We present a design for a low-footprint optical interconnect that efficiently couples between two photonic planes with significant vertical separation up to 10 μm and a footprint less than 10 μm × 10 μm. The device may be used to connect between deposited passive waveguide devices in the upper plane and active devices on a substrate. The design is based on a vertical stack of coupled ring resonators. We demonstrate basic feasibility of the design and estimates of device performance based on numerical simulation. A matrix model is presented to estimate spectral performance as a function of several design parameters.
©2013 Optical Society of America
Silicon CMOS-compatible photonic devices such as photodetectors, modulators, waveguides, and lasers have seen great advances in the recent decade [1–3]. As the individual performance of key photonic devices significantly improves, there is an increased interest in addressing the issues of photonic and CMOS devices integration [4,5], and new applications for a converged electronic-photonics IC platform are being conceived and developed [6–10].
The preferred strategy for integration in any application depends strongly on the nature of the market to which the application belongs. For instance, in optical communications applications where the market volume of the components is relatively small, the photonic devices may be optimized for their own preferred platform - the silicon-on-insulator (SOI) substrate has been the most popular - and combined with CMOS electronics through hybrid packaging [6,8]. In contrast, ultra-short reach applications such as intra-chip and inter-chip optical interconnect in the mainstream logic or non-logic semiconductor products [6,8] would require a much higher degree of integration of CMOS electronic and photonic circuits. To achieve this economically at high volume will likely require that a single substrate and a unified silicon processing flow be adopted for chip fabrication. This could pose additional challenges and restrictions on the design of materials, devices, and architecture.
For example, the specifications for SOI wafers in either electronics or photonics applications are typically very different, due to differing requirements for electrical or optical isolation and thermal dissipation. Consider buried oxide (BOX) thickness - optical waveguides with a core in the device layer of SOI typically require more than 2-3 μm BOX to isolate the optical mode from the substrate. However, current 45 nm SOI CMOS uses 200 nm BOX thickness and the predictions for the future fully-depleted SOI insulator thickness include a value between 5 and 10 nm BOX thickness . These conflicting substrate requirements for the optical and electrical components contribute to the challenge of large scale monolithic integration.
One approach to embed photonic devices on non-photonic SOI wafers is a localized modification of the substrate by removal of silicon from beneath a portion of a very thin oxide layer . This provides a lower cladding of air for waveguides formed from Si on top of the thin oxide. Another method to locally modify a bulk substrate demonstrated by Ji et al.  is to crystallize amorphous silicon deposited on oxide trenches by solid-phase epitaxy (SPE), seeding from the crystalline Si substrate adjacent to the oxide trench. Though promising, the waveguide losses in these demonstrations are larger than for the best SOI and deposited dielectric waveguides, and both approaches rely on significant surface area to be reserved at the substrate level for routing photonic circuitry.
As an alternative to integrating photonic devices at the substrate level with CMOS devices, there are recent efforts by some researchers [13–15] to form a photonics-dedicated plane in the upper level of the chip over the back-end-of-line (BEOL) dielectric material. A significant challenge to this approach is the difficulty of fabricating high-quality devices at the upper layer due to the lack of a crystalline substrate and restrictions on processing temperature.
In this paper, we present an optical coupler that can efficiently transfer optical signal between vertically displaced photonic planes with a large separation. The device comprises multiple vertically-stacked, coupled ring resonators, the lowest and uppermost of which are coupled to separate bus waveguides with a large vertical spacing between them. This configuration is depicted in Fig. 1(a). Such a device can be used in an approach that places active photonic devices in the lower substrate level to take advantage of crystalline silicon substrate and high processing temperature, and places passive devices and waveguides on an upper level, avoiding interference with the substrate or surrounding electronic components. An example of this approach is shown in Fig. 1(b).
In microelectronics IC’s, a vertical metallic plug, or via, provides the functionality of signal connection between electronic interconnect layers. A similar optical “via” made from a compact right-angle waveguide might be ideal, but it is difficult to conceive of such a device that remains both low-loss and feasible to fabricate. Using conventional waveguide directional couplers with adiabatic mode transfer , each coupling takes on the order of 100 μm of horizontal linear propagation per transfer. Furthermore, an interconnection between waveguides with several μm or more of vertical separation would need multiple intermediate coupling steps in a cascaded fashion, thus requiring more than several hundred μm of horizontal propagation. Such a device would also require several alternating deposition, alignment, patterning, and etch cycles to complete fabrication. In contrast, the vertically stacked ring resonator interconnect presented here has a relatively low-footprint per vertical rise and should be simpler to fabricate. In the following sections, we will give details on the concept and present numerical analysis that can estimate the device performance and spectral performance as a function of several design parameters.
