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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Henry van Driel
  • Vol. 29, Iss. 11 — Nov. 1, 2012
  • pp: 3157–3169
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Analytical model for metal–insulator–metal mesh waveguide architectures

Charles Lin, Mohamed A. Swillam, and Amr S. Helmy  »View Author Affiliations


JOSA B, Vol. 29, Issue 11, pp. 3157-3169 (2012)
http://dx.doi.org/10.1364/JOSAB.29.003157


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Abstract

Metal–insulator–metal (MIM) waveguide mesh structures utilize X-junctions as power distribution elements to create interference and feedback effects, thereby providing rich device functionality. We present a generalized analytical model for MIM mesh structures by incorporating a modified characteristic impedance model for MIM junctions into the scattering matrix formalism. The modified impedance model accounts for metal absorption and provides accurate prediction of plasmonic field distribution at X-junctions in terms of both magnitude and phase. Closed-form expressions for 2×1 and 2×2 MIM mesh architectures as well as MIM stub structures are then obtained and are dependent only on waveguide geometry and junction configuration. The model does not require numerically extracted parameters, and results agree, within a few percent, with those obtained from finite-difference time-domain method for both two-dimensional and three-dimensional waveguide geometries. The capability of the model for efficient design and optimization of junction-based MIM devices is demonstrated through the development of various filter and resonant devices.

© 2012 Optical Society of America

1. INTRODUCTION

The tradeoff associated with utilizing the nanoscale confinement of MIM waveguides, however, is the significant metal absorption loss. With a sub-100 nm core, the waveguide propagation loss is a few decibels per micrometer, and the propagation length (distance in which the intensity of the transverse field drops to 1/e) is less than 10 μm [8

8. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006). [CrossRef]

]. Thus, a crucial criterion for designing high performance MIM devices is that the footprint can only be on the scale of a few micrometers.

Fig. 1. (a) Top view and (b) cross-sectional view of the total electric field intensity of the propagating MIM fundamental mode at λ=1550nm. The waveguide is 340 nm tall with a 50 nm core.

Numerous optical components based on MIM waveguides have been proposed, such as splitters [9

9. Y. Guo, L. Yan, W. Pan, B. Luo, K. Wen, Z. Guo, H. Li, and X. Luo, “A plasmonic splitter based on slot cavities,” Opt. Express 19, 13831–13838 (2011). [CrossRef]

], interferometers [10

10. Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006). [CrossRef]

], switches [11

11. C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009). [CrossRef]

], and demultiplexer [12

12. F. Hu, H. Yi, and Z. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36, 1500–1502 (2011). [CrossRef]

]. The design of these resonance-based devices is based on the coupling between nonintersecting MIM cavities. In order for the tunneling of the electromagnetic field to take place, the separation between cavities needs to be less than the metal skin depth, which requires stringent fabrication control. In addition, these designs are often several micrometers in size, because the device subcomponents are cascaded in a sequential fashion.

Despite the promise that these mesh structures hold for nanophotonic circuits, the design of MIM junction-based devices is at the moment hindered by the lack of suitable analytical models. The aforementioned junction-based devices are primarily modeled using the finite-difference time-domain (FDTD) technique, which is computationally intensive, has a significant memory requirement, and does not provide up-front physical insights into device characteristics. Moreover, relying on a numerical method to perform device sensitivity analysis and optimization processes is inefficient especially for larger and more complex mesh structures. Because MIM waveguides have nanoscale field confinement, the required simulation mesh size can only be a few nanometers. This means that the required computational resources will scale drastically with the dimensions and complexity of the network. To reduce the computational cost, the scattering matrix (S-matrix) method commonly used for microwave circuit design has been proposed to analyze the interactions between different parts of a MIM network structure [16

16. E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010). [CrossRef]

]. Although closed-form transmission expression for the overall structure can be derived from the S-matrix, modeling individual junctions is still based on a precompiled library of numerically extracted waveguide dispersion characteristics and junction power distribution ratios. In order to achieve realistic design and optimization cycle durations for functional devices using MIM mesh structures, there is still a need for an analytical model that can handle arbitrary combination of junctions without requiring FDTD-extracted parameters.

For the modeling of MIM junctions, a waveguide characteristic impedance model has been proposed with good agreement compared to its numerical counterpart [13

13. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]

,17

17. M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.

,19

19. H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012). [CrossRef]

]. Unlike dielectric waveguides or other SPP waveguide configurations, MIM waveguides have uniform transverse field confinement similar to that of a perfect electric conductor (PEC) parallel-plate waveguide. Therefore, characteristic impedance can be uniquely defined, and transmission line analysis is applicable for analyzing MIM junctions. Providing a basic description of the junction behavior, this impedance model can then be readily incorporated into the S-matrix formulation to analyze a network of cascaded MIM junctions. Studies up until now, however, have focused on utilizing the impedance model for MIM bends, T-junctions [13

13. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]

,19

19. H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012). [CrossRef]

], stub structures [20

20. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010). [CrossRef]

], and waveguide discontinuities [21

21. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

]. Investigation on the suitability of the impedance model for X-junctions with asymmetric arm widths, the most generalized junction configuration, is still absent. Furthermore, loss and phase information in MIM waveguide junctions are also not accounted for. They are critical when the junctions are cascaded into network structures with strong internal interference effect. Finally, as the widths of the junction arms and the operating wavelength are varied, the dielectric discontinuity at the center of the X-junction can cause the breakdown of the equal four-way splitting behavior, potentially affecting the applicability of the impedance model. The impedance model’s region of validity for designing subwavelength devices has not yet been defined.

