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A metal-insulator-metal (MIM) waveguide can support two plasmonic modes. Efficient conversion between the two modes can be achieved by reshaping of both phase and power density distributions of the guided mode. The converters are designed with the assistance of transformation optics. We propose two practical configurations for mode conversion, which only consist of homogeneous materials yielded from linear coordinate transformations. The functionalities of the converters are demonstrated by full wave simulations. Without consideration of transmission loss, conversion efficiency of as high as 95% can be realized.
©2013 Optical Society of America
1. Introduction
Surface plasmon polaritions (SPPs), which are supported by metal-dielectric interfaces, are electromagnetic waves coupled to electron plasma oscillations in the metal [1]. Use of SPPs has become one of the most promising approaches to achieve sub-diffraction guiding and manipulating of light [1–3]. Various metallic structures have been proposed for guiding SPPs, such as strips [4], wires [5], slots [6–8], V-shape grooves [3, 9], wedges [9], etc. Among these, metal-insulator-metal (MIM) type waveguides are able to squeeze light into deep sub-wavelength scales [6], enabling nanoscale plasmonic devices.
In practice, MIM waveguides are often referred to symmetric heterostructures. A planar MIM structure can sustain two plasmonic eigenmodes, termed anti-symmetric bound (ab) mode and symmetric bound (sb) mode according to the symmetry of the longitude electric field [6, 10]. Theoretical works have shown that the ab modes can be maintained in MIM waveguides with arbitrarily thin insulator gaps; while the sb modes cut off for MIM waveguides with thin insulators [6, 10]. The ab modes can be excited by many configurations, such as end-fire excitation [11] and tapered dielectric junctions [12]. Compared with the ab modes, the sb modes exhibit some unique properties, such as slower group velocity and better field confinement [6]. However, the sb modes are difficult to launch because of their anti-symmetric lateral field components (the symmetry of the lateral components is contrary to the longitude component).
The problem of excitation of the sb modes can be addressed if the sb modes can be converted from the ab modes. Previous works have investigated mode converters between different types of waveguides, including couplers between dielectric waveguides and MIM waveguides [12], and converters from insulator-metal-insulator waveguides to MIM waveguides [13]. However, few works deal with the problem of mode conversion between the ab and sb modes in MIM.
In this paper, we design and demonstrate mode converters in MIM waveguides that can convert the ab modes to the sb modes. The converters are made of homogeneous anisotropic dielectrics yielded from linear coordinate transformations [14–16] according to transformation optics (TO) [17]. As a powerful tool for optical designing, TO method enables us to mold light flexibly by applying transformation materials [17]. And the general requirement of inhomogeneity of transformation materials can be released by linear transformations [14–16]. Moreover, SPPs can be well molded when only the dielectric regions are changed to transformation materials without changing the metallic regions [18].
We first review some basic properties of the plasmonic modes in MIM waveguides. Then we explain our design concept of mode conversion and demonstrate the functionalities of the mode converters. By reshaping the phase and the power density of the incident ab mode, 95% of the incident power can be converted to the sb mode in a lossless MIM waveguide. When loss generated from the metals is considered, conversion efficiency of 82% can be achieved.
2. Basic properties of plasmonic eigenmodes in MIM waveguide
Figure 1(a) shows an MIM waveguide, its notations and the coordinate system. The insulator thickness, the relative permittivities of the insulator and the metal are denoted by d, ε1 and εm, respectively. All the materials are assumed to be nonmagnetic. The field components of a plasmonic eigenmode that propagates along + z direction, can be written as
where k0 is the free space wavevector, neff is the effective refractive index and ω is the angular frequency. neff of the ab and sb eigenmodes can be obtained by numerical solving of the characteristic equations, which are given respectively by [6, 10]with U = k0 (neff2 − ε1)1/2 and W = k0 (neff2 − εm)1/2. Once neff is known, the field components can be derived. The expressions of the field components are given in Appendix A.
