Three kinds of compact thin subwavelength cavity resonators containing lefthanded mediarectangular.pdf
arXiv:cond-mat/0402164v1 cond-mat.mtrl-sci 5 Feb 2004 Three kinds of compact thin subwavelength cavity resonators containing left-handed media: rectangular, cylindrical, spherical Jian-Qi Shen 1,2 1 Centre for Optical and Electromagnetic Research, State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou Yuquan 310027, P.R. China 2 Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China (February 2, 2008) In the present paper we investigate the restriction conditions for three kinds of cavity resonators (i.e., the rectangular, cylindrical, spherical resonators).It is shown that the layer of materials with negative optical refractive indices can act as a phase compensator/conjugator, and thus by combining such a layer with another layer made of the regular medium one can obtain a so-called compact thin subwavelength cavity resonator. Keywords: subwavelength cavity resonators, left-handed medium I. INTRODUCTION More recently, a kind of artifi cial composite metamaterials (the so-called left-handed media) having a frequency band where the eff ective permittivity and the eff ective permeability are simultaneously negative attracts considerable attention of many authors both experimentally and theoretically 15. In 19671 , Veselago fi rst considered this peculiar medium and showed from Maxwellian equations that such media having negative simultaneously negative and exhibit a negative index of refraction, i.e., n = 6. It follows from the Maxwells curl equations that the phase velocity of light wave propagating inside this medium is pointed opposite to the direction of energy fl ow, that is, the Poynting vector and wave vector of electromagnetic wave would be antiparallel, i.e., the vector k, the electric fi eld E and the magnetic fi eld H form a left-handed system; thus Veselago referred to such materials as “left-handed” media, and correspondingly, the ordinary medium in which k, E and H form a right-handed system may be termed the “right-handed” one. Other authors call this class of materials “negative-index media (NIM)” 8, “backward media (BWM)” 7, “double negative media (DNM)” and Veselagos media. There exist a number of peculiar electromagnetic and optical properties, for instance, many dramatically diff erent propagation characteristics stem from the sign change of the optical refractive index and phase velocity, including reversal of both the Doppler shift and Cerenkov radiation, anomalous refraction, amplifi cation of evanescent waves 9, unusual photon tunneling 10, modifi ed spontaneous emission rates and even reversals of radiation pressure to radiation tension 1. In experiments, this artifi cial negative electric permittivity media may be obtained by using the array of long metallic wires (ALMWs) 11, which simulates the plasma behavior at microwave frequencies, and the artifi cial negative magnetic permeability media may be built up by using small resonant metallic particles, e.g., the split ring resonators (SRRs), with very high magnetic polarizability 12. A combination of the two structures yields a left-handed medium. Recently, Shelby et al. reported their fi rst experimental realization of this artifi cial composite medium, the permittivity and permeability of which have negative real parts 1. One of the potential applications of negative refractive index materials is to fabricate the so-called “superlenses” (perfect lenses): specifi cally, a slab of such materials may has the power to focus all Fourier components of a 2D image, even those that do not propagate in a radiative manner 9,13. Engheta suggested that a slab of metamaterial with negative electric permittivity and magnetic permeability (and hence negative optical refractive index) can act as a phase compensator/conjugator and, therefore, by combining such a slab with another slab fabricated from a conventional (ordinary) dielectric material one can, in principle, have a 1-D E-mail address: 1Note that, in the literature, some authors mentioned the wrong year when Veselago suggested the left-handed media. They claimed that Veselago proposed or introduced the concept of left-handed media in 1968 or 1964. On the contrary, the true history is as follows: Veselagos excellent paper was fi rst published in Russian in July, 1967 Usp. Fiz. Nauk 92, 517-526 (1967). This original paper was translated into English by W.H. Furry and published again in 1968 in the journal of Sov. Phys. Usp. 6. Unfortunately, Furry stated erroneously in his English translation that the original version of Veselago work was fi rst published in 1964. 