In this paper, a generic analytical model for MIM mesh structures is presented. We incorporate the impedance model into the S-matrix formalism and for the first time generalize this methodology to 2D MIM mesh structures. We first show that the impedance model can be extended to provide both the amplitude and phase information for symmetric as well as asymmetric X-junctions. Then, we demonstrate that by adopting the S-matrix formulation to model the interaction between various junctions, it is possible to parameterize a plasmonic mesh structure and derive a closed-form transmission expression that is dependent only on the mesh topology. This methodology enables the design and optimization of 2D MIM mesh structures without FDTD-extracted parameters. The validity and versatility of our model are then examined by first comparing the analytical and numerical results for 2×1 and 2×2 networks made out of MIM waveguides with uniform core width and then extending this approach to generic 2D architectures such as stub-loaded mesh and resonance-enhanced network structures with varying core width. The capability of this design scheme is further elucidated through discussions on the model’s scalability for accommodating larger and 3D network configurations as well as the ability to provide device sensitivity analysis.

2. ADAPTATION OF IMPEDANCE MODEL TO MIM X-JUNCTIONS

To develop an analytical model that does not rely on FDTD-extract parameters, the junction impedance model first needs to be extended to include waveguide absorption loss and predict both the amplitude and phase of the resulting plasmonic wave in each junction arm. The fundamental unit of MIM mesh structures is the X-junction, which consists of two orthogonally intersecting MIM waveguides (Fig. 2). When the X-junction is excited from one arm, the optical power is distributed among the four junction arms as represented by reflection and transmission coefficients r and ti. The dielectric discontinuity at the center of the junction can excite higher order evanescent modes, which will then decay exponentially in the single-mode junction arms. Thus, given that the metal cladding is thicker than the metal skin depth, each junction arm can be treated as a discrete single-mode waveguide that does not couple to the others except at the center of the junction. For MIM waveguides with dielectric cores that are much smaller compared to the operating wavelength (d/λo0), it has been reported that quasistatic approximation can be applied and the waveguides can be treated as PEC parallel-plate waveguides [13

13. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]

]. Under quasistatic approximation, an MIM X-junction is equivalent to a set of discrete transmission lines, where each junction arm is represented by its own characteristic impedance (Z=V/I). Since the electromagnetic fields of MIM waveguides are confined primarily in the dielectric region, the approximations V=ωEtransverse and IαHtransverse can be applied to uniquely calculate the waveguide characteristic impedance. For MIM waveguides with a low index filling and core width of less than 200 nm, the impedance can be approximated to be [13

13. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]

,19

19. H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012). [CrossRef]

]
Zi(ω,di)βMIM(ω,di)din2ωεo,
(1)
where βMIM is the real part of the MIM propagation constant, di is the width of the junction arm, and n is the refractive index of the core dielectric medium. The waveguide impedance is primarily determined by the core width, since the propagation constant is also dependent on the width of the waveguide.

Fig. 2. Schematic and parameters of an MIM X-junction.

To calculate the power distribution at an X-junction, the input junction arm is represented as a transmission line with impedance ZO, whereas the remaining junction arms are represented as transmission lines connected in series with total load impedance of ZL. The reflection (r^) and transmission (t^i) coefficients for lossless X-junctions are defined to be
r^=|ZLZOZL+ZO|andti^=|2ZoZLZo+ZLPi|=|2ZoZLZo+ZLZiZL|,
(2)
where Zi is the impedance of output waveguide i, and the Pi factor distributes the transmitted light among the output waveguides according to their relative impedance. If the widths of the junction arms are not identical, then an individual set of reflection and transmission coefficients needs to be defined for the junction excitation from each direction.

Under quasistatic approximation, the excitation of an X-junction is analogous to placing a point source at the center of the junction, and thus the propagation distance of the plasmonic wave in each arm can be treated as identical [14

14. E. Feigenbaum and M. Orenstein, “Perfect 4-way splitting in nano plasmonic X-junctions,” Opt. Express 15, 17948–17953 (2007). [CrossRef]

]. As such, the reflection and transmission coefficients can simply be redefined as
r=r^exp((α4+jβ4)(2L4)),ti=t^iexp([(αi+jβi)(Li)+(α4+jβ4)(L4)]),
(3)
where αi is the the total loss factor, βi is the real part of the propagation constant for each junction arm, and Li is the arm length.

To verify the validity of the modified impedance model, it is applied to both symmetric and asymmetric X-junctions and compared with numerical studies from FDTD simulations using commercial Lumerical software. The FDTD simulations are performed with a mesh size of 2.5 nm near the waveguide core and 5 nm in the rest of the computational domain. The perfectly matched layer absorbing boundary condition is used to terminate the simulation region. The Ag material model used is based on a higher order polynomial fitting of experimental data from Palik [22

22. Lumerical, “Multicoefficient material modelling in FDTD,” http://www.lumerical.com/solutions/whitepapers/fdtd_multicoefficient_material_modeling.html.

