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We present a new metal-insulator-metal (MIM)-based plasmonic Bragg reflector (PBR) design that solves the technical problems of conventional step profile MIM PBRs through the use of sawtooth profiles. Our numerical study revealed that the sawtooth PBRs exhibit lower insertion loss, narrower bandgap, and reduced rippling in the transmission spectrum when compared with the step PBRs. The defect mode of the sawtooth PBR also exhibits a higher transmission, narrower linewidth, and higher Q-factor.
©2010 Optical Society of America
1. Introduction
The surface plasmon-polariton (SPP) is a surface wave that propagates along the interface between metallic and dielectric materials as a hybridization of collective charge density oscillations and evanescent waves. The SPP’s evanescent nature enables subwavelength scale confinement of its power, which leads to the realization of ultra-dense photonic integrated circuits. In such systems, the nanoscale plasmonic waveguide will play the role of the fundamental building block while more advanced functionalities, such as filtering and beam splitting, will be derived from it [1–5].
The metal-insulator-metal (MIM) structure is particularly attractive for plasmonic waveguiding owing to its extreme level of confinement [1,4,6]. For wavelength-sensitive operations, the MIM structure can also be configured into a plasmonic Bragg reflector (PBR) by concatenating two MIM sections with different effective indices (neff = β/k0, where β is the propagation constant of the plasmonic mode and k0 the free space wavenumber) and repeating it [4,12]. A unit cell of a typical MIM-based PBR is shown in Fig. 1(a) . The effective index of a MIM section can be modified by changing the insulator layer’s width [4,7–9] or refractive index [4,7,10,11], or both. This leads to the formation of a bandgap in the transmission spectrum. Since the insulator width modulation results in higher reflection within the PBR bandgap than the refractive index-modulated schemes [4], it is preferred for PBR applications.
Fig. 1 The unit cells of (a) a step PBR and (b) a sawtooth PBR. (c) and (d) show the corresponding PBRs realized by concatenating the unit cells.
To be an efficient PBR, the concatenated MIM structure must exhibit not only high reflection within the bandgap but also high transmission outside the bandgap, which leads to low insertion loss. In addition, PBRs are also expected to exhibit well-defined band edges and, beyond the edges, flat transmission spectra with minimal ripples [4]. A narrow bandwidth is also advantageous for plasmonic applications requiring spectral sensitivity.
Many MIM-based PBRs employing the insulator width modulation scheme, however, suffer from high insertion loss and severe rippling in their spectral responses. The main cause of these performance degradations is the abrupt change in the insulator width at the interface between the two MIM sections. The width change is most abrupt in PBRs with step width profiles shown in Fig. 1(a). The consequential mode mismatch induces high loss to propagating SPPs regardless of their wavelengths and decreases the transmission level outside the bandgap (< 60% in [4,9,11]), resulting in an increase in the insertion loss. Moreover, the abrupt width change also excites higher order modes whose interference causes ripples in the transmission spectrum. Such ripples can severely obscure the band edges, widen the bandgap, and attenuate wavelength components intended to be transmitted. These degradations have been difficult to avoid in insulator width-modulated PBRs.
In this work, we propose to employ gradually changing width profiles in the MIM-based PBR design to solve the problems associated with the PBRs with step profiles, or step PBRs. Among many possible gradually changing width profiles, we chose the sawtooth profile largely for its potential ease of fabrication. A unit cell of a sawtooth PBR is shown in Fig. 1(b). We speculate that the smoother transition in insulator width could mitigate the problems caused by the abrupt width change.
2. Plasmonic Bragg reflector with sawtooth width profile
As shown in Figs. 1(c) and (d), the step and sawtooth PBRs to be investigated in this work are realized by concatenating N1 and N2 unit cells, already shown in Figs. 1(a) and (b), respectively. Major structural parameters are specified in Figs. 1(a) and (b). The sawtooth PBR unit cell differs from the step PBR unit cell most prominently by its symmetry. The unit cell of the step PBR consists of two MIM sections with the insulator widths and section lengths set at a, b, L1, and L2, respectively. The section lengths were determined by the Bragg condition: λB = 4⋅L1⋅na = 4⋅L2⋅nb where na, nb, and λB are the effective indices and the Bragg wavelength, respectively. Because the effective indices of the two sections are different, the unit cell must be asymmetric with different values for L1 and L2. In the case of sawtooth PBRs, a and b represent the widest and narrowest width, respectively. Since the second section is a mirror image of the first one, as shown in Fig. 1(b), L1 and L2 are equal.
