Micro-resonator with metallic mirrors coupled to a bus waveguide

We demonstrate a micro-resonator based on a channel waveguide terminated with metallic mirrors side coupled to a bus waveguide. Transmission through such a resonant structure implemented in a siliconon-insulator platform is investigated theoretically and demonstrated experimentally. The resonator is 13.4 μm long, exhibits an unloaded Qfactor of ~2100, and a free spectral range of 21 nm around the wavelength of 1.55 μm. ©2011 Optical Society of America OCIS codes: (230.5750) Resonators; (130.7408) Wavelength filtering devices. References and links 1. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007). 2. S. F. Preble, Q. Xu, and M. Lipson, “Changing the colour of light in a silicon resonator,” Nat. Photonics 1(5), 293–296 (2007). 3. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, “Low threshold gain metal coated laser nanoresonators,” Opt. Lett. 33(11), 1261–1263 (2008). 4. M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Roomtemperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010). 5. L. Zhang, and A. E. Willner, “Microresonators for Communication and Signal Processing Applications”, in Photonic Microresonator Research and Applications, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds, (Springer 2010), pp. 485–505. 6. T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). 7. L. Pavesi, and D. J. Lockwood, eds., Silicon Photonics, (Springer 2004), pp. 51–84. 8. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998). 9. P. Dumon, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, J. Van Campenhout, D. Taillaert, B. Luyssaert, P. Bienstman, D. Van Thourhout, and R. Baets, “Low-loss SOI photonic wires and ring resonators fabricated with deep UV lithography,” IEEE Photon. Technol. Lett. 16(5), 1328–1330 (2004). 10. Q. Xu, and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). 11. P. Koonath, T. Indukuri, and B. Jalali, “Add-drop filters utilizing vertically coupled microdisk resonators in silicon,” Appl. Phys. Lett. 86(9), 091102 (2005). 12. A. Morand, Y. Zhang, B. Martin, K. Phan Huy, D. Amans, P. Benech, J. Verbert, E. Hadji, and J. M. Fédéli, “Ultra-compact microdisk resonator filters on SOI substrate,” Opt. Express 14(26), 12814–12821 (2006). 13. H.-C. Kim, K. Ikeda, and Y. Fainman, “Tunable transmission resonant filter and modulator with vertical gratings,” J. Lightwave Technol. 25(5), 1147–1151 (2007). 14. H.-C. Kim, K. Ikeda, and Y. Fainman, “Resonant waveguide device with vertical gratings,” Opt. Lett. 32(5), 539–541 (2007). 15. K. Vahala, Optical Microcavities, (World Scientific Publishing 2004). 16. J. Heebner, R. Grover, T. Ibrahim, Optical Microresonators. Theory, Fabrication, and Applications, (SpringerVerlag 2008). 17. H. A. Haus, Waves and Fields in Optoelectronics, (Prentice Hall 1984), ch. 8.3, pp. 226–8 and 243. 18. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals: molding the flow of light, (Princeton University Press 1995). 19. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, (Wiley & Sons 1995), ch. 3. 20. S. Wolff, A. R. Giehl, M. Renno, and H. Fouckhardt, “Metallic waveguide mirrors in polymer film waveguides,” Appl. Phys. B 73(5-6), 623–627 (2001). #139290 $15.00 USD Received 7 Dec 2010; revised 19 Dec 2010; accepted 23 Dec 2010; published 25 Jan 2011 (C) 2011 OSA 31 January 2011 / Vol. 19, No. 3 / OPTICS EXPRESS 2417 21. Y. Shibata, T. Suzuki, and H. Tsuda, “Design and Evaluation of an N:N Optical Coupler Using an Integrated Waveguide Mirror,” Opt. Rev. 11(3), 182–187 (2004). 22. S. Zamek, A. Mizrahi, L. Feng, A. Simic, and Y. Fainman, “On-chip waveguide resonator with metallic mirrors,” Opt. Lett. 35(4), 598–600 (2010). 23. A. Yariv, Optical Electronics, 3d Ed. (Holt, Rinehart and Winston 1985), ch. 13.7, p. 432. 24. R. Kazarinov, C. Henry, and N. Olsson, “Narrow Band Resonant Optical Reflectors and Resonant Optical Transformers for Laser Stabilization and Wavelength Division Multiplexing,” IEEE J. Quantum Electron. 23(9), 1419–1425 (1987). 25. H. A. Haus, and Y. Lai, “Theory of Cascaded Quarter Wave Shifted Distributed Feedback Resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992). 26. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add-drop filters,” IEEE J. Quantum Electron. 35(9), 1322–1331 (1999). 27. Y. Xu, Y. Li, R. K. Lee, and A. Yariv, “Scattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62(5 5 Pt B), 7389–7404 (2000). 28. M. Lohmeyer, “Mode expansion modeling of rectangular integrated optical microresonators,” Opt. Quantum Electron. 34(5-6), 541–557 (2002). 29. M. Hammer, D. Yudistira, and R. Stoffer, “Modeling of grating assisted standing wave microresonators for filter applications in integrated optics,” Opt. Quantum Electron. 36(1-3), 25–42 (2004). 30. E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic 1985), p. 294. 31. A. B. Buckman, Guided Wave Photonics, (Saunder College Publishing 1992), pp. 149–154. 32. K. Srinivasan, P. E. Barclay, O. Painter, J. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. 83(10), 1915 (2003). 33. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998). 34. Y.-F. Xiao, C.-L. Zou, B.-B. Li, Y. Li, C.-H. Dong, Z.-F. Han, and Q. Gong, “High-Q Exterior WhisperingGallery Modes in a Metal-Coated Microresonator,” Phys. Rev. Lett. 105(15), 153902 (2010). 35. K. Preston, and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009).

