1. Introduction

Surface plasmon polaritons (SPPs) are electromagnetic waves propagating along the interface between metal and insulator [1], which have applications in nonlinear optics, sensors, nanoimaging and slow light [2–6]. The simplest structure supporting SPPs is a single metal-insulator interface, whereas several derivatives have been investigated in the past [7–13]. Among these structures, metal-insulator-metal (MIM) waveguides have been intensively studied due to stronger field confinement, and most researches have focused on homo-MIM structures with the same metal claddings. However, hetero-MIM structures with different metal claddings were found interesting in many applications in recent years. Shin et al. demonstrated the imaging process for surface plasmons with a hetero-MIM lens [14]. Lezec et al. experimentally realized negative refraction at visible frequency by an asymmetric MIM waveguide [15]. Tai et al. achieved giant angular dispersion due to modal competition in heterogeneously coupled MIM waveguides [16]. Kim studied the guided dispersion relations of Ag-Air-Al for lossless case when varying the gap width, providing an initial picture of the guided dispersion curves of hetero-MIM waveguide in a wide frequency range [17]. We thus intend to investigate the dispersion properties of the hetero-MIM waveguide and explore new phenomena in this work.

In this letter, we reported a flat slow-light band over a wide frequency range in the hetero-MIM waveguide, supporting nearly constant group index and the zero group velocity dispersion (GVD). The dispersion relations of the hetero-MIM and the homo-MIM waveguides were studied and compared. It was found that the zero GVD originates from the dispersion compensation with opposite signs of GVDs of the photonic mode and the plasmonic mode. By changing the permittivity of the middle insulator layer or the difference of two different metallic plasma frequencies, the group index and the bandwidth could be tuned. We used the indium tin oxide (ITO) for the calculations in this study, but results can be similar by applying other plasmonic materials in the hetero-MIM structure. To our knowledge, the dispersionless slow light characteristic and its mechanism in the hetero-MIM planar waveguide, which may find its applications in plasmonic devices, had not been reported previously.

2. Structure model

Figure 1 illustrates the hetero-MIM plasmonic waveguide composed of an insulator layer sandwiched between the semi-infinite substrate and superstrate of different metals. The dielectric permittivity and the thickness of the insulator layer is ε_d and d, respectively. Silicon dioxide (ε_d = 2.25) is chosen as the insulator with thickness d = 150nm. The two different metals are assumed to be indium tin oxide (ITO) with different carrier concentrations. The permittivity of ITO is characterized by the Drude model ${\epsilon }_m(\omega )={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+i\omega \gamma )$, where ε_∞ is the dielectric constant at infinite frequency, ω_p is the plasma frequency, γ is the damping constant, and ω is the angular frequency of the incident light. For ε_m2, we set ω_p2 = 2.03eV, ε_∞ = 4, γ = 0.06eV [18]. The parameters for ε_m1 are assumed to be ω_p1 = 1.2eV, ε_∞ = 4, γ = 0.06eV for the calculations and the plasma frequency can be changed by varying the molar fraction of tin (Sn) [19]. The propagation is in the x direction and the magnetic field is along y-axis. The dispersion relations and field distributions are calculated using the transfer matrix method for the TM eigenmode solution [20], in which the dispersion equation is:

 

Fig. 1 Schematic of the hetero-MIM plasmonic waveguide.

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3. Results and discussions

In Fig. 2(a), we illustrate the real part of the dispersion relations for the hetero-MIM and homo-MIM structures with parameters of ω_p1 = 1.2eV, ω_p2 = 2.03eV, ε_∞ = 4 and γ = 0.06eV. The dispersion relation of a homo-MIM waveguide [7, 21] is well known to have the lower and the upper branches connected through a regime known as the forbidden band where the curve turns over in accompany with extremely large absorption. Figure 2(b) shows the forbidden bands above their surface plasmon resonance frequencies ${\omega }_{sp1}={\omega }_{p1}/\sqrt{{\epsilon }_{\infty }+{\epsilon }_d}$ and ${\omega }_{sp2}={\omega }_{p2}/\sqrt{{\epsilon }_{\infty }+{\epsilon }_d}$ for the ε_m1-ε_d-ε_m1 and ε_m2-ε_d-ε_m2 homo-MIM waveguides, respectively, with large imaginary parts [22]. For the ε_m1-ε_d-ε_m2 hetero-MIM waveguide, the dispersion curve in Fig. 2(a) approximately follows that of the ε_m1-ε_d-ε_m1 homo-MIM waveguide first and then above the forbidden band tends to follow that of the ε_m2-ε_d-ε_m2 waveguide. Interestingly between the forbidden bands of the two homo-MIM waveguides, the curve of the hetero-MIM waveguide becomes flattened, presenting the slow-light feature. A slow-light passband is thus found between the two forbidden bands in addition to conclusions given in [17] for the hetero-MIM waveguide.

 

Fig. 2 (a) Real part and (b) Imaginary part of the dispersion relations for ε_m1-ε_d-ε_m2 (solid line) waveguide with ω_p1 = 1.2eV, ω_p2 = 2.03eV, ε_∞ = 4, and γ = 0.06eV. Results for ε_m1-ε_d-ε_m1 (dashed line) and ε_m2-ε_d-ε_m2 (dashed-dotted line) waveguides are presented for comparison. The dotted line represents the light line in silicon dioxide.

