Tham khảo tài liệu 'wave propagation 2011 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The Nature of Electromagnetic Waves in Metamaterials and Metamaterial-inspired Configurations 167 where the authors identified the SRR with its outer ring at low frequencies and illustrated the simulated currents to explain the resonance phenomenon. Here we can see it more clearly that eyy becomes less than zero when frequency a is larger than the resonance frequency ca0 leading to the imaginary wave number p thus the resonance stop band is achieved. Also Fig. 2e case has the chance to become resonance when ụxx 0 and there is no resonance stop band for the Fig. 2d and Fig. 2f case. From the analysis above we can easily conclude that imaginary wave number actually leads to the SRR resonance. Such result has been noted by Simovski et al. in their previous work when studied the metamaterial parameters 17 18 . Here we provide an alternative means of characterizing the resonance of SRR metamaterials. Floquet modes analysis of the SRR metamterials Consider an electromagnetic wave to be incident on the SRR metamaterial plane with each element distributed periodically along x and y direction as shown in Fig. 3. a b Fig. 3. Geometry of SRR metamterial plane with incident plane wave a Front view b Side view The electromagnetic fields near the SRR elements must satisfy the periodicity requirements imposed by Floquet s theorem. Thus the scattered and the transmitted fields can be expanded as 20 2 w F_ S T ì ì ì ì ì Ì Rmpq Bmpq Kmpq m 1 p -K q -rn 8 where Rmpq and Bmpq stand for the reflection and the transmission coefficients respectively. The TE and TM vector mode functions Wmpq can be written as 1 vm . u W . . 7 - x - ỹ am TE-Floquet mode W1pq dd 1 2 r. cy T q x y p pq 9a 168 Wave Propagation in Materials for Modern Applications V2pq Jv 2 Up-x ụy l p TM-Floquet mode 9b xy tpq tpq The dominant modes have p q 0 and the higher order modes have p 0 or q 0. And . exp -i upqx vpqy YpqZ 10 Suppose the incidence wave in the direction of 9 ộ with the wave number k one has upq .