Waveguide-coupled resonators can be subdivided into two major families: traveling wave and standing wave resonators.Traveling wave resonators are typically based on microdisks, microrings, and micro-toroids [15,16].Standing wave resonators include photonic crystals, quarter-wave-shifted gratings, or, simply, a waveguide terminated with DBR [17,18].DBRs seem to be the most popular ones due to their high reflectivity along with easy fabrication process, however DBRs have several limitations.First, they are polarization sensitive, as transverse-electric (TE) and transverse magnetic (TM) modes of a waveguide exhibit different guided indices and therefore generally do not satisfy Bragg condition for the same wavelength.Second, DBR's smallest feature size is in the deep sub-micron regime, which requires state-of-the-art fabrication facilities and tools with good process repeatability.
Metallic mirrors overcome some of these limitations.The advantages of metallic mirrors are low polarization sensitivity, compactness, and high tolerances to fabrication imperfections.Despite their wide use in the free-space optical components, metallic mirrors found just a few applications in guided wave optics.This is due to the high anticipated losses of such mirrors when integrated into a high-index waveguides [19].In chip-scale devices, metallic mirrors have only been integrated with low-index materials such as polymer [20] and silica [21] waveguides.In our previous work [22] we demonstrated the feasibility of high Q-factors in silicon micro-resonators constructed with metallic mirrors.Such mirrors were shown to allow dense chip-scale integration of resonators and to relax the fabrication requirements.In this work we demonstrate a micro-resonator with metallic mirrors made in a SOI platform, coupled to a bus waveguide, as shown in Fig. 1a.An analytical model for a side-coupled resonator is developed and the design tradeoffs are investigated.

Description of the device
A schematic of the device is shown in Fig. 1a.SOI platform was used to realize Si waveguides resting on a SiO 2 substrate.The resonator is based on a channel silicon waveguide with a cross-section identical to the cross section of the bus waveguide.The resonator's waveguide is terminated with two metallic plates, creating a cavity.The resonator is side coupled to a channel (bus) waveguide.The upper cladding of the device is silicon dioxide, which is not shown on Fig. 1a for clarity.The resonator's length, L r , is the distance between the mirrors, and L c is the length of the coupling region, where the gap between the two waveguides is uniform (L r ≥ L c ).
When an optical mode propagates in the bus waveguide, some of its energy will couple into the resonator by the mechanism of co-directional evanescent coupling [23].It will further circulate in the resonator, reflecting off the mirrors.Assuming low coupling between the two waveguides, this will generally have a negligible effect, with most of the power transmitted through the bus waveguide.If the resonance condition is satisfied, even a small coupling will cause a buildup of an intense field in the resonator.Some of this field will couple back into the bus resonator and will destructively interfere with the incident mode propagating in the waveguide.As we show below, at resonance this effect can result in an appreciable dip in the transmission spectrum and strong reflection back into the bus waveguide.It is important to emphasize the differences between the suggested device and traveling wave resonators such as rings, disks, and toroids.Standing-wave resonator, such as the one depicted in Fig. 1a, exhibits high reflectance and low transmittance on the resonance.This is in contrast to traveling-wave resonators with all-pass behavior (see for example chapter 3.3 in Ref 16).The advantages and possible applications of standing-wave resonators were extensively studied in previous works [24][25][26][27][28][29].High-Q resonators employed DBRs, while low-Q resonators used cleaved waveguide facets.So far, metallic mirrors received little or no attention in guided wave optics, despite their extensive use in free-space optics.In the following section we derive the analytical model for a resonator with metallic mirrors side coupled to a bus waveguide.

