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In this paper, we present a new optical range low pass filter based on plasmonic nanostrip waveguides. We calculate the characteristic impedance of plasmonic nanostrip waveguides and compare it with that of microstrip transmission lines. An optical range maximally flat low pass filter with subwavelength dimensions is designed based on the nanostrip waveguide structure. Finite-difference time-domain (FDTD) simulations of the designed optical range filter are presented, which demonstrate subwave-length light confinement as well as acceptable filter cutoff performance.
©2007 Optical Society of America
1. Introduction
At optical range frequencies, electron plasma at metallic surfaces strongly interacts with light in subwavelength regime. The resulting electromagnetic waves are called surface plasmon po-laritons (SPPs) and are localized at metal/dielectric interfaces with exponentially decreasing field profiles into the both dielectric and metallic sides. SPPs bring exceptional opportunities for the nanoscale miniaturization of photonic devices suffering from the diffraction limit, and thereby allow realization of highly integrated full photonic circuits [1, 2, 3]. Several waveg-uiding structures and optical components have been proposed for subwavelength transmission and manipulation of light [2, 4, 5, 6, 7, 8, 9, 10, 11], such as channel waveguides, resonators, microcavities and Bragg filters.
Different mechanisms can be utilized for frequency filtering in the optical range. However, structures suitable for integration are mainly restricted to Bragg filters and ring resonators. Bragg filters usually consist of periodically alternating layers with different refractive index materials. The destructive interference of the reflected radiations from each layer results in zero transmission under the Bragg condition. On the other hand, ring resonators consist of reentrant waveguides in proximity of or in contact with straight waveguides. Under the resonance condition, where the total waveguiding length of the ring is an integer multiple of the input wavelength, the electromagnetic energy couples from the straight waveguide into the ring resulting in a drop in the amplitude of the transmission coefficients. Both Bragg filters and ring resonators only provide bandpass filtering mechanisms that are capable of rejecting certain narrow frequency bands. Therefore, they can not be utilized to build low/high pass optical filters.
Fig. 1. (a) Schematic of the nanostrip waveguide structure. FDFD simulation results for nanostrip waveguide, t = 20 nm, w = 100 nm and d = 25 nm at λ =1.55 μm, (b) power density profile (Pz), (c) electric (dashed lines) and magnetic (solid lines) field lines.
Fig. 2. Effective index and modal size versus nanostrip width for t = 20 nm and different d values at λ=1.55 μm. Red and blue (circles) lines indicate d = 35 nm and d = 25 nm, respectively.
Additionally, the dimensions of these structures have to be in the order of a few operating wavelengths, and they can not be miniaturized to less than a few microns in the optical range.
In this paper, we present a numerical analysis of the plasmonic nanostrip waveguide. We show that nanostrip waveguides can provide subwavelength light confinement. Then, a new plasmonic filter structure is presented that is also compatible with the planar fabrication technology. The presented nanostrip optical filter is a maximally flat low pass filter with 3 dB attenuation at the optical communication wavelength (λ=1.55 μm). Not only the light confinement, but also the total dimensions of the designed filter are subwavelength. The Finite-Difference Time-Domain (FDTD) simulation results show a great agreement between the presented filter’s response and the ideal frequency response.
2. Nanostrip Plasmonic Waveguide Based Filter Design
Among the waveguiding structures supporting plasmonic modes propagation, those that focus the electromagnetic energy into the dielectric region of a Metal-Insulator-Metal (MIM) configuration allow for subwavelength light propagation. MIM based plasmon slot waveguides have been shown to provide both long range propagation and subwavelength spatial confinement [1]. MIM waveguides are potentially important in providing an interface between conventional optics and subwavelength optoelectronic devices [12]. In analogy with microwave components in the GHz frequency range, the concept of characteristic impedance can be used to analyze and design bends and splitters that are necessary for realizing optical interconnects. The quasistatic approximate characteristic impedance of the fundamental MIM mode has been shown to be a reliable parameter for designing plasmonic bends and splitters [12].
