1. Introduction

Plasmonic waveguiding structures have recently gained much attention due to their potential for confining light at sub-wavelength scales, thereby enabling extremely compact devices to be realized [1]. Among the different waveguide structures proposed to date which can support plasmonic modes [2, 3], the metal-insulator-metal (MIM) structure is the most promising because they can provide true sub-diffraction confinement [4]. Due to the strong lateral confinement provided by the metallic sidewalls, MIM waveguides are especially attractive for realizing whispering gallery mode plasmonic resonators, in which the bending loss inherent in conventional dielectric microring resonators is virtually eliminated. As a result, extremely compact plasmonic ring resonators with submicron radii and sub-diffraction modal volumes can potentially be achieved. These resonators can be used as building blocks to provide many important plasmonic device functionalities at the nanoscale such as optical signal processing, spectral engineering and nonlinear switching, as have been demonstrated in conventional photonics with various material systems [5, 6].

To date, however, very few studies on MIM plasmonic ring resonators have been reported, primarily due to the lack of an efficient coupling mechanism between the ring resonator and an external bus waveguide. Previous studies [7, 8] have suggested evanescent coupling through the metallic sidewall separating the bus waveguide and the plasmonic ring. Since the decay length of the field in the metal layer is typically around 20nm, very small gaps (<20nm) are required to achieve any noticeable coupling. Thus evanescent coupling is both inefficient and challenging to realize even with state-of-the-art electron beam or ion beam lithography. MIM plasmonic resonators employing the vertical coupling scheme where the bus and ring waveguides reside on two separate layers may be an alternative, although the fabrication of such a multi-level plasmonic waveguide device would also be very challenging. In this paper we propose and investigate MIM plasmonic ring resonators in which efficient coupling between the ring cavity and bus waveguide is achieved through small apertures in the metallic sidewalls of the waveguides. The coupling strength can be controlled by varying the aperture width and depth, with typical dimensions between 50–100nm, which are within the capability of electron beam lithography. Also, by proper design of the aperture coupler, both standing-wave and traveling-wave ring resonators can be achieved, with the former exhibiting reflection spectra of the Fano resonance type. Due to the strong confinement of the MIM waveguide, the roundtrip cavity length of these plasmonic resonators can be made much shorter than the propagation length of the MIM waveguide, yielding cavities with low transmission loss and high field enhancement. An analytical model of the device based on Bethe’s small-aperture coupling theory [9] allows simple expressions for the spectral responses of the plasmonic ring resonator to be derived. The theoretical device responses obtained were also verified by rigorous 2D and 3D finite-difference time-domain (FDTD) simulations, which showed that ring resonators with radius as small as 500nm can be achieved with bandpass loss less than 2dB and intensity enhancement in the cavity near 10dB.

2. Analytical model

Fig. 1(a) shows a schematic of an MIM plasmonic ring resonator with radius R side-coupled to a plasmonic bus waveguide via a small aperture of width w in the ring sidewall. The ring and bus are separated by a gap g, which is also the aperture depth. The ring and bus waveguides consist of a dielectric core layer of thickness d sandwiched between two parallel metallic layers. The waveguides are assumed to support the fundamental TM mode, with propagation length L_p due to absorption loss in the metal. Using Bethe’s small-hole diffraction theory [9] and neglecting any plasmon-mediated coupling, we model the small aperture in the ring sidewall by equivalent electric and magnetic dipole moments. These dipoles radiate into the four ports of the coupling junction, giving rise to the forward and backward scattered waves. FDTD simulations of the coupling junctions with various aperture widths and depths showed that the forward coupling and backward coupling differ only slightly, so to simplify our analysis we assume that these couplings are equal, κ_f=κ_r=κ. With reference to Fig. 1(a), the scattering matrix of the four-port aperture coupler can be expressed as

where t=τ-jκ and for a lossless coupling junction we must have τ²+4κ²=1. Letting A⁺ ₃=0, $A_3^{+}=0,A_2^{+}=A_4^{-}{\alpha }_{\mathrm{rt}}{e}^{-j{\varphi }_{\mathrm{rt}}}$ and $A_4^{+}=A_2^{-}{\alpha }_{\mathrm{rt}}{e}^{-j{\varphi }_{\mathrm{rt}}}$, where ϕ_rt is the roundtrip phase in the ring and a_rt=exp(-πR/L_p) is the roundtrip amplitude attenuation, we solve (1) for the transmitted response T_t=A ^- ₃/A ⁺ ₁ and reflected response T_r=A ^- ₁/A ⁺ ₁ of the device to obtain