Fig. 1 Basic properties of the plasmonic eigenmodes (the ab and sb modes) in MIM. (a) Schematic and notations of MIM waveguide; (b) Cutoff properties; (c) Typical field profiles (red lines) and phase distributions [arg(Ez), blue lines] of Ez at z = 0; (d) Typical profiles of power density, where the powers of the two modes are normalized equally. In (b)-(d), solid and dashed lines correspond to the ab and sb modes, respectively. In (c) and (d), the insulator thickness is 300 nm.
Throughout the paper, we set the operating wavelength λ0 as 1550 nm. The metal is silver with permittivity given by εAg = −126 + 2.9i [19] and the refractive index of the insulator is given by n1 = 3.48 (silicon). Figure 1(b) shows neff as a function of the insulator thickness. The cutoff thickness of the sb mode appears at d ≈181 nm. Typical profiles and phase distributions of Ez (at z = 0, d = 300 nm) are shown in Fig. 1(c). Across the waveguide median, the phase of Ez changes π for the ab mode but remains unchanged for the sb mode. Figure 1(d) shows the typical profiles of power density of the plasmonic modes. The power density is defined by p = Re(E × H*)z / 2 = Re(ExHy*) exp(−2 Im(neff) k0 z) / 2. As shown, the power of the sb mode is more tightly confined at the metal surfaces than that of the ab mode.
Accordingly, efficient conversion between the plasmonic modes requires that the phase profile and the power distribution should be reshaped simultaneously. In order to illustrate this concept and make clear the effect of power density reshaping, in the next two sections we first demonstrate mode conversion by only phase reshaping, then design mode converters based on both phase reshaping and power density reshaping.
3. Mode conversion by only phase reshaping
Conversion from the ab mode to the sb mode can be achieved by only phase reshaping. In the design of the converter, a rectangular region at the lower half of the insulator is replaced with transformation material, through which the lower half fraction of the wave evolves additional π phase compared with the upper half.
We denote the physical space by xi = {x, y, z}, and the virtual space by xi' = {x', y', z'}. According to TO, the dielectric tensors of the transformation material yielded from a coordinate transformation are given by [17]
where is the Jacobian tensor. Considered that the plasmonic modes are of transverse magnetic polarization, the transformation material can be normalized to nonmagnetic material by fixing εz,x μy (εz,x is the permittivity tensor in the z, x-plane) [20]. In addition, according to Eq. (3), homogeneous transformation materials can be yielded by linear transformations [14–16]. For the sake of practical applications, in this paper the transformation materials are designed to be nonmagnetic, homogeneous and anisotropic materials, which can be realized by nano-gratings according to the effective medium theory [16, 21, 22].Figure 2(a) shows the shapes and sizes of the physical and virtual spaces for phase reshaping. The corresponding linear transformation is given by
yielding nonmagnetic transformation material with the permittivity tensor given byFig. 2 Mode conversion by only phase reshaping. (a) Physical and virtual spaces of converter region. The horizontal and vertical grids in both the spaces show constant x' and constant z', respectively. (b) Power flows of the ab and sb component, P(a) and P(s), and the reflected power flow as functions of the length difference between the virtual and physical spaces, (l' – l), at z = l + 600 nm. Here l is fixed at 500 nm, d = 300 nm, and incident power is assumed to unit. (c) Snapshot of Ez field when (l' – l) = 230 nm. White line shows where P(a) and P(s) are extracted in (b).
We use the finite element method to simulate the performance of the transformation material. The incident ab mode is launched at the leftmost boundary of the structure. A perfect matched layer is placed at the right end of the computing domain to absorb the guided wave and suppress the reflection of the domain boundary. Here the metal claddings of MIM are assumed to be lossless, i.e., the imaginary part of the metal permittivity is ignored. Since the conversion efficiency cannot reach 100%, a wave that transmitted through the converter contains both ab and sb components. The total transmitted power flow is the summation of the power flows of the ab and sb components, denoted by P(a) and P(s) respectively. The coupling of the two components disturbs the profiles of the fields and the power density, but does not contribute to the transmittance. Details of the power flow are discussed in Appendix B.
