## Abstract

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 (*a _{b}*) mode and symmetric bound (

*s*) mode according to the symmetry of the longitude electric field [6, 10]. Theoretical works have shown that the

_{b}*a*modes can be maintained in MIM waveguides with arbitrarily thin insulator gaps; while the

_{b}*s*modes cut off for MIM waveguides with thin insulators [6, 10]. The

_{b}*a*modes can be excited by many configurations, such as end-fire excitation [11] and tapered dielectric junctions [12]. Compared with the

_{b}*a*modes, the

_{b}*s*modes exhibit some unique properties, such as slower group velocity and better field confinement [6]. However, the

_{b}*s*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).

_{b}The problem of excitation of the *s _{b}* modes can be addressed if the

*s*modes can be converted from the

_{b}*a*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

_{b}*a*and

_{b}*s*modes in MIM.

_{b}In this paper, we design and demonstrate mode converters in MIM waveguides that can convert the *a _{b}* modes to the

*s*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].

_{b}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 *a _{b}* mode, 95% of the incident power can be converted to the

*s*mode in a lossless MIM waveguide. When loss generated from the metals is considered, conversion efficiency of 82% can be achieved.

_{b}## 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*, *ε*_{1} and *ε _{m}*, 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

*k*

_{0}is the free space wavevector,

*n*

_{eff}is the effective refractive index and

*ω*is the angular frequency.

*n*

_{eff}of the

*a*and

_{b}*s*eigenmodes can be obtained by numerical solving of the characteristic equations, which are given respectively by [6, 10]

_{b}*U*=

*k*

_{0}(

*n*

_{eff}

^{2}−

*ε*

_{1})

^{1/2}and

*W*=

*k*

_{0}(

*n*

_{eff}

^{2}−

*ε*)

_{m}^{1/2}. Once

*n*

_{eff}is known, the field components can be derived. The expressions of the field components are given in Appendix A.

Throughout the paper, we set the operating wavelength *λ*_{0} as 1550 nm. The metal is silver with permittivity given by *ε*_{Ag} = −126 + 2.9i [19] and the refractive index of the insulator is given by *n*_{1} = 3.48 (silicon). Figure 1(b) shows *n*_{eff} as a function of the insulator thickness. The cutoff thickness of the *s _{b}* mode appears at

*d*≈181 nm. Typical profiles and phase distributions of

*E*(at

_{z}*z*= 0,

*d*= 300 nm) are shown in Fig. 1(c). Across the waveguide median, the phase of

*E*changes

_{z}*π*for the

*a*mode but remains unchanged for the

_{b}*s*mode. Figure 1(d) shows the typical profiles of power density of the plasmonic modes. The power density is defined by

_{b}*p*= Re(

**E**×

**H***)

*/ 2 = Re(*

_{z}*E**) exp(−2 Im(

_{x}H_{y}*n*

_{eff})

*k*

_{0}

*z*) / 2. As shown, the power of the

*s*mode is more tightly confined at the metal surfaces than that of the

_{b}*a*mode.

_{b}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 *a _{b}* mode to the

*s*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

_{b}*π*phase compared with the upper half.

We denote the physical space by *x ^{i}* = {

*x*,

*y*,

*z*}, and the virtual space by

*x*= {

^{i'}*x'*,

*y'*,

*z'*}. According to TO, the dielectric tensors of the transformation material yielded from a coordinate transformation are given by [17]

*ε*

_{z}_{,}

*(*

_{x}μ_{y}*ε*

_{z}_{,}

*is the permittivity tensor in the*

_{x}*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 byWe use the finite element method to simulate the performance of the transformation material. The incident *a _{b}* 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

*a*and

_{b}*s*components. The total transmitted power flow is the summation of the power flows of the

_{b}*a*and

_{b}*s*components, denoted by

_{b}*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 *a _{b}* mode is normalized to unit. Hence the conversion efficiency is equal to the transmitted power flow of the

*s*mode,

_{b}*P*

^{(}

^{s}^{)}. Figure 2(b) shows the transmitted power flows of the

*a*and

_{b}*s*modes as functions of the length difference between the virtual and the physical spaces, (

_{b}*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(*E _{z}*) distribution when (

*l'*–

*l*) = 230 nm. Since about 30% of the incident power flow is not converted to the

*s*mode, the coupling effect of the

_{b}*a*mode and the

_{b}*s*mode can be obviously seen in the transmission region (

_{b}*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 *a _{b}* mode to the

*s*mode, the power density of the incident

_{b}*a*mode should be concentrated as tightly confined at the metal surfaces as the

_{b}*s*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 <

_{b}*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) .

