Field Tunnelling and Losses in Narrow Waveguide Channel

Transcription

1 Mkrotalasna revja Decembar 1. Feld Tunnellng and Losses n Narrow Wavegude Channel Mranda Mtrovć, Branka Jokanovć Abstract In ths paper we nvestgate the feld tunnellng through the narrow wavegude channel formed by reducng the heght of a rectangular wavegude. Geometrcal parameters of the channel and materal propertes strongly affect the energy tunnellng and feld densty n the channel. The role of materal losses s systematcally examned as an mportant practcal ssue lmtng the maxmum achevable tunnellng transmsson level. Keywords ENZ metamateral, narrow wavegude channel, energy tunnellng, ncreased feld densty, loss. I. INTRODUCTION In the past two decades there has been great nterest n metamaterals whose electromagnetc propertes were frst predcted n 1968 by Veselago [1]. Metamaterals show characterstcs that are not avalable n ordnary materals found n nature and therefore can be used n desgnng novel mcrowave and optcal devces wth enhanced propertes. Untl recently resea n ths area was focused on double negatve (DNG) metamaterals wth permttvty and permeablty less than zero leadng to negatve ndex of refracton, but n the past few years there has been an ncreased nterest n metamaterals whose relatve delectrc permttvty s close to zero (epslon-near-zero, ENZ). Poneers n ths area of study were Slvernha and Engheta wth ther theoretcal work on energy tunnellng through subwavelength channels flled wth ENZ metamateral n 6 []. They used two sectons of parallel plate wavegudes connected by a very narrow channel and demonstrated that wavefront s reproduced on the second end of the channel wth very small losses. They suggested that ENZ materals can be used n mprovng the transmsson effcency of wavegudes wth sharp bends or dscontnutes and n concentratng energy n a small subwavelength cavty. Later on, several experments were conducted to confrm ths theory n mcrowave regme [3, 4]. It can be seen that there are two approaches for realzaton of ENZ metamaterals, one developed by Lu and Chang et al [3] usng splt-rng resonator meda nserted n narrow channel connectng two parallel plate wavegude sectons, and the other developed by Edwards et al [4] usng dsperson characterstc of a rectangular wavegude near ts cut-off frequency. Varous applcatons have been suggested for these materals, such as cloakng devces [5], confnement of energy beyond dffracton lmt [], wavegude couplng wth narrow channel [4], and desgn of hgh drectvty small antenna [6]. Mranda Mtrovć and Branka Jokanovć are wth the Insttute of Physcs Belgrade, Pregrevca 118, 118 Beograd, Serba and Montenegro, E-mal: mranda@pb.ac.rs, brankaj@pb.ac.rs In ths work, we systematcally nvestgate how geometrcal parameters of the channel and characterstcs of materals fllng wavegude and channel affect tunnellng of energy and feld densty nsde ENZ channel. We also examne how materal losses depend on the channel thckness. Our smulatons reveal the strong mpact of losses both n metal and delectrc when the channel thckness s very small. II. THEORETICAL ANALYSIS Rectangular wavegude of wdth a and heght b (a>b) supports travellng of TE and TM modes. Cut-off frequences for both modes are gven by Eq. (1). f c( m, n) c m a n b Here ε s the relatve delectrc constant of the delectrc n rectangular wavegude, m and n are numbers assgned to dfferent modes. The sequence of these modes n the case of b=a/ can be seen on Fg. 1. The frequency range between TE 1 and TE represents the pass band of a rectangular wavegude. Reducng the heght b of a rectangular wavegude shfts the cut-off frequences of the hgher modes toward hgher frequences, whle ts pass band stays unchanged. Fg. 1. The sequence of propagatng modes n a rectangular wavegude for b=a/ It s ntutvely obvous that reducng the heght of rectangular wavegude (Fg. ) drastcally affects the reflecton ( ) and transmsson (S 1 ) coeffcents of the wavegude, causng a poor transmsson due to strong msmatchng. (1) 8