1 cavity resonator whose dispersion relation may not depend on the sum of thicknesses of the interior materials fi lling this cavity, but instead it depends on the ratio of these thicknesses. Namely, one can, in principle, conceptualize a 1-D compact, subwavelength, thin cavity resonator with the total thickness far less than the conventional 2 14. Enghetas idea for the 1-D compact, subwavelength, thin cavity resonator is the two-layer rectangular structure (the left layer of which is assumed to be a conventional lossless dielectric material with permittivity and permeability being positive numbers, and the right layer is taken to be a lossless metamaterial with negative permittivity and permeability) sandwiched between the two refl ectors (e.g., two perfectly conducting plates) 14. For the pattern of the 1-D subwavelength cavity resonator readers may be referred to the fi gures of reference 14. Engheta showed that with the appropriate choice of the ratio of the thicknesses d1to d2, the phase acquired by the incident wave at the left (entrance) interface to be the same as the phase at the right (exit) interface, essentially with no constraint on the total thickness of the structure. The mechanism of this eff ect may be understood as follows: as the planar electromagnetic wave exits the fi rst slab, it enters the rectangular slab of metamaterial and fi nally it leaves this second slab. In the fi rst slab, the direction of the Poynting vector is parallel to that of phase velocity, and in the second slab, however, these two vectors are antiparallel with each other. Thus the wave vector k2is therefore in the opposite direction of the wave vector k1 . So the total phase diff erence between the front and back faces of this two-layer rectangular structure is k1d1|k2|d2 14. Therefore, whatever phase diff erence is developed by traversing the fi rst rectangular slab, it can be decreased and even cancelled by traversing the second slab. If the ratio of d1and d2is chosen to be d1 d2 = |k2| k1 , then the total phase diff erence between the front and back faces of this two-layer structure becomes zero (i.e., the total phase diff erence is not 2n, but instead of zero) 14. As far as the properties and phenomena in the subwavelength cavity resonators is concerned, Tretyakov et al. investigated the evanescent modes stored in cavity resonators with backward-wave slabs 15. II. A RECTANGULAR SLAB 1-D THIN SUBWAVELENGTH CAVITY RESONATOR To consider the 1-D wave propagation in a compact, subwavelength, thin cavity resonator, we fi rst take into account a slab cavity of three-layer structure, where the regions 1 and 2 are located on the left- and right- handed sides, and the plasmon-type medium (or a superconductor material) is between the regions 1 and 2. The above three-layer structure is assumed to be sandwiched between the two refl ectors (or two perfectly conducting plates) 14. Assume that the wave vector of the electromagnetic wave is parallel to the third component of Cartesian coordinate. The electric and magnetic fi elds in the region 1 (with the permittivity being 1and the permeability being 1) are written in the form Ex1= E01sin(n1k0z),Hy1= n1k0 i1 E01cos(n1k0z)(0 z d1),(2.1) where k0stands for the wave vector of the electromagnetic wave under consideration in the free space, i.e., k0= c, and in the region 2, where d1+ a z d1+ d2 + a, the electric and magnetic fi elds are of the form Ex2= E02sinn2k0(z d1 d2 a),Hy2= n2k0 i2 E02cosn2k0(z d1 d2 a),(2.2) where d1, d2and a denote the thicknesses of the regions 1, 2 and the plasmon (or superconducting) region, respectively. The subscripts 1 and 2 in the present paper denote the physical quantities in the regions 1 and 2. Note that here the optical refractive indices n1and n2 are defi ned to be n1= 11and n2= 22. Although in the present paper we will consider the wave propagation in the negative refractive index media, the choice of the signs for n1and n2will be irrelevant in the fi nal results. So, we choose the plus signs for n1and n2no matter whether the materials 1 and 2 are of left-handedness or not. The choice of the solutions presented in (2.1) and (2.2) guarantees the satisfaction of the boundary conditions at the perfectly conducting plates at z = 0 and z = d1+ d2+ a. The electric and magnetic fi elds in the plasmon (or superconducting) region (with the resonant frequency being p) take the form Exs= Aexp(z) + B exp(z),Hys= i Aexp(z) B exp(z),(2.3) where the subscript s represents the quantities in the plasmon (or superconducting) region, and = 2 p2 c . To satisfy the boundary conditions 2 Ex1|z=d1= Exs|z=d1,Hy1|z=d1= Hys|z=d1(2.4) at the interface (z = d1) between the region 1 and the plasmon region, we should have E01sin(n1k0d1) = Aexp(d1) + B exp(d1), n1k0 1 E01cos(n1k0d1) = Aexp(d1) B exp(d1).(2.5) It follows that the parameters A and B in Eq.(2.3) are given as follows A = 1 2 exp(d1)E01 ? sin(n1k0d1) + n1k0 1 cos(n1k0d1) ? , B = 1 2 exp(d1)E01 ?n 1k0 1 cos(n1k0d1) sin(n1k0d1) ? .(2.6) In the similar fashion, to satisfy the boundary conditions Ex2|z=d1+a= Exs|z=d1+a,Hy2|z=d1+a= Hys|z=d1+a(2.7) at the interface (z = d1+ a) between the region 2 and the plasmon region, one should arrive at E02sin(n2k0d2) = Aexp(d1+ a) + B exp(d1+ a), n2k0 2 E02cos(n2k0d2) = Aexp(d1+ a) B exp(d1+ a).