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

*q*(

*q*= 1, 2, 3) are given respectively by

*A*=

*l'*/

*l*,

*B*

_{1}= − (

*d*/ 2 −

*d*

_{1}) /

*l'*,

*B*

_{2}= (

*d*/ 2 −

*d*

_{1}) /

*l*and

*C*=

*d*/ 2

*d*

_{1}.

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 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 *a _{b}* mode to those of the

*s*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.

_{b}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, *d*_{1}, is optimized.

At the maximum conversion efficiencies, Re(*E _{z}*) 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

*a*components carry very small fractions of the transmitted power flows (

_{b}*P*

^{(}

^{a}^{)}is equal to 0.3% and 1% for the two configurations respectively), nearly no field distortion resulted from the coupling of the

*a*and

_{b}*s*modes can be observed at the transmitted region (

_{b}*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 *d*_{1} = 80 nm, *l* = 500 nm and (*l'* – *l*) = 224 nm. The Re(*E _{z}*) field distribution of an incident

*a*mode transmitted through the converter is shown in Fig. 4(a) . Observed at 500 nm away from the converter (

_{b}*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

*E*field is strictly symmetric in the transmission region.

_{z}In particular, the configuration also works in MIM waveguide with nearly cutoff insulator thickness for the *s _{b}* mode. Figure 4(b) shows the conversion from the

*a*mode to the

_{b}*s*mode in an MIM waveguide with insulator thickness

_{b}*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

*d*

_{1}= 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

*s*mode (

_{b}*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 *s _{b}* mode. In addition, the same converters can also convert the

*s*modes to the

_{b}*a*modes when the incident

_{b}*s*modes are launched from the rightmost boundaries of the structures and propagate along –

_{b}*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 *d*_{1} is 80 nm. The height of the converter is equal to the height of the insulator. The in-plane permittivities (*ε _{z}*

_{,}

*) 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*

_{x}*ε*does not influence the wave propagation [20]. In the simulations,

_{uu}*ε*is set to be equal to the permittivity of the insulator,

_{uu}*n*

_{1}

^{2}. 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

*a*mode in the slot waveguide. In the simulations, a plane wave polarized in the

_{b}*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.

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 *a _{b}* mode is well converted to the

*s*mode after transmitted through the converter.

_{b}## 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 *a _{b}* mode can be converted to the

*s*mode by the converter in lossless MIM waveguide (

_{b}*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

*s*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.

_{b}The large energy density generated by the *s _{b}* modes can find applications in a variety of areas. This work facilitates exploring and utilizing of the

*s*modes in MIM waveguides and may enable potential applications by using the

_{b}*s*modes, such as light harvesting in large volume gaps, enhancing the sensitivity of plasmonic sensors and generation of slow light in MIM waveguides.

_{b}## Appendix. Field components of plasmonic eigenmodes in MIM

The field components of the eigenmodes are given by

*A*is the amplitude coefficient, and

*m*is equal to −1 for

*a*mode and + 1 for

_{b}*s*mode.

_{b}## 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 *a _{b}* and

*s*components. According to Eq. (11), the

_{b}*E*values at the boundaries are given by

_{z}*E*|

_{z}

_{x}_{= 0}=

*A*

^{(}

^{a}^{)}+

*A*

^{(}

^{s}^{)}and

*E*|

_{z}

_{x}_{=}

*= −*

_{d}*A*

^{(}

^{a}^{)}+

*A*

^{(}

^{s}^{)}. Here

*A*

^{(}

^{a}^{)}and

*A*

^{(}

^{s}^{)}denote the amplitude coefficients of the

*a*and

_{b}*s*components, respectively, which in turn can be expressed by

_{b}*R*=

*ε*/

_{m}U*ε*

_{1}

*W*. Thus the real part of the right hand side of Eq. (15) multiplied by the damping term exp(−2Im(

*n*

_{eff})

*k*

_{0}

*z*) is the time averaged power flow of the eigenmode component. The power density that arises from the coupling of the

*a*and

_{b}*s*components is given by

_{b}*a*and

_{b}*s*modes, the coupling effect does not contribute to the total time averaged power flow (${\int}_{-\infty}^{+\infty}{p}_{c}dx=0$). Thus the time averaged power flow of an arbitrary guided plasmonic wave is given by

_{b}*P*

^{tot}=

*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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