SURFACE MODES AT THE INTERFACES BETWEEN ISOTROPIC MEDIA AND UNIAXIAL PLASMA

Transcription

1 Prgress In Electrmagnetics Research, PIER 76, 1 14, 2007 SURFACE MODES AT THE INTERFACES BETWEEN ISOTROPIC MEDIA AND UNIAXIAL PLASMA H. Huang and Y. Fan Schl f Electrical Engineering Beijing Jiatng University Beijing , P. R. China B. Wu Research Labratry f Electrnics Massachusetts Institute f Technlgy Cambridge, MA , USA F. Kng Schl f Infrmatin Science and Engineering Shandng University Jinan , P. R. China J. A. Kng Research Labratry f Electrnics Massachusetts Institute f Technlgy Cambridge, MA , USA Abstract A detailed study f surface TM mdes at the interface between an istrpic medium and a uniaxial plasma is presented. Fur cases fr the istrpic medium, including nrmal, Left-handed, magnetic, and metallic media, are cnsidered. The cnditins fr the existence f surface mdes in each case are analyzed, shwing that the existence is determined by the parameters f media, wrking frequency, and the directin f the principle axis. The Pynting vectr alng the prpagating directin is als calculated. Depending n the media parameters and the frequency, the surface mde can have time-average Pynting vectr in the ppsite directin f the mde phase velcity. Als with Research Labratry f Electrnics, Massachusetts Institute f Technlgy, Cambridge, MA , USA.

2 2 Huang et al. 1. INTRODUCTION Surface waves (SWs) have attracted the attentin f many scientists since the wrk f Jnathan Zenneck in 1907 [1] and Lrd Rayleigh n elastic slids [2]. They can exist under certain cnditins n an entirely free surface bunded by air, r at the interface separating tw semiinfinite half-spaces. They prpagate alng the interface and decay in the transverse directin expnentially while having the field maximum n the interface [3 8]. They are useful fr studying f the physical prperties f the surfaces [9, 10]. Thus the investigatins f SWs are imprtant frm bth a scientific pint f view and a practical ne. The general case f surface waves at the interface between a uniaxial crystal and istrpic medium were analyzed in detail in previus publicatins [11 13]. They cncluded that surface wave prpagatin is pssible nly fr psitive uniaxial materials and that t fr a narrw angular range f prpagatin directins abut the bisectr f the angle between the ptical axes. The case between a uniaxial crystal and a magnetic istrpic medium has als been studied [14]. Furthermre, Walker et al. analyzed surface wave prpagatin at the interface f an istrpic material and an arbitrarily riented uniaxial r biaxial material [4]. They shwed that if bth ptical axes f the biaxial material are tangential t the interface, surface wave can ccur ver a large range f prpagatin directins and will cnfine wave mre tightly than biaxial materials with ptical axes in ther rientatins r uniaxial materials. And mre extended cases f the interface between istrpic media and biaxial by Alshits et al. [15] and between tw uniaxial crystals by Darinskii and Furs et al. respectively [16, 17] were als studied. And quite recently, Sudarshan et al. investigated surface wave at the interface f identical biaxial crystals with a relative twist abut the axis nrmal t the interface [18]. They shwed that the selected type f surface wave is pssible nly fr a restricted range f the twist angle, which depends n the rati f the maximum and the minimum f the principal refractive indexes and the angle between the ptical axes. Characteristics f surface mdes at the interface between an istrpic medium and an indefinite medium that has a dispersin relatin f hyperblic frm are studied by Yan et al. [19]. They cnsidered fur cases fr the istrpic medium, and gave the cnditins fr the existence f surface mdes in each case, indicating that the existence f surface mdes is determined by the nature f the indefinite medium as well as the rientatin f the bundary surface f this anistrpic medium. T ur knwledge, hwever, n detailed analysis abut surface

