1. Introduction

Metamaterials are artificially structured materials whose electromagnetic properties are expressed in terms of effective material parameters. Those parameters are artificially designed to form the unit cell by conducting elements. Using the specific design of these elements, it is able to control the electromagnetic responses. Metamaterials have attractive electromagnetic properties such as negative refraction, super lensing and invisibility cloaking [1–4].

The negative refraction can be realized in case that the effective permeability and permittivity are both negative. The most successful structure that has negative refractive index in the microwave region is a split ring resonator combined with metal wire structure. The split ring resonator shows negative permeability and metal wire provides negative permittivity. In recent years, with the scale down of the size of the unit cell, the various designs of thin layer structures, which have negative refractive index in optical wavelengths, were proposed such as cut-wire pair structure and double fish net structure [5, 6]. Optical metamaterials which have negative refractive index provide potential application for optical elements, such as optical lithography, optical lenses and waveguides.

Negative refraction at visible frequencies has been directly observed in metal-insulator-metal (MIM) plasmonic waveguides [7]. In addition, super lensing has been reported for plasmonic waveguides coated with a thin insulator layer (an insulator-insulator-metal, or IIM geometry). By reducing the thickness of insulator layer, a huge enhancement of refractive index of MIM plasmonic waveguides was reported at optical wavelengths [8]. For designing the optical metamaterials, these types of plasmonic waveguides are beneficial to make the structures in the wavelength which is below the diffraction limit.

MIM plasmonic waveguides with stub resonators are analyzed in some previous reports [9–11]. The stub resonators are considered as an additionally attached transmission line. The resonant wavelength is mainly determined by the height and width of the stub resonator. The electromagnetic properties can be modulated at this resonance wavelength which is much longer than the size of the unit cell. The plasmonic waveguide with periodic stub resonators provide negative group velocity (V_g) dispersion curves originated from the Brillouin Zone (BZ) folding in the optical region. This mode is similar to the cases of photonic crystals. However, by designing the unit cell, we can realize this mode in the wavelengths far longer than the size of the unit cell. Therefore, these modes can be characterized by using the material parameters. This type of the plasmonic waveguides can be considered as metawaveguide. Therefore, to analyze the effective material properties of MIM plasmonic waveguides with periodic stub resonators is interesting issue. In addition, the effective material properties were investigated by changing the size of the unit cell. The propagating modes can be characterized by the coupling between waveguide mode and stub resonance mode. Considering the periodicity of this structure, the coupling can be classified to the 2 types. One is a coupling between waveguide mode, which has positive group velocity, and stub resonance. Another is a waveguide mode, which has negative group velocity originated from the BZ folding. Therefore, effective material properties are dependent on the types of coupling between waveguide mode and stub resonance. In this paper, the effective material properties of Au/Al₂O₃/Ag plasmonic waveguides with periodic serial stub resonators were numerically investigated by using Finite-Time-Domain (FDTD) simulations. The effective material properties of the propagating modes that have negative refractive index originated from the BZ folding were especially focused on.

In section 2, the basic refractive index properties of Au (thickness 150 nm)/Al₂O₃ (thickness 10~120 nm)/Ag (thickness 150 nm) plasmonic waveguides are shown and the same characteristic which has previously reported are introduced. In addition, the calculation results of the impedance of the stub resonator, which is analyzed according to the way of previous report [9], are presented. In section 3, the dispersion calculation results of the Au/Al₂O₃/Ag plasmonic waveguide metamaterial with serial periodic stub resonators are shown. In section 4, the properties of the effective permittivity and permeability are discussed.

2. The properties of Au/Al₂O₃/Ag plasmonic waveguide and stub resonator

The general properties of MIM plasmonic waveguide have been reported [12–15]. The refractive index of MIM plasmonic waveguide is much larger than the general semiconductor materials. The dispersion relations of Au (thickness 150 nm)/Al₂O₃ (thickness 10~120 nm)/Ag (thickness 150 nm) plasmonic waveguides were calculated by using FDTD simulation. The commercial software OPTIFDTD was used in our calculation [16]. In Fig. 1(a) , the schematic of the calculation set up is shown. In FDTD simulations, the computational domain containing Yee cells with Δx = Δy = 2 or 5 nm were used and the whole region was surrounded by a perfect absorbing boundary. Plasmonic waveguides were set in the vacuum medium and input pulse was positioned in the Al₂O₃ center layer for x electric field polarization. The thicknesses of metal layers were set to be 150 nm. Thicknesses of insulator ware ranging from 10 nm to 120 nm. Reflection from the structure’s output interface is negligible because propagation modes are completely absorbed before getting to the structure’s output interface. Therefore, the observed values are not affected by the reflection from the structure’s interface. To calculate the dispersion relation of the waveguide mode, the two observation points inside the Al₂O₃ center layer were set and the phase differences (θ) of the y component of magnetic field between two observation points were derived. θ can be described by the real part of wave vector and the gap between two observation points. In Fig. 1(b), the calculated dispersion relations for various thicknesses of Al₂O₃ center layer (t) are shown.

