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Kleinman, G.F. Roach, and S.E.G. Stro ¨ m, The null field equations for the exterior problems of acoustics, Proc R Soc Lond A 394 (1984), 121–136. 16. A. Bostro ¨m, G. Kristensson, and S. Stro ¨m, Transformation properties of plane, spherical and cylindrical scalar and vector wave functions, In: V.V. Varadan, A. Lakhtakia, and V.K. Varadan, (Eds.), Field repre- sentations and introduction to scattering, Elsevier, Amsterdam, 1991, pp. 165–210. © 2009 Wiley Periodicals, Inc. SLOW MICROWAVE WAVEGUIDE MADE OF NEGATIVE PERMEABILITY METAMATERIALS Wentao T. Lu, 1 Savatore Savo, 1,2 B. Didier F. Casse, 1 and Srinivas Sridhar 1 1 Electronic Materials Research Institute, Department of Physics, Northeastern University, Boston, Massachusetts 02115; Corresponding author: w.lu@neu.edu 2 CNR-INFM Coherentia, Department of Physics, University “Federico II”, Napoli, Italy Received 25 May 2009 ABSTRACT: The framework for designing a slow light waveguide structure which operates in the GHz and up to THz frequencies is out- lined. The design for the structure consists of a dielectric core layer cladded with negative permeability metamaterials. The parameter space for the metamaterial has been identified for the waveguide to stop light and the negative permeability is achieved by split-ring resonator (SRR) metallic elements. A prototype structure operating at 8.5 GHz is pro- posed. Numerical simulations of electromagnetic waves interacting with SRRs have been performed to extract scattering parameters and a pa- rameter retrieval method has been used to verify the designed window for negative permeability. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2705–2709, 2009; Published online in Wiley In- terScience (www.interscience.wiley.com). DOI 10.1002/mop.24727 Key words: slow light; negative permeability metamaterials; guided modes; split-ring resonators; parallel plate waveguide 1. INTRODUCTION Recently there is a strong interest in slow light [1] for optical buffers [2] and memories. This is partly driven by the progress in optical integrated circuits where optical buffers are one of the most important components. Without slow light architectures, only an optical fiber with a length of a few kilometers can provide enough delay time for optical networking switching. However in micro- waves, the use of coaxial cable for wave delay is impractical due to the much longer wavelength in the radio frequency (RF) and microwave wavelengths. A long pulse delay is normally achieved by electronic circuits. Nevertheless, it is still very desirable to have RF and microwave delay lines without electronic circuits. Slow light has been demonstrated in many systems by using very different methods, such as the electromagnetic induced trans- parency [3] and the coupled resonator structures [4, 5]. Slow light in optical fibers has also been achieved [6]. Only very recently the idea of using double-negative metamaterial (DNM) [7] as waveguide core layer to slow down or stop light was proposed by Tsakmakidis et al. [8] and the trapped rainbow phenomenon was illustrated. The drawback with delay lines made of these types of structures [8, 9] is the narrowness of the bandwidth of DNM, which will yield a very small delay-bandwidth product [4]. The scheme to design broad bandwidth slow light waveguide [10] utilizing indefinite metamaterials [11] has been developed [12, 13]. It was also reported that other types of metamaterials such as metallic gratings in THz range [14] can reduce the group velocity by a factor of 10 3 –10 4 . Interestingly, the intrinsic loss in negative- index metamaterials will render stopping light impossible [15, 16]. To avoid the difficulty of realizing DNM with small loss, slow light waveguide with dielectric core layer and single-negative metamaterial cladding was subsequently proposed [17]. This de- sign of slow light waveguide also opens the door of using gain media to compensate loss and bring light to a standstill possible. The idea of electromagnetic metamaterials started in the field of RF and microwaves with the so-called artificial dielectrics [18]. Active research on negative refraction [19] was fueled by the realization of DNM by using split-ring resonators (SRR) [20] and wires [21] structure in microwaves [22]. Thus, single-negative metamaterials in microwaves is a natural platform to demonstrate the realization of stopping light. Microwaves are often guided in hollow or dielectric filled waveguide cladded with metal. It is well known that near the cutoff thickness or diameter, the group velocity of microwaves can be very small. However, the problem is that at the cutoff thickness or wavelength, both group velocity and phase velocity are zero, leading to complete reflection. In this article, we focus on trapping transverse electric (TE) waves by using negative permeability metamaterials. We show that zero group velocity can be realized at finite phase velocity. The article is organized as follows: In Section 2, we give exact solutions for guided TE modes in a planar waveguide and derive the condition on parameter space for stopping light. In Section 3, SRRs are designed to achieve negative permeability below 10 GHz. Numerical simulations on the SRRs were performed to extract the effective permittivity and permeability. A prototype slow-light microwave structure is theoretically proposed in Section 4. We conclude in Section 5. 2. TE MODES IN PLANAR WAVEGUIDE We consider a two-dimensional (2D) structure which is formed by a parallel plate waveguide (PPW). The vertical direction or the normal to the PPW is along the y-direction. If the thickness of the PPW is h, then for frequency below c/2h, the waveguide can only support TE waves whose electric field is in the y-direction. For example for h = 1 cm, microwaves below 15 GHz will be only TE waves. Therefore, we only need to consider TE wave propagation in the xz-plane. We further consider a slab waveguide made of a dielectric with  D cladded by a negative permeability metamaterial with  M  0 and  M  0 as shown in Figure 1. The dielectric core layer has a thickness d in the x-direction. The guided mode DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 11, November 2009 2705