3 Prgress In Electrmagnetics Research, PIER 76, mdes at bundary f uniaxial medium, which has dispersin characteristics, has yet been given. In this paper, we cnsider the uniaxial medium as an anistrpic plasma externally applied with an infinitely strng DC magnetic field B 0. We study fur kinds f istrpic media: nrmal, Left-handed (with negative permeability and permittivity), magnetic (with negative permeability), and metallic medium (with negative permittivity), and btain the existence cnditins f surface mdes. Mrever, t get deeper insight int the physics f such waves, we als calculate their Pynting vectr and the energy flw alng the prpagating directin. Regin 1 Istrpic medium η z Regin 2 Uniaxial plasma B 0 ξ θ x Figure 1. Diagram fr an interface between a semi-infinite istrpic medium and a uniaxial plasma. The ξ axis indicates principle axis, als the directin f the external magnetic field which is infinitely strng in the uniaxial plasma. 2. SURFACE MODE DISPERSION LAWS We cnsider a special case f uniaxial medium: an electrn plasma applied with an external infinitely strng magnetic filed. In this case, ω c and the medium becmes a uniaxial plasma which has dispersin characteristics, and the permittivity tensr ε 2, takes the fllwing frm in the principal crdinate system: [20] [ ε(ω) 0 0 ] ε 2ξζη = 0 ε 0 0, (1) 0 0 ε 0

4 4 Huang et al. where ε = ε 0 (1 ω2 p ) and ξ axis is in the same directin f the external ω 2 magnetic filed B 0. Furthermre, we cnsider electrmagnetic waves prpagating alng the interface between an istrpic medium and the uniaxial plasma, as shwn in Fig. 1. The ξζη crdinate system is defined accrding t the principle axis f the uniaxial plasma, als the directin f the external magnetic field B 0. Regin 1 is the istrpic medium, with permittivity ε 1 and permeability µ 1 ; regin 2 is the uniaxial plasma, with permeability µ 2 and permittivity tensr ε 2. T get the slutin f the fields in the tw media, we establish a xyz crdinate system s that the interface is in the y-z plane and the x axis is perpendicular t it. Fr the xyz crdinate system, the permittivity tensr can be transfrmed t [ ] εxx 0 ε xz ε 2 = 0 ε y 0, (2) ε zx 0 ε zz with ε xx = ε cs 2 θ + ε 0 sin 2 θ, (3.1) ε xz =(ε ε 0 ) sin θ cs θ, (3.2) ε xz = ε zx, (3.3) ε zz = ε sin 2 θ + ε 0 cs 2 θ, (3.4) ε y = ε 0, (3.5) where θ is the angle made by the magnetic field and the nrmal f the interface (see Fig. 1) and π/2 θ π/2. The surface waves prpagate alng the z-axis with a wave number k z, and the attenuatin f the waves in the x-directin are defined by the quantities α 1, α 2. The wave vectr and electrical field in the tw regins can be written as } k 1 =ˆx( iα 1 )+ẑk z Ē 1 =(ˆxE 1x +ŷe 1y +ẑe 1z )e ikzz+α 1x (x <0) (4) } k 2 =ˆx(iα 2 )+ẑk z Ē 2 =(ˆxE 2x +ŷe 2y +ẑe 2z )e ikzz α 2x (x >0) (5) Fr the uniaxial plasma, ntice that the ff-diagnal permittivity tensr, shwn in Eq. (2), is symmetrical and there is nly ne element in y-directin. Accrding t the Maxwell Equatins, the electrical field cmpnents depend upn each ther in the fllwing way ( )( ) k 2 z + ω 2 µ 2 ε xx iα 2 k z + ω 2 µ 2 ε xz E2x iα 2 k z + ω 2 µ 2 ε zx α2 2 + = 0 (6.1) ω2 µ 2 ε zz E 2z (α 2 2 k 2 z + ω 2 µ 2 ε y )E 2y = 0 (6.2)