]. The source and power monitors are placed 200 and 400 nm away from the junction center, respectively, to allow higher order modes to diminish. Moreover, the loss factor αi is assumed to be only the MIM waveguide propagation loss.

Fig. 3. Comparison between FDTD and S-matrix modeling results of a symmetric X-junction with 100 nm cores and 400 nm arms. The junction is excited from the left horizontal arm, and both the normalized transmission and phase of the plasmonic waves in each junction arm are plotted.

The junction phase calculated using our model also demonstrates good agreement with FDTD simulations, as shown from Fig. 3(e) to 3(h). The phase in each arm is continuous across the spectrum and the discontinuities observed in the plot are not actually physical. Instead, they are the result of wrapping the phase information between the π to π range. The agreement further validates the applicability of the modified impedance model and indicates that under quasistatic approximation, the dielectric discontinuity at the center of the junction does not induce phase change and the plasmonic waves in all arms can also be assumed to have propagated over equal distances. In other words, phase accumulation is only attributed to propagation along the junction arms. Although the symmetric junction exhibits equal power distribution, the plasmonic wave in the upper arm has a π phase shift compared to that of other arms. This phenomenon is observed independent of waveguide configurations or the locations of the source and monitors in FDTD simulations.

The modified impedance model is also verified for an asymmetric X-junction where d1=d3=100nm and d2=d4=50nm (Fig. 4). Similar to the case of the symmetric X-junction, the discrepancy is less than 1% in the wavelength regime above 0.8 μm. Thus, with a subwavelength waveguide core, the asymmetry in X-junction arms does not affect the performance of the modified impedance model. In other words, the modified impedance model is confirmed to be applicable for modelling both intensity and phase of generalized X-junctions and can be incorporated into the S-matrix formulation.

Fig. 4. Comparison between FDTD and S-matrix modeling results of an asymmetric X-junction with 50 nm horizontal cores, 100 nm vertical cores, and 400 nm arms. The junction is excited from the left horizontal arm, and both the transmission and phase of the plasmonic waves in each junction arm are plotted.

Fig. 5. Model limitation demonstrated by the percent deviation from equal power splitting (25%) in a symmetric X-junction at (a) λ=1100nm, (b) λ=1550nm, and (c) λ=2000nm.

With a sub-120 nm core width, the perfect equal splitting behavior of an X-junction also breaks down. As the width of the waveguide core decreases, the SPP waves become even more tightly confined to the metal–dielectric interfaces. Therefore, due to the continuity of the metal–dielectric interfaces, the MIM mode will preferentially divide among the sideways and backward junction arms. With a sub-120 nm waveguide core, quasistatic approximation still holds, but the assumption that the dielectric discontinuity at the center of the junction induces negligible loss and phase change is no longer valid. From Fig. 5, it can be observed that the normalized optical power in all of the arms is less than 25%, indicating the existence of additional loss mechanisms, such as metal absorption, scattering, and the excitation of single surface SPP modes and higher order MIM modes. Furthermore, since the amount of power in different junction arms are not identical, the assumption that the plasmonic wave in each arm has propagated over an equal distance is not valid over the entire spectrum. In order for the above assumptions to be valid and the model to provide analysis of MIM junctions with ±2% discrepancy in comparison to FDTD results, the core width needs to be less than 200 nm and the MIM mesh device should operate in the near-IR regime. Within the framework of designing compact optical sensors, circuitry, and interconnects near 1550 nm, these constraints are already satisfied. Thus, the modified characteristic impedance model does not impose additional constraints on junction-based optical device design.

On the other hand, the improvement provided by the exact formulation is also determined to be minimal in the sub-200 nm regime. This is because the exact and approximated impedance formulations match well when the SPP waves are strongly coupled [23

23. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009). [CrossRef]

]. To examine the effect of using the approximated formulation, the discrepancies between numerical and analytical results for symmetric and asymmetric X-junctions are compared (Figs. 3 and 4). The model’s performance should degrade in the case of the asymmetric junction, since the approximated impedance formulation should introduce a slight error when calculating the relative impedance between junction arms. Nonetheless, the discrepancy is around 1% for both symmetric and asymmetric junctions. This suggests that although using the approximated impedance model does introduce errors, the discrepancies between numerical and analytical results are dominated by the additional loss mechanisms of the MIM waveguides. Therefore, within the region of validity of our modified impedance model, the approximated impedance formulation already provides sufficient accuracy for analyzing MIM mesh structures.

Although the exact impedance formulation does not offer significant improvement in accuracy for modeling generalized X-junctions, it may be appropriate for other junction geometries. For mesh structures that are made out of bends, T-junctions, and/or waveguide discontinuities, the coupling between orthogonal arms is more efficient and the equal four-way splitting behavior is preserved over a larger spectrum. Thus, the region of validity of the modified impedance model may be extended for specific mesh configurations.

