Magnetism from Conductors, and Enhanced NonLinear Phenomena


 Donald Goodwin
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1 Mnetism from Conductors, nd Enhnced NonLiner Phenomen JB Pendry, AJ Holden, DJ Roins, nd WJ Stewrt Astrct  We show tht microstructures uilt from nonmnetic conductin sheets exhiit n effective mnetic permeility, µ eff, which cn e tuned to vlues not ccessile in nturlly occurrin mterils, includin lre iminry components of µ eff. The microstructure is on scle much less thn the wvelenth of rdition, is not resolved y incident microwves, nd uses very low density of metl so tht structures cn e extremely lihtweiht. Most of the structures re resonnt due to internl cpcitnce nd inductnce, nd resonnt enhncement comined with compression of electricl enery into very smll volume retly enhnces the enery density t criticl loctions in the structure, esily y fctors of million nd possily y much more. Wekly nonliner mterils plced t these criticl loctions will show retly enhnced effects risin the possiility of mnufcturin ctive structures whose properties cn e switched t will etween mny sttes. Index Terms  effective permeility, nonlinerity, photonic crystls JB Pendry is with The Blckett Lortory, Imperil Collee, London, SW7 BZ, UK. AJ Holden, DJ Roins, nd WJ Stewrt re with GECMrconi Mterils Technoloy Ltd, Cswell, Towcester, Northmptonshire, NN 8EQ, UK. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
2 I. INTRODUCTION In sense every mteril is composite, even if the individul inredients consist of toms nd molecules. The oriinl ojective in definin permittivity, ε, nd permeility, µ, ws to present n homoeneous view of the electromnetic properties of medium. Therefore it is only smll step to replce the toms of the oriinl concept with structure on lrer scle. We shll consider periodic structures defined y unit cell of chrcteristic dimensions. The contents of the cell will define the effective response of the system s whole. Clerly there must e some restrictions on the dimensions of the cell. If we re concerned out the response of the system to electromnetic rdition of frequency ω the conditions re esy to define: << λ= πc ω () If this condition were not oeyed there would e the possiility tht internl structure of the medium could diffrct s well s refrct rdition ivin the me wy immeditely. Lon wvelenth rdition is too myopic to detect internl structure nd in this limit n effective permittivity nd permeility is vlid concept. In the next section we shll discuss how the microstructure cn e relted to ε, µ. eff eff In n erlier pper [] we showed how structure consistin of very thin infinitely lon metl wires rrned in 3D cuic lttice could model the response of dilute plsm, ivin netive ε eff elow plsm frequency somewhere in the ihertz rne. Theoreticl nlysis of this structure hs een confirmed y experiment []. Sievenpiper et l hve lso investited plsmlike effects in metllic structures [3,4]. Idelly we should like to proceed in the mnetic cse y findin the mnetic nloue of ood electricl conductor: unfortuntely there isn t one! Nevertheless we cn find some lterntives which we elieve do ive rise to interestin mnetic effects. Why should we o to the troule of microstructurin mteril simply to enerte prticulr µ eff? The nswer is tht toms nd molecules prove to e rther restrictive set of elements from which to uild mnetic mteril. This is prticulrly true t frequencies in the ihertz rne where the mnetic response of most mterils is einnin to til off. Those mterils, such s the ferrites, tht remin modertely ctive re often hevy, nd my not hve very desirle mechnicl properties. In contrst, we shll show, microstructured mterils cn e desined with considerle mnetic ctivity, oth dimnetic nd prmnetic, nd cn if desired e mde extremely liht. There is nother quite different motivtion. We shll see tht stron mnetic ctivity implies stronly inhomoeneous fields inside the mteril. In some instnces this my result in locl field strenths mny orders of mnitude lrer thn in free spce. Dopin the composite with non liner mteril t the criticl loctions of field concentrtion ives enhnced nonlinerity, reducin power requirements y the field enhncement fctor. This is not n option ville in conventionl mnetic mteril. We show first how to clculte µ eff for system, then we propose some model structures which hve mnetic ctivity nd ive some numers for these systems. Finlly we show how electrosttic enery cn e stronly concentrted in these structures nd hence the demonstrte potentil for enhncin non liner effects. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