 

Fig. 1 (a) Schematic of simulation set up. Two observation points were positioned for calculating the phase difference and input pulse was set for x polarization. (b) Dispersion relation of the plasmonic waveguide for various Al₂O₃ layer thicknesses, whose values are described by the parameter t.

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In Fig. 1(b), the negative group velocity dispersions are confirmed near the plasma frequency (about 2eV). These modes have negative refractive index, which was observed by the previous report [7]. These modes show blue shift with reducing the thickness of the Al₂O₃ center layer (t). This tendency was confirmed by investigating the analytical formula from Ref [12]. Above the plasma frequencies, the modes which propagate inside the thinner Al₂O₃ center layer (t < 70) tend to have leaky nature because the dispersion curves get close to the light lines. In fact, the propagating modes localized at the interface between metal and vacuum medium. Therefore, the calculated results show noisy curves because observation points were located inside the Al₂O₃ center layer. Below the plasma frequency, the modes localized within the Al₂O₃ center layer. These propagating modes are similar to the fundamental TEM mode. In Fig. 2 , the calculated refractive indexes of this system are shown for various thickness of the Al₂O₃ center layer (t). Reducing the Al₂O₃ layer thickness, the increase of the refractive index was confirmed. These values are much larger than the general semiconductor materials because the modes penetrate inside the metal layer. This causes the increasing of the refractive index.

 

Fig. 2 The wavelength dependence of refractive index of the plasmonic waveguides derived from the results of Fig. 1. Al₂O₃ layer thicknesses are ranging from 10 nm to 120 nm.

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According to the some previous reports [9, 10], a stub resonator, incorporated within the Au/Al₂O₃/Ag plasmonic waveguide, can be regarded as an additionally attached transmission line that acts as a Fabry-Perot resonator, which is shown in Fig. 3(a) and 3(b). Stub resonance modes oscillate along the x direction. In this case, this additional transmission line is terminated by Au layer. The value of impedance of stub resonator (Z_stub) can be obtained from transmission line theory and given by Ref [9]. Z_L and Z_S correspond to the impedance of the termination and waveguide impedance inside the stub respectively. d means height of the stub and β_stub corresponds to the wave vector inside the stub resonator. The formula of Z_stub is as follows [9,17]:

 

Fig. 3 (a)Schematic of Au/Al₂O₃ /Ag plasmonic waveguides with a stub resonator (b) The equivalent transmission-line representation of the stub resonator. Z_L means the termination of the stub resonator. In this case, this can be expressed by the impedance of Au. Z_MIM is a characteristic impedance of the Au/Al₂O₃ /Ag plasmonic waveguides. (c) The calculation results of the absolute value of the Z_stub/Z_MIM for various stub width ranging from 30 ~70 nm, where the stub height was fixed at 150 nm.　The peak was obtained at the resonance energy of stub resonator.

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In Fig. 3(c), schematic of stub resonator and the calculation results of Z_stub/Z_MIM are shown. In this case, Z_MIM stands for the impedance of the Au (thickness 150 nm)/Al₂O₃ (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide that doesn’t have stub resonator. It is confirmed that, with increase of the stub width, the resonance peak shows blue shifts due to the decreasing of the wave vector inside the stub. The wavelengths of the stub resonances can be determined from these calculations. In addition, the impedance of stub resonator goes higher at the resonance wavelength so that the reflection from the stub increases.

3. Analysis of MIM plasmonic waveguide with serial periodic stub resonators

The dispersion relation of Au (thickness 150 nm)/Al₂O₃ (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide with serial periodic stub resonators was also calculated by using FDTD simulation. By deriving the phase differences (θ) between two observation points at a distance of a, the real parts of the wave vector (k_r) were calculated. And also, from the decrease of magnetic field amplitude (H) between observation points, the imaginary parts of the wave vector (k_i) were calculated. For the calculations of k_r and k_i, these formulas were used:

L is about 2 μm, which is adequate length to obtain appropriate value of k_i. For calculation of k_r, the range of θ was assumed from –π to + π considering 1st Brillouin zone (BZ). This assumption is valid because the waveguides have periodicity. Propagating modes are characterized by the dispersion relation inside the 1st BZ. In Fig. 4(a) and 4(b), the schematic of the waveguide and the dispersion relation of our system are shown. And, in Fig. 4(c), the calculation results of dispersion relation, which is limited to positive group velocity dispersion curve, are shown. In addition, in Fig. 4(d), the imaginary parts are shown.