3. SCATTERING MATRIX MODEL OF MIM MESH STRUCTURES

In order to model the interaction between X-junctions in an MIM mesh structure, the junction impedance model needs to be incorporated into the S-matrix formulation. The S-matrix method breaks down a network of structures into subcomponents, and the matrix elements relate the electric fields entering and exiting the various subcomponents. Since the mesh structure can be treated as a network of interconnected junctions, the S-matrix method is a suitable mathematical framework for analyzing the distribution of electromagnetic energy within the mesh as well as the resulting transmission responses at different output ports. The matrix elements are complex, since both the magnitude and phase of the electromagnetic waves are modified by the network topology. By solving the S-matrix, a closed-form analytical expression can be obtained for designing and optimizing mesh functionalities as well as analyzing the physics and effects of modifying mesh parameters. In the following section, the capability of the S-matrix methodology is demonstrated by comparing the performance of the derived closed-form expressions against the results obtained through FDTD for 2×1 and 2×2 mesh devices, which are the fundamental building blocks for creating larger MIM mesh structures.

A. Closed-Form Expression for a 2×1 MIM Mesh

A 2×1 MIM mesh structure consists of two cascaded X-junctions, as shown in Fig. 6. Since a portion of the plasmonic wave is reflected every time it reaches a junction, each pair of X-junctions will create a localized resonance effect [16

16. E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010). [CrossRef]

]. This weak resonance is controlled by the relative power distribution at each junction and the phase accumulated as the plasmonic wave propagates between junctions. To model the 2×1 mesh, the electric field entering and exiting the structure can be described by a 2×2 S-matrix:
[Y1Y2]=[S11S12S21S22][X1X2],
(4)
where
S11=rJ1+(tJ1,2rJ2tJ1,2exp(ϕ))1rJ1rJ2exp(2ϕ),S12=S21=tJ1,2tJ2,2exp(ϕ)1rJ1rJ2exp(2ϕ),andS22=rJ2+tJ2,2rJ1tJ2,2exp(ϕ)1rJ1rJ2exp(2ϕ).
Fig. 6. Schematic and S-matrix setup 2×1 MIM mesh structure.
The coefficients rJ1 and tJ1,2 denote the reflection and transmission coefficients for the first X-junction, and rJ2 and tJ2,2 are for the second X-junction, all of which are calculated using the modified characteristic impedance model. The phase accumulated between two X-junctions is described by ϕ=(α+jβ)L, which is dependent on the propagation constant and length of the MIM junction interconnect. The electromagnetic wave exiting this MIM junction resonator can be treated as the summation of an infinite series described as
Eout=[exp(ϕ)+rJ1rJ2exp(3ϕ)+rJ12rJ22exp(5ϕ)+rJ14rJ24exp(7ϕ)+]Ein=exp(ϕ)1rJ1rJ2exp(2ϕ)Ein.
(5)

Therefore, all the S-matrix elements have a common multiplier exp(ϕ)/1rJ1rJ2exp(2ϕ) to account for the infinite reflections within the structure. Assuming single input and identical X-junctions, the transmission expression of a 2×1 mesh structure is derived and simplified to be
T=Y2X1=t22exp(ϕ)1r2exp(2ϕ).
(6)

A 2×1 MIM mesh with a 50 nm air core and 750 nm waveguide interconnect is simulated in FDTD and the transmission spectrum closely resembles that of the S-matrix model shown in Fig. 7. In addition, localized resonance inside the X-junction pair is observed and can potentially be utilized to create feedback and interference effects within a larger MIM mesh structure. Since the S-matrix method is only a mathematical framework that relates X-junctions represented by the impedance model, it does not provide correction or introduce additional error into the overall performance of the analytical model. Therefore, similar to the case of a single X-junction, the analytical prediction deviates from numerical results at lower wavelengths due to the breakdown of equal four-way splitting behavior.

Fig. 7. Comparison between the S-matrix model and FDTD simulation results of a 2×1 MIM resonator. The structure consists of MIM waveguides with 50 nm cores and a 750 nm long interconnect.

B. Closed-Form Expression for a 2×2 MIM Mesh

In order to solve the closed-form expression for this S-matrix, constitutive relationships are defined between X2, Y2, X3, and Y3:
X2=x1Y2+y1Y3,X3=x2Y3+y2Y2,
(8)
where
x1=rJ2exp(2ϕv)+tJ2,3rJ3tJ2,1exp(2ϕv2ϕh)1rJ2rJ3exp(2ϕh),x2=rJ3exp(2ϕv)+tJ3,1rJ2tJ3,3exp(2ϕv2ϕh)1rJ2rJ3exp(2ϕh),y1=tJ3,1tJ2,1exp(2ϕvϕh)1rJ2rJ3exp(2ϕh),andy2=tJ2,3tJ3,3exp(2ϕvϕh)1rJ2rJ3exp(2ϕh).

Based on the S-matrix and the constitutive relationships, the transmission expression is determined to be
T=Y4X1=S41+S21(S42x1+S43y2)1S22x1S23y2+[S42y1+S43x2+(S42x1+S43y2)(S22y1+S23x2)1S22x1S23y2][S31+S21(S32x1+S33y2)1S22x1S23y21S32y1S33x2(S32x1+S33y2)(S22y1+S23x2)1S22x1S23y2].
(9)

This closed-form expression is verified with a homogeneous plasmonic 2×2 mesh structure where the core width is 50 nm and Lh=Lv=750nm (Fig. 9). The analytical prediction and FDTD simulation matches well, with a discrepancy of less than 1% despite the breakdown of quasistatic approximation at lower wavelengths. Compared to that of the X-junction pair, the transmission spectrum of this network also displays multiple resonance peaks, but at different wavelengths. Therefore, the interference effect within the mesh network has significantly altered the dispersion characteristics of the structure, and this is useful for engineering the functionality of the overall device.