2. Vertically stacked multiple ring interconnect
Microring resonators have been used extensively to couple light between waveguides, both laterally within planes [17,18], and between planes with vertical separation [14,19,20]. Several resonators can be coupled in series or parallel to engineer the spectral response of a coupling channel, for instance to sharpen transmission resonances or to modify group delay [18,21]. Large systems of coupled resonators, commonly referred to as coupled-resonator optical waveguides (CROWs) , have been studied and fabricated for filter applications and applications benefitting from drastically altered the group velocity, such as slow light experiments. These structures tend to be coplanar microrings with weak coupling between neighboring rings at exchange points where the rings are closest.
In the concept presented here, multiple ring resonators are stacked atop one another in planes between two widely separated bus waveguides, according to the basic design illustrated in Fig. 1(a). Light is coupled evanescently from the input waveguide to the first ring, from which it couples to next, and so on to the last ring, from which it is coupled evanescently to the output waveguide. The proposed device can be simply fabricated by depositing alternating layers of waveguide core and cladding materials with only minimal patterning and etch steps. A final step to deposit cladding material on the ring sidewalls may be included in the process. In this paper, we consider waveguides made with a deposited Si core and a deposited SiO2 cladding. This is a photonic material system that affords high index contrast, minimizing bend radius for a low footprint, and similar materials have been used to fabricate devices with low propagation loss using traditional silicon processing tools [23–25].
Figure 2 illustrates how a ring stack couples light from one bus to another separated by a substantial vertical distance. In that figure, the intensity of a TE optical mode at 1.55 μm, as computed by a 3D-FDTD simulation, is shown as a function of position in the x-y plane of each ring and bus in a 5 ring stack of α-Si rectangular channel waveguide microrings with SiO2 cladding. The channel cross-section for each ring and bus is 200 nm by 400 nm, and the rings have a 4 μm radius. These dimensions were chosen as a reasonable compromise between competing factors of bending loss, total foot-print, and inter-ring coupling strength. Modal confinement needs to be high enough to prevent excess bending loss in a tight radius for small footprint, but not so high to prevent effective vertical inter-ring coupling. It can be seen that the light is coupled from the input bus into the topmost ring where it couples smoothly into the ring below. At resonance, the central rings (uncoupled to any bus) are uniformly filled with circulating light. The bottom ring transfers light from the rings above it to the output bus, passing along more than 95% of the optical energy. The stack structure efficiently couples light from one plane to another with a small footprint, on-par with a single ring area.
If this device is used as a component of an optical communications channel, it is important that the cavity lifetime is sufficiently small to accommodate the required bandwidth of the modulated data signal. Based on the decay envelope of the time-domain signal in this 3D-FDTD simulation, the cavity lifetime of the composite structure was shown to be relatively short, less than 600 fs, suggesting that it can support a modulation bandwidth of several hundred GHz without compromising signal integrity.
In general, the transmission is highly spectral, determined by three contributing factors: coupling of light from the straight sections to the ring sections, inter-ring coupling during propagation around the ring stack, and feedback with wavelength dependent phase delay. The design of an efficient interconnect relies on understanding how these factors interact, and how they are affected by parameter selection.
3. Device structure and analytical modeling
The full 3D-FDTD simulation verifies the feasibility of a ring-stack vertical coupler. However, it is computationally expensive to optimize a design using a full simulation for each design iteration, so an alternative method is presented here, based on a combination of matrix algebra and coupled-mode theory (CMT). Similar methods exist to analyze coupled resonator devices [18,19,26–29]. The typical approach is to track the slowly time-varying complex amplitudes of the optical modes propagating in each resonator. The modes in each resonator are coupled to those in the neighboring resonators by some coupling constant, and the mode amplitudes form a system of coupled first-order differential equations in time. This system of equations, with appropriate boundary conditions, can be solved in the steady state for a given optical frequency to determine the spectral transfer function of the complete structure.