3 II. DEFINING AN EFFECTIVE PERMEABILITY We re seekin to uild structures with effective epsilon nd mu, B D = µ µ H ve eff ve = ε ε E ve eff ve () where we ssume tht the structure is on scle much shorter thn the wvelenth of ny rdition so tht we cn sensily spek of n vere vlue for ll the fields. A key question is how do the veres differ? Clerly if the structure is mde of thin wires or sheets of metl then if the veres were tken over the sme reions of spce, ε, µ would lwys e unity. However we oserve tht Mxwell s equtions, eff eff H = + D / t E = B / t my e pplied in the interl form, zc zc H dl = + E dl = t t zs zs D. ds B. ds (3) (4) where the line interl is tken over loop c which encloses n re s. This form of the equtions immeditely suests prescription for verin the fields. For simplicity we shll ssume tht the periodic structure is descried y unit cell whose xes re orthoonl s shown in fiure elow. Some of the ruments used in this section re similr to those we used in derivin finite difference model of Mxwell s equtions [5]. Fiure. Unit cell of periodic structure. We ssume tht the unit cell dimensions re much smller tht the wvelenth of rdition, nd vere over locl vritions of the fields. In the cse of the B  field we vere over the fces of the cell nd in the cse of the H  field, over one of the edes. We choose to define the components of H ve y verin the H field lon ech of the three xes of the unit cell. If we ssume simple cuic system, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 3
4 H H H ve x ve y ve z z zz r=,, = H dr r=,, r=,, = H dr r=,, r=,, = H dr r=,, (5) There is only one cvet concernin the definition of the unit cell: its edes must not intersect with ny of the structures contined within the unit cell. This leves us free to cut the structure into whole numer of unit cells when we come to crete surfce nd ensures tht the prllel component of H ve is continuous cross the surfce s required in consistent theory of n effective medium. To define B ve we vere the B field over ech of the three fces of the unit cell defined s follows: S x is the surfce defined y the vectors y, z S y is the surfce defined y the vectors x, z S z is the surfce defined y the vectors x, y Hence we define, B B B ve x ve y ve z = B ds S = B ds z zz S = B ds S x z y The rtio defines the effective epsilon nd mu from (), d i d i d i µ eff = B µ x ve x Hve x µ eff = B µ y ve y Hve y µ eff = B µ z ve z Hve z (6) (7) Thus if we seek lre effect we must try to crete fields tht re s inhomoeneous s possile. We shll explore vrious confiurtions of thin sheets of metl, derive µ eff, nd discuss the results with view to mkin the effect s lre s possile. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 4
5 III. EXAMPLES OF MAGNETIC MICROSTRUCTURES A. An Arry of Cylinders Fiure. Model A consists of squre rry of metllic cylinders desined to hve mnetic properties in the direction prllel to the xes of the cylinders. We strt with very simple structure for the purposes of illustrtion. Let us pply n externl field, H, which we shll tke to e prllel to the cylinders. We ssume tht the cylinders hve conductin surfce so tht current, j, per unit lenth, flows. The field inside the cylinders is, H = H + j j (8) where the second term on the riht hnd side is the field cused directly y the current, nd the third term is the result of the depolrisin fields with sources t the remote ends of the cylinders. If the cylinders re very lon the depolrisin field will e uniformly spred over the unit cell, ut will hve the sme numer of lines of force in it s the direct field inside the cylinders. We now clculte the totl emf round the circumference of cylinder: L M N M L N O P Q O P Q r emf = π r µ π H + j j t M P = + iωπ r µ H + j j M P σj σj where σ is the resistnce of the cylinder surfce per unit re. The net emf must lnce nd therefore, (9) j = iωπ r µ H = L r i r O r M L π ωπ µ P π σ M N Q N H O r i P + σ ω r µ Q () We re now in position to clculte the relevnt veres. The vere of the Bfield over the entire unit cell is, Bve = µ H () /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 5