Fig. 9. Comparison between the S-matrix model and FDTD simulation results of a 2×2 MIM resonator network. The structure consists of MIM waveguides with 50 nm cores and a 750 nm long interconnect.

Because of the symmetry of the structure, by modifying the S-matrix elements, Eq. (9) can also be used to obtain the transmission spectrum for output ports connected to the X-junctions J2 and J4. The closed-form expressions for the remaining output ports can be derived by following the same procedure, but with modifications to the constitutive relations in addition to the S-matrix elements. The modelling of multiple-input and multiple-output responses can also be achieved by superimposing transmission spectra generated by the closed-form expressions for excitations from various input ports.

Since the S-matrix method used to derive the closed-form expressions is an exact mathematical formulation, any discrepancy between numerical and theoretical results is only attributed to the breakdown of the characteristic impedance model. The error of the impedance model is amplified when the junctions are cascaded to form larger network structures. This increase in mismatch is evident when comparing the discrepancies between analytical and simulation results for a single X-junction (Fig. 3), a 2×1 network (Fig. 7), and then a 2×2 structure (Fig. 9).

4. MIM MESH ATTRIBUTES AND S-MATRIX MODEL VERSATILITY

The topology of an MIM mesh structure can be engineered in many ways to match the output transmission response to a desired function. By terminating some of the output ports within a network of interconnected MIM junctions, various stub structures can be created. By introducing width differences between the network output ports and the internal junction interconnects, the localized resonance in the network can be fine-tuned for wavelength-selective applications. This resonance effect can be even further enhanced by designing width discontinuities along the interconnects. The generalized S-matrix impedance model can accommodate these different design freedoms with a minimal increase in the required computational cost.

A. Stub-Loaded Filter Using a 2×1 MIM Mesh

MIM stub structures have been proposed for various sensing and demultiplexing applications [20

20. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010). [CrossRef]

,24

24. X. Lin and X. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33, 2874–2876 (2008). [CrossRef]

]. By treating stubs as junctions with terminating arms, our S-matrix design model is also applicable to generic MIM stub structures.

To demonstrate our model’s capability for designing stub-loaded devices, a 2×1 MIM mesh consisted of two double-sided stubs [Fig. 10(b)] is analyzed for a band-reject filter application. The structure is similar to the cascaded X-junction pair displayed in Fig. 6, but with all the vertical ports terminated. Therefore, the analytical expression for the power distribution in an X-junction with terminating vertical ports [Fig. 10(a)] needs to be derived first, and then Eq. (6) can be applied to obtain the transmission spectrum of the overall stub structure.

Fig. 10. Schematic and S-matrix setup of (a) a MIM X-junction with closed vertical ports and (b) a MIM mesh with two double-side stubs.

The powers entering and exiting the arms of a single input and single output (X'1=1 and X'4=0) closed X-junction are described by the following 4×4 S-matrix and constitutive relationships:
[Y1Y2Y3Y4]=[S11=rS12=t3S13=t1S14=0S21=t1S22=rS23=t2S24=0S31=t3S32=t2S33=rS34=0S41=t2S42=t1S43=t3S44=0][X1X2X3X4],
(10)
X2=rtopY2,X3=rbottomY3,
(11)
where rtop=exp(2ltop(αtopstub+jβtopstub)) and rbottom=exp(2lbottom(αbottomstub+jβbottomstub)).

The elements of this S-matrix are the power distribution ratio of a regular X-junction with four open ports, and rtop and rbottom represent the phase accumulation as well as the power attenuation acquired as plasmonic wave propagates within the stubs. Both reflection coefficients have a negative sign, since the reflection at the end of the stub results in a π-phase shift.

By solving the S-matrix using the constitutive relationships, the transmission expression of the closed X-junction is derived to be
T=S41+S42S21rtop1S22rtop+rbottom(S42S23rtop1S22rtop+S43)(S31+S32S21rtop1S22rtop1rbottomS33+S32S23rtop1S22rtop).
(12)
Moreover, the same expression can be used to obtain the reflection coefficient of the closed X-junction by replacing S41, S42, and S43 with S11, S12, and S13.

Once the transmission and reflection coefficients for both closed X-junctions are obtained and substituted back into Eq. (6), the closed-form transmission expression can be solved and then plug into MATLAB optimization procedure to create band-reject filter response. In Fig.11, the comparisons between the transmission and phase spectra obtained from the S-matrix model and FDTD simulation are displayed. The optimized double stubbed MIM structure has parameters L1=L2=325nm, L3=L4=100nm, Larm=200nm, and a core width of 100 nm. Our model demonstrates good agreement with simulation results and the device has a 250 nm wide bandpass response centered at 900 nm and a 300 nm wide band-reject response centered at 1500 nm.

Fig. 11. (a) Intensity and (b) phase comparison between FDTD and S-matrix modeling results for a homogeneous double-stubbed MIM structure where L1=L2=325nm, L3=L4=100nm, and Larm=200nm. The structure is optimized for a band-reject response centered at λ=1550nm.