3. Simulation settings, approaches, and method
The PBR device parameters, defined in Fig. 1, were chosen mainly to facilitate the comparison between the step and sawtooth PBRs. First, a step PBR was designed based on the well-established procedure [4,12]. The Bragg wavelength λB was set to the telecom wavelength 1.55 μm. SiO2 (n = 1.46) was used as the insulator material and Ag as the metal for its low loss and high plasmonic activity. The dielectric constant of Ag was taken from the experimental data in Ref [13]. The loss of the dielectric material was not considered in this study. We fixed a = 100 nm and used two different values of b to study the impact of having a strong (b = 30 nm) or a weak (b = 70 nm) width modulation. Transfer matrix method [14] was used to calculate the effective indices na and nb for each MIM section of the step PBRs. Then we chose the section lengths L1 and L2 based on the Bragg condition. For a = 100 nm and b = 30 nm, na and nb were 1.76 and 2.29, respectively. The corresponding values for L1 and L2 were 220 nm and 170 nm, respectively. For b = 70 nm, nb and L2 became 1.86 and 208 nm, respectively. These step PBR parameters (na, nb, a, and b) were utilized in the sawtooth PBR design directly. The total length of the sawtooth PBR unit cell was set to L = L1 + L2 to match the Bragg wavelength of the step PBR. Finally, we adjusted the number of unit cells N1 and N2 to make the two PBRs exhibit the same transmission levels at λB.
Once the transmission levels of the two PBRs at the Bragg wavelength got equalized, they were compared with each other in terms of the bandgap width, the transmission levels outside the bandgap, and the levels of rippling in the transmission spectrum. We also studied the defect modes of the two PBRs.
A commercial finite-element method (FEM) solver Comsol MultiphysicsTM was used to simulate the propagation in two-dimensional computational domains. We assumed that a plane wave with a known power level was incident from the input side of the PBR structure as shown in Figs. 1(c) and (d). TM waves with magnetic fields oriented in the z-direction were used. For both the input and output boundaries, we used the non-reflecting port boundaries. In the transversal directions, perfectly matched layers (PMLs) were used as the boundaries. Reflected and transmitted power were calculated by integrating the power flow over the input and output ports of the PBRs. Adaptive meshing was employed for grid generation, allowing the mesh size to be finer for smaller features. The mesh sizes are less than 15 - 20 nm depending on the minimum feature size in the computational domain. We confirmed that these mesh sizes were sufficiently fine for accurate simulations by repeating some of the simulations with a < 5 nm mesh size and obtaining similar results.
4. Simulation results
4.1 Transmission levels within the bandgap
As the first step of our investigation, we calculated the transmission levels, defined as the ratio between the PBR output power to the input power, for different values of N1 and N2. For the step PBRs, the wavelengths at which the transmission levels reached their minima coincided almost exactly with the Bragg wavelength λB = 1.55 μm. When a = 100 nm and b = 30 (70) nm, the transmission minimum missed λB by only 50 (30) nm. The corresponding differences in the transmission levels were < 0.35 dB. The Bragg condition also worked well with the symmetric, sawtooth profile. In the sawtooth PBR with a = 100 nm and b = 30 nm, the transmission minimum missed λB = 1.55 μm by only 50 nm. The corresponding difference in transmission was < 0.5 dB. For b = 70 nm, the two points coincided. Based on these observations, we kept using λB = 1.55 μm as the wavelength at which the transmission within the bandgap is measured for both PBR types. The results are plotted in Fig. 2 . The imaginary parts of infinite MIM sections with 30, 70, and 100 nm-thick SiO2 layers are 0.00924, 0.00480, and 0.00358, respectively.
Fig. 2 The transmission levels at 1.55 μm for the step and sawtooth PBRs (a = 100 nm) with different number of unit cells.
We used the transmission results in Fig. 2 to find the N1 and N2 values which will bring the two PBRs’ transmission levels as close to each other as possible so that the comparisons on the transmission characteristics outside the bandgap would be fair and meaningful. We found the pertinent values of N1 and N2 by drawing a horizontal line from a preset transmission level and reading the nearest integers that correspond to the intersection points on the x-axis. For example, when b = 70 nm, the transmission levels come within 1 dB difference when N1 = 12 and N2 = 18.
4.2 Transmission characteristics outside the bandgap
For the first set of simulations with equalized transmission levels at λB, we chose PBRs with a = 100 nm and b = 30 nm. The corresponding N1 and N2 were 6 and 8, respectively. As shown in Fig. 2, the difference in their transmission levels at λB falls within 4 dB (−40.7 dB for step and −37.0 dB for sawtooth PBRs). Figure 3(a) shows the two PBRs’ transmission spectra. The 3 dB bandwidth of the sawtooth PBR bandgap was 694 nm which was 43.8% narrower than the 1234 nm bandwidth of the step PBR. In the case of the sawtooth PBR, the modulation depth of the ripples beyond the band edges was < 40% of that of the step PBR, except for the one closes to the band edge. Figure 3(a) shows that the reduction in rippling in the transmission spectrum led to highly enhanced transmission levels outside the bandgap.
