Superluminal Group Velocity of Electromagnetic Near-fields *
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1 Supeluminal Goup Velocity of Electomagnetic Nea-fields * WANG Zhi-Yong( 王智勇 )**, XIONG Cai-Dong( 熊彩东 ) School of Physical Electonics, Univesity of Electonic Science and Technology of China, Chengdu Supeluminal phenomena have been epoted in many expeiments of electomagnetic wave popagation, whee the supeluminal behavios of evanescent waves ae the most inteesting ones with the genuine physical significances. Conside that evanescent waves ae elated to the nea-one fields of electomagnetic souces, based on the fist pinciples, we study the instantaneous goup velocities of electomagnetic fields in nea-field egion, and show that they can be supeluminal, which can povide a heuistic undestanding fo the supeluminal popeties of evanescent waves. Keywods: nea fields, evanescent waves, goup velocity, supeluminal PACC: 40H, 40. Intoduction A seies of ecent expeiments have evealed that electomagnetic wave was able to tavel at a goup velocity faste than the velocity of light in vacuum, o at a negative goup velocity. Fo example, these phenomena have been obseved in dispesive media [-4], in electonic cicuits [5], and in evanescent wave cases [6-3]. In fact, ove the last decade, the discussion of the tunneling time poblem has expeienced a new stimulus by the esults of analogous expeiments with evanescent electomagnetic wave packets [4-9], and the supeluminal effects of evanescent waves have been evealed in photonic tunneling expeiments in both the optical domain and the micowave ange [6-3]. All the expeimental esults have shown that the phase time do descibe the baie tavesal time [3,0]. * Poject suppoted by the Doctoal Pogam Foundation of Institution of Highe Education of China (Gant No ). ** Coesponding autho. ywang@uestc.edu.cn
2 In view of the fact that the evanescent waves ae actually attibuted to the nea-one fields of electomagnetic souces, in this aticle, we shall show that the goup velocities of electomagnetic nea-fields can be supeluminal.. Geneal expessions of phase velocity and goup velocity Thee has seveal diffeent methods to intoduce the concepts of phase velocity and goup velocity, hee we will choose a most geneal method. In the most geneal case, one can expand a field quantity Φ(, t) as the supeposition of diffeent fequency components with the usual Fouie-tansfom as its special case ( = ( x, y, )): + 3 = ω 3/ (π) k k k Φ(,) t Ψ (,, t, )exp[i Θ(,, t, ω )]d () whee i=, 3 d k dkxdkydk =, ω = ω(k) is the fequency, k = ( k, k, k ) the wave-numbe x y vecto, and Ψ (,, t k, ω) the amplitude of the k-th component of Φ(,) t. Let =, k = k and so on. Only expand Φ(,) t as the supepositions of monochomatic plane waves can one wite Φ(, t) as the usual Fouie-tansfom fom. While, if we expand Φ(,) t as the supeposition of monochomatic cylindical (o spheical, etc.) waves, the amplitude Ψ (,, t k, ω) would depend on space (o even time) coodinates. Howeve, the validity of ou late conclusions has nothing to do with the fact whethe Equ. () coesponds the usual Fouie-tansfom o not. The quantity Θ(,, t k, ω) in Equ. () epesents a phase. Following the definition of phase velocity, we let Θ(,, t k, ω ) = C (C is a constant) and have dθ d t = ( Θ )(d d t) + Θ t = 0, then the phase velocity is ( a = / ) v d p ( Θ = a = )/( Θ dt a t ) On the othe hand, we may take goup velocity (say, v g () ) as the move velocity of the peak of wave packet, which is valid fo both defomed and undefomed wave packets, and in ageement with the phase time theoy of tunneling time (note that the phase time has nothing to do with the concept of phase velocity) [5-7]. In fact, in quantum mechanics, the classical velocity of a paticle coesponds to the goup velocity of a wave packet that epesents the paticle, in spite of the fact that the wave packet is
