Radiation from a current sheet at the interface between a conventional medium and anisotropic negative refractive medium

Transcription

1 Bull Mater Sci, Vol 3, No 4, August 9, pp Indian Academ of Sciences Radiation from a current sheet at the interface between a conventional medium and anisotropic negative refractive medium YUAN YOU School of Phsics and lectronics, Yancheng Teachers College, Yancheng 4, Jiangsu, China MS received 9 Jul 8; revised 3 April 9 Abstract In this paper we investigate the radiation from a current sheet at the interface between semiinfinite isotropic positive refractive medium and anisotropic negative refractive medium The distribution of the electric and magnetic fields in two regions and Ponting vectors associated with propagating and evanescent waves are calculated The reasons for the singularit of the electric or magnetic field are briefl provided if the waves are evanescent in two media Kewords Negative refractive medium; radiation; Ponting vector Introduction With the realiation of composite medium consisting of intercalated SRRs and wires which ehibit simultaneousl negative ε and (left-handed material) in eperiments (Smith et al ; Shelb et al, ), interests on left-handed materials are beginning to soar (Smith and Kroll ; Maro and Sououlis ; Smith et al ; Fan et al 8; Fang and Xu 8; Li et al 8; Lin et al 8; Mandatori et al 8; Powell et al 8; Wang et al 8; Yan et al 8) Such media are denoted as negative refractive medium (NRM) NRM permits electromagnetic wave to propagate with unusual effects, such as negative inde of refraction, anomalous Snell s law, Doppler effect etc (Veselago 968) The main character of such isotropic media is that the direction of Ponting vector and phase velocit are antiparallel With the eception of some brief discussion, the emphasis of theoretical studies have centred on the case where the NRM is isotropic Alu and ngheta () investigated the radiation from a travelling-wave current sheet between two isotropic media, one of which is conventional media and the other is NRM The radiation of a current sheet is also analsed to demonstrate frequenc regions where the sign of refractive inde is negative (Smith and Kroll ) However, in practice, the structures investigated in eperiment are strongl anisotropic (Smith et al ) Compared with isotropic NRM, anisotropic NRM with ε and tensors ehibited some different behaviours, such as anomalous reflection and refraction at the interface of isotropic regular media and uniaiall (ctcouuan@63com) anisotropic left handed media, especiall anomalous total reflection that occurs under the condition that the incident angles are smaller but not larger than a critical angle (Hu and Chui ) Reflection shows frequenc selective total oblique transmission that is distinct from the Brewster effect at a planar dispersive negative inde interface (Zhou et al 3) Smith and Schurig (3) investigated such anisotropic media and provided a wa with such media to accomplish near-field focusing which is similar to perfect lens but with man advantages As what is mentioned above, NRM is anisotropic in eperiment, so we should eamine how to radiate from a current sheet at the interface between semi-infinite isotropic positive refractive medium (PRM) and anisotropic NRM In this paper, we will present a detailed analsis of the radiation from a current sheet between two aforementioned media (ie PRM and anisotropic NRM) The distribution of the electric and magnetic fields and Ponting vectors associated with propagating and evanescent waves are provided We find that if the waves are evanescent in two media, the electric and magnetic fields appear in singularit due to the presence of surface polaritons at the interface between two media If certain condition is satisfied, the total power flow ma be equal to ero Formulation We consider a semi-infinite anisotropic NRM surrounded b a isotropic PRM, divided b the infinite plane and assume two media are lossless and nondispersive The geometrical structures are shown in figure, where PRM occupies the upper half space > in figure (a) ( > in figure (b)), and the region of < in figure (a) ( < in figure (b)) is anisotropic NRM The permittivit and 437