Analytical Model
Analytical approaches for the analysis of waveguide-coupled resonators include rigorous solution of four coupled modes equations [24], coupling of modes in time [25], and scatteringtheory formalism [27].Here we took an approach based on multiple reflections model, analogous to the one used for the analysis of Fabry-Perot interferometer.The advantage of this model is simplicity and an insight into the features of the transmission spectrum.
The formalism is illustrated in Fig. 1b.A mode with amplitude E i is launched into the input port of the bus waveguide, as shown in the figure.Some part of it, a, couples into the resonator, while part b propagates to the output port.The mode coupled into the resonator with amplitude aE i , will undergo multiple reflections, while in each cycle some of it (a) will couple back into the bus waveguide.The coefficients a and b can be obtained from the coupled mode theory [23]: Similarly to the derivation of a transmission through a Fabry-Perot interferometer [17], the output field E t is given by an infinite sum of the fields coupled out from the resonator into the output port ( ) where κ is the coupling coefficient between the two waveguides, L c and L r , are the lengths of the coupling section and the resonator (as shown in Fig. 1a), β is the wave number of the guided mode, and R = r 0 2 is the reflectance of the mirror.The expression in Eq. ( 1) was confirmed with 2D simulations based on the Finite Element Method (FEM).In these simulations we considered a TE-mode, 200 nm wide waveguide, side-coupled to a resonator with the coupling coefficient of κ = 560 cm −1 .Three resonators with Lc = Lr = 5,10, and 15µm, terminated with mirrors with reflectivity R = 0.95 were simulated and full agreement between Eq. ( 1) and the simulation results was obtained.
To obtain the Q-factor of the resonator, we expand the propagation constant in the vicinity of a resonance as 0 / g v , where g v is the group velocity of the mode.After the result is substituted into Eq.( 1), the transmittance is obtained: On the other hand it is well known that a waveguide side-coupled to a resonator exhibits a transmission spectrum of the general form [27]: Here Γ 0 and Γ C are the mode decay rates associated with the internal resonator losses, and losses due to the coupling of the resonator to a waveguide.Comparison of Eqs. ( 2) and (3) reveals the factor cos 2 (κL c ), which appears in Eq. ( 2) but not in Eq. ( 3).This factor is due to the difference between previous research [27] and this work in the choice of reference planes, where E i and E t are defined (see Fig. 1b).If the input and the output reference planes were chosen at the center of the resonator, as it was done in Ref 26, the factor would disappear from Eq. ( 2).Aside from this factor, Eqs. ( 2) and ( 3) are equivalent, where the decay rates Γ 0 and Γ C and the associated Q-factors are given by: ( ) Here k 0 = ω/c, and n g is the group index of the waveguide mode.We note that Q 0 is the unloaded Q-factor independent of the coupling losses.The total Q-factor is calculated as It is interesting to notice, that the two Q-factors associated with the coupling and the mirror losses are both dependent on the resonator length, but in an opposite manner, as shown in Fig. 2a.For extremely short cavities the coupling is negligible and the Q-factor is dominated by the mirror losses.For long cavities, the coupling happens over a longer length, leading to a larger mode decay rate, and the Q-factor is dominated by the coupling.To confirm the results obtained in Eqs. ( 4) and ( 5) we performed 2D FEM simulations as shown in Figs.2b-d.The Q-factors were calculated from the eigenmodes found from the simulations, with the simulation results shown in Fig. 2a by black cross-marks.Good agreement between simulations and analytical results was found, confirming the validity of our model.
Another important characteristic of the coupled micro-resonator, is the extinction ratio (ER).ER is defined as the ratio of the transmittance in the pass-band to the one on the resonance.To derive it, we should notice here that at resonance exp(2iβL r ) = 1 in Eq. ( 1).This happens when βL r = πm, where m is an integer describing the longitudinal mode number.Exactly in the middle between two resonances βL r = π(m + 0.5) and therefore exp(2iβL r ) = −1.By substitution of these two into Eq.( 1), the ratio between the two transmittances can be calculated and the ER is found: It is interesting to notice that for no coupling between the resonator and the bus waveguide, i.e. κ = 0, ER approaches 1, as anticipated.This case is equivalent to a straight waveguide, for which the transmission equals 1 over the entire spectrum above the waveguide's cutoff frequency.On the other hand, when the reflectivity R approaches 1, the ER tends to infinity.From this discussion it becomes evident that the coupling loss must predominate over the internal resonator loss to achieve high ER.