(2.8) It follows that the parameters A and B in Eq.(2.3) are given as follows A = 1 2E02 exp(d1+ a) ? sin(n2k0d2) n2k0 2 cos(n2k0d2) ? , B = 1 2 exp(d1+ a)E02 ?n 2k0 2 cos(n2k0d2) + sin(n2k0d2) ? .(2.9) Thus, according to Eq.(2.6) and (2.9), we can obtain the following conditions E01 ? sin(n1k0d1) + n1k0 1 cos(n1k0d1) ? + E02exp(a) ? sin(n2k0d2) n2k0 2 cos(n2k0d2) ? = 0, E01 ?n 1k0 1 cos(n1k0d1) sin(n1k0d1) ? E02exp(a) ? n2k0 2 cos(n2k0d2) + sin(n2k0d2) ? = 0.(2.10) In order to have a nontrivial solution, i.e., to have E016= 0 and E026= 0, the determinant in Eq.(2.10) must vanish. Thus we obtain the following restriction condition exp(a) + exp(a) ?n 1 1 tan(n2k0d2) + n2 2 tan(n1k0d1) ? + k0 exp(a) exp(a) ? tan(n1k0d1)tan(n2k0d2) + n1n2k2 0 212 ? = 0(2.11) for the electromagnetic wave in the three-layer-structure rectangular cavity. If the thickness, a, of the plasmon region is vanishing (i.e., there exists no plasmon region), then the restriction equation (2.11) is simplifi ed to n1 1 tan(n2k0d2) + n2 2 tan(n1k0d1) = 0.(2.12) In what follows we will demonstrate why the introduction of left-handed media will give rise to the novel design of the compact thin cavity resonator. If the material in region 1 is a regular medium while the one in region 2 is the left-handed medium, it follows that 3 tan(n1k0d1) tan(n2k0d2) = n12 n21 .(2.13) According to Engheta 14, this relation does not show any constraint on the sum of thicknesses of d1and d2. It rather deals with the ratio of tangent of these thicknesses (with multiplicative constants). If we assume that , d1and d2 are chosen such that the small-argument approximation can be used for the tangent function, the above relation can be simplifi ed as d1 d2 2 1 .(2.14) This relation shows even more clearly how d1and d2should be related in order to have a nontrivial 1-D solution with frequency for this cavity. So conceptually, what is constrained here is d1 d2, not d1 + d2. Therefore, in principle, one can have a thin subwavelength cavity resonator for a given frequency 14. So, one of the most exciting ideas is the possibility to design the so-called compact thin subwavelength cavity resonators. It was shown that a pair of plane waves travelling in the system of two planar slabs positioned between two metal planes can satisfy the boundary conditions on the walls and on the interface between two slabs even for arbitrarily thin layers, provided that one of the slabs has negative material parameters 15. In the following let us take account of two interesting cases: (i) If a 0, then the restriction equation (2.11) is simplifi ed to n1 1 tan(n2k0d2) + n2 2 tan(n1k0d1) + 22a k0 ? tan(n1k0d1)tan(n2k0d2) + n1n2k2 0 212 ? = 0,(2.15) which yields tan(n1k0d1) = k0 n1 1 ,tan(n2k0d2) = k0 n2 2 .(2.16) If both n1k0d1and n2k0d2are very small, then one can arrive at d1 . = 1 1, d2 . = 1 2 from Eq.(2.16), which means that the thicknesses d1and d2depend upon the plasmon parameter . (ii) If the resonant frequency pis very large (and hence ), then it follows from Eq.(2.10) and (2.11) that n1 1 tan(n2k0d2) = 0, n2 2 tan(n1k0d1) = 0,(2.17) namely, regions 1 and 2 are isolated from each other., which is a result familiar to us. In conclusion, as was shown by Engheta, it is possible that when one of the slab has a negative permeability, electromagnetic wave in two adjacent slabs bounded by two metal walls can satisfy the boundary conditions even if the distance between the two walls is much smaller than the wavelength 14. III. A CYLINDRICAL THIN SUBWAVELENGTH CAVITY RESONATOR Here we will consider the restriction equation for a cylindrical cavity to be a thin subwavelength cavity resonator containing left-handed media. It is well known that the Helmholtz equation 2E + k2E = 0 in an axially symmetric cylindrical cavity (with the 2-D polar coordinates and ) can be rewritten as 2E 1 2 E 2 2 E + k2E= 0, 2E 1 2 E+ 2 2 E + k2E= 0, 2Ez+ k2Ez= 0.(3.1) One can obtain the electromagnetic fi eld distribution, E, Eand H, H, in the above axially symmetric cylindrical cavity via Eq.(3.1). But here we will adopt another alternative way to get the solutions of electromagnetic fi elds in the cylindrical cavity. If the electromagnetic fi elds are time-harmonic, i.e., E(,z,t) = E(,)expi(hz kct) and H(,z,t) = H(,)expi(hz kct), then it follows from Maxwell equations that 4 ikcE= 1 ? 1 Hz ihH ? , ikcE= 1 ? ihH Hz ? , ikcEz= 1 ?H + 1 H 1 H ? ,(3.2) and ikcH= 1 ? 1 Ez ihE ? , ikcH= 1 ? ihE Ez ? , ikcHz= 1 ?E + 1 E 1 E ? .(3.3) Thus it is demonstrated that the electromagnetic fi elds E, Eand H, Hcan be expressed in terms of Ezand Hz, i.e., E= i k2 h2 ? hEz + k2 Hz ? ,E= i k2 h2 ? h 1 Ez k2 Hz ? ,(3.4) and H= i k2 h2 ? hHz k2 Ez ? ,H= i k2 h2 ? h 1 Hz + k2 Ez ? .(3.5) As an illustrative example, in what follows, we will consider only the TM wave (i.e., Hz= 0) in the axi- ally symmetric double-layer cylindrical thin subwavelength cavity resonator.Assume that the