Fig. 3 Transmission spectra for the step and sawtooth PBRs with (a) a = 100 nm, b = 30 nm and (b) a = 100 nm, b = 70 nm. The mode patterns for points (A) (at 1.6 μm) and (B), (C) (at 2 μm) marked in (a) are shown in Fig. 6.
We also investigated the transmission characteristics of PBRs with smoother transition profiles by increasing the value of b to 70 nm. Again, N2 and N1 were chosen to make the transmission levels at λB as close to each other as possible. This time, setting N1 = 12 and N2 = 18 resulted in transmission levels of −25.2 dB and −25.5 dB for the step and sawtooth PBR, respectively. Figure 3(b) shows the calculated transmission spectra. The width of the sawtooth PBR bandgap was 282 nm which was 39.7% narrower than the 468 nm observed from the step PBR. The level of rippling in the transmission spectrum and the level of transmission outside the bandgaps were also more favorable in the case of the sawtooth PBR.
To further confirm the observed trend, we calculated the transmission spectra of the two PBRs for intermediate values of b and plotted the results in Figs. 4(a) and (b) . The corresponding transmission levels and 3 dB bandwidths were also retrieved and plotted in Figs. 5(a) and (b) , respectively. N1, N2, and a were fixed at 6, 8, and 100 nm, respectively. Figure 5(a) shows that this set of parameters ensures fair comparisons by keeping the two PBRs’ transmission levels inside the bandgap within 4 dB from each other for all values of b. The transmission spectra of Fig. 4(a) and (b) clearly show that the sawtooth PBR’s advantages observed at two extreme values of b, the higher transmission levels and lower spectral rippling outside the bandgap, are consistently maintained for all intermediate values as well.
Fig. 4 Transmission spectra for (a) the step PBR and (b) the sawtooth PBR with different values of b, while a is fixed at 100 nm.
Fig. 5 The impact of b on (a) the attenuation at λB and (b) the 3 dB bandwidth when the values of a, N1 and N2 were fixed.
Figure 5(b) shows that the bandwidths of the sawtooth PBRs were 40 - 44% narrower than those of the step PBRs across the whole range of b. The transmission levels outside the bandgap were also higher in the case of the sawtooth PBR due largely to the reduction in the level of rippling. The latter two features contribute to the decrease in the insertion loss of the PBR. Outside the bandgap, the propagation lengths of the PBRs are manifested in the levels of transmission shown in Figs. 3 and 4 which indicate that even after 8 (b = 30 nm) to 18 (b = 70 nm) periods, the sawtooth PBRs retained >75% of the original input power.
4.3 Mode propagation patterns
From the numerical simulations of the PBRs, we obtained the snapshots of the propagating mode’s field patterns and plotted them in Fig. 6 . The operation wavelengths are marked in Fig. 3(a) as A, B and C. Figures 6(a) and (b) contrast the sawtooth PBR’s mode field patterns within (A at λo = 1.6 μm) and outside (B at λo = 2.0 μm) the bandgap. At λo = 2.0 μm, the step PBR is still in the bandgap and the propagation inhibited, as shown in Fig. 6(c). The mode energy was well confined within the < 100 nm insulator layers.
Fig. 6 Field profiles |Hz|2 at different wavelengths for the sawtooth PBR with N2 = 8, (a) λo = 1.6 μm (b) λo = 2.0 μm, and for the step PBR with N1 = 6, (c) λo = 2.0 μm.
5. Defect modes of sawtooth plasmonic Bragg reflectors
The defect mode represents the high-level transmission occurring within a PBR’s bandgap due to a defect in the PBR’s periodicity. Optical power will be concentrated around the defect due to the formation of resonance modes. Because its resonance frequency can be tuned by modifying its structural parameters [15], the defect mode has been widely utilized for sensing, lasing, and filtering [15].
In this Section, we investigate the characteristics of the sawtooth PBR’s defect mode. For comparison, we simulated both step (N1 = 6) and sawtooth (N2 = 8) PBRs. a and b were set to 100 nm and 30 nm, respectively. In the absence of the defect, the two PBRs would result in a similar transmission level.