Design
As a proof-of-concept, we designed a micro-resonator coupled to a bus waveguide, based on channel Si waveguides with SiO 2 cladding, terminated with Au mirrors.To do so, we used a SOI wafer with top silicon thickness of 250nm and thermal oxide layer ~3 µm thick.The design was carried out in the following steps.First we analyzed the reflectance of the mirror for different waveguide widths.Having set the width of the guide, we chose the length of the resonator as a tradeoff between the unloaded Q-factor, Q 0 , and the FSR.For the chosen waveguide cross section and resonator's length, we analyzed the coupling coefficient and the external Q-factor Q C , as a function of the separation distance between the resonator and the bus waveguide.This distance was set to obtain the desired extinction ratio in the transmission spectrum.
Reflectance of a mirror was analyzed for the basic TE-like mode of the waveguide as a function of its width, W, based on three dimensional (3D) FEM simulations, in a configuration shown in Fig. 3a.For most waveguide geometries, the decay length of the mode into the cladding is only several hundreds of nanometers or less.Therefore a mirror that extends sufficiently into the cladding behaves as an infinite metallic plate for any practical purpose.For the gold we used dielectric constant ε = −132-12.7iat the wavelength of 1.55 µm [30].The obtained reflectance R is shown in Fig. 3b by a solid black line.The total loss, 1-R, is composed of the material loss in the metal and scattering due to the waveguide discontinuity introduced by the mirror.The relative contribution of scattering to the total loss is shown by broken blue line on the same figure.To understand the result we consider two extremes.In the first extreme, W approaches zero, the mode is poorly confined, and most of the power flows in the cladding.This case resembles reflection of a plane wave normally incident onto the Au-SiO 2 interface, for which the reflectance equals ~0.975.In the other extreme, where W is sufficiently large, the mode is strongly confined to the core of the waveguide, and the reflectance approaches ~0.95, which is the reflectance at the Au-Si interface.For W = 280 nm, the reflectance reaches its minimum value of ~0.78, associated strong scattering.For this geometry most of the loss is attributed to the scattering, as shown by the broken blue line in Fig. 3b.A width W = 400 nm was chosen to provide both high reflectance R = 0.9, and high group index n g = 4.27, as shown in Figs.3b and 3c.
The length of the resonator, L R , is a tradeoff between the Free Spectral Range (FSR) and the Q-factor.The FSR is given by λ 2 /2L r n g [22].We chose the length to be L r = 13.4 µm, leading to FSR of 21 nm, making it possible to observe several resonant lines with the setup described below.Such length resulted in an unloaded Q-factor of Q 0 = 2100 at the wavelength of λ = 1.55 µm, as given by Eq. ( 2).The total Q-factor is dominated by the external Q-factor, Q c , which is calculated below.
Next we consider two waveguides with a cross section 250nm × 400nm, separated by a distance G, as shown in the inset of Fig. 3d.To analyze the coupling coefficient, κ, we separately used the expressions provided in Refs 17 and 31.The field profile used in the calculations was obtained from FEM simulations.Both results designated 'H' and 'B' obtained from each of the references are shown in Fig. 3d correspondingly.For the chosen distance G = 240nm, the coupling is κ = 1400cm −1 , the external Q-factor Q c ~300, and the extinction ratio of the transmission spectrum is approximately 20 dB, as obtained from Eq. ( 6).To summarize, our design targeted the following parameters of the micro-resonator coupled to a bus waveguide: Q 0 = 2100, Q C = 300, FSR = 21nm, and ER = 20dB at the wavelength of 1.55µm.