Although the S-matrix model is utilized for the optimization of a 2×1 double-sided MIM stub network, it can be extended to analyze MIM stub-loaded mesh in general. Single-sided stubs can be modeled by setting the S-matrix elements S13, S23, S31, S32, and S33 to zero. Moreover, Eq. (6) can be utilized to include additional stubs in the structure. Overall, the model lends itself for efficient design of not only MIM junctions but also MIM stub structures.

B. Resonance in a 2D MIM Mesh

From our closed-form model, some insight can be inferred about how one can utilize the localized resonance within MIM mesh structures for wavelength-selective applications. The resonant wavelength within a network is determined by the length and the width of the interconnects between the various junctions. To enhance the resonance effect within the network, the widths of the interconnects can be made wider than those of the output waveguide ports such that the amount of internal reflection is increased (Fig. 12). The higher reflectivity can be utilized to reinforce the resonance effect while keeping the dimensions of the MIM mesh structures in the nanoscale so that the amount of metal absorption loss can be minimized.

Fig. 12. Schematic of a 2D MIM mesh structure with widened internal network waveguides. The structure can be modeled analytically using the S-matrix setup illustrated in Fig. 8.

Fig. 13. Normalized transmission at different output ports of a 2D MIM mesh with widened internal waveguide interconnects. The network output arms have 100 nm cores, and the internal interconnects are 800 nm long with 150 nm cores.
Fig. 14. Normalized transmission of output port 7 of a 2D MIM mesh with (a) 150 nm wide network interconnects with varying lengths and (b) 800 nm long network interconnects with varying widths. The widths of the output arms are kept constant at 100 nm.

C. Modified 2D Mesh with Enhanced Characteristics

In addition to using the relative width of the junction arms to minimize the amount of light coupling out of the network structure, the resonance effect can be even further enhanced by creating width discontinuities along the interconnects. For sensing and nonlinear applications, the mesh network may need to be filled with polymers or quantum dots at designated locations. The width of the discontinuity relative to the size of nanoparticles can potentially be used as the control parameter for selective filling.

The S-matrix model can incorporate the interconnect width discontinuity and thus adds another degree of design freedom to MIM mesh devices. Similar to the homogeneous 2×2 MIM mesh, where the MIM core width is constant, the new structure can be treated as four cascaded X-junctions [Fig. 15(a)]. However, one of the junction arms would include a widened MIM section [Fig. 15(b)]. To determine the power distribution ratio of this modified junction, the widened waveguide section [Fig. 15(c)] can be treated as a Fabry–Perot cavity, and the transmission and reflection coefficients are determined based on the characteristic impedance model to be
R=R1+R3T1T2exp(2ϕB)1R2R3exp(2ϕB)andT=1R2,
(13)
where
R1=|ZBZAZB+ZA|,R2=|ZAZBZB+ZA|,R3=|ZCZBZB+ZC|,T1=1R12,T2=1R22.

Fig. 15. Schematic of a 2×2 MIM mesh with (a) width discontinuity in the interconnects, (b) a modified X-junction that includes width discontinuity, and (c) an interconnect discontinuity.

The factors ZA, ZB, and Zc are the impedances for MIM waveguide sections A, B, and C, respectively and ϕB=(αB+jβB)LB is the phase accumulated in section B.

With excitation from the left horizontal arm, the power distribution ratio for the modified X-junction as shown in Fig. 15(b) can then be redefined as
r=r+t22Rexp(2ϕA)1Rr,t1=t1+t2Rt3exp(2ϕA)1Rr,t2=t2T+t2RrTexp(3ϕA)1Rr,t3=t3+t2Rt1exp(2ϕA)1Rr,
(14)
where r, t1, t2, and t3 are the power distribution ratio of an X-junction without the cascaded discontinuity section and ϕA=(αA+jβA)LA is the phase accumulated in waveguide section A. Since the junction arms are not identical, four sets of transmission and reflection coefficients associated with the different direction of junction excitation will need to be obtained individually using the same procedure.

The effect of introducing additional cavities is demonstrated in Fig. 16 by the comparison between FDTD-generated transmission spectrum for a regular 2D mesh with constant interconnect width (Design 1) and that of a resonance-enhanced 2D mesh with interconnect width discontinuity (Design 2). The parameters of Design 1 are d1=d2=d3=d4=40nm, L1=L2=820nm, and L3=L4=0nm, whereas the parameters of Design 2 are d1=d2=40nm, d3=d4=100nm, L1=L2=960nm, and L3=L4=700nm. For both networks, output ports 3, 4, 7, and 8 are closed to conserve the optical power in the structure. Although the optimized design is slightly larger, the amount of transmitted power does not degrade and the full-width half-maximum of the localized resonance is reduced by half [Fig. 16(c)]. This enhancement is the result of additional resonant cavities created by the widened interconnect section that has lower propagation loss. It can also be observed that the resonance peaks at outputs 2 and 6 are shifted when the discontinuities are introduced. Therefore, the length of these discontinuities is an additional control parameter for fine-tuning the location of the resonance peaks. Finally, our S-matrix model demonstrates good agreement with FDTD results, with discrepancies of less than 2%. Thus, it is confirmed to be capable of accurately modeling generalized MIM mesh structures with stubs, closed ports, and width discontinuities.