In the simulations, the power flow of incident ab mode is normalized to unit. Hence the conversion efficiency is equal to the transmitted power flow of the sb mode, P(s). Figure 2(b) shows the transmitted power flows of the ab and sb modes as functions of the length difference between the virtual and the physical spaces, (l' – l). Here the insulator thickness is 300 nm, and the physical space length l is fixed at 500 nm. The maximum of conversion efficiency P(s) appears at (l' – l) ≈230 nm and is more than 70%.
Figure 2(c) shows Re(Ez) distribution when (l' – l) = 230 nm. Since about 30% of the incident power flow is not converted to the sb mode, the coupling effect of the ab mode and the sb mode can be obviously seen in the transmission region (z > l). On the other hand, due to impedance mismatch between the MIM structure and the converter region (0 < z < l), reflection of guided waves can never be avoid. Less than 10% of the incident power is reflected within our parameter range [Fig. 2(b)]. At the maximum conversion efficiency, the reflectance is 4.4%.
4. Mode conversion by reshaping both phase and power density
In order to improve the efficiency of conversion from the ab mode to the sb mode, the power density of the incident ab mode should be concentrated as tightly confined at the metal surfaces as the sb mode besides proper phase reshaping. While phase reshaping requires a length difference between the two rectangles in virtual space that associate with the lower and upper halves of the converter region (0 < z < l, 0 < x < d) in physical space respectively, field concentration requires tapered shapes in physical space. In our design, the physical space consists of two symmetrically placed trapezoids, as shown in Fig. 3(a) .
Fig. 3 Mode conversion by reshaping both phase and power density. (a) and (d) show two divisions of the physical and virtual spaces. In each panels, the horizontal and vertical grids show contours of x' and z', respectively. (b) and (e) show the power flows of transmitted ab and sb components, P(a) and P(s), as well as the reflected power, as functions of (l' – l), corresponding with (a) and (d) respectively. l = 500 nm, d = 300 nm and d1 = 75 nm. The incident power flow is unit. The power flows are extracted at z = l + 500 nm. (c) and (f) show the snapshots of Ez fields at the maximum conversion efficiencies of (a) and (d) respectively. In (c) and (f), (l' – l) is equal to 226 and 220 nm respectively. White lines show where the data of power flow are extracted.
In order to apply linear transformations, the quadrangles in both the physical space and the virtual space are divided into triangles by their diagonal lines. Two different divisions of the virtual and physical spaces are shown in Figs. 3(a) and 3(d). In order to transform the virtual space to the physical space, for the configuration shown in Fig. 3(a), regions 1' and 2' shrink in the z' direction and regions 2' and 3′ are linear compressed in the x' direction. Region 4' keeps unchanged as region 4 in physical space. The transformations from regions q' to q (q = 1, 2, 3) are given respectively by
According to Eq. (3), the permittivities of nonmagnetic transformation materials of regions q (q = 1, 2, 3) are given respectively bywhere A = l' / l, B1 = − (d / 2 − d1) / l', B2 = (d / 2 − d1) / l and C = d / 2d1.For the configuration shown in Fig. 3(d), in the virtual space, regions 1' and 2' are compressed in z' direction; regions 1' and 4' are compressed in the x' direction; and regions 2' and 3′ slide along x' direction to fit the shape of the physical space. The transformations from regions q' to q (q = 1, 2, 3, 4) are given respectively by
The permittivities yielded from the transformations are given respectively byThe designs of the converter shown in Figs. 3(a) and 3(d) are theoretically equivalent to each other. They are both able to reshape the phase profile [Fig. 1(c)] and the power density distribution [Fig. 1(d)] of the ab mode to those of the sb mode. We simulate the performances of the converters using the same simulation setups as the previous section. The geometrical parameters are given in the caption of Fig. 3. Also the metal claddings are assumed to be lossless.
