Effect of dipole location on profile properties of symmetric surface plasmon polariton mode in Au/Al₂O₃/Au waveguide

Abstract

This study uses a dipole embedded in Al₂O₃ layer to excite a symmetric surface plasmon polariton (SPP) mode in Au/Al₂O₃/Au waveguide to investigate its profile properties by using finite-difference time-domain (FDTD) method. The excited dipole decay radiatively direct near-field coupling to SPP mode owing to thin Al₂O₃ layer of 100 nm. The effects of electric and magnetic field intensity profiles and decay length have been considered and characterized. It is found that dipole location is an important factor to influence the horizontal and vertical profile properties of symmetric SPP mode in Au/Al₂O₃/Au waveguide. The amplitudes of electric and magnetic field intensity and the wavelengths of metal-insulator-metal (MIM) SPP resonance mode can be tuned by varying dipole location. The horizontal and vertical decay lengths are 19 and 24 nm, respectively. It is expected that the Au/Al₂O₃/Au waveguide structure is very useful for the practical applications of designing a SPP source.

Key words： waveguide; surface plasmon polariton (SPP); profile properties

Introduction

Surface plasmon polariton (SPP) is electromagnetic waves coupled to free electron oscillation on metal surface and propagates along the metal-dielectric interface with the amplitudes decaying into both sides exponentially [1]. One of the most attractive aspects of SPP is the way in which they help concentrate and channel light beyond diffraction limit [2]. Some novel photonic devices based on SPP, such as waveguide [3], reflector [4] and beam splitters [5], have gained an increasing research interest recently. Of available SPP waveguides, metal-insulator-metal (MIM) configuration [6–10] is very promising for compact photonic integration. This is the reason that such a MIM geometry serves as a plasmonic slot waveguide, “squeezing” the SPP field into the dielectric core [11]. Until now, different MIM SPP waveguide have been demonstrated, such as various passive (e.g., filters [12]) and active plasmonic devices (e.g., emitters [13]).

In recent years, the excition of SPP has been a promising way to squeeze optical energy in nanometric cross section [14]. There are a lot of literatures on SPP excitation, such as electron-induced [15] or light-induced [16] SPP excitation. As the insulator layer of MIM waveguide is too thin to support photonic modes, the excited dipole decay radiatively only by direct near-field coupling to plasmonic transverse magnetic (TM) modes. So, the MIM waveguide geometry is particularly promising for light-induced SPP excitation. Most recently, the SPP sources based on MIM waveguide have been demonstrated experimentally, including an electrical source of SPP using organic semiconductors by Koller et al. [17] and using silicon nanocrystals embedded in Al₂O₃ by Walters et al. [18]. Although the pioneering work [18] already observed and testified that the SPP mode can occur in the MIM waveguide under a certain condition, they did not offer physical origin of such a MIM SPP mode theoretically and an exact condition for its existence. Moreover, the physical origin of such a MIM SPP mode and the exact condition for its existence have important theoretical reference value for the design of a SPP source. However, these vital electromagnetic filed profile properties of SPP mode have been left untouched. It is also noticed that the numerical study of the light (dipole)-induced SPP excitation is an area in urgent need of further work.

In this paper, we present the first study of horizontal and vertical profile properties of symmetric SPP mode in Au/Al₂O₃/Au waveguide, which only supports tightly confined coupled SPP mode within the thin Al₂O₃ layer by using finite-difference time-domain (FDTD) method. The aim of this paper is to give a numerical analysis of light (dipole)-induced SPP excitation. Attention will be focused on the fundamental physics and the electromagnetic field profile properties of MIM SPP mode. The simulation results show that both the amplitudes of electric and magnetic field intensity and the wavelengths of MIM SPP resonance mode can be tuned by varying dipole location, owing to the role of coupling between dipole and Au/Al₂O₃/Au waveguide. This phenomenon may provide guidance in designing a SPP source by using Au/Al₂O₃/Au waveguide structure.