Fig. 16. Comparison of transmission spectra between FDTD simulation and the S-matrix model for an enhanced 2×2 mesh with width discontinuity along the interconnect (Design 2) for output ports (a) 1, (b) 2, (c) 5, and (d) 6. The FDTD spectrum for a regular 2×2 mesh without discontinuity is also plotted as a reference (Design 1).

5. DISCUSSION

The combination of the modified impedance model and the S-matrix formulation addresses the need for a purely analytical method for the investigation of junction-based MIM structures. Although the impedance model has previously been incorporated into the S-matrix microwave design methodology for the design of one-dimensional MIM Bragg reflectors [25

25. A. Hosseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef]

] and mode converters [21

21. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

], it has not been extended for 2D network structures. In addition, our model accounts for phase accumulation and the resonance effect inherent in junction pairs such that interference, resonance, and feedback effects can all be accurately and analytically investigated. The feedback effect in an MIM network [18

18. M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012). [CrossRef]

] was previously modeled with a similar mathematical treatment, but the study did not take into account the multiple reflection effect between T-junctions, which is considerably less than the internal reflection between X-junctions. Our model demonstrates higher accuracy by including the contribution of wave reflections, which can be significant for generalized mesh structures.

The significance of formulating closed-form expressions is the ability to analyze MIM mesh structures without the need to perform FDTD simulations and/or create a precompiled parameter space. This leads to significantly shorter computational cycle and smaller memory requirement. The closed-form expressions capture the interrelations of various S-matrix elements and therefore can provide physical insights into the effects of manipulating network topologies. These physical effects are concealed if numerical simulations are performed or the exact solutions of Maxwell’s equations are solved. The proposed model is an all-purpose analytical framework that can be adapted to model MIM structures for specific applications. Therefore, the required computational resources for device optimization do not scale dramatically with the complexity or dimension of the MIM structure. Modifications to the mesh structure only require changes to the S-matrix elements, which are encapsulated in the overall transmission expression that remains unchanged. Thus, an efficient design process is possible by treating the MIM mesh architecture as a black box with a set of input and output parameters. One can envision potentially incorporating such encapsulation into microwave or electronic circuit design software for examining optoelectronic circuits.

In addition to reducing the design cycle duration, the closed-form model can also be utilized for device sensitivity analysis. This can be done by taking the derivative of transmission expressions such as Eqs. (6) and (9) with respect to each parameter. The critical parameters and dominant physical interactions within a multi-variable network device can then be identified analytically.

Another advantage of employing our closed-form model is that individual MIM networks can be replaced with respective analytical expressions and then easily integrated to analyze larger and more complex network configurations. The 2×1 and 2×2 networks can be treated as the building blocks for mesh architectures in general, since larger structures can be broken down into a combination of either component. Thus, Eq. (6), which describes the interaction of two cascaded X-junctions, can be applied to larger networks by replacing the transmission response of individual X-junctions with that of 2×1 and/or 2×2 networks. Equations (1014) then allow the manipulation of topological parameters to create more complex mesh configurations. However, despite the scalability of our model, the significant metal absorption loss prohibits the device footprint from expanding into the microscale. The rich resonance and feedback effects between X-junction pairs effectively increase the overall distance that the plasmonic waves propagate. This in turn limits the amount of transmitted output power as shown in Figs. 9, 13, and 16. Thus, extending the mesh structures beyond their current 2×2 configurations may not be practical.

Although our investigation focuses on 2D network devices that are truncated in the vertical dimension, the closed-form model remains effective for realistic network structures that extend into the third dimension. Similar to its 2D counterpart, a subwavelength 3D MIM waveguide supports quasi-TEM modes with electric field predominantly in the direction parallel to the waveguide core. However, the mode profile now includes edge effects that disrupt the uniformity and the confinement of the electromagnetic field. Therefore, the exact formulation of waveguide characteristic impedance is required. In this case, the effective voltage and current need to be calculated by a direct integral of field components along the waveguide’s cross section [26

26. W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010). [CrossRef]

]. Nonetheless, the S-matrix formulation would remain the same, and therefore the transmission expressions derived for the various mesh structures in this study remain valid for 3D MIM waveguide configuration.

Fig. 17. Model limitation demonstrated by the percent deviation from equal power splitting (25%) in a symmetric 50 nm X-junction at (a) λ=1100nm, (b) λ=1550nm, and (c) λ=2000nm. The data are extracted from 3D FDTD simulations.

6. CONCLUSION

A generic analytical model for 2D MIM mesh structures is presented by incorporating the characteristic impedance model for MIM junctions into the S-matrix formalism. We modified the impedance model to include waveguide loss and phase accumulation and demonstrated for the first time that it can provide a full description of the behavior of generalized X-junctions. It was determined that when the width of the waveguide core is close to 120 nm, the dielectric discontinuity at the center of the junction does not induce attenuation or phase changes. This enables an S-matrix representation of MIM networks that takes into account the interference, resonance, and feedback effects within the structure. Closed-form expressions were derived for 2×1 and 2×2 mesh structures, thereby eliminating the need for computationally intensive simulations or precompiled parameter space while showing good agreement with numerical results. Since the 2×1 and 2×2 networks are the building blocks for MIM network structures, the existing model can be extended to accommodate more complex mesh structures and offers the ability to handle arbitrary combinations of junctions. Therefore, our model serves as an all-purpose analytical framework that is scalable and can be adapted to model specific MIM structures and perform device sensitivity analysis. We also suggested potential device design techniques such as closing output ports, designing width differences between junction arms, and engineering width discontinuity along the junction interconnect. Although we focused on 2D MIM mesh structures in this work, the same design methodology can be applied to larger, nonorthogonal, or even 3D configurations of MIM mesh structures.