2 December, 1 Fg.. Reducng heght b of a rectangular wavegude In a dfference to the structure wth a sngle step, Fg. 3 shows the structure wth Π-channel ( <<b) nserted between two wavegude sectons, whch can support transmsson n a short frequency range. Mcrowave Revew should be greater than n the channel (Eq. (5c)) to ensure transmsson wthn the pass band. w w f f (5a) fte tun 1 TE c c c (5b) a a a (5c) 4 Consderng that ε reff s near zero n the narrow channel around cut-off frequency, wave vector β s also near zero, whch means that wavelength approaches nfnty. Ths knd of behavour s analogous to a balance case for LH (lefthanded) metamaterals, or zeroth-order resonance () [7]. As a consequence, energy s capable of tunnellng through the narrow channel at ths frequency, and a great densty of feld proportonal to b/ whch s constant along the channel s acheved. A. ENZ Channel III. SIMULATION RESULTS Fg. 3. Wavegude sectons connected wth a narrow П-channel ( <<b) for mpute matchng In the case of <<b, t s possble to descrbe the propagaton of TE 1 mode n a rectangular wavegude as a propagaton of TEM mode n parallel-plate wavegude wth effectve permttvty ε reff [3]: where f ch reff k TEM, (a) a ch TE1 c f k (b) c From prevous equatons we can derve an expresson for ε reff : c (3) reff 4 f a Here c s the speed of lght n vacuum and ε s a relatve delectrc constant n the channel. It s seen that ε reff equals zero at the cut-off frequency of the channel, whch gves us the opportunty to consder ths structure as an ENZ metamateral near ths frequency. Ths s the frequency where tunnellng of energy occurs (Eq. 4). ch c f tun f (4) TE1 a The frequency of tunnellng should be wthn the pass band of two wavegudes (Eq. (5a)) and from ths condton we can see that relatve delectrc constant n the wavegude sectons Here we consder the structure from Fg.. wth followng dmensons: a=11.6mm (wavegude wdth), b=a/=5.8mm (wavegude heght), and channel thckness wth dfferent values, 1 =b, =b/=5.4mm, 3 =b/8=6.35mm and 4 =b/64=.8mm. Delectrc constant n both wavegude sectons s ε =, and cut-off frequences for TE 1 and TE modes are 1.44GHz and.88ghz, respectvely. The smulated transmsson and reflecton coeffcents are gven n Fgs. 4 (a) and (b) respectvely. Smulatons are performed usng Ansoft HFSS software. S 1 v 1, 1, 1,4 1,6 1,8, 1, 1, 1,4 1,6 1,8, (a) (b) Fg. 4. Transmsson (a) and reflecton (b) coeffcents for dfferent values of : () 1 =b=5.8mm, () =5.4mm, () 3 =6.35mm, (v) 4 =.8mm As we can see n Fg. 4, even f the heght of the rectangular wavegude s reduced to a half of ts orgnal value, the transmsson wth small attenuaton s stll possble. Consderable smaller transmsson coeffcent s notceable only after wavegude heght reducton of =b/8. In order to compensate nfluence of dscontnuty between wavegude and channel (whch behaves as an equvalent capactance), mpedance matchng should be accomplshed. Ths can be acheved by ntroducng transton matchng area between wavegude sectons and ENZ channel that forms a Π- channel, (Fg. 3). Dmensons for a and b are the same as n the prevous case, relatve delectrc constant n the channel s ε =1 (ar) and n the wavegude sectons ε = (Teflon). v 9