In the analysis of coplanar microring structures, coupling is typically considered to be weak, limited to nearest-neighbor interactions, and restricted to a point where the two waveguides almost touch [18,26,27]. For vertically coupled, concentrically stacked resonators, the interaction length is along the whole circumference of the ring, which means that the coupling constants are strongly dependent on the ring radii. The transmission characteristics for nearest-neighbor coupled, vertically stacked ring resonators has been studied for uniformly spaced rings  and for ring stacks with a defect  in the context of abstract coupling coefficients. It was demonstrated that the strong vertical coupling can have a marked effect on the transmission characteristics of vertically coupled ring resonators as compared with coplanar coupled ring resonators.
Our approach is to derive coupling constants numerically directly from device parameters by employing CMT calculations based on the solutions obtained from a numerical 2D mode solver. In this way, we modify the coupling constants by changing the physical device parameters directly, and yet we separate independent calculations to enable rapid iteration over the available physical parameter space. It should be noted that we do not restrict our analysis to nearest-neighbor coupling, since the long interaction length associated with vertically coupled ring resonators suggests that in the general case of closely spaced rings, interaction with more distant rings should be considered.
The transmission from one waveguide to the other can be modeled using a 2 × 2 scattering matrix whose elements are determined by the overlap integrals of the waveguide and mixed ring resonator modes. Feedback from the ring resonator stack is governed by the coupled delay of the several ring modes. Figure 3 shows a representation of the vector model used to approximate the expected transmission spectrum of a stack structure. Here, a is a 2 × 1 vector combining the complex amplitude of the input and output buses propagating from left to right, prior to scattering from the ring stack. The vector b is a 2 × 1 vector combining the complex amplitudes the input and output buses propagating from left to right, subsequent to scattering from the ring stack. The vectors r- and r+ are N × 1 vectors, where N is the total number of ring modes. They represent the complex amplitudes of the clockwise propagating modes just prior to and just after interaction with the bus waveguides, respectively. Using this simple representation, the complex amplitude of the forward propagating bus modes before and after the stack can be related by a linear system of equations, which we will briefly derive here.
Let us define four partial scattering matrices T, U, W, and V by the following:Eq. (3) into Eq. (2) to giveEquations (3) and (5) are then substituted into Eq. (1) to give the resultEq. (6).
There are two general benefits of this approach. The first is that the calculation of the various matrix elements in Eqs. (1)-(3) is far less computationally demanding than any full 3D-FDTD simulation. The second benefit is that the matrices T, U, V, and W can be calculated independently of Δ, and that they partially depend on different parameters of the design. Together, these allow for rapid iteration of coupler design.
The matrices T, U, V, and W can be estimated for a given waveguide cross section and bus-ring spacing by running a very short FDTD simulation (either 2D or 3D) of a bus guide coupled to a single (open) ring guide. Only a single ring needs to be considered for these calculations if the ring-to-ring spacing is large enough, since the interaction length is very small, and almost none of the light scatters from bus to any rings in another plane.
For the matrix Δ, the interaction length is much longer - the entire circumference of the ring - and in general all the rings are coupled. To avoid running long simulations for every parameter set, a coupled mode theory (CMT) approach is taken [30–32]. The matrix Δ is obtained by solving the linear system of coupled equationsEq. (3) are derived using CMT [22,23]:
Under the assumption that the matrices are independent of l, Eq. (3) can be solved by diagonalizing (B + P−1K) such that
If all the rings have the same cross section, E, H and β need only be computed once (by a 2D mode solver) per wavelength. Ei and Hi are found by translation, according to the chosen ring guide separation. The elements of B, P, and K can then be calculated using Eqs. (8-10).
Note that vertical spacing of the rings does not affect T, U, V, or W, and the bus-ring separation does not affect Δ. The partial scattering matrices can be rapidly recalculated after adjusting bus-to-ring spacing. The fields E and H together with β only need recalculating after wave-guide cross-section or material system is changed.
Figure 4 compares the calculation of transmission spectra for two coupler designs performed using Eqs. (1)-(5) to calculations performed with a full 3D-FDTD simulation (the software package used for the simulations was Lumerical FDTD Solutions). Transmission spectra are given for optical power exchanged between buses (Texchange) and optical power remaining on the same bus (Tsame). There is good agreement between the two calculations with respect to the shape of the transmission curves. The CMT method seems to estimate a slightly larger total width of each transmission window when compared to FDTD simulations for both designs, perhaps indicating a slight overestimation of the inter-ring coupling matrix elements of K.
