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We use coupled mode theory (CMT) to analyze a metal-insulator-metal (MIM) plasmonic stub structure, to reveal the existence of asymmetry in its transmittance spectra. Including the effect of the near field contribution for the stub structure, the observed asymmetry is interpreted as Fano-type interference between the quasi-continuum T-junction-resonator local-modes and discrete stub eigenmodes. Based on the asymmetry factor derived from the CMT analysis, methods to control transmittance asymmetry are also demonstrated.
©2011 Optical Society of America
1. Introduction
The stub structure [1], a short- or open-circuited section attached to a transmission line, plays an important role in microwave engineering, primarily for impedance matching and signal filtering purposes. Due to their compact size and ease of fabrication, stub structures also have received much attention in the nano-photonic area [2–8], including plasmonics applications. For example, utilizing a stub structure in plasmonic MIM waveguide platform, a nanoscale wavelength selective filter [5,9–11], absorption switches [12,13], and demultiplexers and waveguide bends of high transmittivity [4,14] have been demonstrated. Especially, a resonant-less signal filtering function of the stub also has received much attention, alleviating the metallic loss experienced in plasmonic resonator based filters.
With the need for precise analysis of MIM stub structures and related emerging devices, different analytical modeling approaches have been suggested [15–19], including scattering matrix method and microwave transmission-line model. Plasmonics, often treated as the optical analogue of microwave engineering, there also has been good success in extending the microwave transmission-line theory for a plasmonic stub [4,17–19]. For example, Pannipitiya [19] successfully predicted the characteristics of MIM stub structures with relatively simple formulation, modifying transmission line theory to include metal loss. Still, for the transmission-line stub-model treating / including the transmission mode only, the effect of real metal through the near field contribution has not been properly explored so far.
Here, by assuming a resonator in the cross-section junction of the stub-waveguide structure - to incorporate the contributions from local modes -, we reveal the existence of Fano-type asymmetry [20] in the transmission spectra of the stub, for MIM plasmon waveguides. Solving the CMT for the stub structure by assuming a low-Q resonator in the stub-waveguide T-junction, to account for local modes, we explain the observed asymmetry as the Fano-type mixing between the broadband resonator mode and discrete stub eigenmodes. Deriving the asymmetry factor and key physical parameters from the CMT analysis, we then demonstrate the control of spectral asymmetry in the stub transmittance response. The MIM stub structure, providing controllable Fano-type spectral asymmetry in a convenient waveguide platform, will be useful in the realization of future plasmonic devices, such as low-power, high-contrast optical switches [21–24].
2. CMT analysis of MIM stub
Figure 1 illustrates the schematics of the simplest MIM single mode (W << λspp) plasmonic waveguide stub, used in the study. CMT analysis was carried out, assuming an effective low-Q junction resonator [25,26] in the cross section region of the T-junction (in between the plasmonic stub and main waveguide). With the resonator in the T-junction, the stub-waveguide system is now equivalent to T-branch waveguides weakly coupled to the junction resonator. As a result, the transmittance property of the stub becomes affected by the presence of assumed resonator and its resonance characteristics.
Fig. 1 (a) Schematics of stub structure. W is the width of stub / waveguide, and L is stub length. (b) Analytic equivalent model for CMT with an effective low-Q resonator in the junction region
Denoting κ1 = (2/τ1)1/2, κ2 = (2/τ2)1/2, and κ3 = (2/τ3)1/2 as the coupling coefficients between the left / right / stub waveguide branches and the junction resonator, and also writing 1/τ1, 1/τ2, 1/τ3 as the corresponding decay rates from the resonator to coupling waveguides, the time evolution of field amplitude a for the junction resonator can be written in following form [27],
where ωR is the resonance frequency of the T-junction resonator, and si ± being the field amplitudes in each waveguide (i = 1, 2, 3, for outgoing (-) or incoming ( + ) from the resonator).With the conservation of total power, we also dictate [27],
Now, further assuming perfect reflection at the end of the stub of length L, we get,
where ϕ = 2π∙(2L/λ) is the phase change between the outgoing and incoming wave to the stub (λ represents λspp for real metal structure, and λair /n dielectric for PEC structures).Using Eqs. (1)–(5), we then write the transmittance equation for the simplest MIM stub,
where, ω is the operation frequency. Now, for a sufficiently narrow (W << λspp) and long (L > λspp) stub, in good approximation we set τ1 ~τ2 ~τ3 as τ0, and use κ0 = (2/τ0)1/2 to get;For Eq. (7), here we define the asymmetry factor AR, as;where Γ0 is the full width at half maximum bandwidth for the resonator-to-stub coupling.Using AR, we then rewrite the transmittance coefficient T for the stub structure, to arrive to the result of our CMT analysis;
As can be seen in Eq. (9), depending on the sign of the phase term sinϕ = sin(2L∙(2π/λ)) around the stub operation frequency, the stub transmittance becomes to have spectral asymmetry - for its asymmetry strength determined by AR. It is worth noting that AR can be approximated as a constant for a given structure around the operation frequency of interest ω ~ωT, since Γ0 of the low-Q junction resonator is much larger than the bandwidth of the stub, and then AR = 2(ω − ωR)/ Γ0 ~2(ωT − ωR)/ Γ0 + 0(δω/ Γ0). Most importantly, noting that AR measures the Γ0-normalized distance between the operation frequency ω ~ωT and the resonant frequency ωR of the broadband junction resonator, the asymmetry factor increases if the phase deviation between two frequencies involved in the Fano process increases.3. Interpretation / control of Fano-type asymmetry in MIM stub
Figure 2 shows the FDTD (Finite Difference Time Domain) obtained transmittance spectra from the stub, overlaid to the analytical CMT solution (Eq. (7)) with the fitting parameter AR. MIM waveguides, composed of Ag or PEC (Perfect Electric Conductor) sandwiching Si layer (W = 30nm) have been assumed in the analysis. To incorporate the dispersion of Ag permittivity, Drude model (reasonably accurate within 1~2μm wavelength range) was used. For the operation frequency near ωT = 193.5THz (λair = 1550nm), two stub structures with different stub length L (440nm, 810nm) returned perfect fit with an identical asymmetry factor of AR = − 0.8 for those assumed identical T-junction resonators, justifying the use of resonator local modes (determined by the T-junction structure) in the analysis. Compared to the symmetric (AR = 0) spectra from the PEC-Si-PEC waveguide stub (note that τ0 = 0 and Γ0 = ∞ for PEC, as local SPP modes are absent and only propagation modes exist, leading to AR = 2(ω − ωR)/Γ0 = 0), the asymmetric spectral profile of the Ag-Si-Ag stub is evident, derived from the Fano-type mixing of quasi-continuum resonator local modes and discrete eigenmodes of the stub.