3 defomed with time (povided that the paticle has noneo mass). Accoding to Ref. [], the goup velocity may be meaningful even fo boad band pulses and when the goup velocity is supeluminal o negative. Using Equ. (), and conside that the supeposition of diffeent fequency components gives an extemum at peak location (say, ) of the wave packet Φ(,) t, namely, in the stationay-phase c appoximation, the peak location is given by Θ k = 0, accodingly the goup velocity is c v = a. Let Δ Θ k = 0, then we have dδ dt = 0, that is, g dc dt ( Δ )(d d t) + Δ t =0, one obtains Δ Δ Θ Θ vg = a ( ) ( ) = a ( ) ( ) (3) t t k k Equ. () and Equ. (3) ae the geneal expessions fo calculating the phase velocity and goup velocity, espectively. As an example, let Θ = ωt k, one has vp = a ω k and v g = a ω k. Futhemoe, let us conside that the phase and goup velocities along a given diection, say, the -axis diection. Let k denotes the pojection of k in the -axis diection, it is easy to show that the phase and goup velocities along the -axis diection ae, espectively v Θ Θ p = a ( ) ( ) t, v Θ Θ g = a ( ) ( ) t k k (4) whee a is the unit vecto along the -axis diection. Equ. (4) ae the most geneal expessions fo calculating the phase and goup velocities. It is highly impotant to note that, even though in the absence of dispesion, one has v p v g povided that k k. 3. Goup velocities of the nea-one fields of antennas To gain physics insight, we estict ouselves to the fields of an electic dipole antenna without loss of geneality. Fo a system of chages and cuents vaying in time we can make a Fouie analysis of the time dependence and handle each Fouie component sepaately []. We theefoe lose no geneality by consideing the fields of an electic dipole antenna with cuents vaying sinusoidally in time: J( x, t) = J( x )exp(i ωt) (5) 3
4 As usual, the eal pat of such expessions is to be taken to obtain physical quantities. In the Loent gauge, the coesponding vecto potential A( x, t) is μ expi( ωt k ) 3 Ax (, t) = ( ) d x 4π J x (6) Whee = = x x, k = k, k is the wave-numbe vecto, μ is the vacuum pemeability. Futhemoe, we assume that the electic dipole antenna, as a linea antenna, is oiented along the axis, extending fom = d / to = d /, and its length d satisfies d λ, whee λ = πc / ω is the wavelength. The magnetic induction is B = A while, outside the souce, the electic field is E = i B k. On the basis of Ref. [-3], in spheical coodinate system (, θ, ϕ), k = k, and the non-eo components of the electomagnetic field of the antenna ae: sinθ Hϕ = H0 ( + )exp[i( ωt k)] ik cosθ E = E ( + )exp[i( ωt k)] (7) ik sinθ Eθ = E ( + )exp[i( ωt k)] ik k Whee θ is the angle between the diection of obsevation (along ) and the polaiation diection of the electic dipole moment,,, E ae puely eal o puely imaginay constants. In the nea H 0 E one, 0< k, such that Hϕ E 0 and Hϕ Eθ 0, thus the non-eo components of the electomagnetic field of the antenna can be witten as cosθ E = E ( + )exp[i( ωt k)] E exp(i Θ + C) ik sinθ Eθ = E ( + )exp[i( ωt k)] E exp(i Θ C θ + ) ik ik whee C and C ae two constants, and the phase facto Θ is: (8) Θ = ωt k+ actan( k) (9) whee actan( x ) is the invese tangent function of x. Using Equs. ()-(3) and a = /, we obtain the phase velocity v p and goup velocity v g as follows: 4
5 vp = a[ + ] c = a v p (0) ( k) 4 ( k) + ( k) + vg = a[ ] c = a 4 vg () ( k) + 3( k) What we ae inteested in is only the goup velocity. We give the diagammatic cuve of v g c (instead of v ) vs k. g.4 v g c The cuve of goup velocity vs distance k Fig. The cuve of goup velocity vs distance Based on Fig., we summaie the main chaactes of the goup velocities of the nea fields as follows: ) As k >, the goup velocities of the nea fields satisfy 8c 9 0.9c < v < c, and, as g k + subluminal; one has v c. Theefoe, in the fa one, the goup velocities of the nea fields ae g ) As 0< k, in contast to the case ), the goup velocities of the nea fields ae geate than o equal to the velocity of light in vacuum. Theefoe, in the nea one, the goup velocities of the nea fields ae supeluminal. 4. Nea-one fields and evanescent waves As an example, we show that thee exists a close similaity between the evanescent waves inside an undesie waveguide and the nea-one fields of antennas: a) Distinct fom the tavelling waves inside an odinay waveguide, the evanescent waves inside an undesie waveguide, attenuate exponentially along the diection of popagation, and thei aveage powe flows do not exist (but exist as an enegy stoage), which due to the fact that the impedance is puely 5