3 Mkrotalasna revja Decembar 1. Transmsson and reflecton coeffcents n case of =b s =.8mm s shown n Fg. 5. The frst transmsson peak occurs at frequency f tun =1.464GHz whch s assgned to the zeroth-order resonance (). Tunnellng of energy occurs at ths frequency, snce effectve permttvty n the channel s equal to zero. The second transmsson peak f =1.816GHz s Fabry-Perot resonance whch s hghly dependent on length of ENZ channel. That s not the case wth, as long as the condton <<b s fulflled [8]., S 1 - f tun f 1,4 1,5 1,6 1,7 1,8 1,9 Fg. 5. Transmsson coeffcent (a=11.6mm, b=a/=5.8mm, =.8mm, b s =, ε =, ε =1) Feld dstrbuton and a real part of Poyntng vector n the channel are shown n Fgs. 6 (a) and 6 (b) respectvely. It can be seen from H-feld bar that energy densty s ncreased n channel by factor whch s very close to b/ =64. The dstrbuton of real part of Poyntng vector shows the energy flow along the channel and s concentrated and greatly enhanced n the mddle of the channel. wth reducton of channel heght and transmsson s perfect. In realty, feld densty n channel s restrcted by break down voltage n delectrc and transmsson s lowered due to fnte conductvty of metallc walls and delectrc losses. Further dscusson on ths subject wll be presented later on n ths paper. S ,5 1,5 1,75, Fg. 7. Shftng of the second transmsson peak (Fabry-Perot resonance) due to varaton of channel length: () L 1 =95.5mm, () L =17mm, () L 3 =19.5mm As t was ponted out before, the second transmsson peak as a product of Fabry-Perot resonance s hghly dependent on the channel length. Ths varaton has no effect on the poston of the zeroth-order resonance () frequency at whch tunnellng of energy s occurred. Change of Fabry-Perot resonance for varous lengths of narrow channel ( =b/64=.8mm, L 1 =95.5mm, L =17mm and L 3 =19.5mm) can be seen n Fg. 7. Ths property can be used to manpulate second transmsson peak n order to remove t or leave t wthn the pass band of the wavegude. B. Losses n ENZ Channel (a) (b) Fg. 6. (a) Feld dstrbuton n narrow channel shows H-feld densty nsde the channel enhanced by factor b/ n comparson to feld n wavegude sectons; feld densty s constant along the channel; (b) Real part of Poyntng vector shows energy flow through the channel; energy s concentrated n n the mddle of the channel and s gradually descendng toward edges. In approxmaton of perfectly conductng metallc walls and lossless delectrcs, feld densty n the channel s ncreased In order to study materal losses we wll consder structure wth copper claddng (σ Cu =58MS/m) and real lamnated delectrcs: Plexglas wth propertes ε r =3.4, tgδ=1*1-4 n wavegude secton), Rogers RT/durod 587 wth ε r =.33, tgδ=1*1-4 and Arlon AD 5 wth ε r =.5, tgδ=3*1-4 n channel area. To preserve the smlar operatng band, dmensons of the prevous structure (Fg. 3) are changed: wdth a=7mm, heght b=35mm, channel thckness varyng between.5 and 4 mm, length of the channel L=49mm. Cut-off frequency n wavegude sectons s f w TE1=1.16GHz, frequences are GHz whle Fabry-Perot resonances vary n range.17.7ghz wth the change of channel thckness. In Fgs. 8. (a) and (b) transmsson and reflecton coeffcents for two values of channel thckness are gven. As t can be seen from Fg. 8, delectrc losses for both (a) and Fabry-Perot resonances (b) ncrease wth decreasng the channel heght. Also, change n resonant frequency s much greater for Fabry-Perot resonance n comparson to. 1

4 December, 1 Mcrowave Revew a), S 1, S ,5 1,3 1,35 1,4 1,45 1,5 1,55 -,1,,3,4,5 b) Fg. 8. Transmsson and reflecton coeffcents for a) and b) Fabry-Perot resonance n the case of: () =.5mm and () =3mm. Materals used are: Plexglas (ε=3.4, tgδ=1*1-4 ) n wavegude sectons, Rogers RT/durod 587 (ε r =.33, tgδ=1*1-4 ) n the channel, and perfect conductor for metal plates. In general, losses n narrow channel consst of two components: losses n metallc plates and losses n delectrcs fllng the channel and wavegude sectons. Frstly, losses n metallc plates are nvestgated usng n smulatons copper claddng wth a real conductvty and delectrcs wthout