6 However if we vere the Hfield over line lyin entirely outside the cylinders, Hve = H j = H L NM H O r i P + σ ω r µ Q = H L NM σ + i ω rµ O i P + σ ω rµ Q () Hence we define, µ eff σ + i B ve ω rµ = = µ Hve σ + i ω rµ L NM = + σ i ω rµ O QP (3) For n infinitely conductin cylinder, or in the hih frequency limit, µ eff is reduced y the rtio of the cylinder volume to the cell volume. This rtio of volumes will turn out to e the key fctor in determinin the strenth of the effect in ll our models. Evidently in the present model µ eff cn never e less thn zero, or reter thn unity. It should lso e mentioned tht to mximise the effect we could hve replced the metllic cylinders with prisms of squre cross section to mximise the volume enclosed within the prism. If the resistivity of the sheets is hih then the dditionl contriution to µ eff is iminry ut lwys less thn unity, µ eff 3 ω µ + i, σ >> ω rµ (4) σ A further point tht should e noted is tht ll the structures we discuss hve electricl s well s mnetic properties. In this prticulr cse we cn crudely estimte for electric fields perpendiculr to the cylinders, ε eff F HG = F = I KJ where F is the frction of the structure not internl to cylinder. In derivin (5) we ssume tht the cylinder is perfect conductor, nd nelect depolrisin fields risin from interction etween cylinders. Inclusion of ε eff in our clcultions removes one difficulty y ensurin tht, (5) lim ω d i (6) liht eff eff ω c = lim c ε µ = c Evidently without ε eff the velocity of liht in the effective medium would hve exceeded tht in free spce. Most of the structures discussed in this pper hve similr ε eff. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 6
7 B. A Cpcittive Arry of Sheets Wound on Cylinders The previous structure showed limited mnetic effect. Now we show how to extend the rne of mnetic properties ville to us y introducin cpcittive elements into the structure. We tke the sme structure of cylinders s efore except tht the cylinders re now uilt in split rin confiurtion shown elow in fiure 3. Fiure 3. Model B consists of squre rry of cylinders s for model A ut with the difference tht the cylinders now hve internl structure The sheets re divided into split rin structure nd seprted from ech other y distnce d. In ny one sheet there is p which prevents current from flowin round tht rin. The importnt point is tht there is p which prevents current from flowin round ny one rin. However there is considerle cpcitnce etween the two rins which enles current to flow, Fiure 4. When mnetic field prllel to the cylinder is switched on it induces currents in the split rins s shown in the fiure. The reter the cpcitnce etween the sheets, the reter the current. Detiled clcultions ive, µ eff = F σi 3 + ω rµ 3 π µ ω Cr (7) where F is the frctionl volume of the cell side occupied y the interior of the cylinder, r F = π (8) nd C is the cpcitnce per unit re etween the two sheets, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 7
8 ε C = = d dc µ (9) Hence, µ eff = σi 3dc + ω rµ 3 π ω r () Becuse we now hve cpcitnce in the system which cn lnce the inductnce present, µ eff hs resonnt form which we sketch elow in fiure 5. µ eff µ eff = ω ω mp ω Fiure 5. The effective mnetic permeility for model B shows resonnt structure dictted y the cpcitnce etween the sheets nd the mnetic inductnce of the cylinder. We sketch the typicl form for hihly conductin smple, σ. Below the resonnt frequency µ eff is enhnced, ut ove resonnce µ eff is less thn unity nd my e netive close to the resonnce. Fiure 5 illustrtes the eneric form of µ eff for ll the structures we present here. We define ω to e the frequency t which µ eff diveres, 3 3dc ω = = 3 3 π µ Cr π r () nd ω mp to e the mnetic plsm frequency ω mp = π µ 3 3 Cr F = π 3 r 3dc F HG I KJ () Note tht the seprtion etween ω ndω mp, which is mesure of the rne of frequencies over which we see stron effect, is determined y /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 8