An ideal MIM waveguide is infinite in the transverse direction and is not suitable for practical implementations. A practical waveguide structure with MIM configuration is shown in Fig. 1(a). This structure, which we refer to as nanostrip waveguide, consists of a metallic strip of width w and thickness t lying on a dielectric thin film (thickness d) supported by a metallic layer underneath. The dielectric layer is assumed to be silicon dioxide, SiO 2, and silver is considered for all the metallic materials. We characterize silver using the experimental data from [13]. In order to characterize the structure shown in Fig. 1(a), we utilize a Finite-Difference Frequency-Domain (FDFD) analysis [14]. Figure 1(b) shows that the power density (Pz) in the nanostrip waveguide is mainly concentrated in the dielectric region between the metallic strip and the metallic thin film. Therefore, the nanostrip waveguide is expected to have similar specifications as MIM waveguides. The electric and magnetic field lines for the fundamental mode of the nanostrip waveguide are shown in Fig. 1(c).
Fig. 3. Characteristic impedance (Z0) versus metal strip width (w) for t = 20 nm and different d values at λ=1.55 μm. The blue lines show the impedance values calculated from the FDFD simulation results using Eq. (1), and the red lines (circles) show the impedance values calculated for a microstrip transmission line [15] with the same dimensions.
Figure 2 depicts the variation of the effective index (neff = β/k 0) and the modal size of the fundamental mode versus nanostrip width, where β is the propagation constant and k 0 is the vacuum wavenumber. Note that the fundamental mode is hybrid. However, as w increases, the fundamental mode becomes more transverse magnetic polarized (TM) similar to the case of planar MIM waveguides (Ey ~ 0, Hx ~ 0 and Hz ~ 0) [1].
In order to quantify the ability of a waveguide structure to spatially confine the light, we define modal size as square root of the area, in which the power density is more than 1/e 2 times its maximum value. The roughly linear dependence of modal size on the nanostrip width implicitly confirms that the electromagnetic energy is concentrated in the dielectric region below the metallic strip. Although the modal size increases with width, all the modal size values shown in Fig. 2 are subwavelength. Therefore, the nanostrip waveguide can effectively confine the light in subwavelength dimensions. In addition, modal size increases as effective index decreases, which is a behavior inherent to all MIM based waveguides.
As the electromagnetic energy is confined to dimensions much smaller compared to the wavelength, the quasistatic condition holds and the characteristic impedance (Z0) can be defined as
where s and c are an arbitrary path connecting the metallic film to the metal strip and an arbitrary closed loop around the metal strip, respectively. We found that the denominator in Eq. 1 has negligible dependence on the choice of c. In the case of a nanostrip waveguide with w = 100 nm, d = 25 nm and t = 20 nm, the numerator changes about 20% when s, which is taken to be a vertical line connecting the metallic film to the metal strip, moves along the x-direction from the center of the metal strip toward each of the metal strip corners, due to the fringing effects around the metal strip corners. As the fringing effects decrease with increasing w, the variation of ∫s E.dl with s is reduced. In order to take into account the fringing effects, we average on the integrals calculated along several different paths. Figure 3 depicts the variation of Z0 values versus w for t = 20 nm and d = 25 nm and 35 nm. By increasing w, the equivalent current increases resulting in a less impedance. In contrast, as the dielectric layer thickness increases, the equivalent voltage increases resulting in a higher impedance. This behavior is similar to that in microstrip transmission lines, whose characteristic impedances are given as follows for w/d ≥ 1 [15],
Fig. 4. (a) Equivalent LC ladder circuit model for the 6th order maximally flat low pass filter. (b) Top view of the presented nanostrip 6th order maximally flat low pass filter. Dimensions are in nanometer.
where
We assume that ε r = ε SiO2 = 2.25 for the waveguide structure shown in Fig. 1(a). The calculated characteristic impedance for the nanostrip dimensions using Eq. (2) is close to the results from Eq. (1), as shown in Fig. 3. Note that the neff values from Fig. 2 are used in these calculations. As w or d decrease, the light confinement increases resulting in smaller modal size (Fig. 2). Therefore, the plasmonic effect becomes more dominant and the electric and magnetic field lines differ from those in the case of nonplasmonic fundamental mode of microstrip transmission lines.