Theoretic model and FDTD modeling method

MIM waveguide is composed of two identical Au films (200 nm) separated by an insulator layers Al₂O₃ (100 nm), and the horizontal length of the MIM waveguide is infinite, as shown in Fig. 1. When the width (w) of dielectric layer is reduced below the diffraction limit, conventional guiding modes cannot exist. In this case, a TE 90 polarization incident light (dipole) will be transformed into the coupled SPP on the metal surfaces and propagates along the waveguide. The dispersion relation of the SPP in the MIM waveguide can be deduced from the Maxwell equations as [19]

(1){\displaystyle \frac{p{ϵ}_{\mathrm{d}}}{k{ϵ}_{\mathrm{m}}}}={\displaystyle \frac{\mathrm{1}-{\mathrm{e}}^{\mathit{k}\mathit{w}}}{\mathrm{1}+{\mathrm{e}}^{kw}}},

where

(2)k=k_{\mathrm{0}}\sqrt{{\left({\displaystyle \frac{{\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}}{k_{\mathrm{0}}}}\right)}^{\mathrm{2}}-{ϵ}_{\mathrm{d}}},

(3)p=k_{\mathrm{0}}\sqrt{{\left({\displaystyle \frac{{\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}}{k_{\mathrm{0}}}}\right)}^{\mathrm{2}}-{ϵ}_{\mathrm{m}}},

and

(4){\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}=n_{\mathrm{e}\mathrm{f}\mathrm{f}}{k}_{\mathrm{0}},

where k_{\mathrm{0}} is the wave vector in free space, {\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}} and n_{\mathrm{e}\mathrm{f}\mathrm{f}} are the complex propagation constant and the effective refractive index of SPP, respectively. {ϵ}_{\mathrm{m}} and {ϵ}_{\mathrm{d}} are the relative dielectric constant of the metal and the dielectric material, respectively. Gold for the metallic cladding layers is characterized by the Drude model [20]

(5){ϵ}_{\mathrm{m}}=\mathrm{1}-{\displaystyle \frac{{\omega }_{\mathrm{p}}^{\mathrm{2}}}{\omega (\omega +\mathrm{i}\gamma )}},

where {\omega }_{\mathrm{p}}=\mathrm{1.2}\times {\mathrm{10}}^{\mathrm{16}}\mathrm{H}\mathrm{z} and \gamma =\mathrm{1.2}\times {\mathrm{10}}^{\mathrm{14}}\mathrm{H}\mathrm{z} are the bulk plasma and damping frequencies, respectively, and Al₂O₃ with refractive index n_{\mathrm{d}}=\mathrm{1.7}. As Al₂O₃ layer is too thin (100 nm) to support photonic modes, the excited dipole decay radiatively only by direct near-field coupling to a symmetric SPP mode, which is highly confined within the insulating region. The characteristic equation of symmetric SPP mode is given by [21]:

(6){\displaystyle \frac{{\kappa }_{\mathrm{m}}}{{ϵ}_{\mathrm{m}}}}+{\displaystyle \frac{{\kappa }_{\mathrm{d}}}{{ϵ}_{\mathrm{d}}}}\tanh \left({\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\kappa }_{\mathrm{d}}w\right)=\mathrm{0},

where {\kappa }_{\mathrm{m}}=\sqrt{{\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}^{\mathrm{2}}-{ϵ}_{\mathrm{m}}} and {\kappa }_{\mathrm{d}}=\sqrt{{\beta }_{\mathrm{S}\mathrm{P}\mathrm{P}}^{\mathrm{2}}-{ϵ}_{\mathrm{d}}} denote the transverse wavenumber of the plasmonic mode in the metal and dielectric, respectively.

Fig.1 Schematic structure of proposed Au/Al₂O₃/Au waveguide composed of two identical Au films separated by an insulator layers Al₂O₃, and a dipole is embedded in Al₂O₃ layer to excite a MIM SPP mode for forward propagating

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Au/Al₂O₃/Au waveguide two dimensional (2D) structure is designed and modeled by a commercial FDTD package<FootNote>

The commercial software FDTD solutions was employed for the numerical simulation

</FootNote> that supports ununiform meshing and eigenmode calculation. Spatial mesh cells are set to \mathrm{\Delta }x=\mathrm{\Delta }y=\mathrm{5}\mathrm{n}\mathrm{m} and the time step is taken as \mathrm{\Delta }t=\mathrm{\Delta }s/\mathrm{2}c=\mathrm{8.3333}\times {\mathrm{10}}^{-\mathrm{18}}\mathrm{s}. The 2D FDTD method with perfectly matched layer (PML) as boundary condition is used in this work. The fundamental TM mode (E_x;E_y;H_z) of the Au/Al₂O₃/Au waveguide is excited by a dipole source being located in the intermediate Al₂O₃ layer and the mesh grid size is set to 1 nm in order to maintain convergence. And then the MIM SPP mode is excited and two different power monitors PD1 (horizontal) and PD2 (vertical) are set to detect the incident power A1. In this simulation experiment, the dipole in the proposed waveguide structure is located in Al₂O₃ at distances of 10, 15, 20, 25, 30, 35, 40, 45 and 50 nm from the bottom Al₂O₃/Au interface and the thickness of Al₂O₃ is fixed at 100 nm. It is obvious that there is a symmetric plasmonic Hz mode (as shown in Fig. 1) supported in the MIM SPP mode, which arises from an asymmetric coupling of SPP at inner interfaces of Au layers. The simulated electric field and magnetic field intensity profiles of MIM SPP mode with different MIM geometry (t_{\mathrm{A}\mathrm{u}(\mathrm{t}\mathrm{o}\mathrm{p})}=t_{\mathrm{A}\mathrm{u}(\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{m})}=\mathrm{200}\mathrm{n}\mathrm{m}, w=\mathrm{100}\mathrm{n}\mathrm{m}) by changing the dipole distance are shown in Figs. 2(a) and 3(a).