5 Prgress In Electrmagnetics Research, PIER 76, Equatins (6.1) and (6.2) shw that the TE mde (E y, H x, H z ) is nt cupled t the TM mde (H y, E x, E z ) and it is the TM mde that is affected by the dispersin. That is why we fcus n the TM wave in this paper. Furthermre, by setting the determinant f the matrix in Eq. (6.1) equal t zer, we btain the dispersin relatin f the TM wave in the uniaxial plasma, which is ω 2 µ 2 = (iα 2 cs θ + k z sin θ) 2 + ( iα 2 sin θ + k z cs θ) 2. (7) ε 0 ε And it is knwn that the dispersin relatin f the istrpic medium is kz 2 α1 2 = ω 2 µ 1 ε 1. (8) Fr TM wave, the magnetic field in the tw regins can be written as H 1 =ŷae ikzz+α1x (x<0), (9) H 2 =ŷae ikzz α2x (x>0). (10) S, substituting Eqs. (9) and (10) int Maxwell Equatins, the electrical fields can be btained as Ē 1 = 1 (ˆxk z +ẑiα 1 )Ae ikzz+α1x (x<0), (11) ωε 1 ( Ē 2 = A ˆx ε zzk z iε xz α 2 ω ε 2 +ẑ ε xzk z + iε xx α 2 xz ε xx ε zz ε 2 e ikzz α2x (x>0). (12) xz ε xx ε zz By matching the bundary cnditins at interface f x = 0, i.e., E 1z = E 2z, we btain the relatin f prpagatin cnstant k z and α 1, α 2, which is iα 1 = ε xzk z + iε xx α 2 ε 1 ε 2 = ε xzk z + iε xx α 2. (13) xz ε xx ε zz εε 0 Finally, slutins f equatins (7), (8) and (13) are [19] kz 2 = ω 2 ε 1(ε 1 ε xx µ 2 εε 0 µ 1 ) ε 2 1 εε, (14) 0 α 1 = ω ε 1 ) (ε xx µ 2 ε 1 µ 1 ) ε 2 1 εε, (15) 0 α 2 = α 2r + iα 2i = α 1εε 0 ε 1 ε xx where bth α 2r, α 2i are real. + i ε xzk z ε xx, (16)

6 6 Huang et al. 3. CONDITIONS FOR EXISTENCE OF TM SURFACE MODES The necessary cnditin f surface mde existence is kz, 2 α 1 and α 2r must be psitive. We will discuss the cnditin f surface TM mde existence fr fur cases f istrpic medium: (i) ε 1 > 0, µ 1 > 0; (ii) ε 1 < 0, µ 1 < 0; (iii) ε 1 > 0, µ 1 < 0; (iv) ε 1 < 0, µ 1 > 0, which crrespnd t nrmal, left-handed, magnetic, and metallic media respectively. Withut lsing any generality, we assume that the permeability f the uniaxial plasma is psitive, i.e., µ 2 > 0. The cnditins f surface TM mde existence fr the fur cases are shwn in Table 1. And the critical angle θ c1 and θ c2 are defined as ( cs 2 θ c1 = 1 ε )( ) 1µ 1 ω 2 ε 0 µ 2 ωp 2 (17) [ ( )] (ω ) cs 2 2 θ c2 = (18) 1 ε 0µ 1 ε 1 µ 2 1 ω2 p ω 2 ω 2 p Table 1. Cnditins fr the existence f the surface TM mde. Case Parameter Cnditin ω θ (i) ε 1 > 0, ε 1µ 1/(ε 0µ 2) < 1 ω<ω p π/2 > θ >θ c1 µ 1 > 0 (ii) ε 1 < 0, ε 1 >ε 0,ε 1µ 1/(ε 0µ 2) < 1 ω>ω p π/2 > θ >θ c1 µ 1 < 0 ε 1µ 1/(ε 0µ 2)<1 ω (ω p,ω pε 0/ ε 2 0 ε 2 1 ) π/2 > θ >θc1 ε 1 <ε 0 ε 1µ 1/(ε 0µ 2) > 1/[1 (ω 2 p/ω 2 )] ω>ω pε 0/ ε 2 0 ε2 1 θ <θ c1 (iii) ε 1 > 0, ε 0µ 1/(ε 1µ 2)>1/[1 (ω p/ω 2 2 )] ω<ω p π/2 > θ >θ c2 µ 1 < 0 (iv) ε 1 < 0, ε 1 >ε 0 ω>ω p π/2 > θ > 0 µ 1 > 0 ε 1 <ε 0 ω (ω p,ω pε 0/ ε 2 0 ε2 1 ) π/2 > θ > 0