 

Fig. 4 (a)the design of the plasmonic waveguide with serial periodic stub resonators (b) Schematic of dispersion relation of plasmonic waveguide mode and stub resonance, where only positive group velocity dispersion curve is shown. (c) The calculation results of the dispersion relation for the condition of the size of the unit cell of 250 nm and the stub width of 30 nm. Only positive group velocity dispersion relation is shown. Near the 1.55 eV, the stub resonance is confirmed. (d) The imaginary part plot of the dispersion relation is presented. It is confirmed that the increases of the imaginary part of the wave vector are confirmed at the 1st BZ boundary, stub resonance energy and the Γ point.

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As the size of the unit cell is set to be 250 nm, the wavelengths of negative group velocity region in Fig. 4(c) are longer than 500 nm, which is twice as long as the size of the unit cell. Therefore, the characterizing by using material parameters is meaning because there is no diffraction in those wavelength range. The stub resonance is confirmed at the 1.55 eV and its resonance couples to the waveguide mode which is folded at the 1st BZ boundary. In Fig. 4(d), the imaginary part of wave vector increases at the boundary of 1st BZ (near the 1.15 eV), the Γ point (near the 1.9 eV) and the stub resonance energy point. These results preserve the assumption that the phase differences between observation points ranges from –π to + π.

By reducing the stub-resonator width (w), shown in Fig. 5(a) , these modes are shift towards the lower energy because the slope of the dispersion relation inclines. From Fig. 5(b), the imaginary part of wave vector increases at the boundary of 1st BZ (near the 1.15 eV), the Γ point (near the 1.9 eV) and the stub resonance wavelength (ranging 1.55 ~1.75 eV). For the 30 nm stub resonator width, however, the large increase of the imaginary part could not be confirmed clearly at the boundary of 1st BZ. This suggests the probability that the structure which has the stub width of 30 nm can’t be dealt as a periodic structure. Considering that the reflection from the stub goes weaker with the narrowing of the stub width and this waveguide is absorptive, periodicity of this waveguide doesn’t emerge in case that the reflection from the stub is too weak. Therefore, to investigate the condition that the band gap vanishes near the boundary of 1st BZ is needed. The dispersion relation for this waveguide can be written as Ref [9–11]:

 

Fig. 5 The calculation results of the dispersion relations, where the size of the unit cell is 250 nm and the width of the stub is ranging from 30 nm to 80 nm. (a) The real part plot and (b) Imaginary part plot of the dispersion relations.

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k is a wave vector of Au/Al₂O₃/Ag plasmonic waveguide with serial periodic stub resonators, which can be expressed by using the data of dispersion calculation results in Fig. 5. Z_MIM and β_MIM means impedance and wave vector of the Au/Al₂O₃/Ag plasmonic waveguide. The value of $\left|\cos (ka)\right|$ was calculated. Generally, the band gap condition is$\left|\cos (ka)\right|>2$, so that the evaluation of this value for various width of stub resonator provides the information whether this waveguide can be regarded as a periodic structure. In Fig. 6 , the calculation results of $\left|\cos (ka)\right|$ for several stub widths ranging from 10 nm to 30 nm are shown.

 

Fig. 6 The calculation result of |cos(ka)|. Stub width ranges from 10 nm to 30 nm. Stub width ranges from 10 nm to 30 nm. in the case that the stub width is less than 20nm, the energy gap closes because the value is less than 2.

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In Fig. 6, energy gap is confirmed at the stub resonance wavelength, whose values are over 2. In addition, near the 1st BZ boundary, the rapid decrease is confirmed with reducing the stub width. As for the structure which has stub width of 30 nm, the energy gap is formed at the 1st BZ boundary. For the structure whose stub width is less than 20 nm, however, energy gap vanishes because $\left|\cos (ka)\right|<2$. Therefore, validity of BZ folding is limited for specific stub widths and that is dependent on the relation between the reflection intensity from the stub and the absorption loss of the waveguide. Next, the dispersion relation for changing the size of the unit cell of 100 nm was calculated, where the stub resonator width is fixed on the 20 nm. In this case, the stub resonance mode couples to the waveguide mode whose dispersion is characterized by $V_g\cdot k>0$. In Fig. 7 , the calculation results are shown. The sharp resonance is confirmed. The absolute value of the refractive index at this energy is over 4, which is quite different from the previous results of the structure with the size of the unit cell of 250 nm.

 

Fig. 7 The calculation results of the dispersion relations, where the size of the unit cell is 250 nm and the width of the stub is 20 nm. (a) The real part plot and (b) Imaginary part plot of the dispersion relations.