As shown in Figs. 3(b) and 3(e), the performances of the converters are enhanced compared with the converter using only phase reshaping [Fig. 2(b)]. Conversion efficiency (P(s)) of over 90% is found for both the configurations shown in Figs. 3(a) and 3(d) with proper length difference between the lower and upper halves of the virtual space. The maximum conversion efficiencies are as high as 93% in Fig. 3(b) and 95% in Fig. 3(e), which are found at (l' – l) = 226 nm and 220 nm, respectively. Note that the maximum conversion efficiencies will increase if the base length of the trapezoids in physical space, d1, is optimized.
At the maximum conversion efficiencies, Re(Ez) distributions of the two configurations are shown in Figs. 3(c) and 3(f), respectively. The field distributions within the converter region are well adapted to the shapes of the trapezoids, confirming that a guided wave becomes gradually more tightly confined at the metal surfaces as it transmits through the converter. Since the ab components carry very small fractions of the transmitted power flows (P(a) is equal to 0.3% and 1% for the two configurations respectively), nearly no field distortion resulted from the coupling of the ab and sb modes can be observed at the transmitted region (z > l). Moreover, reflection of the converter can be rendered to be less than 10% within our parameter range [Figs. 3(b) and 3(e)]. 7% and 4% of the incident powers are reflected at the maximum conversion efficiencies of the two configurations respectively.
5. Mode conversion in MIM with real metals and in metal slot waveguide
We have considered mode conversion in lossless MIM waveguides, which allow us to accurately estimate the transmittance and reflectance of the converters. Now we demonstrate that the same concept of mode conversion also works in MIM waveguides with real metals.
First we apply the configuration of Fig. 3(a) to an MIM waveguide with insulator thickness d = 300 nm. The optimized geometrical parameters are given by d1 = 80 nm, l = 500 nm and (l' – l) = 224 nm. The Re(Ez) field distribution of an incident ab mode transmitted through the converter is shown in Fig. 4(a) . Observed at 500 nm away from the converter (z = l + 500 nm), the conversion efficiency is 81.9% and the transmittance (P(a) + P(s)) is 82.3%. Compared with the converter in lossless MIM waveguide, the conversion efficiency is reduced because of the transmission loss of the guided mode. Nevertheless, the functionality of the converter is entirely preserved, and the Ez field is strictly symmetric in the transmission region.
Fig. 4 Mode conversion in MIM waveguides with real metal. (a) Ez snapshot with d = 300 nm, d1 = 80 nm, l = 500 nm and (l' – l) = 224 nm. (b) Ez snapshot with d = 200 nm, d1 = 34 nm, l = 500 nm and (l' – l) = 210 nm.
In particular, the configuration also works in MIM waveguide with nearly cutoff insulator thickness for the sb mode. Figure 4(b) shows the conversion from the ab mode to the sb mode in an MIM waveguide with insulator thickness d = 200 nm, which is ~19 nm greater than the cutoff thickness [Fig. 1(b)]. The converter length l is 500 nm, and the associated optimized parameters are given by d1 = 34 nm and (l' – l) = 210 nm. As seen in the reflection region (z < 0), the incident field is disturbed seriously due to the interference with the reflected field. The reflection occupies over 33% of the incident power, and the conversion efficiency is as low as 64%. Nevertheless, the majority of the power in the transmitted region is carried by the sb mode (P(s) / P(a) = 38).
We note that a converter designed by only phase reshaping will fail to achieve mode conversion in an MIM waveguide near the cutoff point of the sb mode. In addition, the same converters can also convert the sb modes to the ab modes when the incident sb modes are launched from the rightmost boundaries of the structures and propagate along –z direction. More simulations show that the maximum conversion efficiency can be improved by decreasing the length of the physical space, which also requires larger anisotropy of the transformation materials.