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Figure 3 shows the dispersion relations for the ε_m1-ε_d-ε_m2 waveguide in the lossless and lossy cases with above parameters. As shown in Fig. 3(a), finite damping results in modification of the curve shape at ω_sp1 and ω_sp2. However, in both cases, the lineshape keeps the same in regime ω₁~ω_sp2 with ${\omega }_1={\omega }_{p1}/\sqrt{{\epsilon }_{\infty }}$, i.e. with opposite curvatures toward both ends, and thus is not affected by the damping constant. Corresponding to the imaginary part in Fig. 3(b), in the lossy case, the FOM = ∣Real(k_x)/Imag(k_x)∣in the flat slow-light band can reach 5. The small FOM is limited by the large damping constant γ of ITO. In contrast, the FOM becomes infinity due to the zero imaginary part in the lossless case, which means low loss can be achieved with low-loss plasmonic materials.

 

Fig. 3 (a) Real part and (b) Imaginary part of the dispersion relations for the ε_m1-ε_d-ε_m2 waveguide in the lossless (dash line) and lossy cases (solid line) with ω_p1 = 1.2eV, ω_p2 = 2.03eV, and ε_∞ = 4 and FOMs (dashed-dotted line) in the lossy case.

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The slow light characteristics of the ε_m1-ε_d-ε_m2 hetero-MIM waveguide were further investigated in Fig. 4, which shows the group index n_g = c/ν_g = cdk/dω as the slow down factor and the GVD β₂ = d²k/dω² as a measure of pulse distortion. At ω = 0.68eV, there are n_g ≈8.37 and β₂ = 0. For greater or less frequency, n_g increases while β₂ changes its sign. That means a pulse with center frequency ω = 0.68eV can propagate in the waveguide with almost no distortion. For 10% variation with respect to the mean value of n_g and the normalized bandwidth Δf/f [23], the bandwidth here is around Δf = 16.9 THz and Δf/f = 0.102, in contrast to Δf = 8.6THz and Δf/f = 0.024 in MIM waveguide with two stubs based on analogue of electromagnetically induced transparency (EIT) [24].

 

Fig. 4 Dispersion relation (solid line), group index (dashed line), and GVD (dashed-dotted line) in the ε_m1-ε_d-ε_m2 waveguide. The parameters are the same as that in Fig. 2(a).

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The evolution of the magnetic field distribution in the hetero-MIM waveguide for varying frequency is illustrated in Fig. 5. At ω = 0.62eV, according to the Drude model, the real part of ε_m1 becomes positive, so no SPPs exist at the ε_m1-ε_d interface. The ε_m1-ε_d-ε_m1 homo-MIM waveguide shows a field of photonic mode at this frequency, which has negative GVD as seen in Fig. 2(a). In the hetero-MIM waveguide, the field is modified from the photonic mode due to the existence of SPPs at the ε_m2-ε_d interface with positive GVD. With opposite signs of GVDs for the photonic mode and the plasmonic mode, dispersion compensation occurs and reaches balance at ω = 0.68eV, at which the hetero-MIM waveguide presents dispersionless slow light characteristic. For ω > 0.68eV, the plasmonic mode effect becomes dominant and the field distribution, i.e. at ω = 0.78eV as shown in Fig. 5, is merely of SPPs at the ε_m2-ε_d interface, so the GVD becomes positive. In contrast, the dispersionless characteristic due to dispersion compensation does not exist in the homo-MIM structure. Such dispersionless slow light mechanism is also different from that in the plasmonic waveguide formed by single metal and several insulator layers [9, 12], where the mode moves asymptotically to different SPPs supported between the metal and different insulator layers.

 

Fig. 5 Field distribution profiles for various frequencies in the ε_m1-ε_d-ε_m2 waveguide. The field at ω = 0.62eV in the ε_m1-ε_d-ε_m1 waveguide is also presented.

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The group index n_g and the bandwidth Δf versus the permittivity of insulator ε_d for zero GVD are plotted in Fig. 6(a). As we know, ω_sp2 decreases with increasing ε_d and thus the separation between the two forbidden bands shrinks, leading to more flattened dispersion curve and higher n_g. At the same time, the bandwidth Δf decreases, which shows the trade-off relationship between delay and bandwidth [25]. Changing the difference value Δω_p = ω_p1- ω_p2 will produce the similar effect, as shown in Fig. 6(b). Although n_g obtained in this study is smaller than the values reported in [24], the bandwidth here is much greater and the structure is quite simple.

 

Fig. 6 (a) Group index n_g and bandwidth Δf versus ε_d with ω_p1 = 1.2eV and ω_p2 = 2.03 eV. (b) Group index n_g and bandwidth Δf versus Δω_p with ω_p1 = 1.2eV and ε_d = 2.25. The zero GVD is kept.

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4. Conclusion

In summary, we investigate the flat slow-light band over a wide frequency range with nearly constant group index and zero GVD in the hetero-MIM waveguide. The zero GVD property originates from dispersion compensation by the photonic mode and the plasmonic mode. Such an effect is unique in the hetero-MIM structure while does not exist in the homo-MIM structure. In addition, by changing dielectric permittivity of the insulator or the difference of plasma frequencies of the two plasmonic materials, the group index and the bandwidth can be tuned. The hetero-MIM structures may raise interest in applications that appreciate the slow light effect.