REFERENCES

1.

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003). [CrossRef]

2.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006). [CrossRef]

3.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef]

4.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

5.

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31, 2133–2135 (2006). [CrossRef]

6.

P. Berini, “Bulk and surface sensitivity of surface plasmon waveguides,” New J. Phys. 10, 105010 (2008). [CrossRef]

7.

B. Lau, M. A. Swillam, and A. S. Helmy, “Hybrid orthogonal junctions: wideband plasmonic slot-silicon waveguide couplers,” Opt. Express 18, 27048–27059 (2010). [CrossRef]

8.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006). [CrossRef]

9.

Y. Guo, L. Yan, W. Pan, B. Luo, K. Wen, Z. Guo, H. Li, and X. Luo, “A plasmonic splitter based on slot cavities,” Opt. Express 19, 13831–13838 (2011). [CrossRef]

10.

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006). [CrossRef]

11.

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009). [CrossRef]

12.

F. Hu, H. Yi, and Z. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36, 1500–1502 (2011). [CrossRef]

13.

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]

14.

E. Feigenbaum and M. Orenstein, “Perfect 4-way splitting in nano plasmonic X-junctions,” Opt. Express 15, 17948–17953 (2007). [CrossRef]

15.

F. Hu and Z. Zhou, “Wavelength filtering and demultiplexing structure based on aperture-coupled plasmonic slot cavities,” J. Opt. Soc. Am. B 28, 2518–2523 (2005). [CrossRef]

16.

E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010). [CrossRef]

17.

M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.

18.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012). [CrossRef]

19.

H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012). [CrossRef]

20.

A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010). [CrossRef]

21.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

22.

Lumerical, “Multicoefficient material modelling in FDTD,” http://www.lumerical.com/solutions/whitepapers/fdtd_multicoefficient_material_modeling.html.

23.

D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009). [CrossRef]

24.

X. Lin and X. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33, 2874–2876 (2008). [CrossRef]

25.

A. Hosseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef]

26.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010). [CrossRef]

27.

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(240.6680) Optics at surfaces : Surface plasmons
(240.6690) Optics at surfaces : Surface waves
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(250.5403) Optoelectronics : Plasmonics
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 29, 2012
Revised Manuscript: September 4, 2012
Manuscript Accepted: September 5, 2012
Published: October 29, 2012

Citation
Charles Lin, Mohamed A. Swillam, and Amr S. Helmy, "Analytical model for metal–insulator–metal mesh waveguide architectures," J. Opt. Soc. Am. B 29, 3157-3169 (2012)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-29-11-3157


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References

  1. J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003). [CrossRef]
  2. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006). [CrossRef]
  3. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006). [CrossRef]
  4. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]
  5. L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31, 2133–2135 (2006). [CrossRef]
  6. P. Berini, “Bulk and surface sensitivity of surface plasmon waveguides,” New J. Phys. 10, 105010 (2008). [CrossRef]
  7. B. Lau, M. A. Swillam, and A. S. Helmy, “Hybrid orthogonal junctions: wideband plasmonic slot-silicon waveguide couplers,” Opt. Express 18, 27048–27059 (2010). [CrossRef]
  8. R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006). [CrossRef]
  9. Y. Guo, L. Yan, W. Pan, B. Luo, K. Wen, Z. Guo, H. Li, and X. Luo, “A plasmonic splitter based on slot cavities,” Opt. Express 19, 13831–13838 (2011). [CrossRef]
  10. Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006). [CrossRef]
  11. C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009). [CrossRef]
  12. F. Hu, H. Yi, and Z. Zhou, “Wavelength demultiplexing structure based on arrayed plasmonic slot cavities,” Opt. Lett. 36, 1500–1502 (2011). [CrossRef]
  13. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005). [CrossRef]
  14. E. Feigenbaum and M. Orenstein, “Perfect 4-way splitting in nano plasmonic X-junctions,” Opt. Express 15, 17948–17953 (2007). [CrossRef]
  15. F. Hu and Z. Zhou, “Wavelength filtering and demultiplexing structure based on aperture-coupled plasmonic slot cavities,” J. Opt. Soc. Am. B 28, 2518–2523 (2005). [CrossRef]
  16. E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010). [CrossRef]
  17. M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.
  18. M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012). [CrossRef]
  19. H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012). [CrossRef]
  20. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18, 6191–6204 (2010). [CrossRef]
  21. S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]
  22. Lumerical, “Multicoefficient material modelling in FDTD,” http://www.lumerical.com/solutions/whitepapers/fdtd_multicoefficient_material_modeling.html .
  23. D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009). [CrossRef]
  24. X. Lin and X. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33, 2874–2876 (2008). [CrossRef]
  25. A. Hosseini, H. Nejati, and Y. Massoud, “Modeling and design methodology for metal-insulator-metal plasmonic Bragg reflectors,” Opt. Express 16, 1475–1480 (2008). [CrossRef]
  26. W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010). [CrossRef]
  27. J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009). [CrossRef]

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