Fabrication
The step-by-step fabrication process is illustrated in Fig. 4. We first used electron beam lithography (EBL) to pattern Hydrogen SilsesQuioxane (HSQ) resist on a SOI wafer.The created pattern was used as a mask for dry etching of silicon, resulting in Si waveguides on top of a SiO 2 layer.The residual HSQ was removed by 10 sec wet etching in a diluted (1:10) buffered oxide etch (1:6 BOE solution).A layer of polymethyl methacrylate (PMMA) resist was patterned on top using an overlay EBL.The patterned PMMA was used as a mask for the second dry etching of silicon, and a successive dry etching of silicon dioxide, opening gaps for the mirrors in the Si waveguides.Electron beam evaporation of gold with a consecutive liftoff in acetone left gold plugs inserted into the Si waveguide.This created a resonator composed of a Si waveguide side coupled to a bus waveguide as shown in Fig. 5a.A layer of SiO 2 with a thickness of 2 µm was deposited on top using plasma-enhanced chemical vapor deposition (PECVD).The fabricated device and its parameters are shown in Fig. 5a.

Experiment
To measure the transmission spectrum we used a telecom-grade linearly polarized tunable laser source.Its output was coupled an on-chip waveguide through a tapered polarization maintaining fiber.The fiber was affixed to a 5-axis mechanical stage to allow precise alignment between the fiber and the waveguide.The output port of the waveguide was imaged through a polarizer onto the detector, connected to a power meter.The polarizer was set to pass the electric field parallel to the plane of the sample, to efficiently excite TE-like mode of the waveguide.The reading of the power meter was recorded as the wavelength of the source was scanned in steps of 20 pm.The obtained power spectrum of the transmission through the bus waveguide is plotted in Fig. 5b by red dots.The solid black curve shows the analytical result as calculated from Eq. ( 1) for the parameters used in the design.
The obtained transmission spectrum exhibits several interesting features.First, it shows a ripple of approximately 2 dB, which is a result of interference created by mode reflection off the cleaved facets of the waveguide.Second, it shows a FSR~21 nm, which is in agreement with the analytical prediction.The ER grows towards longer wavelengths, due to poorer mode confinement and larger coupling between the two waveguides.The experimentally obtained extinction ratio ER~10 dB is significantly lower than the analytical value (~20 dB).This is due to several factors, listed here in the order of their significance: mirror reflectivity R~0.8 lower than anticipated (~0.9) due to fabrication imperfections, smaller coupling coefficient κ (~1200cm −1 instead of 1400cm −1 ) due to narrower waveguides, depolarization of the mode due to the coupling between TE-and TM-like modes at waveguide bends and perturbations created by the mirrors, reduced refractive index of the upper cladding created by PECVD process, and polarization extinction ratio in the excitation of the waveguide by a tapered fiber.To summarize, the fabricated device exhibited Q C = 360 instead of 300 (by design) and Q 0 = 900 instead of 2100, providing a total Q-factor of 260.According to these parameters, the insertion loss is ~0.7dB as estimated from Eq. ( 1).
The mirror quality, which is the dominant factor among those listed above, is compromised by the roughness, tilt, and conformity to the waveguide facet.The roughness is introduced in the dry-etching of Si through a mask of PMMA, with its typical values on the order of ~10 nm peak-to-peak.The roughness of the deposited metal is significantly increased outside of the silicon waveguide due to the lift-off process.However the rough surface of metal is believed to play minor role as most of the power flows in the Si waveguide, where the roughness is minimal since no contact between metal and the resist was established.The tilt of the mirror is about 7 degrees as estimated from the SEM images.This tilt is due to the shadowing of the evaporated metal by the PMMA resist.