However, accurate simulations using FDTD require several hours of computation per design, even on a modern workstation with 8 CPU cores. In contrast, the computer code used here to execute the CMT matrix method requires about a minute of computation time for the 14 ring design (using MATLAB scripts) – almost all of that time is used evaluating the matrix elements of P and K. Consequently, the CMT matrix calculation method is a valuable design tool for rapid evaluation of the effect of parameter choice on the transmission spectrum of stacked ring interconnects. This is briefly demonstrated in the next section.
4. Parameter selection and coupler design
As for any spectrally dependent interconnect, we can evaluate a particular design with a number of important metrics, including maximum transmission power, bandwidth, and pass-band ripple. For the application which motivated this design – a back-end to front-end photonic interconnect for dense photonic-electronic CMOS integration – we would like the transmission window to have a maximum magnitude near 100%, a wide bandwidth, and a reasonably flat pass-band. The rapid iteration of the CMT matrix method presented here provides a powerful tool with which to explore the impact of parameter choice on these metrics.
There are two factors that primarily determine the shape and features of the transmission spectrum. The first is the coupling strength of the bus to rings, represented by T, U, V, and W in the CMT matrix model, and the second is the relative coupling strength between the rings in the stack represented by the B, P, and K matrices. In this section, we will examine the contribution of these two sets of coupling factors to the shape of the transmission spectrum. To simplify the discussion, we will limit our analysis to a fixed waveguide cross-section and individual ring radius for the time-being, and focus on the effects on the relative positioning of ring and bus waveguides (i.e., bus-to-ring spacing and ring-to-ring spacing). The qualitative results obtained from this analysis will hold, even as other parameters are varied, providing insight into the design and development of vertically stacked microring waveguides.
First we consider the ring-to-ring spacing and its influence on the shape of the transmission spectrum. In Fig. 5, the transmission spectra of two 5-ring designs with uniform ring spacing and a fixed ring radius are compared. In the first design, the ring-to-ring period is 800 nm, and in the second it is 900 nm. In general, the spectral transmission windows are each made up of several peaks, one for each separate resonance of the ring stack. The primary differences between the two spectra are the widths of the pass bands and the degree of merging between the centermost peaks. This can be understood by the splitting of effective propagation index of the “super-modes” associated with the coupled ring structure. As the rings get further apart, the difference in effective indices of the “supermodes” gets less pronounced, and the peaks are pulled together. In the limit of vanishingly weak inter-ring coupling, all the peaks collapse onto the same wavelength, and only a single resonance remains (per free spectral range) – this is the limit of a single microring resonator coupled to a bus waveguide.
One feature common to all spectra of stacks with uniform ring spacing is that the depth of the valleys between peaks increases towards the edge of the transmission window, such that the band ripple is small at the center of the window than at the edge. This phenomenon can be partially mitigated by varying the vertical ring spacing. In Fig. 6, the spectra of two different designs are compared. The first spectrum belongs to a 5-ring stack of uniformly spaced rings with a ring-to-ring period of 900 nm. The second spectrum belongs to a stack with the same average spacing, but with the vertical position of the 2nd and 4th rings having been shifted closer to the lowermost and uppermost rings by 16 nm. With this modified design, the ripple of the pass-band has been evened out across the entire transmission window, extending the useful bandwidth of the device.
This exercise demonstrates one benefit of the CMT matrix model used in this paper, which is the ease and speed with which the ring spacing can be arbitrarily adjusted to alter the transmission spectrum. The bus-to-ring spacing is equally well addressed by the modeling approach. Figure 7 shows transmission spectra for three similar designs, with identical waveguide cross section, total vertical rise, ring number, and ring spacing. The bus-to-ring separation, measured as the distance between the inner edge of the bus and the outer edge of the ring, was assigned a different value in each design. If the distance is too large or small, the ripple has a large amplitude. Figure 8 depicts the ripple minimum for bus-to-ring spacing between 0 nm and 100 nm. Apparently there is a critical coupling distance which gives rise to the flattest pass-band.
In these simple analyses, we demonstrate how the sensitive interaction of parameter selections can be explored by this design methodology to quickly converge on promising parameter sets. This makes a rapid-iteration method like the CMT matrix model an attractive choice for design analysis.