2 438 Yuan You permeabilit of PRM are characteried b ε and, both of them are larger than ero, while the permittivit, ˆε and permeabilit, ˆ, of anisotropic NRM are denoted as the following tensor forms (Smith et al 3): ε εˆ ε, ε ˆ For anisotropic NRM, not all principle elements of permittivit and permeabilit possess the same sign (Smith and Schurig 3) Due to the special anisotropic properties, there are two cases corresponding to two different orientations of interfaces, ie the interface between isotropic positive refractive medium and anisotropic negative refractive material is at plane (or plane) to the interface (Zhou et al 3) Figure Geometrical structures where PRM occupies the upper half space > (a) and > (b) The regions of < (a) and < (b) are anisotropic NRMs () () Interface at plane We first stud the case with the interface between isotropic PRM and anisotropic NRM at plane The position of the interface is at, where an infiniteletent thin sheet of current is located A time harmonic dependence ep( iω t) is assumed throughout The densit of the current sheet is epressed as follows i J J e δ ( ) e, (3) where J stands for the strength of the surface current densit, δ is the Dirac delta function and the propagating constant in the direction We consider the case of s-polariation (ie the electric field is transverse to the plane of incidence) In PRM and anisotropic NRM, the wave vectors satisf the following relations ω ε, (4) (5) ω ε, where The epressions of the electric field in the two regions are displaed as i ωε iei ωε Ae e Be > i i ωε i i ωε Ae e Be e < (6) We now that for PNM ( > ), the direction of phase velocit and Ponting vector are parallel The energ flow should propagate awa from the source, so the phase velocit is propagating awa from the source If ω ε >, the wave is in propagating mode, we have ω ε, and if ωε < (evanescent mode) we must choose i ωε to ensure all fields to vanish at For lossless anisotropic NRM, if ω ε >, the wave in NRM is in propagating mode, while ω ε <, the wave is evanescent one although the signs of the components of permittivit and permeabilit need not be full negative, it will have great influence on the modes of waves Smith and Schurig (3) investigated such media and identified four classes of media (cut-off, anti-cutoff, never cutoff and alwas cutoff media) We are interested in the case which includes propagating and evanescent modes, so we choose ε <, < and <, the signs of the other components are arbitrar, which belongs to the first media which Smith and Schurig defined There eists the critical wave vector c ω ε, ie if > c, the wave in NRM is propagating mode, while < c, the wave is evanescent one Now we have ω ε for

3 Radiation at interface between conventional medium and anisotropic negative refractive medium 439 propagating mode and i ω ε for evanescent mode For the latter choice, it still ensures the fields to disappear at According to the above analsis, the electric fields in the two regions are given as i ωε Ae e > i i ωε Ae e < and the magnetic fields are ωε i i ωε Ae e e ω > i i ωε Ae e e ω H ωε i i ωε Ae e e ω < i i ωε Ae e e ω (8) The coefficients A and A can be given as J A A ω ε ω ε ω ω b considering the boundar conditions at, ie e ( e e) (9) e ( H H ) J For convenience, we denote ω ε and ω ε Then the electric and magnetic fields are obtained in different regions rigorousl Furthermore, the Ponting vectors of propagating modes, together with evanescent modes, can also be epressed in the two regions,, (7) S ( ) J ω ωε e e < ( ) ωε e e > () From (), we can easil find that the evanescent wave propagates along the direction and decas along the or direction in the two regions Due to the different signs of and, the direction of Ponting vectors in two media is antiparallel, which ma cause the net energ flow to be ero if some condition can be satisfied If the waves are propagating modes in the two media, ie < min{ ω ε, ω ε }, the power flu densit per unit area of current source can be calculated as P S e S e J ω rad > < ( ) () which equals to the power densit emitting from the current sheet, and P sou ( ) J* (3) For the case in which waves are in evanescent modes in two media, the total time average power can be given b (ie > ma{ ω ε, ω ε }) J ω > < 4 ( ) P S d S d ωε ωε, (4) according to this epression, we find that the net power flow equals to ero if the following equation ω ε ω ε (5), J ω J ω ( ) ( ) e e > S e e <, J ω J ω ( ) ( ) () can be satisfied Let us note the forms of the electric and magnetic fields For propagating modes, their denominator will not be equal to ero, which indicates the electric and magnetic fields have finite values However, there are different cases for evanescent modes If the following epression ω ε ω ε (6), for propagating modes Similarl, the Ponting vectors of evanescent modes are given as is satisfied, it will lead to the singularit of the electric or magnetic field We would lie to stress that the condi-

