ad liig the side walls of the cavity with commercial microwave absorber (ARC Tech DD10017D) which provides about 5 db of reflectio loss betwee 6 ad

3 1 GHz (Fig.1.(c)). We measure the radiatio scatterig coefficiet S for the cavity, approximatig the situatio where the side walls are moved to ifiity. I this case S does ot deped o the chaotic ray trajectories of the cavity, ad thus gives a determiistic (i.e. o-statistical) characterizatio of the couplig idepedet of the chaotic system. Havig measured the cavity S ad S, we the trasform these quatities ito the correspodig cavity ad radiatio impedaces ( Z ad Z ) ad determie the ormalized impedace z as discussed above. I order to test the validity of the theory for systems with varyig loss, we create differet cavity cases with differet degrees of loss. Loss is cotrolled ad parameterized by placig 15. cmlog strips of microwave absorber alog the ier walls of the cavity. These strips cover the side walls from the bottom to top lids of the cavity. We thus geerate differet loss scearios by icreasig the umber of 15. cm -log absorber strips placed alog the ier cavity walls ad defie the absorber perimeter ratio α as the ratio of absorber legth to the total cavity perimeter (147.3 cm). 3. Data: We first examie the degree to which esemble averagig to estimate S ad Z, as employed i the Poisso Kerel, ca reproduce the radiatio cases S ad Z. Fig. shows typical data for the magitude of the cavity impedace versus frequecy for several cases. The dots show the cavity impedace for oe particular reditio of the cavity ad its perturbers i the case of o added absorber. The dashed lie shows the result of averagig the complex impedace of 100 reditios of the cavity. The thick solid lie is the measured radiatio impedace Z, which should be equivalet to the mea of the cavity impedace Z. It is clear that eve after averagig the properties of 100 cavities i the esemble, the mea value of measured Z has ot yet approached the radiatio case. This demostrates the importace of obtaiig very high quality statistics before the Poisso Kerel ca be used o real data. It also illustrates the relative ease with which the radiatio impedace ca be used to characterize the o-ideal couplig of real wave chaotic systems. Fig. 3: The magitude of a sigle reditio of the cavity impedace with (0-dots, 1-dashed lie, 4-solid triagles) absorbig strips is show as a fuctio of frequecy. The solid lie is the magitude of the measured radiatio impedace for the same atea ad couplig detail as show i Fig. 1(b). As losses withi the cavity icrease, the cavity resoaces are dampeed out ad the measured cavity impedace approaches the radiatio impedace. We ext examie the depedece of impedace statistics o the global cotrol parameter k /( k. This parameter depeds o the frequecy, the volume of the scatterig system (through the resultig mode desity k ), ad o the losses (parameterized by the scatterer quality factor Q ). It determies the probability desity fuctios (pdf) for the real ad imagiary parts of the ormalized impedace z, as well as the pdf of s. Oe ca determie the value of k /( k from the variace of the Re[z ] ad Im[z] pdfs, as show i [14], sice the variace σ = (1/ π ) /( k /( k ) for systems with time-reversal symmetry. Fig. 3 demostrates how the cavity impedace evolves with icreasig loss. Show are impedace magitude data versus frequecy for 3 cavities with differet umbers of microwave absorber strips iside (0, 1, or 4), but otherwise idetical. These data sets 3

4 are for a sigle reditio of the cavity. Also show is the measured radiatio impedace magitude for the same atea. As losses icrease, the fluctuatios i Z clearly decrease, ad approach the radiatio case. This qualitative observatio is substatiated by quatitative tests of the Re[z] ad Im[z] pdf variaces, ad their depedece o system loss [14]. Fig. 4: The relatioship betwee the loss parameter k /( k ad the absorber perimeter ratio ( α ) is show betwee 7. ad 8.4 GHz. The symbols represet (hollow star-0 ; hollow circle- 1; hollow triagle- ; hollow square- 3; solid star- 4 ) absorbig strips withi the cavity. The best liear fit to all the data poits is show as the solid lie. The x-itercept of this lie idicates the α required to make a loss-less cavity have the same k /( k as the empty experimetal cavity of Fig. 1(a). Figure 4 further examies the depedece of the experimetally determied value of k /( k versus the umber of absorber strips placed alog the periphery of the cavity walls. The k /( k values were determied by the variaces of the Re[z ] ad Im[z] pdfs. Fig. 4 shows a clear liear relatioship of k /( k o the absorber perimeter ratio. This is expected because 1/ Q is proportioal to the dissipated power i the cavity, which scales with the amout of microwave absorber placed i the cavity. A liear fit of the data is quite accurate ad shows a zero-crossig for k /( k at α = This suggests that the empty cavity losses correspod to coverig the walls of a perfectly coductig cavity with 3.5% coverage of microwave absorber. 4. Discussio ad Coclusios: Our work has illustrated the beefits of usig the impedace, rather tha scatterig matrix, to determie the uiversal scatterig properties of wave chaotic systems coected to the outside world. The measuremet of radiatio impedace to characterize the o-ideal couplig to the system is a very simple ad powerful tool, ad is more reliable tha a average statistical quatity. The evolutio of wave chaotic scatterig systems with iteral losses has bee illustrated i this paper. The impedace approach also reveals other uiversal properties of the cavity, such as the geeralized variace ratio, related to the Hauser-Feshbach relatio [1]. Ackowledgemets: We ackowledge useful discussios with R. Prage ad S. Fishma, as well as commets from Y. Fyodorov, D.V. Savi ad P. Brouwer. This work was supported by the DOD MURI for the study of microwave effects uder AFOSR Grat F ad a AFOSR DURIP Grat FA Refereces: [1] E. P. Wiger, A. Math. 53, 36 (1951); 6, 548 (1955); 65, 03 (1957); 67, 35 (1958). [] F. Haake, Quatum Sigatures of Chaos (Spriger-Verlag, 1991). [3] H. -J Stockma, Quatum Chaos (Cambridge Uiversity Press, 1999), ad refereces therei. [4] Y. Alhassid, Rev. Mod. Phys. 7, 895 (000). [5] R. Hollad ad R. St. Joh, Statistical Electromagetics (Taylor ad Fracis, 1999), ad refereces therei. [6] P.A. Mello, P. Peveyra, ad T.H. Selgima, A. Of Phys. 161, 54 (1985). [7] P. W. Brouwer, Phys. Rev. B 51, (1995). 4

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