7 Prgress In Electrmagnetics Research, PIER 76, A. Case (i): ε 1 > 0, µ 1 > 0 (nrmal medium) First, we cnsider the case f ε 1 > ε 0. As (ε 2 1 εε 0) = [ε 2 1 ε2 0 (1 ω2 p/ω 2 )] > 0, accrding t Eq. (14), the psitiveness f kz 2 requires ε xx > εε 0 µ 1 /(ε 1 µ 2 ). Accrding t Eqs. (15) and (16), the psitiveness f α 1 and α 2r requires ε xx > ε 1 µ 1 /µ 2, ε xx ε < 0 respectively. Because ε is the functin f the wrking frequency ω, the existence cnditin is based n the frequency. When ω>ω p, then ε>0, psitiveness f α 2r requires ε xx < 0, while psitiveness f α 1 requires ε xx >ε 1 µ 1 /µ 2 > 0, thus there is n surface wave under this cnditin, n matter what the angle θ is. When ω<ω p, ε<0, the slutin f these inequalities is ε xx >ε 1 µ 1 /µ 2. Accrding t Eq. (3.1), the slutin can be transfrmed as (ε cs 2 θ + ε 0 sin 2 θ) >ε 1 µ 1 /µ 2, i.e., (ε ε 0 ) cs 2 θ>(ε 1 µ 1 /µ 2 ε 0 ). Hence, the cnditin fr existence ( )( ) can be expressed as cs 2 θ< 1 ε 1µ 1 ω 2 ε 0 µ 2. Since 0 cs 2 θ 1, ωp 2 there ( is a necessary )( ) cnditin fr the surface wave existence, which is 1 ε 1µ 1 ω 2 ε 0 µ 2 > 0, i.e., ε ωp 2 1 µ 1 /(ε 0 µ 2 ) < 1. It means that if ε 1 µ 1 /(ε 0 µ 2 ) > 1, there is n surface wave at all, n matter what the angle θ is. Secnd, we cnsider the ther case, i.e., 0 < ε 1 < ε 0. When ω<ω p, then ε<0 and (ε 2 1 εε 0) > 0, the cnditin fr existence is the same as ω<ω p in case f ε 1 >ε 0. When ω p <ω<ω p ε 0 / ε 2 0 ε2 1, we find ε>0and (ε 2 1 εε 0) > 0, the same as ω>ω p in case f ε 1 >ε 0, s there is n surface wave either. When ω>ω p ε 0 / ε 2 0 ε2 1, we get ε > 0 and (ε 2 1 εε 0) < 0. Based n the Eq. (14), (15) and (16), the cnditin is ε xx < 0, i.e., (ε cs 2 θ + ε 0 sin 2 θ) < 0, which can nt be satisfied because ε>0. S, there is n surface wave under this cnditin. T sum up, the cnditin f surface TM wave existence fr case (i) is fund as ( cs 2 θ< 1 ε )( ) 1µ 1 ω 2 ε 0 µ 2 ωp 2, (19) with the necessary cnditin f ω<ω p, ε 1 µ 1 /(ε 0 µ 2 ) < 1. Figure 2 is the plt f critical angle θ c fr the existence f the surface TM mde in case (i). Fig. 3 is the dispersin characteristics fr the surface TM mdes in case (i). In Fig. 2(a), we can see that the line θ c =45 has intersectins with lines ε 1 µ 1 /(ε 0 µ 2 )=0.1, 0.2, 0.3 and 0.4. That means that when ε 1 µ 1 /(ε 0 µ 2 ) equals t thse values, there must exist surface mde, and the frequency f the intersectin in Fig. 2 is the cut-ff frequency under such situatin. Thse can be verified clearly by Fig. 3(a).

8 8 Huang et al. 100 critical angle θc (in degree) ε 1 µ 1 /( ε 0 µ 2 )= frbidden area ω / ωp (a) 1 θ=90 parameter rati θ=80 θ=70 θ=60 θ=50 θ=40 θ=30 20 frbidden area ω / ωp (b) Figure 2. Cntur plt f critical angle θ c fr the existence f the surface TM mde in case (i). The surface mde can exist when ε 1 µ 1 /(ε 0 µ 2 ) < 1 and π/2 > θ >θ c fr (a) different parameter rati ε 1 µ 1 /(ε 0 µ 2 ); (b) different external magnetic field directin (θ is the angle between the magnetic field and the nrmal f the interface). Shaded area in each figure indicates the frequency dmain where the surface waves are frbidden.