Simulation results and discussion

To study the influence of dipole location on the performance of symmetric SPP mode in Au/Al₂O₃/Au waveguide, the horizontal profiles of the electric field and magnetic field intensity profiles are calculated for the structures as shown in Figs. 2(a) and 3(a). Figure 2(a) shows the horizontal profiles of the electric field intensity (i_{\mathrm{E}}^x) of MIM SPP mode at the interface of Al₂O₃/Au along the x-axis for the frequency corresponding to the free space wavelength of dipole from 0.6 to 1.8 μm. In addition, the amplified part of MIM SPP resonance model is shown in the inset. By varying dipole distances from 10 to 50 nm in steps of 5 nm with the thickness of Al₂O₃ layer of 100 nm, i_{\mathrm{E}}^x is decreased, due to the different coupling strengths between dipole and Au/Al₂O₃/Au waveguide. This is the reason that dipole distance is critical to the SPP coupling strength. As the dipole distance gets larger, the SPP coupling strength becomes weaker, leading to decrease of i_{\mathrm{E}}^x. In addition, we have plotted two curves based on the data from i_{\mathrm{E}}^x of MIM SPP resonance mode. i_{\mathrm{E}}^x and SPP resonance wavelength ({\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}}) as a function of dipole distance are shown in Fig. 2(b). It is clearly seen that {\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}} is red shifted consistently and i_{\mathrm{E}}^x is exponential decay. The exponential relationship between i_{\mathrm{E}}^x and the dipole distance (d) becomes obvious. i_{\mathrm{E}}^x can be fitted using a exponential function to determine the horizontal decay properties of SPP mode along the x-axis. The fitting curve of the FDTD simulation results for i_{\mathrm{E}}^x is i_{\mathrm{E}}^x=\mathrm{5}\exp (-d/\mathrm{19})+\mathrm{11}. So, the horizontal decay length is 19 nm. The physical concept of the horizontal decay length arises from the fact that the change of dipole distance resulting in i_{\mathrm{E}}^x being diminished by 1/e. This is very useful for the designing a SPP source with a desirable luminescence center (19 nm) by merely adjusting dipole distance to make best SPP coupling between top and bottom Au films in Au/Al₂O₃/Au waveguide. The redshift of {\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}} with the augment of dipole distances results from the asymmetric coupling of SPP between dipole and Au/Al₂O₃/Au waveguide. It is the reason that the asymmetric coupling of SPP becomes strong when the location of dipole is near the middle position.

Fig.2 (a) Horizontal profiles of electric field (E) intensity at bottom Al₂O₃/Au interface as a function of dipole distance; (b) electric (E) field peak intensity and SPP resonance wavelength as function of dipole distance

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Figures 3(a) shows the horizontal profiles of the magnetic field intensity (i_{\mathrm{H}}^x) of MIM SPP mode at the interface of Al₂O₃/Au along the x-axis. i_{\mathrm{H}}^x is also decreased. As the dipole distance gets larger, the SPP coupling strength becomes weaker which results in a decrease of i_{\mathrm{H}}^x. i_{\mathrm{H}}^x and {\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}} as a function of dipole distance are shown in Fig. 3(b). It is clearly seen that {\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}} is red shifted consistently and i_{\mathrm{H}}^x is exponential decay. The fitting curve of the FDTD simulation results for i_{\mathrm{H}}^x is i_{\mathrm{H}}^x=\mathrm{2.4}\times {\mathrm{10}}^{-\mathrm{4}}\exp (-d/\mathrm{19})+\mathrm{5.7}\times {\mathrm{10}}^{-\mathrm{4}}. The horizontal decay length is 19 nm.

