A new method for confining and guiding two-dimensional (2D) optical waves is proposed using low-refractive-index materials as cores of metal-gap waveguides. A proper choice of the refractive index of claddings prohibits 2D optical waves from propagating inside the claddings, resulting in strong confinement of waves inside the low-refractive-index cores. We numerically demonstrate that this new method for guiding 2D optical waves can achieve stronger confinement than the conventional one using high-refractive-index cores. This strong confinement leads to efficient guiding of 2D optical waves not only at straight path, but also at sharp 90° bends with negligible radiation losses.
©2006 Optical Society of America
Decreasing the size of optical elements is limited by the diffraction of light. As one of the promising ways to guide optical waves beyond the diffraction limit, low-dimensional optical waveguides have been receiving much attention in recent years . Based on the definition of the dimension of optical waves , surface plasmon polaritons (SPP’s) propagating at a planar metal-dielectric interface are classified as two-dimensional (2D) optical waves. Typical 2D optical waveguides are a metal thin film and a metal-gap structure which has a dielectric thin film sandwiched between two semi-infinite metals. When the metal or dielectric film is thin enough, the SPP’s associated with each metal-dielectric interface become coupled, resulting in two Fano modes [3, 4]. One of the Fano modes can be applied to produce nanometer-sized optical beams beyond the diffraction limit . SPP guiding in regions much smaller than that achievable with conventional optics is also possible using other metallic subwavelength structures, such as nanowires [2, 6], nanoparticle arrays  and triangular grooves , etc. These metal waveguides are now widely expected to become basic building blocks of future integrated nano-optical devices .
We have recently demonstrated that 2D optical waves can be guided by index-guided metal-gap waveguides, in which a middle dielectric film has a high-refractive-index core and claddings with a lower index . Vertically localized 2D optical waves have been found to be laterally confined and guided by the index difference. Besides this method, metal heterowaveguides  and metal ridge waveguides  have been studied to confine and guide 2D optical waves. All these methods rely on the principle that optical waves are confined in a high-refractive-index region, or in other words, in a region where light has lower phase velocity than its adjacent regions.
In this paper, we propose a new method for confining and guiding 2D optical waves using low-refractive-index materials as cores of metal-gap waveguides. Intuitively, 2D optical waves are thought to leak into adjacent high-refractive-index claddings. However, 2D optical waves are forbidden from propagating in the claddings by properly choosing the refractive index of claddings, resulting in strong confinement of waves inside the low-refractive-index cores. We numerically demonstrate that 2D optical waves are well confined and guided along a core, and stronger confinement can be achieved compared with the case of the previous method using high-refractive-index cores .
2. Mechanism of confining 2D optical waves in low-refractive-index materials
We describe the new mechanism for confining 2D optical waves inside low-refractive-index materials. We consider here metal-gap structures in which a homogeneous dielectric film with a refractive index n is sandwiched between two semi-infinite metals (Au) as shown in the inset of Fig. 1. Optical waves with a wavelength in vacuum λ0 of 632.8 nm propagate in the z direction with the propagation constant β and we assume that the optical field is uniform in the y direction. When the gap distance t is thin enough, SPP’s associated with each metal-dielectric interface become coupled, resulting in 2D optical waves.
Figure 1 shows dependence of the normalized propagation constant β/k 0 on the gap distance t for two different cases with n=1.0 and n=4.0, where k 0 is the wave number in vacuum. For the simplicity, Au considered here is a lossless metal with relative permittivity ε m=-9.51. It is shown that there is only one TM (2D) mode in the case of n=1.0 in the whole gap distance range of 0 nm<t<200 nm. The other modes are not seen due to cutoff and only 2D optical waves can propagate. In contrast, one TM (2D) mode and five other modes are seen in the case of n=4.0. Although the existence condition of surface plasmons at a planar interface (n<|ε m|1/2) is not satisfied in this case, the coupling between the two interfaces allows the existence of the new 2D mode. A notable point is that there is a gap for TM modes shown as a gray area in Fig. 1. The 2D optical waves in metal-gap waveguides are TM modes, and hence we found from this analysis that 2D optical waves cannot exist in the case of t=100 nm and n=4.0. Therefore, 2D optical waves are expected to be strongly confined inside the core in 2D optical waveguides with t=100 nm when the refractive indices of a core and claddings are set to be 1.0 and 4.0, respectively.
3. Guiding properties by finite-difference time-domain simulations
Figure 2(a) shows a schematic diagram of proposed 2D optical waveguides with a low-refractive-index core. The middle dielectric thin film between two semi-infinite metals (Au) has a low-refractive-index core and high-refractive-index claddings. Figure 2(b) shows an inside view of the middle dielectric film, indicating a path of 2D optical waves. The refractive indices of the core and the claddings are n 1=1.0 and n 2=4.0, respectively. 2D optical waves propagate toward the positive z direction, and the region z<0 is a free space. A gap distance t is chosen to be t=100 nm so that 2D optical waves propagate only in the core.
