JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 30, NO. 2, JANUARY 15, 2012 273 Effect of Gyrotropic Substrates on the Surface Plasmon Polaritons Guided by Metal Films of Finite Width Gishamol Mathew, Bhagyaraj C, Anju Babu, and Vincent Mathew, Member, IEEE Abstract—The propagation characteristics of surface plasmon polariton modes in metal-strip plasmonic waveguides with gy- rotropic (magneto-optic) substrate material has been numerically studied. The method of line formalism has been adopted for computing the dispersion of fundamental symmetric and anti- symmetric modes supported by the structure. The magneto-optic substrate has been chosen such that the resulting propagating modes exhibit nonreciprocal dispersion, enabling their appli- cation in many optical devices. Fundamental pair modes of surface plasmon polaritons supported by metal strips, which have degeneracy properties with isotropic dielectric substrates, have been found to be nondegenerate in our case. These modes show different trends in the dispersion with variation of metal film thickness, compared to that in an asymmetric strip waveguide with isotropic substrate. A nonreciprocal phase shift of 0.5–14 rad/mm is achievable for antisymmetric modes that are localized near the gyrotropic interface. Index Terms—Gyrotropy, method of lines (MoL), metal-strip waveguide, surface plasmon polaritons (SPPs). I. INTRODUCTION I N response to the need for miniaturization and chip level electronic–photonics integration, there has been an emer- gence of waveguiding structures based on the surface plasmon polaritons (SPPs) [1]–[4]. Many different plasmon waveguide structures that can offer the capability for operation within length scales less than the diffraction limit have been inves- tigated, including infinitely wide metal–insulator–metal and insulator–metal–insulator structures, metal strips (or stripes) in a dielectric environment, and metal particle waveguides [5]–[8]. In the design of plasmon waveguiding systems, the major concern is of ensuring sufficient energy propagation length while overcoming the strong damping in metal. In integrated optoelectronic applications, nonreciprocal propagation effects are incorporated by using various mag- neto-optical materials [9]. Optical gyrotropy of iron garnet Manuscript received August 20, 2011; revised November 17, 2011; accepted November 29, 2011. Date of publication December 13, 2011; date of current ver- sion January 25, 2012. This work was supported by the Department of Science and Technology, Government of India, through Project SR/S2/CMP-0012/2009. G. Mathew, Bhagyaraj C, and A. Babu are with Postgraduate and Research Department of Physics, St. Thomas College Palai, Kottayam, Kerala 686574, India (e-mail: mathew.gishamol@gmail.com; cbhagyaraj.c@gmail.com; anjubabu05@gmail.com). V. Mathew is with the Department of Physics, Central University of Kerala, Kasaragod, Kerala 671328, India (e-mail: vincent.mathew.palai@gmail.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2011.2179288 films is employed in the fabrication of functional photonic crystals for providing magnetically controlled optical switching [10]. Nonreciprocal SPP modes guided by magnetized semi- conductors has been studied in the past [11]. Recently, various subwavelength waveguide components based on magneto-optic activity of gyrotropic materials have been studied, enabling the possibility of subwavelength guiding and tunable nonreciprocal plasmon propagation [12]–[14]. In these theoretical studies, SPPs supported by wide films have been considered. However, the SPP modes supported by infinitely wide metal film struc- tures (hereafter called slab modes) offer field confinement only in 1-D, perpendicular to the direction of wave propagation. At the same time, metal films of finite width and thickness offer 2-D field confinement compared to the slab modes. These SPP modes along thin metal strips upon isotropic substrate have been studied both theoretically and experimentally [15]–[17], and such a waveguide structure is considered to be the basic building block of integrated optical elements incorporating plasmon polaritons. Hence, it is important to study the effect of gyrotropy in such waveguides with metal films of finite width (strips) to be useful for practical implementation. Metal strip (film of finite width) embedded in symmetric or asymmetric dielectric background supports four fundamental SPP bound modes as reported by Berini [18]. In the above paper [18], the dispersion of fundamental modes with film thickness is studied and the behavior is compared with that of the purely bound modes supported by the infinite wide metal film wave- guide structure. These fundamental modes have been designated as , and modes as described in [18] and is adopted widely for designating them. In this paper, we study an asymmetric waveguide structure comprised of a thin metal film of finite width deposited on a semi-infinite, homogeneous gyrotropic substrate material ex- hibiting magneto-optic activity, and covered by a different semi- infinite, isotropic dielectric material, proposing the feasibility of tunable plasmonic waveguides based on optical gyrotropy. Ini- tially, an infinite wide metal film supported by a gyrotropic sub- strate and covered with an isotropic cladding has been consid- ered. A dispersion relation has been derived in analytical form for the structure by the application of boundary conditions at layer interfaces. However, obtaining a completely analytical so- lution for a waveguide structure with metal film of finite width is practically difficult. Therefore, the dispersion relation for the metal strip waveguide with gyrotropic substrate has been nu- merically constructed using the method of lines (MoL). The MoL is an accurate and flexible frequency domain numerical method that does not generate any nonphysical modes [19]. The 0733-8724/$26.00 © 2011 IEEE