9 Prgress In Electrmagnetics Research, PIER 76, kz/k ε1 µ 1/( ε0µ 2)=0.4 ε1µ 1/( ε0µ 2)=0.3 ε1 µ 1/( ε0µ 2) =0.2 ε1µ 1/( ε0µ 2)=0.1 frbidden area ω / ωp (a) θ= kz/k θ=80 θ=70 frbidden area X: Y: ω / ωp (b) Figure 3. Dispersin characteristics fr the surface TM mdes in case (i): (a) when the directin f the external magnetic field is fixed (θ =45 ); (b) when the parameter rati is fixed (ε 1 µ 1 /(ε 0 µ 2 )=0.8). Each case has a cut-ff frequency except θ =90. Shaded area in each figure indicates the frequency dmain where the surface waves are frbidden.

10 10 Huang et al. Similarly, in Fig. 2(b), the line ε 1 µ 1 /(ε 0 µ 2 )=0.8has intersectins with curves θ =80, θ =70. Accrding t the existence cnditins, i.e., θ>θ c, we can see that when ε 1 µ 1 /(ε 0 µ 2 )=0.8, if θ =70 r 80, there must exist SW with the cut-ff frequency equals t that f the intersectin. When θ =90, there is SW withut cut-ff frequency. Als, thse can be verified by Fig. 3(b). B. Case (ii): ε 1 < 0, µ 1 < 0 (left-handed medium) Taking the prcedure similar t that in case (i), we can find that the cnditin fr the existence f surface wave, shwn in Table 1. The cnditin depends n the cmparisn f abslute value f ε 1 with ε 0. Fr different situatin, there is different frequency dmain and angle range needed. C. Case (iii): ε 1 > 0, µ 1 < 0 (magnetic medium) When the regin 1 is a magnetic medium, (see Table 1) the frequency must be lwer than ω p, and there is als further restraint t the medium parameter and angle, similar t case (i), except the expressin f critical angle and medium parameter cnditin. D. Case (iv): ε 1 < 0, µ 1 > 0 (metallic medium) In this case, the existence f the surface wave is independent f the directin f the magnetic field, i.e., there is n restraint t the angle θ, shwn in Table 1. That means that nce the parameter and frequency cnditins are satisfied, there will be a surface mde, n matter what the directin f the principle axis f the uniaxial plasma is. 4. POYNTING VECTOR AND ENERGY In rder t get deeper insight int the physics f wave prcess, we prceed with calculatin f energy flw alng the surface. Withut lsing any generality, the prpagating directin f the wave is assumed t be z-directin, i.e., k z > 0 in the fllwing. The time-averaged Pynting vectrs f surface wave are < S 1 > =ẑ< S 1z >=ẑ k z A 2 e 2α 1x (x <0), (20) 2ωε 1 < S 2 > =ẑ< S 2z >=ẑ k z A 2 e 2α2rx (x>0). (21) 2ωε xx Furthermre, the crrespnding ttal energy flw assciated with