To demonstrate that the proposed method is effective in guiding of 2D optical waves, we have employed the finite-difference time-domain (FDTD) method . The Drude model is used to describe the complex relative permittivity of Au (ε m=-9.51+1.22i) . Because 2D optical waves in the metal-gap waveguides are TM modes, we have considered linearly polarized incident plane waves (λ 0=632.8 nm) at which the electric field is polarized parallel to the x direction. The incident plane wave propagating toward the positive z direction is excited just above the entrance of the waveguide at z=-0.05 µm to generate 2D optical waves by endfire excitation inside the core. The incident beam is centered at x=y=0 µm, and its electric field is confined within a rectangular aperture of t (=100 nm)×d.
Figure 3 shows the simulation result for the core width d=350 nm. The electric field |E| distribution on the y-z plane at x=0 is shown in the figure. This field distribution is an instantaneous snapshot of |E| at an arbitrary moment. Although the field distribution on the x-z plane is not shown here, 2D optical waves are well confined in the x direction as in the case of index-guiding by high-refractive-index cores . We found from Fig. 3 that the optical beam is strongly confined in the low-refractive-index core. It should be noted that the 2D optical wave in the core is not a purely TM mode, but rather a hybrid mode with all components of the electric and magnetic fields present. However, it has been found from our simulation that dominant field components are only E x and H y, and thus the 2D optical waves in the core can be regarded as a TM mode. Thus, the leakage into the TE mode in claddings can be neglected.
Figure 4 shows core width dependence of beam width obtained by FDTD simulations, where the beam width is defined as the width at which the amplitude of electric field |E| is reduced to 1/e of its maximum value. It is seen that beam width becomes smaller with decreasing core width. In the case of index-guiding by high-refractive-index cores, the minimum beam width is obtained at a certain core width and beam width becomes larger at both above and below the core width. Thus, metal-gap waveguides with low-refractive-index cores reported here have an advantage over the previous method in making optical beams with small diameter.
Core width dependence of propagation loss is also plotted in Fig. 4. We found that the beam width can be decreased at the cost of high propagation loss. It is known that a metal-gap waveguide is a highly lossy waveguide due to its metal-induced loss. In our simulations, the dielectric constant of Au (ε m=-9.51+1.22i) has an imaginary part which induces loss. Propagation loss is not considered to be a serious problem because long propagation length is not needed in highly-integrated nano-optical devices. Moreover, as one of methods to compensate the loss, gain-assisted propagation of SPP have been recently proposed by incorporating optically active media [15, 16]. Although no material at present can compensate the loss to achieve lossless propagation for our waveguide, the idea of using active material is helpful in developing future applications of metal-gap waveguides. We also found from Fig. 4 that propagation loss increases rapidly when the core width becomes smaller. The core-cladding interfaces can be considered as perfect electric conductor walls for 2D optical waves because amplitudes of the electric field components are almost zero at the interfaces. Thus, 2D optical waves cannot exist as a propagation mode when the core width is reduced far below λ 2D/2, where λ2D is the wavelength of 2D optical waves in the core. In this case, λ2D is calculated to be about 480 nm by dividing λ 0 by β/k 0 in Fig. 1. Therefore, the propagation loss increases rapidly as the core width approaches λ 2D/2.
Finally, we have investigated guiding properties of the waveguide with a low-refractive-index core around a sharp 90° bend. The simulation result is shown in Fig. 5 for the waveguide with core width d=350 nm It is obvious that 2D optical waves are well guided around the sharp bend, which is difficult to achieve using index-guiding by high-refractive-index cores. No power is radiated out of the core as 2D optical waves travel around the bend, since there is no mode into which the 2D optical waves can couple. The simulation results obtained in this paper clearly shows that we have achieved efficient guiding of 2D optical waves not only at straight lines, but also at sharp 90° bends.
There is an analogy between metal-gap waveguides with low-refractive-index cores and 2D photonic crystal waveguides. In both waveguides, there is no extended modes in claddings into which propagating modes can couple, which leads to strong confinement of optical waves. It should be also noted that magnetic fields as well as electric fields can be confined inside cores for waveguides reported here. Thus, the guiding mechanism is totally different from the recently proposed method of guiding light in low-refractive-index materials which utilizes large discontinuity of the electric field at high-index-contrast interfaces and only electric fields can be confined in the low-refractive-index region .
There are also other recent proposals of air-core surface plasmon waveguides [18, 19]. Slot waveguides in a thin metallic film  utilize the difference of wave number between a core and claddings to confine light, which is different from the method reported here. Guiding of light in air by plasmonic modes at a single metal-dielectric interface  is based on the same principle that light is prohibited from propagating in claddings. However, our method is effective for coupled plasmon modes between two metal-dielectric interfaces, and thus the mechanism to prohibit light in claddings is different from that for a single interface.
In conclusion, we have proposed a new method for guiding 2D optical waves by metal-gap waveguides with low-refractive-index cores. 2D optical waves have been found to be strongly confined in the core because there is no extended mode in the claddings into which propagating modes can couple. We have also shown that 2D optical waves can be well guided around sharp 90o bends due to the strong confinement. Not being restricted by radiation losses to moderate curvature bends, this new method has an advantage in miniaturization of future metal-based nano-optical waveguides.
This research is supported in part by the 21st Century COE Program (G18) “Core Research and Advanced Education Center for Materials Science and Nano Engineering” of the Japan Society for the Promotion of Science.
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