Fig.3 (a) Horizontal profiles of magnetic field (H) intensity at bottom Al₂O₃/Au interface as function of dipole distance; (b) magnetic field (H) peak intensity and SPP resonance wavelength as function of dipole distance

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To achieve an in-depth understanding of the effect of dipole distance on the profile properties of MIM SPP mode, we used a 2D FDTD numerical simulation to model the vertical profiles of the electric field and magnetic field intensity of the corresponding MIM SPP resonance mode, as shown in Figs. 4 and 5 along the y-direction. Figure 4 shows the vertical profiles of the electric field intensity (i_{\mathrm{E}}^y) of the corresponding MIM SPP resonance mode along the y-axis. i_{\mathrm{E}}^y is decreased exponentially with increasing the dipole distance from 10 to 50 nm in steps of 5 nm, due to the SPP coupling strength becoming weaker. The MIM SPP resonance mode effect clearly appears at vertical profiles along the y-axis (peak and dip). Furthermore, the positions of peaks and dips in the electric field intensity profiles are almost not affected, which are centered around ±0.044 and 0 μm, respectively, and the width between dips is 0.01 μm. In addition, the exponential relationship between i_{\mathrm{E}}^y and d becomes obvious as shown in the inset. i_{\mathrm{E}}^y can be fitted using a exponential function to determine the vertical decay properties of SPP mode in the Au/Al₂O₃/Au MIM waveguide along the y-axis. The fitting curve of the FDTD simulation results for i_{\mathrm{E}}^y is i_{\mathrm{E}}^y=\mathrm{14}\exp (-d/\mathrm{24})+\mathrm{34}. So, the vertical decay length is 24 nm. The physical concept of the vertical decay length arises from the fact that the change of dipole distance resulting in i_{\mathrm{E}}^y being diminished by 1/e. This is very useful for designing an SPP source with a desirable luminescence center (24 nm) by merely adjusting the locations of dipole to make the coupled SPP model being more gathered and propagated in Al₂O₃ layer.

Fig.4 Vertical profiles of electric field (E) intensity of corresponding MIM SPP resonance mode along y-axis as function of dipole distance. Inset, dependence of electric field dip intensity on dipole distances

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Figure 5 shows the vertical profiles of the magnetic field intensity (i_{\mathrm{H}}^y) of the corresponding MIM SPP resonance mode along the y-axis. i_{\mathrm{H}}^y is also decreased exponentially with increasing the dipole distance from 10 to 50 nm in steps of 5 nm, due to the SPP coupling strength becoming weaker. Furthermore, the positions of peaks and dips in the magnetic field intensity profiles are almost not affected, which are centered around ±0.044 and 0 μm, respectively, and the width between dips is 0.01 μm. The exponential relationship between i_{\mathrm{H}}^y and d becomes obvious as shown in the inset. The fitting curve of the FDTD simulation results for i_{\mathrm{H}}^y is i_{\mathrm{H}}^y=\mathrm{1.9}\times {\mathrm{10}}^{-\mathrm{4}}\exp (-d/\mathrm{24})+\mathrm{4.8}\times {\mathrm{10}}^{-\mathrm{4}}. So, the vertical decay length is 24 nm.

Fig.5 Vertical profiles of magnetic field (H) intensity of corresponding MIM SPP resonance mode along y-axis as function of dipole distance. Inset, dependence of magnetic field dip intensity on dipole distances

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Conclusions

In conclusion, the horizontal and vertical profile properties of symmetric SPP mode in Au/Al₂O₃/Au MIM waveguide are investigated by varying dipole location yet the thickness of Al₂O₃ layer being fixed at 100 nm through the FDTD simulation. Based on the calculated electromagnetic field intensity profile properties, the dipole location is an important factor to the horizontal and vertical profile properties of symmetric SPP mode. The horizontal profiles of i_{\mathrm{E}}^x and i_{\mathrm{H}}^x decrease and {\lambda }_{\mathrm{S}\mathrm{P}\mathrm{P}} are red shifted consistently when the dipole distances increase. The vertical profiles of i_{\mathrm{E}}^y and i_{\mathrm{H}}^y are also decreased exponentially with increase of the dipole distances. However the positions of peaks and dips in the electric field and magnetic field intensity are not affected. The horizontal and vertical decay lengths are 19 and 24 nm respectively. It indicated that the resonant coupling effect of SPP plays a crucial role in the Au/Al₂O₃/Au MIM waveguide. These properties will undoubtedly facilitate the application of Au/Al₂O₃/Au waveguides in designing an SPP source.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 60907024 and 61036011), the New Teachers’ Fund for the Doctoral Program of Higher Education (No. 20100001120024), Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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