9 F = (3) the frction of the structure not internl to cylinder. As for the cse A, the simple cylinder, the hih frequency limit is iven y, lim µ eff ω ω = (4) We mention in pssin tht the system sustins lonitudinl mnetic modes t the mnetic plsm frequency, the nloue of the plsm modes of s of free electricl chres [6,7]. Of course, we hve no free mnetic poles, only the ppernce of such s currents round the cylinders mke the cylinder ends pper to support free mnetic poles in the fshion of r mnet. Toether with ε eff iven in eqution (5), which is lso pplicle here, we cn illustrte eneric dispersion reltionship shown elow in fiure 6. Fiure 6. Generic dispersion reltionship for resonnt structures with µ eff. The solid lines represent twofold deenerte trnsverse modes, the dotted line sinle lonitudinl mnetic plsmon mode. The relevnt points to note re: (i) wherever µ eff is netive there is p in the dispersion reltionship, i.e. for, ω ω ω < < mp (5) (ii) lonitudinl mnetic plsm mode, dispersionless in this pproximtion, is seen t ω = ω mp. (iii) The dispersion reltion converes symptoticlly to the free spce liht cone. s discussed ove. In fct metllic structures in enerl represent fresh pproch to the photonic insultor concept introduced independently y Ylonovitch [8,9] nd John []. If we tke the followin vlues, 3 r =. m 3 = 5. m 4 d =. m (6) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 9
10 we et,. Hz (7) f mp = π ω mp = 47 9 f F HG I 9 = fmp 94 KJ =. Hz (8) Note tht the frequency t which the structure is ctive corresponds to free spce wvelenth of cm, much reter tht the.5cm seprtion etween cylinders. This will e typicl of these cpcittive structures nd implies tht the effective medium pproximtion will e excellent. C Swiss Roll Cpcitor We tke the sme rrnement of cylinders on squre lttice s efore except tht the cylinders re now uild s follows: Fiure 7. In model C metllic sheet is wound round ech cylinder in coil. Ech turn of the coil is spced y distnce d from the previous sheet. The importnt point is in tht no current cn flow round the coil except y virtue of the self cpcitnce, Fiure 8. When mnetic field prllel to the cylinder is switched on it induces currents in the coiled sheets s shown in the fiure. Cpcitnce etween the first nd lst turns of the coil enles current to flow. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
11 In this instnce we find for the effective permeility, µ eff = F σi + ω rµ N 3 π r µ N ω C (9) = σi dc + ω rµ N 3 π r N ω where F is s efore the frction of the structure not internl to cylinder, nd the cpcitnce per unit re etween the first nd the lst of the coils is, ε C = = d N µ dc N (3) The criticl frequencies re, ω = dc = 3 π r µ C N π r N 3 (3) ω mp = dc = 3 F π r µ C N F I 3 π r N HG KJ (3) If we tke the vlues we used efore in (6), 3 r =. m 3 = 5. m 4 d =. m N = (33) we et, π ω. Hz (34) 9 f = = 38. (35) f mp = π ω mp = i.e. there is much more cpcitnce in this model nd the rne of ctive frequencies is n order of mnitude lower thn it ws in model C which used only two overlppin sheets. Choosin n even smller scle nd reducin the numer of turns in order to drive up the frequencies to our rne of interest, /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
12 4 r =. m 4 = 5. m 5 d =. m N = 3 (36) we et, 9 f = 85. Hz (37) f mp =. 5 9 Hz (38) Note tht the free spce wvelenth t the plsm frequency is round 3cm nd compre this to the very much smller spcin etween cylinders of.5cm. We shll now clculte the dispersion of µ eff for vrious prmeters. First let us tke the prmeters iven in eqution (36). The resultin dispersion of µ eff is shown elow in fiure 9 Fiure 9. Dispersion with frequency of µ eff for Swiss roll structure, clculted for the prmeters shown in eqution (36), ssumin tht the metl hs zero resistivity. Next we enquire wht is the effect of mkin the sheets resistive? Below we present series of clcultions for vrious vlues of the resistivity, σ, iven in Ω. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
13 Fiure. Dispersion with frequency of µ eff for Swiss roll structure, clculted for the prmeters shown in eqution (36), for vrious vlues of the resistivity of the sheets:.ω,.ω, 5.Ω,.Ω. In fiure we increse the resistivity from. Ω to. Ω. Note the rodenin of the resonnce, the complementry ehviour of µ rel nd µ im, dictted y Krmers Kroni, nd how resistivity limits the mximum effect chieved. Next we explore the dependence on the rdius of the cylinders. In fiure the rdius of the cylinders is decresed, reducin the volume frction occupied y the cylinders, nd risin the resonnt frequency y fctor of two. We lso decrese d, the spcin etween the sheets, incresin the cpcitnce in the system nd rinin the resonnt frequency ck down to its oriinl vlue. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 3