Firstly, we designed and simulated the step PBRs with defects introduced by inserting short rectangular sections between their 3rd and 4th unit cells as shown in Fig. 7(a) . The defect length Ld was set commonly to 195 nm. Two values of t were used: (1) a narrower section with t = 30 nm and (2) a wider one with t = 100 nm. The calculated transmission spectra, shown in Fig. 8 , indicate that the defect mode with t = 100 nm exhibited higher transmission and Q-factor than those for t = 30 nm. So we set t to 100 nm in all subsequent simulations.
Fig. 7 The defect mode pattern |Hz|2 for (a) the step PBR at 1.552 μm and (b) the sawtooth PBR at 1.45 μm with the same defect: Ld = 195 nm and t = 100 nm. (a) and (b) share the same color bar for better comparison.
Fig. 8 Transmission spectra of PBRs with defect modes. The black dotted line shows only the defect mode, not the whole spectrum for better visualization.
Then, we introduced a defect to the sawtooth PBR by inserting a rectangular section with t = 100 nm between the 4th and 5th unit cells, as shown in Fig. 7(b), and simulated it. The defect length Ld was also set to 195 nm. The transmission spectrum was superimposed in Fig. 8. For both PBRs, the transmission levels of the defect modes were higher than 10%, which can be deemed high considering the severe attenuation (> 35 dB) within the bandgap. The field patterns of the defect modes at their respective resonance frequencies, i.e., 1.45 μm for the sawtooth PBR and 1.552 μm for the step PBR, are also shown in Fig. 7. While the energy was concentrated at the defect in both schemes, the sawtooth DBR’s defect mode exhibits a factor of 2.5 higher peak power than that of the step PBR defect mode for the same input level.
Then, we varied Ld of the sawtooth PBR to bring the resonance wavelengths of the two defect modes to a common wavelength point, which was achieved when Ld was set to 260 nm, as shown in Fig. 8. At this point, the defect mode of the step PBR exhibited 12.3% transmission and 16.0 nm linewidth, resulting in a Q-factor of 96.8. For the sawtooth PBR defect mode, the results were 19.2%, 14.5 nm, and 107.10, respectively, which corresponds to an improvement of 56% in the transmission, 12% in the linewidth, and 11% in the Q-factor. As the direct consequence of the sawtooth PBR’s lower propagation loss, the level of peak power localized within the sawtooth DBR’s defect was also 2.3 times higher than that of the step DBR. These results indicate that employing the slowly changing, sawtooth width profile in the MIM design could lead to improvement in defect mode characteristics as well.
6. Discussion
The bandgap narrowing and ripple suppression shown above indicate that employing the sawtooth profile satisfied the requirements for PBR stated in Section 1. In this Section, we plan to discuss the underlying mechanism enabling such improvements in PBR performance. We interpret the bandgap narrowing and ripple suppression as a consequence of phase scrambling due to the non-uniform PBR unit cell geometry. To facilitate the analysis of the Bragg reflection by the gradually changing sawtooth geometry, we will approximate the PBR profile with multiple step-transitions.
The inset of Fig. 9(a) shows a four-step approximated version of the sawtooth PBR with b = 30 nm. To better retain the sawtooth profile and the symmetry, one half of the 390 nm unit cell was divided into two 65 nm sections and two 32.5 nm sections. The step height between these sections is 11.6 nm. The simulated transmission spectrum for N = 8 is shown in Fig. 9(a). Transmission spectra of the original sawtooth (N = 8) and step (N = 6) PBRs, previously given in Figs. 3 and 4, are superimposed. Comparison of the transmission spectra shows that the four-step PBR, despite the crude approximation, already exhibits bandgap narrowing and ripple suppression, thus validating the multiple step-transition approximation.
Fig. 9 (a) Transmission spectrum of a 4-step approximated sawtooth PBR. The transmission spectra of regular step and sawtooth PBRs are superimposed for comparison. The high level of bandgap narrowing and ripple suppression observed from the 4-step structure indicates the validity of multiple step-transition approximation. (b) The schematic diagram of a sawtooth PBR unit cell approximated by a 3-step geometry involving a regular step PBR perturbed by a small third step. Δϕi represent the phase change incurred by the reflection at each junction Ji.