Discussion
The described fabrication process can be improved to reduce the roughness and the tilt of the mirror.Evaporation of metal at an angle can be done to reduce the tilt and to increase the conformity of the mirror to the waveguide.PMMA can be replaced with other resists that offer higher sidewall smoothness.Adding post-fabrication processing, such as rapid thermal annealing, can smooth the rough surface.Also other fabrication techniques can be investigated.
High mirror reflectivity R is critical to obtain high finesse, narrow resonance line width, and high extinction ratio, desirable in many applications.Our design was constrained by the height of the Si waveguide (250 nm), set by the availability of SOI wafers.In practice, the guide cross section can be optimized to achieve reflectivity of 0.95, as suggested by Fig. 3b.Even higher reflectivity (up to 0.975) can be obtained on the expense of mode confinement by significantly decreasing the cross section of the waveguide, and pushing the mode out into the cladding, as suggested by the same figure.Nevertheless, the inherent material loss limits the reflectivity to ~97.5%, corresponding to the maximum finesse of ~120.Such losses do not exist in DBRs, whose reflectivity approaches unity for a sufficiently large number of periods.
The reflectivity of metallic mirrors cannot compete with the conventional DBRs.Nor can the resonator achieve the Q-factors of the state-of-the-art resonators with small mode volumes [3,[32][33][34][35].The major strength of metallic mirrors, however, rests with their ability to provide reasonably high reflectivity over a broad band of wavelengths and low polarization sensitivity.In addition, the optical skin depth in metal is on the order of 10 nm, so that a very thin layer of metal is sufficient to achieve the desired reflectivity.Hence the proposed metallic mirrors can attain extremely small footprints, by an order of magnitude smaller than those of a typical DBR.DBR's smallest feature size is in the deep sub-micron regime and requires a high degree of fabrication accuracy.Metallic mirrors, in contrast, impose no such constraints and exhibit low sensitivity to fabrication imperfections.

Summary
To summarize, we demonstrated a micro-resonator based on silicon waveguide terminated with metallic mirrors.The mirrors are compact and highly reflective.The geometry was optimized to achieve an unloaded Q-factor of 2100 for a resonator 13.4 µm long.Its measured transmission spectrum was in good agreement with the developed analytical model.The device may be used to construct high-order inline filters, spectrum shapers, true-time delays, modulators, channel add-drop multiplexers, and biochemical sensors.

Fig. 1 .
Fig. 1.(a) Schematic of the studied device.(b) Illustration of the formalism used to construct the analytical model.

Fig 2 .
Fig 2. Confirmation of the model.(a) Q-factors as a function of normalized length of the resonator L/λ calculated from Eqs 4 and 5. Q-factor associated with the external coupling (Q C ), mirror losses (Q 0 ), and the total Q-factor (Q T ) are shown by broken blue, dashed black, and solid black correspondingly.The black cross-marks show FEM simulation results of the coupled resonator.(b-d) Electric field |Ez| distribution as obtained from FEM simulations for L/λ=0.22,1.33, and 3.54 (where L=L r =L c ).All simulation used TE polarization with the following parameters: mirror reflectivity R=0.98, normalized coupling coefficient κλ=0.114,normalized separation distance between the resonator and the bus waveguide G/λ=0.26, and normalized thickness of the waveguides d/λ=0.13.

Fig. 3 .
Fig. 3. Theoretical investigation of the micro-resonator.(a) Schematic of an Au mirror inserted into a Si waveguide.(b) Mirror's reflectance, R, (solid black) and the ratio of scattering loss to the total loss (broken blue) as a function of guide's width, W.An inset shows the considered cross-section of the waveguide.The shaded area corresponds to W below 150nm, for which the results are inaccurate due to the high numerical error of FEM, associated with low mode confinement.(c) Effective and group index, n e and n g , as a function of waveguide's width, W. (d) Coupling coefficient, κ, as a function of the gap, G, between the guides, for the geometry shown in the inset.All results pertain to TE-like mode and a wavelength of λ = 1.55µm.

Fig. 4 .
Fig. 4. Conceptual illustration of the fabrication process.The two panes show two different cross-sections of the device (XY in the upper pane and ZY in the lower pane, as shown).The dimensions are not up to scale.

5 .
Experimental results.(a) Micrograph obtained with a scanning electron microscope (SEM) of the fabricated device right after the lift-off and prior to the deposition of top cladding with PECVD.The device is tilted at 45 degrees with respect to the imaging axis.(b) Experimental (red dots) and simulated (solid black) transmission spectra of TE-like mode of the fabricated device in dB.