14 Fiure. Dispersion with frequency of µ eff for Swiss roll structure. Top: clculted for the prmeters shown in eqution (36), except tht the resistivity of the sheets is now.ω, nd the rdius of the cylinders hs een reduced from. 4 m to 6. 4 m, thus risin the resonnt frequency y fctor of two. Bottom: d, the spcin etween the sheets, hs een reduced to. 5 5 m rinin the resonnt frequency ck to the oriinl vlue. Usin cpcittive cylindricl structures such s the Swiss roll structure we cn djust the mnetic permeility typiclly y fctor of two nd, in ddition if we desire, introduce n iminry component of the order of unity. The ltter implies tht n electromnetic wve movin in such mteril would decy to hlf its intensity within sinle wvelenth. This presumes tht we re seekin rodnd effects tht persist over the reter prt of the GHz reion. However if we re prepred to settle for n effect over nrrow rne of frequencies spectculr enhncements of the mnetic permeility cn e chieved, limited only e the resistivity of the sheets nd y how nrrow nd we re willin to tolerte. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 4
15 III. AN ISOTROPIC MAGNETIC MATERIAL The structures shown ove ive mnetic properties when the field is lined lon the xes of the cylinders, ut hve essentilly zero mnetic response in other directions. They suffer from nother potentil prolem: if the lternte polristion is considered where the electric field is not prllel to the cylinders, the system responds like n effective metl ecuse current is free to flow lon the lenth of the cylinders. For some pplictions this hihly nisotropic ehviour my e undesirle. Therefore we redesin the system with view to restorin isotropy, nd minimisin purely electricl effects. To this end we need sic unit tht is more esily pcked into rrys thn is cylinder, nd which voids the continuous electricl pth provided y metl cylinder. We propose n dpttion of the split rin structure in which the cylinder is replced y series of flt disks ech of which retins the split rin confiurtion ut in slihtly modified form: see fiure. First we shll clculte the properties of disks stcked in squre rry s shown if fiure 3. This structure is still nisotropic, prolem we shll ddress in moment, ut y elimintin the continuous conductin pth which the cylinders provided, it elimintes most of the electricl ctivity lon this direction. Fiure. Left: pln view of split rin showin definitions of distnces. Riht sequence of split rins shown in their stckin sequence. Ech split rin comprises two thin sheets of metl. The rin shown is scled up version defined y the prmeters shown elow in fiure 3. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 5
16 Fiure 3. Pln view of split rin structure in squre rry, lttice spcin. The two dimensionl squre rry shown in fiure 3 cn e mde y printin with metllic inks. If ech printed sheet is then fixed to solid lock of inert mteril, thickness, the locks cn e stcked to ive columns of rins. This would estlish mnetic ctivity lon the direction of stckin, the z xis. The unit cell of this structure is shown in fiure 4 on the left. How do we mke symmetricl structure? Strt from the structure just descried comprisin successive lyers of rins stcked lon the z xis. Next cut up the structure into series of sls thickness, mkin incisions in the y z plne nd ein creful to void slicin throuh ny of the rins. Ech of the new sls contins lyer of rins ut now ech rin is perpendiculr to the plne of the sl nd is emedded within. Print onto the surfce of ech sl nother lyer of rins nd stck the sls ck toether in. The unit cell of this second structure is shown in the middle of fiure 4. In the next step third set of sls is produced y cuttin in the x z plne, printin on the surfce of the sls, nd ressemlin. Finlly we now hve structure with cuic symmetry whose unit cell is shown on the riht of fiure 4. Fiure 4. Buildin 3D symmetry: ech successive restckin of the structure dds rin to nother side of the unit cell. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 6