Fig. 2 Field patterns of plasmonic MIM (Ag-Si-Ag) stub structures, at the operation frequency ωT = 193THz. Waveguide width W was set to 30 nm. Stub length L = (5/4λspp and 9/4λspp) - δskin-depth for FDTD was set at, (a) 440 nm and (b) 810 nm. Corresponding transmittance spectra are shown in (c) and (d), calculated either with CMT (marks) or FDTD numerical analysis (lines). AR = 0 for PEC and −0.8 for real metal.
Also interesting to investigate is the functional form of the asymmetry factor AR, from which we can devise a means to enhance the asymmetry of the resonance, for example, by adjusting ω ~ωT, ωR, or Γ0. Directly modifying the T-junction resonator (ωR or Γ0), the asymmetry of the transmittance can be controlled. Figures 3(d) –3(f) show the transmittance spectra obtained from the same plasmonic stub-waveguide structure (L = 440 nm, W = 30 nm, ω ~ωT ~193 THz), yet with different refractive index (n = 1, 3.46, and 5 for Figs. 3(a), 3(b), and 3(c)) in the junction resonator region. The asymmetry factor AR was found to be 0, −0.8, and −1.7, for the junction resonator index of n = 1, 3.46, and 5, respectively. Stronger asymmetry in the transmittance spectra, accompanied by the pronounced emergence of local modes (insets in Fig. 3, calculated with COMSOL) was evident for the index-raised-resonator stub structures.
Fig. 3 Stub structures (Ag-Si-Ag) with different refractive index n in the junction resonator region; (a) n = 1, (b) n = 3.46, and (c) n = 5. Transmittance spectra from stub (a)~(c) are shown in (d)~(f). Insets show the zoomed-in local mode profiles around the junction resonator. The values of AR used in the CMT was 0 (for n = 1), −0.8 (for n = 3.46), and −1.7 (for n = 5), respectively.
Instead of modifying the resonator structure as above, the ωT dependence of AR can be used to control the asymmetry. By tuning the stub length L, it is possible to adjust the stub operation frequency ωT. Figure 4 shows the structure and transmittance spectra of the stubs, for ωT = 150 THz (L = 580 nm), 200 THz (L = 440 nm), and 300 THz (L = 260 nm), respectively. Increased asymmetry was observed from AR = −0.5 to −0.8 and −1.2, with the increase of ωT.
Fig. 4 Stub structures (Ag-Si-Ag) and field profiles at different operation frequencies; (a) ωT = 150 THz, (L = 580 nm), (b) ωT = 193 THz (L = 440 nm), and (c) ωT = 300 THz (L = 260 nm). Transmittance spectra from the stub structures of (a) ~(c) are shown in (d) ~(f). The values of asymmetry factor AR used in the CMT calculation were −0.5, −0.8, and −1.2, respectively.
4. Summary
By including the local field contribution for the exact analysis of real-metal plasmonic MIM stub, we develop a temporal coupled mode theory to reveal the existence of Fano-type asymmetry in the stub transmittance spectra. Perfect fit with the results of FDTD numerical analysis was observed for various stub geometries and operation conditions, validating the theoretical analysis. We also derive the asymmetry factor AR for the MIM stub from the analytical CMT model, which could be used to control the degree of asymmetry by various means. Tuning of the asymmetry factor AR, from 0 to −1.7 was achieved by employing different refractive index materials in the junction resonator, or by changing the stub operation wavelength. Our analytical model for the exact analysis of the MIM plasmonic stub, and findings for the existence of, controllable asymmetry in their transmittance spectra will stimulate the design of low power optical switching devices, in convenient plasmonic waveguide platforms.
Acknowledgments
This work was supported by the National Research Foundation (NRF, Global Research Laboratory Project, K20815000003), and in part by a Korea Science and Engineering Foundation (KOSEF, SRC, 2010-0001859) grant funded by the Korean government (MEST).
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