A vertical optical coupler device that comprises multiple vertically-stacked ring resonators has been presented and analyzed for the purpose of transferring optical signals between waveguides in different photonic planes with significant vertical separation. The design has a small footprint and a high vertical rise per area ratio, and is fully compatible with traditional Si CMOS and proven photonics technology. The device can be simply fabricated by deposition of alternating layers of waveguide core and cladding materials followed by minimal patterning, alignment, and etching steps. Numerical 3D-FDTD simulations indicate a design using α-Si rings with SiO2 claddings can achieve greater than 90% transmission with a FWHM bandwidth of 10 nm at a central wavelength of 1550 nm. Over 10 μm vertical rise is achieved with a footprint less than 10 μm × 10 μm. A matrix model using coupled-mode theory to derive a delay matrix has been developed to facilitate rapid iteration of parametric design and analysis. The coupler device can be used as a vertical interconnect, or optical “via”, between an active device layer at the surface of a silicon substrate and low-loss, deposited waveguides in the upper level above the metal layer stack of typical silicon CMOS process. With further development, the device could facilitate closer integration of high-performance photonics with established silicon CMOS electronics processing.
References and links
1. J. Michel, J. Liu, and L. Kimerling, “High-performance Ge-on-Si photodetectors,” Nat. Photonics 4(8), 527–534 (2010). [CrossRef]
3. R. E. Camacho-Aguilera, Y. Cai, N. Patel, J. T. Bessette, M. Romagnoli, L. C. Kimerling, and J. Michel, “An electrically pumped germanium laser,” Opt. Express 20(10), 11316–11320 (2012). [CrossRef] [PubMed]
4. M. Beals, J. Michel, J. F. Liu, D. H. Ahn, D. Sparacin, R. Sun, C. Y. Hong, L. C. Kimerling, A. Pomerene, D. Carothers, J. Beattie, A. Kopa, A. Apsel, M. S. Rasras, D. M. Gill, S. S. Patel, K. Y. Tu, Y. K. Chen, and A. E. White, “Process flow innovations for photonic device integration in CMOS,” Proc. SPIE 6898, 689804, 689804-14 (2008). [CrossRef]
5. S. Sahni, A. Narasimha, A. Mekis, B. Welch, C. Bradbury, C. Sohn, D. Song, D. Martinez, D. Foltz, D. Guckenberger, G. Masini, J. Eicher, J. Dong, J. Schramm, J. White, J. Redman, K. Yokoyama, M. Tlalka, M. Harrison, M. Peterson, M. Saberi, M. Mack, M. Sharp, P. De Dobbelaere, R. LeBlanc, S. Leap, S. Abdalla, S. Gloeckner, S. Hovey, S. Jackson, S. Yu, T. Pinguet, W. Xu, and Y. Liang, “Silicon photonic integrated circuits,” in CLEO: Science and Innovations, OSA Technical Digest (online) (Optical Society of America, 2012), paper CM3A.3.
6. J. S. Orcutt, B. Moss, C. Sun, J. Leu, M. Georgas, J. Shainline, E. Zgraggen, H. Li, J. Sun, M. Weaver, S. Urošević, M. Popović, R. J. Ram, and V. Stojanović, “Open foundry platform for high-performance electronic-photonic integration,” Opt. Express 20(11), 12222–12232 (2012). [CrossRef] [PubMed]
7. H.-C. Ji, K. H. Ha, K. W. Na, S. G. Kim, I. S. Joe, D. J. Shin, K. H. Lee, S. D. Suh, J. K. Bok, Y. S. You, Y. W. Hyung, S. S. Kim, Y. D. Park, and C. H. Chung, “Bulk silicon photonic wire for one-chip integrated optical interconnection,” in Proceedings of 2010 7th IEEE International Conference on the Group IV Photonics (GFP) (2010), pp. 96–98. [CrossRef]
8. K.-H. Lee, D. J. Shin, H.-C. Ji, K. Na, S. G. Kim, J. Bok, Y. You, S. Kim, I. Joe, S. D. Suh, J. Pyo, Y.-H. Shin, K. Ha, Y. D. Park, and C. H. Chung, “10Gb/s silicon modulator based on bulk-silicon platform for DRAM optical interface,” in the National Fiber Optic Engineers Conference (2011), paper JThA033. C. [CrossRef]
9. A. Batten, A. Joshi, J. Orcutt, A. Khilo, B. Moss, C. W. Holzwarth, M. A. Popovic, H. Li, H. I. Smith, J. L. Hoyt, F. X. Kartner, R. J. Ram, V. Stojanovic, and K. Asanovic, “Building many-core processor-to-DRAM networks with monolithic CMOS silicon photonics,” Micro, IEEE 29(4), 8–21 (2009). [CrossRef]
10. S. Jatar, Y. Malinge, Z. Zhou, H. Liang, S. Liao, Z. Li, C. Bushyakanist, D. C. Lee, R. Shafiiha, J. Luff, D. Feng, and M. Asghari, “Performance of parallel 4×25 Gbs transmitter and receiver fabricated on SOI platform,” presented at the Group IV Photonics (GFP), 2011 8th IEEE International Conference on, 2012, pp. 159–161.