In Fig. 1(b) we plotted the theoretical response of a plasmonic ring with a 500nm radius for the ideal case of no loss and the practical case where the plasmonic waveguide has a propagation length L_p=50µm, which is typical for MIM waveguides at infrared wavelengths. The aperture power coupling strength is assumed to be κ²=0.1, which, as shown by FDTD simulations below, can be achieved with an aperture width of 50nm. In both cases the reflection response has resonance peaks while the transmission spectrum exhibits resonance dips. The reflection response is also seen to have a highly asymmetric spectral shape with a reflection null near the resonant frequency. This behavior is characteristic of a Fano resonance, which is caused by the interference between the back-scattered wave from the aperture and the discrete resonance modes of the ring cavity. The dip in the reflection spectrum occurs at the frequency where the back-scattered wave and the wave coupled out from the ring interfere destructively with each other. For the lossless device, complete power reflection and zero transmission is obtained at resonance. For the lossy case, even when the propagation length of the ring waveguide is only 50µm (corresponding to an intrinsic Q ~400), the device responses are only slightly degraded, showing a 2dB insertion loss in the reflected signal. Inside the ring, forward and backward scattered waves from the coupling aperture give rise to counter-propagating modes, which interfere to form standing wave patterns at the resonant frequencies. In Fig. 1(b) we also plotted the intensity spectrum inside the lossy ring, which shows that an intensity enhancement near 10dB can be achieved at resonance.

 

Fig. 1. (a) Schematic of an MIM plasmonic ring resonator side-coupled to a bus waveguide via a small aperture of width w and depth g. (b) Theoretical spectral response of a 500nm-radius plasmonic ring resonator with κ²=0.1 for the ideal lossless case (dashed lines) and the lossy case (solid lines) when the MIM waveguide has a propagation length of 50µm.

Download Full Size | PDF

3. Numerical results

To verify the results of the analytical model, we performed 2D FDTD simulations of a 500nm-radius metallic ring resonator coupled to a bus waveguide via a small aperture of width w=50nm. The separation gap g between the bus and ring was chosen to be 50nm to eliminate evanescent coupling between the ring and bus waveguides. Both the ring and bus MIM waveguides have width d=100nm, with a core index of 1.6. The metallic sidewalls were assumed to be silver, whose dielectric function was described by the Drude model ε_r=ε_∞-ω² _p/(ω²+jγω), with ε_∞=3.7, ω_p=1.3826×10¹⁶ rad/s and γ=2.7348×10¹³ s^-1 (parameters obtained by fitting the experimental data [10] at the infrared frequencies). We first performed simulations of the aperture coupling junction in order to determine the forward and backward power coupling strengths, κ² _f and κ² _r, of the coupler. The results are shown in Fig. 2(a) as functions of the aperture width w for a fixed gap separation g of 50nm. It is seen that the forward and backward couplings are nearly identical to each other and strong couplings can be achieved with practical aperture widths between 50nm and 100nm. Setting the aperture width to 50nm, which corresponds to a power coupling value of κ²=0.1, we next performed FDTD simulations of the plasmonic ring resonator by applying a broadband Gaussian pulse to the bus waveguide and analyzing the transmitted and reflected signals with fast Fourier transform. The wavelength range of the signals was chosen to be in the infra-red region since we are primarily interested in device applications for optical communication. Fig. 2(b) shows the 2D FDTD simulation results for the reflected and transmitted responses, which are in agreement with the analytical responses in Fig. 1(b). The device has a wide reflection bandwidth of 30nm, an insertion loss of 1.7dB and a wide FSR of 460nm at the 1.51µm resonance. The intensity enhancement in the ring at this resonance wavelength is 8.2dB, which is comparable to the value predicted by the theoretical model. In Fig. 2(c) we show the time-averaged intensity distribution of the magnetic field, |H_y|², in the plasmonic ring at the 1.51µm resonance, which corresponds to cavity mode number m=4. The standing-wave resonance pattern is evident from the figure, as well as the enhanced intensity in the ring and total field extinction at the output port at resonance.