losses. Tpcal value for conductvty and rougheness of electrodeposted copper are σ Cu =58MS/m and RMS=.4. Dsspated power s calculated n two cases: perfectly smooth, and metallc walls wth roughness RMS=.4. As t can be seen from Fg. 9, dsspated power n metallc plates wth surface roughness s almost two tmes greater than n a case of perfectly smooth metal claddng. Also, reflecton takes part n dsspated power smaller then 8% n all cases except for the mnmum value of channel thckness at frequency. Dsspated power starts to grow rapdly above 1% for the channel thckness =.43 ( =1.5mm) for resonance and =.9 ( =1mm) for Fabry-Perot resonance. Ths s understandable havng n mnd that power densty n channel takes much greater values at tunnellng frequency than at Fabry-Perot resonance. Dsspated power n metal, 1- - S Fg. 9. Dsspated power n metal versus channel thckness - lnes marked wth crcles represent perfectly smooth copper claddng and lnes wth trangles represent copper claddng wth RMS=.4; Results for versus channel thckness are gven n rght upper corner. Comparatve results for two delectrcs fllng the channel wth dfferent loss tangents (Rogers RT/durod 587 (ε r =.33, tgδ=1*1-4 )) and Arlon AD 5 (ε r =.5, tgδ=3*1-4 )), whle delectrc n wavegude sectons s Plexglas (ε r =3.4, tgδ=1*1-4 ). Smulatons are performed wth perfect conductor used for metal claddng. Dsspated power n delectrc, 1- - S 1,5,4,3,,1,,8,6,4,,,,4,6,8,1,1,,4,6,8,1,1 Fg. 1. Dsspated power n delectrc n dependence of channel thckness - lnes marked wth crcles represent delectrc wth tgδ=1*1-4 and lnes wth trangles represent delectrc wth tgδ=3*1-4 ; In the rght upper corner the results for n dependence of channel thckness are gven Accordng to the slope of the curves for the small channel thcknesses (Fg. 1), t s obvous that dsspated power n delectrcs s less senstve to the change of channel thckness n comparson to dsspated power n metal. On the other hand, f we look at dsspated power n the whole range t can be seen that t s hghly dependent on the choce of delectrc. Next we want to show comparatve results for attenuaton n metal claddng and delectrcs. The worst scenaro was used n smulatons: copper claddng wth RMS=.4 and Arlon AD 5 (ε r =.5, tgδ=3*1-4 ) as delectrc n the channel. Wavegude sectons are flled wth Plexglas (ε r =3.4, tgδ=1*1-4 ). Dagrams for attenuaton at and Fabry- Perot frequences are dsplayed separately n Fgs. 11 (a) and (b), respectvely. 11

5 Mkrotalasna revja Decembar 1. Attenuaton, [db/mm] Attenuaton, [db/mm],,15,1,5,,1,8,6,4,, (a) - -3 Delectrc -35,,4,6,8,1,1 Delectrc,,4,6,8,1, ,,4,6,8,1,1 Delectrc,,4,6,8,1,1 Delectrc (b) Fg. 11. Attenuaton versus channel thckness for (a) and Fabry- Perot resonances (b); Losses n delectrc are domnant n channels greater than =1mm ( =.9) for and =.8mm ( =.3) for Fabry-Perot resonance. Dashed lne denotes added up attenuaton for metal and delectrc and full lne shows total smulated attenuaton. In the rght upper corner the results for n dependence of channel thckness are gven. As t can be seen from Fg. 11, attenuaton due to delectrc losses s domnant n channels wth greater thckness values, whle losses n metal are domnant for narrow channels. Ths s especally vsble at frequency. For frequency attenuaton n metal and delectrc becomes equal for the channel thckness =1mm ( =.9) and for Fabry-Perot resonance that happens for the narrower channel =.8mm ( =.3). Along wth attenuaton n delectrc and metal, the total smulated attenuaton s gven and compared wth the sum of delectrc and metal attenuaton. Good agreement s observed between the last two. It can be seen that attenuaton at frequency s approxmately as twce as hgh as at Fabry-Perot resonance. As t was stated before, the reason for ths s a