6 capacitive (fo the TM mode) o inductive (fo the TE mode). As fo the enegy stoage in the undesie waveguide, the electic enegy is moe than magnetic enegy fo the TM mode and on the contay fo the TE mode. b) Similaly, the nea-one fields of an antenna ae also shaply attenuated as compaed with the fa-one fields of the antenna, and the aveage powe flows of the nea-one fields do not exist because the impedance is puely capacitive (fo the electic dipole antenna) o inductive (fo the magnetic dipole antenna), it is only an enegy stoage that exists. As fo the enegy stoage of the nea-one fields, the electic enegy is moe than magnetic enegy fo the electic dipole antenna and on the contay fo the magnetic dipole antenna. In fact, in Ref. [4], Feynman has given a detailed analysis fo the equivalence between the evanescent waves inside a waveguide and the nea fields of a souce 5. Conclusions and pospects On the one hand, the supeluminal behavios of evanescent waves have been evealed in photonic tunneling expeiments. On the othe hand, in this aticle we have shown that the goup velocity of the nea-one fields is supeluminal. Owing to the fact that evanescent waves actually coespond to the nea-one fields of a souce, the esults obtained in this aticle can povide a heuistic undestanding fo the supeluminal phenomena of the evanescent waves. In fact, accoding to quantum mechanics, the position-momentum uncetainty would become emakable as a measuement is pefomed in the nea one of a souce, such that the supeluminal behavios of the nea-one fields take place without violating causality pinciple. In ou next wok, we shall povide a moe quantitative analysis fo the elations between the supeluminal behavios of the nea-one fields and those of the evanescent waves, and popose that these supeluminal phenomena, without violating causality pinciple, have a common theoetical foundation in quantum field theoy. Acknowledgments The fist autho wishes to thank William D. Walke, Pof. Ole Kelle as well as Pof. G. Nimt fo 6
7 valuable discussions. Refeences [] Bigelow M S, Lepeshkin N N and Boyd R W 003 Science [] Kumich A, Dogaiu A and Wang L J et al 00 Phys. Rev. Lett [3] Wang L J, Kumich A and Dogaiu A 000 Natue [4] Solli D R, McComick C F and Ropes C et al 003 Phys. Rev. Lett [5] Mitchell M W and Chiao R Y 998 Am. J. Phys [6] Steinbeg A M, Kwiat P G and Chiao R Y 993 Phys. Rev. Lett [7] Endes A and Nimt G 99 J. Phys. I (Fance) 693. [8] Endes A and Nimt G 993 Phys. Rev. E [9] Mugnai D, Ranfagni A and Schulman L S 997 Phys. Rev. E [0] Mugnai D, Ranfagni A and Ronchi L 998 Phys. Lett. A [] Balcou Ph and Dutiaux L 997 Phys. Rev. Lett [] Alexeev I, Kim K Y and Milchbeg H M 00 Phys. Rev. Lett [3] Nimt G and Heitmann W 997 Pog. Quant. Elect. 8. [4] Büttike M and Landaue R 98 Phys. Rev. Lett [5] Hauge E H and Støvneng J A 989 Rev. Mod. Phys [6] Olkhovsky V S and Recami E 99 Phys. Repots [7] Landaue R and Matin Th 994 Rev. Mod. Phys [8] Kwiat P G, Chiao R Y and Steinbeg A M 99 Physica B [9] Matin Th and Landaue R 99 Phys. Rev. A [0] Th. Hatman Th 96 J. Appl. Phys [] Peatoss J, Glasgow S A and Wae M 000 Phys. Rev. Lett [] J. D. Jackson J D 975 Classical Electodynamics (New Yok: John Wiley & Sons, Inc.) pp [3] Gue B S and Hiioglu H R 998 Electomagnetic Field Theoy Fundamentals (New Yok: JPWS Publishing Company) pp [4] Feynman R P, Leighton R B and Sands M 964 Feynman Lectues on Physics Vol. (New Yok: 7
8 Addison-Wesley Publishing Company), Chapte 7-5 and Chapte
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