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In Fig. 8(a) and 8(b). the magnetic amplitude distributions at the stub resonance energy are shown, where the sizes of the unit cell were set to be 250 nm and 100 nm. It is confirmed that the dissipation of the magnetic amplitude along the z direction in Fig. 8(a) is much larger than that in Fig. 8(b). Considering that the impedance increases at the stub resonance energy, the propagating modes inside the Al₂O₃ center layer shows Fabry-Perot resonance along the z direction. In case that the unit cell size of 250 nm, however, the distance between stub resonators is so large that the effect of Fabry-Perot resonance along the z direction is small because the propagating mode is absorbed before getting to the next resonator. In case that the unit cell size of 100 nm, however, the distance between stub resonators is small so that the effect of Fabry-Perot resonance along the z direction is large. Therefore, the dispersion relation, where the size of the unit cell is 100nm, is strongly modulated at the stub resonance energy.

 

Fig. 8 The magnetic amplitude distributions at the stub resonance energy. (a) the size of the unit cell is 250 nm and the width of the stub resonator is 30 nm (b) the size of the unit cell is 100 nm and the width of the stub resonator is 20 nm.

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4. The calculation of effective permittivity and permeability

The effective permittivity (ε_eff) and permeability (μ_eff) of Au (thickness 150 nm)/Al₂O₃ (thickness 50 nm)/Ag (thickness 150 nm) plasmonic waveguide with serial periodic stub resonators were investigated. The ε_eff and μ_eff provide the information whether the responses are magnetic or electric nature. Considering the calculation results of the dispersion relation which can be obtained in Fig. 4, it is worthwhile to study the dependence of those parameters on the size of the unit cell. To derive the effective permittivity and permeability, the calculation of the Bloch impedance (Z_B) is needed because those parameters are expressed as${\epsilon }_{eff}=n/Z_B$,${\mu }_{eff}=n{Z}_B$. In the periodic system, the impedance is expressed by Bloch impedance. In the transmission line theory, the Bloch impedance is expressed by using the voltage (V) and current (I) flow: Z_B = V/I. The voltage and current correspond to the electric field which has the x polarization and magnetic field which has y polarization. For calculation, the electromagnetic fields obtained from one of the observation points were used, as is shown in Fig. 4(a). From these correspondences, the Bloch impedance was calculated. In Fig. 9 , the effective permittivity and permeability calculation results are shown, where the size of the unit cell is 250 nm and stub width is 30 nm.

 

Fig. 9 (a)The calculation result of effective permittivity and (b)Effective permeability of Au/Al₂O₃ /Ag plasmonic waveguides with stub resonator. The sizes of the unit cell and stub width are 250 nm and 30 nm respectively.

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In Fig. 9, the calculation results suggest the energy dependence of ε_eff and μ_eff which is originated from the BZ folding of waveguide mode. The sign of the wave vector matches well with the both of the sign of Re (ε_eff) and Re (μ_eff). At the stub resonance energy, however, the Re (ε_eff) and Re (μ_eff) have opposite value. This characteristic matches well with the previous results in Fig. 6 because the energy gap was formed at this resonance energy. In addition, the discontinuity at the boundary of 1st BZ is confirmed. Near the 1st BZ boundary, the Re (ε_eff) and Re (μ_eff) have opposite value so that the energy gap is formed. This tendency is suited with the analogy of the general band theory. Next, ε_eff and μ_eff for the size of the unit cell 100 nm with stub width 20 nm were calculated. In Fig. 10(a) and 10(b), Re (ε_eff) abruptly decreases near the stub resonance wavelength. And the sharp resonance of Re (μ_eff) are obtained. In this case, the real parts of both parameters have the same sign. This is quite different from the case of Fig. 9.

 

Fig. 10 (a)The calculation result of ε_eff and (b)Effective μ_eff of Au/Al₂O₃ /Ag plasmonic waveguides with stub resonator. The sizes of the unit cell and stub width are 100 nm and 20 nm respectively. In this case, both signs of Re (ε_eff) and Re (μ_eff) are same at the stub resonance energy.

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5. Conclusion

The dispersion relations of the Au/Al₂O₃/Ag plasmonic metawaveguide with serial periodic stub resonators were numerically investigated. In these analyses, the dependences on the size of the unit cell and the width of the stub resonators were focused. In the case that size of the unit cell is 250 nm, the negative group velocity dispersion was confirmed in the condition that the width of stub is over 30nm. Reducing the width of stub less than 30 nm, this negative group velocity dispersion vanishes because energy gap closes at the 1st BZ boundary. As for the stub resonance mode, the modulation of dispersion relation could not be confirmed clearly in the size of the unit cell of 250 nm because the dissipation of the waveguide mode is significant. On the contrary, the large modulation was confirmed in the cases that the size of the unit cell is 100 nm and stub width is 20 nm because the distance between stub resonators is small. Therefore, the propagating mode doesn’t so dissipate before getting to the next stub resonator. In this case, the stub resonance provides sharp response of refractive index, ε_eff and μ_eff. From these analyses, it is found that this type of design has the property to strongly modulate the propagating characteristics of light.