4 44 Yuan You tions for ero net power flow and the singularit of electric or magnetic fields are different The latter is the dispersion relation of surface polaritons at the interface between PRM and anisotropic NRM The singularit of the electric and magnetic fields is the result of the eistence of surface polaritons Their wave vectors are obtained as ε ε sur ω ± (7) In order to eplain the singularit of the electric and magnetic fields, we assume the line current is along direction, ie i I i π Jline I e δ( ) δ( ) e e δ( ) de The corresponding electric field can be obtained as I line π J d (8) (9) The magnetic field can also be obtained For instance, we just calculate the electric field in PRM (ie > ) sur i ω ε Iω e π ω ε ω ε i I ωe i ω ε ω ε ω ε sur () Residue theorem has been applied during the course of the calculation The same procedure can be applied to the magnetic field For ever line source, there eists surface wave with sur For this reason, now a current sheet is located at, the sum of all surface waves leads to the singularit of the electric and magnetic fields naturall The interface at plane In this section, we consider the second case, ie the interface between isotropic PRM and anisotropic NRM is plane The position of the interface is assumed to be at, where now an infinitel-etent thin sheet of current is located at the interface The densit of the current sheet is i J J e δ ( ) e, () where stands for the propagating constant in the direction We still consider the case of s-polariation Now in PRM and anisotropic NRM, the wave vectors satisf the following relations ω ε, () (3) ω ε, where The epressions of the electric field in two regions are displaed as i ωε i i ωε Ae e Be e > i i ωε i i ωε Ae e Be e < (4) For the sign choice of wave vector, similar considerations are given For PNM ( > ), we must ensure that the direction of phase velocit and Ponting vector are parallel So, we have ωε for propagating mode ( ωε > ), and i ωε for evanescent mode ( ωε < ) to ensure all fields to vanish at For anisotropic NRM, we appl similar method as Smith and Schurig to give the critical wave vector for the present case, ie c ω ε If ε > and / >, the waves in anisotropic NRM are propagating modes if is larger than the critical wave vector, on the contrar, the waves are evanescent ones If ε < and / <, there eist propagating ( < c ) and evanescent modes ( > c ) While ε > and / <, there eist propagating modes for all the real and if ε < and / >, there eist evanescent modes for all real Considering our choice for the signs of the components of permittivit, ˆε and permeabilit, ˆ, ω ε for propagating modes and ωε i for evanescent modes are provided We still ensure the fields to disappear at in the present configuration Now the electric fields in two regions are given as i ωε Ae e > i i ωε Ae e < (5) Similarl, the magnetic fields in two regions can also be given according to the Mawell equation Combined with the boundar conditions at, the electric and magnetic fields can be obtained eplicitl Ponting vectors of propagating and evanescent modes are also obtained as

5 Radiation at interface between conventional medium and anisotropic negative refractive medium 44 J ω J ω e e > ( ) ( ) S e e <, S J ω J ω ( ) ( ) ( ) J ω ωε e e < ( ) ωε e e >, (6) where ω ε and ωε Here, the power flu densit of propagating waves also equals to the power densit emitting from the current sheet, P rad Psou ( ) (7) For the case in which the waves are in evanescent modes in two media, the total time average power can be given (ie > ma{ ω ε, ω ε }) b > < P S d S d 4 ( ) J ω ω ε ω ε (8) Basing on (8), we can easil obtain that the net power flow equals to ero if the equation ω ε ω ε (9), can be satisfied Similarl, the dispersion relation of surface polaritons for the present case is ω ε ω ε (3), which will also lead to the singularit of the electric and magnetic fields Because similar eplanations are available in, we will not give further illustrations Another case is that the distribution of current sheet has the form i J J e δ ( ) e (3) Similar considerations can be implemented for this case, we will also not give further analsis 3 Conclusions In this paper, we give the detailed analsis about the radiation from a current sheet at the interface between semi-infinite isotropic positive refractive medium and anisotropic negative refractive medium The distribution of the electric and magnetic fields in two media and Ponting vectors associated with propagating and evanescent waves are displaed The eistence of surface polaritons leads to the singularit of the electric or magnetic fields if the waves are evanescent ones in two media Snchronousl, the net power flow ma be equal to ero if certain conditions are satisfied Acnowledgement This project was supported b Jiangsu Provincial Natural Science Basic Research of China under Grant No 6KJD499 and the Natural Science Foundation in Yancheng Teachers College References Alu Andrea and ngheta Nader Microwave Opt Technol Lett Fan C Z, Gao Y and Huanga J P 8 Appl Phs Lett Fang Weihai and Shanjia Xu 8 Int J Infrared Milli Waves Hu Liangbin and Chui S T Phs Rev B Li T Q et al 8 Appl Phs Lett 9 3 Lin Xian Qi, Cui Tie Jun, Chin Jessie Yao, Yang Xin Mi, Cheng Qiang and Liu Ruopeng 8 Appl Phs Lett Mandatori A, Sibilia C, Bertolotti M and Haus J W 8 Appl Phs Lett 957 Maro P and Sououlis C M Phs Rev B Powell David A, Shadrivov Ila V and Kivshar Yuri S 8 Appl Phs Lett 9644 Shelb R A, Smith D R and Schult S Science 9 77 Shelb R A, Smith D R, Nemat-Nasser S C and Schult S Appl Phs Lett Smith D R and Kroll Norman Phs Rev Lett Smith D R and Schurig D 3 Phs Rev Lett Smith D R, Padilla W, Vier D C, Nemat-Nasser S C and Shult S Phs Rev Lett Smith D R, Schult S, Maro P and Sououlis C M Phs Rev B Veselago V G 968 Sov Phs Usp 59 Wang Z P, Wang C and Zhang Z H 8 Opt Commun 8 39 Yan Changchun, Cui Yiping, Wang Qiong, Zhuo Shichuang and Li Jianin 8 Appl Phs Lett 948 Zhou Lei, Chan C T and Sheng P 3 Phs Rev B68 544