11 Prgress In Electrmagnetics Research, PIER 76, the whle mde is determined by integratin ver the x-crdinate < S >= 0 =ẑ k z A 2 4ω < S 1 >dx+ ( ) ε 1 α 1 ε xx α 2r 0 < S 2 >dx =ẑ k z A 2 εε 0 ε 2 1 (22) 4ωα 1 ε 1 εε 0 Accrding t Eq. (16), if there exists surface wave, bth α 1 and α 2r are psitive, then ε 1 ε xx ε<0. Ntice that in bth case (i) and (iii), ε 1 > 0 and ω<ω p, which causes ε<0. S we can get ε xx > 0. Hence, frm Eqs. (20) and (21), we can see that the Pynting vectrs in bth media are in the same directin, i.e., the prpagating directin. They are bth frward waves, and the guided energy flws in bth regins are always in the same directin, which can be seen bviusly in Eq. (22). Fr case (ii) and case (iv), ε 1 < 0 and ω > ω p (see Table 2), s ε > 0. Then we have ε xx > 0. S S 1z < 0 crrespnding t ε 1 < 0, and S 2z > 0 crrespnding t ε xx > 0. That means that in the istrpic medium f regin 1, the Pynting vectr is in the ppsite directin f prpagatin, and the surface wave is a backward wave; while in the uniaxial plasma f regin 2, the directins f Pynting Table 2. Directin f the pynting vectr and energy flw f the surface TM mde (The directin f the wave prpagating is assumed t be z-directin.) Case Parameter and Frequency Cnditin < S 1> < S 2> < S> (i) ε1>0, ε 1µ 1/(ε 0µ 2) < 1, ω < ω p ẑ ẑ ẑ µ 1>0 (ii) ε1 > ε 0, ε 1µ 1/(ε 0µ 2) < 1, ω > ω p ẑ ẑ ẑ ε1<0, ε 1µ 1/(ε 0µ 2) < 1, ω (ω p,ω pε 0/ ε 2 0 ε2 1 ε ) ẑ ẑ ẑ 1 <ε 0 µ 1<0 ε 1µ 1/(ε 0µ 2)<1/[1 (ωp/ω 2 2 )],ω>ω pε 0/ ε 2 0 ε2 1 ) ẑ ẑ ẑ (iii) ε 1 > 0, ε 0µ 1/(ε 1µ 2) > 1/[1 (ωp/ω 2 2 )], ω < ω p ẑ ẑ ẑ µ 1<0 (iv) ε 1 >ε 0, ω > ω p ẑ ẑ ẑ ε 1<0, ε 1 < ε 0, ω (ω p,ω pε 0/ ε 2 0 ε2 1 µ ) ẑ ẑ ẑ 1>0

12 12 Huang et al. vectr and prpagatin are the same, and the surface wave is a frward wave. Based n Eq. (22), we can see that if ε 1 ε(εε 0 ε 2 1 ) < 0, i.e., (εε 0 ε 2 1 ) > 0, the ttal energy flw is in the ppsite directin f prpagatin. If ε 1 >ε 0, there must be (εε 0 ε 2 1 ) < 0 n matter what the frequency is. That means the ttal energy flw is in the same directin f prpagatin. If ε 1 <ε 0, nly when ω>ω p ε 0 / ε 2 0 ε2 1, is (εε 0 ε 2 1 ) psitive, therwise it is negative. S, we can see frm Table 2 that in mst cases f (ii) and (iv), the ttal energy flw is in the prpagatin directin, but it is ppsite t that f prpagatin when regin 1 is a Left-Handed Material, which has the parameter ε 1 <ε 0, and the frequency larger than ω p ε 0 / ε 2 0 ε2 1 is needed. 5. CONCLUSION This paper gives an investigatin n hw surface TM mdes respnse in different frequency dmains at the interface between an istrpic medium and a uniaxial plasma which has dispersin characteristics. We cnsider a special case f uniaxial medium: an electrn plasma applied with an external infinitely strng magnetic filed. Fur cases fr the istrpic medium, including nrmal, LHM, magnetic, and metallic media, are cnsidered. The cnditins fr the existence f surface mdes in each case are analyzed, shwing that the existence is determined by the parameter f media, wrking frequency, and the directin f the external magnetic field. The Pynting vectr in the prpagating directin is als calculated and we have fund that if regin 1 is nrmal r magnetic medium, i.e., ε 1 > 0, the surface waves in bth regins are frward waves and the pwer flw alng the prpagatin directin; if regin 1 is metallic medium r LHM, the surface mde in regin 1 is backward wave but frward in regin 2, and the pwer mstly flw in the prpagatin directin except the case when regin 1 is a Left-Handed Material which has the parameter ε 1 <ε 0, and the frequency larger than ω p ε 0 / ε 2 0 ε2 1 is needed. ACKNOWLEDGMENT This wrk is spnsred by the Office f Naval Research under Cntract N , the Department f the Air Frce under Air Frce Cntract F C-0002, the Chinese Natinal Fundatin under Cntract , and the Chinese 863 Fundatin under Cntracts 2002AA and 2004AA

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