11. SOITEC, “Questions and answers on fully depleted SOI technology,” pp. 1–17 (2010). Available online: http://www.soitec.com/pdf/SOIconsortium_FDSOI_QA.pdf
12. C. Holzwarth, J. Orcutt, H. Li, M. Popovic, V. Stojanovic, J. Hoyt, R. Ram, and H. Smith, “Localized substrate removal technique enabling strong-confinement microphotonics in bulk Si CMOS processes,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (Optical Society of America, 2008), paper CThKK5. [CrossRef]
13. K. Preston, S. Manipatruni, A. Gondarenko, C. B. Poitras, and M. Lipson, “Deposited silicon high-speed integrated electro-optic modulator,” Opt. Express 17(7), 5118–5124 (2009). [CrossRef] [PubMed]
14. A. Biberman, K. Preston, G. Hendry, N. Sherwood-Droz, J. Chan, J. S. Levi, M. Lipson, and K. Bergman, “Photonic network-on-chip architecture using multilayer deposited silicon materials for high-performance chip multiprocessors,” ACM J. Emer. Tech in Comp. Syst. 7 (2011).
15. K. A. McComber, X. Duan, J. Liu, J. Michel, and L. C. Kimerling, “Single-crystal germanium growth on amorphous silicon,” Adv. Funct. Mater. 22(5), 1049–1057 (2012). [CrossRef]
16. R. Sun, M. Beals, A. Pomerene, J. Cheng, C. Y. Hong, L. Kimerling, and J. Michel, “Impedance matching vertical optical waveguide couplers for dense high index contrast circuits,” Opt. Express 16(16), 11682–11690 (2008). [CrossRef] [PubMed]
17. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. K. Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012). [CrossRef]
18. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
21. C. Ciminelli, C. E. Campanella, F. Dell’Olio, and M. N. Armenise, “Fast light generation through velocity manipulation in two vertically-stacked ring resonators,” Opt. Express 18(3), 2973–2986 (2010). [CrossRef] [PubMed]
23. S. K. Selvaraja, E. Sleeckx, M. Schaekers, W. Bogaerts, D. V. Thourhout, P. Dumon, and R. Baets, “Low-loss amorphous silicon-on-insulator technology for photonic integrated circuitry,” Opt. Commun. 282(9), 1767–1770 (2009). [CrossRef]
24. R. Sun, K. McComber, J. Cheng, D. K. Sparacin, M. Beals, J. Michel, and L. C. Kimerling, “Transparent amorphous silicon channel waveguides with silicon nitride intercladding layer,” Appl. Phys. Lett. 94(14), 141108 (2009). [CrossRef]
25. S. Zhu, G. Q. Lo, and D. L. Kwong, “Low-loss amorphous silicon wire waveguide for integrated photonics: effect of fabrication process and the thermal stability,” Opt. Express 18(24), 25283–25291 (2010). [CrossRef] [PubMed]
26. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12(1), 90–103 (2004). [CrossRef] [PubMed]
28. M. Shafiei, M. Khanzadeh, M. Agha-Bolorizadeh, and R. F. Moghaddam, “Linear transmission properties of a vertically stacked multiring resonator with a defect,” Appl. Opt. 48(31), G148–G155 (2009). [CrossRef] [PubMed]
30. A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973). [CrossRef]
31. H. Haus, W. Huang, S. Kawakami, and N. Whitaker, “Coupled-mode theory of optical waveguides,” J. Lightwave Technol. 5(1), 16–23 (1987). [CrossRef]
32. B. E. Little and W. P. Huang, “Coupled-mode theory for optical waveguides,” Prog. Electromagnetics Res. 10, 217–270 (1995).