17 Of course n lternte method of mnufcturin this structure would e to strt from set of cues of the inert mteril nd loriously stick rins to their sides efore ssemlin the cues into lttice. The cut nd pste method we suest ove is much more efficient. Now let us clculte the effective permeility. First we need to clculte the cpcitnce etween the two elements of the split rin. We shll ssume: r >> c, r >> d (39) l < r (4) ln c d >> π (4) Under these conditions we cn clculte the cpcitnce etween unit lenth of two prllel sections of the metllic strips: ε c C c = ln = π d πµ c d ln (4) The effective mnetic permeility we clculte on the ssumption tht the rins re sufficiently close toether tht the mnetic lines of force due to currents in the stcked rins, re essentilly the sme s those in continuous cylinder. This cn only e true if the rdius of the rins is of the sme order s the unit cell side. We rrive t: µ eff = = σ + i 3 l l σ c ω rµ + l i 3 l 3 π µ ω C r ω rµ c πω d r 3 ln where σ is the resistnce of unit lenth of the sheets mesured round the circumference. (43) To ive some exmples let us choose convenient set of prmeters:  =. m 3 c =. m 4 d =. m 3 l =. m 3 r =. m (44) Fiures, 3 show the rins drwn to scle. These prmeters do not quite stisfy ll the inequlities, which is difficult to do with resonle numers, ut note tht the inequlities re only importnt to the ccurcy of our formule, not to the functionin of the structure. The resonnt frequency t which µ eff diveres is iven y, 3lc ω = = 7. c 3 π ln d r (45) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 7
18 or, ω = π 35. GHz (46) If we choose to mnufcture the split rins from lyer of copper, it is esily possile to chieve σ.. Evidently from fiure 5, this produces hihly resonnt structure. Fiure 5. Plot of µ eff for the cuic split rin structure clculted usin the chosen prmeters. Left: for copper rins, σ =. ; riht: for more resistive rins, σ =.. In order to see sustntil effect we hve to increse the resistnce either y incresin the resistivity of the mteril of which the rins re mde, or y mkin them thinner. The sclin of frequency with size cn e deduced from (45) we see tht the resonnt frequency scles uniformly with size: if we doule the size of ll elements in iven structure, the resonnt frequency hlves. Nerly ll the criticl properties re determined y this frequency. IV. ENHANCED NONLINEAR EFFECTS We hve seen how the ddition of cpcitnce to the structure ives fr richer vriety of mnetic ehviour. Typiclly this hppens throuh resonnt interction etween the nturl inductnce of the structure nd the cpcittive elements, nd t the resonnt frequency electromnetic enery is shred etween the mnetic fields nd the electrosttic fields within the cpcittive structure. To put this more explicitly: tke the split rin structure descried in fiures, 3: most of the electrosttic enery of the cpcitor is locted in the tiny p etween the rins. Concentrtin most of the electromnetic enery in this very smll volume will results in n enormously enhnced enery density. If we wish to enhnce the nonliner ehviour of iven compound, we locte smll mount of the sustnce in the p where the stron electrosttic fields re locted. Since the response scles s the cue of the field mplitude, we cn expect enhncements of the order of the enery density enhncement squred. Furthermore not only does the structure enhnce the nonlinerity, it does so in mnner tht is very economicl with the mteril: less tht % of the structure need e filled with the nonliner sustnce. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 8
19 Note tht there is symmetry etween, on the one hnd the present structures desined to enerte mnetic permeility nd within which we find enhnced electrosttic fields, nd on the other hnd the erlier thin wire structures [,] desined to enerte netive electricl permittivity, nd within which we find enhnced mnetic fields. We shll now clculte the enery density in the cpcitnce etween the two split rins in fiures,3. First we clculte the volte etween the two rins s function of the incident mnetic field, H. V in V out s= s= s= V out V in Fiure 6. The emf ctin round one of the sheets of the split rin in fiure s function of the distnce, s, round the rin. V in denotes the emf on the inner rin, nd V out tht on the outer rin. Note tht this rin is cut t s = so tht the emf is discontinuous. The electric field etween the two hlves of the rin is then of the order, E rin V d (47) We clculte tht, 3li H V = ωcπ r L i 3 O r 3 M P + lσ l ω µ π µ ω C r N Q (48) Hence on sustitutin from (4) nd (47) into (48): E rin 3lµ c i H c dr L ω ln i c d O r c NM QP + lσ 3l ω µ 3 πω r ln d (49) Now we rue tht the electrosttic enery density in the incident electromnetic field is equl to the mnetic enery density, which in turn cn e relted to the electrosttic enery density in the rin. Hence, 3l ε E rin ε π r d ωcπ r = µ µ H lσ i 3 l ve + ω rµ 3 π µ ω C r If we evlute this formul on resonnce we et much simplified formul, (5) /word/pp/wecs/mwires.doc t 3 Ferury 999 pe 9