greater power densty n the channel at frequency whch makes the channel more senstve to losses. At the end, the resonant frequences along wth Q-factors for total smulated losses from the prevous example are gven n Table I for dfferent channel thcknesses. TABLE 1 RESONANT FREQUENCIES AND LOADED Q-FACTORS f f Q L Q L [mm] [GHz] [GHz] It can be seen from Table I that loaded Q-factors have greater values for than for Fabry-Perot resonance. IV. CONCLUSION Metamaterals wth relatve delectrc permttvty close to zero (epslon-near-zero, ENZ) can be desgned by reducng the heght of a rectangular wavegude. Feld tunnellng n the channel occurs when the channel and wavegude thckness rato becomes very low, and when delectrc permttvty n the channel s less than delectrc permttvty n wavegude sectons. The frequency of tunnellng s assgned as the zeroth-order resonance () snce t does not depend on change of channel length. That s not the case wth the second peak n transmsson through the channel, whch s Fabry- Perot resonance. In ths paper, we show that geometrcal parameters of ENZ channel and characterstcs of metal claddng and delectrc n the channel greatly affect the transmsson at both resonances, and Fabry-Perot. Dsspated power due to metal losses grows rapdly for channel thcknesses smaller than =1.5mm at resonance, and =1mm at Fabry-Perot resonance. The dfference n channel thckness lmtaton between these two resonances occurs because of the hgher power densty n the channel at tunnellng frequency than at Fabry-Perot resonance. smulated attenuaton for the worst case whch accounts a copper claddng wth roughness RMS=.4 and a lossy delectrc wth tgδ=3*1-4 s n the range db/mm for, whle at Fabry-Perot resonance total attenuaton s about two tmes smaller than for, and n the range.7.78db/mm. Attenuaton due to delectrc losses s domnant for the channel thcknesses greater than =1mm for, whle for Fabry-Perot resonance channel thckness lmtaton s =.8mm. It can be seen that wavegude channel wth epslon-near-zero exhbts the attenuaton consderably greater than n standard rectangular wavegude (=.5dB/m). 1

6 December, 1 ACKNOWLEDGEMENT Ths work s supported by the Serban Mnstry of Scence and Technologcal Development through the project TR- 119 Dual-Band and Trple-Band Metamateral-Based Devces and Antennas for Modern Communcaton Systems. REFERENCES [1] V.G. Veselago, The electrodynamcs of substances wth smultaneously negatve values of ε and µ, Sovet Physcs Uspekh, vol. 1, no. 4, pp. 5914, [] M. G. Slvernha and N. Engheta, Tunnellng of electromagnetc energy through subwavelength channels and bends usng ε-near-zero materals, Physcal Revew Letters, vol. 97, 15743, 6. [3] R. Lu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cu, S. A. Cummer, and D. R. Smth, Expermental demonstraton of electromagnetc tunnellng through an epslon-near-zero Mcrowave Revew metamateral at mcrowave frequences, Physcal Revew Letters, vol. 1, 393, 8. [4] B. Edwards, A. Alù, M. E. Young, M. G. Slvernha, and N. Engheta, Expermental verfcaton of epslon-near-zero metamateral couplng and energy squeezng usng a mcrowave wavegude, Physcal Revew Letters, vol. 1, 3393, 8. [5] M. G. Slvernha, A. Alù, and N. Engheta, Parallel-plate metamaterals for cloakng structures, Physcal Revew E, vol. 75, 3663, 7. [6] S. Enoch, G. Tayeb, P. Sabouroux, N. Guérn, and P. Vncent, A metamateral for drectve emsson, Physcal Revew Letters, vol. 89, 139,. [7] C. Caloz and T. Itoh, Electromagnetc Metamaterals: Transmsson Lne Theory and Mcrowave Applcatons, New Jersey, Wley, 6. [8] M. G. Slvernha and N. Engheta, Theory of supercouplng, squeezng wave energy, and feld confnement n narrow channels and tght bends usng ε-near-zero metamaterals, Physcal Revew B, vol. 76, 4519, 7. 13