In order to design an optical low pass filter based on the nanostrip structure, we utilize a 6th order maximally flat (Butterworth) configuration. Maximally flat filters demonstrate flat passband without any ripple and the transition (between the stopband and the passband) is relatively smooth, which becomes sharper (shorter transition band) by increasing the filter order. A low pass 6th order maximally flat filter has a frequency response with an attenuation more than 20 dB at 2f 0, where f 0 is the 3 dB cut-off frequency, and therefore, this filter is expected to have an observable sharp transition.
A circuit model for the maximally flat filter is shown in Fig. 4(a). The guide-wavelength value λg = λ0/neff is used to find the length of the shunt and series branches, where λ0 is the free space wavelength. The Z0 values from Fig. 3 are used to find the required widths of the different branches in order to fulfill the impedance matching conditions. A top view of the maximally flat low pass filter is depicted in Fig. 4(b). The low pass filter is designed to exhibit a 3 dB cut-off wavelength equal to λc,3 bB=1.55μm. Note that the total length of the designed structure is less than λc,3 bB=1.55μm/2.
3. FDTD Simulation Results
We have used an FDTD method with perfectly matched layers [16] to model light transmission through the designed filter structure. The method utilizes time-marching algorithm to solve Maxwell’s curl equations on a discretized spatial grid and can accurately model plasmonic modes and resonances [17]. An adaptive meshing scheme has been adopted with grid sizes of 5 nm, 2.5 nm and 5 nm in the x, y and z directions at the boundaries of the waveguide and larger grid sizes far from the waveguide.
Fig. 5. FDTD simulation results of the nanostrip based 6th order maximally flat low pass filter. (a) Frequency response, (b) power transmission density (Pz) at λ0 = 0.80 μm, (c) power transmission density (Pz) at λ0 = 2.00 μm.
Figure 5(a) shows the transmission spectrum of the nanostrip based maximally flat filter and compares it with the transmission spectrum of an ideal maximally flat low pass filter. In this figure, the 3 dB cut-off wavelength is λc,3 bB=1.48μm, which corresponds to -4.6% error with respect to the desired λc,3 bB=1.55μm. This relatively small error indicates that our approach to calculate the average Z0 values properly grasps the fringing effects around the metal strip corners. The almost flat transmission in the bandpass and stopband indicates the general behavior of maximally flat filters. Although, the transition of the nanostrip filter is not as sharp as the ideal 6th order maximally flat filter, it can be improved by increasing the order of the filter. Increasing the order of the filter has an inverse effect on the performance of the filter in the passband due to the transmission loss of the additional stages. However, as the lengths of the stages are much shorter than λc,3 bB (Fig. 4), the increase in the total transmission loss is negligible for a few additional stages.
Power density profiles (Pz) of the structure at λ0=0.80 μm and 2.00 μm are shown in Figs. 5(b) and (c), respectively. Wavelength λ0=0.80 μm lays in the stopband of the filter, at which no transmission is expected. Note that in Figs. 5(b) and (c) the incident radiation is in the +y direction. Figure 5(a) indicates 93% transmission at λ = 2.00 μm. In this figure, the lower transmission of the plasmonic filter for λ > λc,3 bB compared to the ideal filter is due to the ohmic losses of the metallic layers.
4. Conclusion
In this paper, we presented an optical range low pass filter based on the plasmonic nanostrip waveguides, which provides subwavelength confinement. An FDFD method was used to calculate the propagation properties and characteristic impedance of the nanostrip waveguides. We modeled the light propagation through the designed nanostrip based maximally flat low pass filter by the FDTD technique. It has been shown that the concept of the characteristic impedance is applicable to the nanostrip waveguides and can be used as a reliable design tool for photonic components in the integrated optical circuits.
Acknowledgments
This research was supported by the Semiconductor Research Corporation NRI program and the National Science Foundation.
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