The two-dimensional (2D) designs of mode converters shown in Fig. 3 can be directly applied in three-dimensional (3D) MIM type plasmonic waveguides. As an example, here we demonstrate mode conversion in a kind of metal slot waveguide [23], whose dimension is finite in the y direction. The cross section of the slot waveguide is shown in the inset of Fig. 5(a) . The insulator thickness (x direction) is 300 nm and insulator height (y direction) is 500 nm. The length of the converter region l is 500 nm. The length of base line d1 is 80 nm. The height of the converter is equal to the height of the insulator. The in-plane permittivities (εz,x) of the materials in the converter region can be obtained from Eqs. (7) and (9). Since the plasmonic modes are transverse magnetic polarized, the value of εuu does not influence the wave propagation [20]. In the simulations, εuu is set to be equal to the permittivity of the insulator, n12. In addition, real lossy metal is used. And the refractive index of the substrate is assumed to be 1.45. The end-fire scheme is used to excite the ab mode in the slot waveguide. In the simulations, a plane wave polarized in the x direction impinges perpendicularly on the input port of the slot waveguide from the free space. Also a perfect matched layer (PML) is used to eliminate light reflection of the exit port.
Fig. 5 Mode conversion in metal slot waveguides. (a) and (b) show Ez snapshots using the configurations shown in Figs. 3(a) and 3(d) respectively. Inset of (a) shows the cross section of the slot waveguide.
Figures 5(a) and 5(b) demonstrate the functionality of the 3D mode converters corresponding with the 2D counterparts shown in Figs. 3(c) and 3(f) respectively. For both the configurations, the length difference of the virtual space and the physical space is given by (l' − l) = 220 nm. As shown, the ab mode is well converted to the sb mode after transmitted through the converter.
6. Conclusions
In conclusion, we have proposed designs of efficient mode conversion in MIM plasmonic waveguides. According to the concept of reshaping both phase and power density, we have designed practical mode converters with the assistance of transformation optics and have demonstrated the functionalities of the converters. As high as 95% of the power of the incident ab mode can be converted to the sb mode by the converter in lossless MIM waveguide (d = 300 nm). And conversion efficiency of over 80% are observed for the converter in MIM waveguide (d = 300 nm) with real metal claddings. The design concept is also valid for MIM waveguide with insulator thickness near the cutoff point of the sb mode. In addition, we have demonstrated mode conversion in 3D metal slot waveguide using the same design concept. This design concept can also be applied to conversions of modes in other types of plasmonic waveguide and conventional waveguides.
The large energy density generated by the sb modes can find applications in a variety of areas. This work facilitates exploring and utilizing of the sb modes in MIM waveguides and may enable potential applications by using the sb modes, such as light harvesting in large volume gaps, enhancing the sensitivity of plasmonic sensors and generation of slow light in MIM waveguides.
Appendix. Field components of plasmonic eigenmodes in MIM
The field components of the eigenmodes are given by
where A is the amplitude coefficient, and m is equal to −1 for ab mode and + 1 for sb mode.B. Time averaged power flow of arbitrary plasmonic wave in MIM
In general, a stable plasmonic wave maintained in MIM structure can be written as superposition of ab and sb components. According to Eq. (11), the Ez values at the boundaries are given by Ez|x = 0 = A(a) + A(s) and Ez|x = d = −A(a) + A(s). Here A(a) and A(s) denote the amplitude coefficients of the ab and sb components, respectively, which in turn can be expressed by
The time averaged power flow of an eigenmode component is given byThe integral can be written explicitly aswhere R = εmU / ε1W. Thus the real part of the right hand side of Eq. (15) multiplied by the damping term exp(−2Im(neff)k0z) is the time averaged power flow of the eigenmode component. The power density that arises from the coupling of the ab and sb components is given byAccording to the symmetries of the field components of the ab and sb modes, the coupling effect does not contribute to the total time averaged power flow (). Thus the time averaged power flow of an arbitrary guided plasmonic wave is given by Ptot = P(a) + P(s).Acknowledgments
This research is supported by the Chinese National Key Basic Research Special Fund (2011CB922003), the National Natural Science Foundation of China (11074132), and the 111 Project (B07013).
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