Based on the previous result, we modeled the sawtooth PBR as a step PBR with a small third step of length Lp added at the position of the vertex of the sawtooth profile, as shown in Fig. 9(b). If the impact of the third step is small, then the Bragg reflection will continue to be achieved through the interference between the two main reflections at J1 and J4. The phase shift in the reflected wave due to the abrupt width change at each junction (Δϕ) depends mainly on the difference between the characteristic impedance ZMIM ≡ β⋅w/εoεi of the two subsections constituting the junction where w and εi denote the width and dielectric constant of the subsection’s insulator layer, respectively [2]. When w << λo/neff, a wave reflected at the narrower-to-wider junction experiences Δϕ = π phase shift while no phase shift is incurred at the wider-to-narrower junction [2]. In addition, along each subsection, the propagating wave will accumulate phase equivalent to β⋅L = (2π⋅neff /λo)⋅L where L and neff are the length and effective index of the corresponding subsection. Under these conditions, the transmission spectrum will also retain the features governed by the Bragg condition λB = 4⋅L1⋅na = 4⋅L2⋅nb.
The impact of the reflections at J2 and J3, however, must be taken into consideration as a perturbation to the original transmission spectrum. In the presence of perturbative reflections that are not in phase with the main reflection components, the Bragg condition, which requires constructive superposition between reflected waves, becomes harder to satisfy, leading to narrowed bandgap and suppressed sidelobes. It is clear from Fig. 9(b) that the perturbation to the Bragg reflection due to the third step becomes minimized when the reflections from J2 and J3 cancel out. This can be realized when 2βp Lp + Δϕ3 = -Δϕ2. If Δϕ2 ~π and Δϕ3 ~ 0 as stated above, a total cancellation of the perturbative reflections occurs when 2βp Lp = π. Since Lp << λo/neff, the requirement becomes increasingly difficult to satisfy for waves with longer wavelengths. As the wavelength decreases, λo/neff becomes comparable to Lp and meeting the phase requirement for total cancellation becomes easier. This argument can be corroborated by comparing the step and sawtooth PBR transmission spectra in Figs. 3(a) and (b). For both values of b, the bandgap narrowing and ripple suppression are more noticeable for λo > λB to which the total cancellation of the perturbations at J2 and J3 becomes more difficult. On the other hand, in short wavelength regime (λo < λB) in which the perturbation can be easily negated, the degree of bandgap narrowing and ripple suppression becomes lower.
In the framework of multiple step-transition approximation, the vertex of the sawtooth PBR profile functions as a wavelength-dependent phase scrambler that perturbs the Bragg reflection process, resulting in narrower bandgap and suppressed ripples. As the step-approximated profiles approach a gradual transition such as the sawtooth profile, the phase relation will become more complicated, further reducing the bandgap and ripples.
A comparison with the S-shaped PBR in Ref [8]. is in order. The S-shaped PBR was implemented by rounding the sharp corners of the wider half of a step PBR, which rendered the profile flat-topped. In contrast, the sawtooth PBR’s gradual transition spans the whole, not half, of the unit cell while the vertex was left to be sharp to generate the phase scrambling effect. Such an arrangement lowered the propagation loss, which is evidenced by >75% out-of-band transmission while the transmission reached −35~25 dB within the bandgap.
Wide bandgap PBRs are useful for ensuring highly efficient reflection to a broad range of wavelength components [7,9]. On the other hand, there also are efforts to narrow the bandgap for filtering applications [5,16]. Currently, the bandgap widths achievable in the MIM format needs further narrowing for filtering densely wavelength-multiplexed signals. However, the MIM-based PBRs can be useful for applications in coarsely wavelength-multiplexed systems, such as sorting mixed signals into S-, C-, and L-bands spanning over hundreds of nanometers.
7. Conclusion
In conclusion, we proposed to employ sawtooth profiles in the design of MIM-based PBRs to solve the technical problems associated with the conventional, step profile PBRs. Using FEM, we numerically investigated the performance of the sawtooth PBR. Central to the investigation was achieving fair performance comparison between the two types of PBRs. To that end, we first assured that the two PBRs exhibit close to equal transmission levels at the Bragg wavelength and then compared their performance and spectral characteristics outside the bandgap.
The results of numerical simulations showed that the sawtooth PBRs exhibit higher transmission levels to wavelength components outside the bandgap than those for step PBRs even when the sawtooth PBRs were longer with more constituent unit cells. Over the whole span of the width modulation depth, the sawtooth PBRs exhibited 40 - 44% narrower 3 dB bandwidth and significantly reduced rippling in the transmission spectrum outside the bandgap when compared with those for step PBRs. Such features lead directly to higher output power levels for the transmitted wavelength components and, hence, result in lower insertion loss. Based on the simulation results, we attributed these advantages in the sawtooth PBRs to the relaxed rate of transition in the insulator width. Comparison of the defect modes of the sawtooth and step PBRs also showed that the sawtooth defect mode exhibit 56% higher transmission, 12% narrower linewidth, and 11% higher Q-factor.
This work was supported by Iowa Office of Energy Independence through Iowa Power Fund Project 08-02-1073.
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