We also performed 3D FDTD simulations of a realistic MIM plasmonic ring with a 500nm radius. For the 3D simulation the waveguide height was set to 400nm and the substrate and cladding index was 1.45. The cross-section of the MIM waveguide structure is shown in the inset of Fig. 2(d). The coupling aperture width w was set to 100nm and the other parameters including the metal permittivity were the same as in the 2D structure. As shown in Fig. 2(d), the resonance spectra of the 3D structure are more degraded than the 2D results in terms of insertion loss and output-port extinction due to extra out-of-plane scattering loss. Nevertheless the reflected response has an insertion loss of only 1.8dB at the 1.84µm resonance, which can be improved with further optimization of the plasmonic waveguide to reduce loss. We also note that the resonance at 1.84µm corresponds to a cavity mode number m of only 3, and the 3D ring resonator has an effective modal volume of only 0.125µm³, or 0.1(λ₀/n _eff)³. FDTD simulations have also shown that even smaller cavity volumes can be achieved by further decreasing the ring radius and the waveguide cross-section, making the MIM plasmonic ring resonator potentially one of the most compact optical cavities with truly sub-diffraction confinement.

 

Fig. 2. (a) Forward κ² _f, blue line) and backward (κ² _f, red line) power coupling coefficients as functions of the aperture width w with fixed aperture depth g=50nm. (b) 2D-FDTD simulation results of the spectral responses of a 500nm-radius plasmonic ring with w=50nm and g=50nm. (c) Time-averaged distribution of the magnetic field, |H_y|², in the ring at the 1.51µm resonance wavelength. (d) 3D-FDTD simulation results of the spectral responses of a 500nm-radius plasmonic ring resonator with coupling aperture width w=100nm and depth g=50nm. The inset shows the cross-section of the MIM waveguide.

Download Full Size | PDF

We can also construct aperture-coupled traveling-wave plasmonic ring resonators by using directional aperture couplers in which the back-scattered waves are suppressed. Such a coupler may be realized with two or more identical apertures separated by a length of λ/4. All-pass and four-port add-drop plasmonic ring devices can then be constructed by coupling a metallic ring to one or two MIM waveguides via these multiple-aperture couplers. In Fig. 3(a) we show the theoretical responses of an all-pass and an add-drop plasmonic ring resonator using two-hole directional couplers. The ring radius was 500nm and the waveguides were assumed to have a propagation length of L_p=50µm. To simplify the analysis we have also neglected dispersion in the theoretical model. It is seen that these devices have spectral characteristics that are much like those of conventional dielectric microring resonators. To verify the theoretical responses, we again performed 2D-FDTD simulation of an all-pass plasmonic ring resonator with a 500nm radius coupled to a bus waveguide via a directional coupler consisting of two apertures separated by a distance of 220nm, or approximately λ/4 at the 1.51µm wavelength. The aperture and waveguide dimensions of the device as well as the material parameters were the same as for the device in Fig. 2(b). The transmission and reflection spectral responses of the all-pass plasmonic ring obtained from the simulation are plotted in Fig. 3(b). From the plot it is evident that the reflected signal is suppressed below -11dB at the 1.5µm resonance and the transmitted spectrum has the characteristic dip of a lossy all-pass ring resonator. Due to the extremely dispersive nature of the plasmonic waveguides and the narrow-band nature of the two-hole coupler, suppression of the back-scattered wave could be achieved only around the 1.5µm resonance. However, this is not a concern since the resonator has a very wide FSR so that the adjacent unsuppressed reflection peaks occur 300–400nm away. We also note that better suppression of the back-scattered wave can be achieved by adding more apertures to the coupler. Due to their similar spectral responses as conventional traveling-wave microring resonators, plasmonic ring resonators with directional aperture couplers can be used as basic building blocks to construct more advanced signal processing plasmonic circuits such as filters, multiplexers, switches and delay elements.

 

Fig. 3. (a) Theoretical spectral responses of an all-pass and an add-drop plasmonic ring resonator using two-hole directional aperture couplers. (b) 2D-FDTD simulation results of the spectral responses of a 500nm-radius all-pass plasmonic ring resonator.

Download Full Size | PDF

4. Conclusion

Aperture-coupled MIM plasmonic ring resonators were proposed as ultracompact metallic optical resonators which combine the advantages of sub-diffraction modal confinement of MIM waveguides with the low loss, high quality factors of conventional microring resonators. A theoretical model of the aperture coupler was presented based on the scattering matrix which allows simple expressions for the responses of the plasmonic ring to be derived. The device responses obtained theoretically were also verified by rigorous FDTD simulations. The proposed aperture-coupling scheme potentially has less stringent fabrication requirements than devices based on evanescent coupling, and can be used to achieve efficient coupling in plasmonic ring resonators as well as other important MIM plasmonic structures such as power combiners, splitters, and interferometers. We expect that these aperture-coupled MIM devices to have broad applications as basic building blocks for constructing ultracompact plasmonic circuits with more advanced functionalities.