20 ε Erin ω resonnt enhncement= Q = = µ H ω 3 πω r µ 4lσ dc (5) Let us tke s n exmple the prmeters used to clculte fiure 5, Hence, d =. m σ ω l =. m r =. m Q = R = =. ct = 7. L NM πω r µ 4lσdc O QP (5) = (53) A more detil picture of enhncement s function of frequency is shown in fiure 7. Fiure 7. Enhncement of the enery density of the electric field within the p etween the split rins (see fiures nd 3) for two different vlues of the resistivity of the metl sheet. The correspondin vlues of µ eff re shown in fiure 5. For exmple: em of microwves t 3.4GHz with power flux of 4 wm hs n electric field strenth of the order of 3 Vm in vcuo. If this em were incident on, nd entirely trnsmitted into, our mnetic structure it would enerte field strenth of the order of /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
21 4 Vm in the spce etween the split rins, or of the order of 6 V etween the edes of the two rins: more thn enouh to cuse electricl rekdown in ir! It is evident tht these structures hve considerle potentil for enhncin nonliner phenomen. Furthermore the nonliner medium need only e present in the smll volume within which the enery is concentrted, openin the possiility of usin smll quntities of expensive mteril, nd reducin ny requirements of mechnicl interity tht lrer structure would impose. In pssin we drw n nloy with surfce enhnced Rmn sctterin (SERS), oserved on rouh metllic surfces, typiclly silver surfces. The Rmn sinl from molecules dsored on these surfces my e enhnced y fctors of the order of 6 over tht seen on insultin surfces. The Rmn effect is proportionl to the second power of the electromnetic mode density t the surfce, 3 4 nd it is known tht rouhness cn enhnce the locl mode density y fctors of up to, hence the spectculr Rmn enhncement (see [] for further detils nd references). A very similr locl enhncement tkes plce in our system nd, we expect, cn e exploited in n nloous fshion. In conclusion: we hve shown how to desin structures mde from nonmnetic thin sheets of metl, which respond to microwve rdition s if they hd n effective mnetic permeility. A wide rne of permeilties cn e chieved y vryin the prmeters of the structures. Since the ctive inredient in the structure, the this metl film, comprises very smll frction of the volume, typiclly : 4, the structures my e very liht, nd reinforced with stron insultin mteril to ensure mechnicl strenth, without dversely ffectin their mnetic properties. It is likely tht the structures will e exploited for their ility to concentrte the electromnetic enery in very smll volume, incresin its density y hue fctor, nd retly enhncin ny nonliner effects present. REFERENCES [] J.B. Pendry, A.J. Holden, W.J. Stewrt, I. Youns, Extremely Low Frequency Plsmons in Metllic Meso Structures, Phys. Rev. Lett. vol. 76, pp , 996. [] J.B. Pendry, A.J. Holden, D.J. Roins, nd W.J. Stewrt, Low Frequency Plsmons in Thin Wire Structures, J. Phys. [Condensed Mtter], vol., pp , 998. [3] D.F. Sievenpiper, M.E. Sickmiller nd E. Ylonovitch, 3D Wire mesh photonic crystls Phys. Rev. Lett., vol. 76 pp , 996. [4] D.F. Sievenpiper, E. Ylonovitch, J.N. Winn, S. Fn, P.R. Villeneuve, nd J.D. Jonnopoulos, 3D MetlloDielectric Photonic Crystls with Stron Cpcitive Couplin etween Metllic Islnds, Phys. Rev. Lett., vol. 8, pp 8983, 998. [5] J.B. Pendry Clcultin Photonic Bnd Structure J. Phys. [Condensed Mtter], vol. 8 pp 858, 996. [6] D. Pines nd D. Bohm, A Collective Description of Electron Interctions: II Collective vs Individul Prticle Aspects of the Interctions, Phys. Rev. vol. 85 pp , 95. [7] D. Bohm nd D. Pines, A Collective Description of Electron Interctions: III Coulom Interctions in Deenerte Electron Gs, Phys. Rev. vol. 9. pp 6965, 953. [8] E. Ylonovitch, Inhiited Spontneous Emission in Solid Stte Physics nd Electronics, Phys. Rev. Lett., vol. 58, pp 596, 987. [9] E. Ylonovitch, Photonic Bnd Gp Crystls, J. Phys.: [Condensed Mtter], vol. 5, pp , 993. [] S. John, Stron Loclistion of Photons in Certin Disordered Lttices, Phys. Rev. Lett., vol. 58, pp , 987. [] F.J. Grci Vidl nd J.B. Pendry, Collective Theory for Surfce Enhnced Rmn Sctterin, Phys. Rev. Lett., vol. 77, pp 6366, 996. /word/pp/wecs/mwires.doc t 3 Ferury 999 pe
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