We present the first prediction of lateral leakage behavior of the TM-like mode in thin-ridge SOI curved waveguides and ring resonators. A simple phenomenological model is first presented which predicts that the lateral leakage in these structures is significantly impacted by both the ring radius and waveguide width. This prediction is verified using full vectorial mode matching and finite element methods. We show that specific combinations of waveguide width and ring radius can lead to very low-loss propagation in the TM-like mode. This finding is critical for the design of high-Q resonators on such waveguide platforms and will have major impact on the field of silicon lasers and sensing applications.
©2010 Optical Society of America
Silicon photonics  has attracted considerable attention due to the potential for dense integration of photonic components while leveraging the mass manufacturing capability of silicon technology. Silicon photonic devices are usually fabricated on a silicon-on-insulator (SOI) wafer. The high index contrast between the silicon core and silica cladding allows for highly confined waveguide modes resulting in low loss, compact devices with extremely tight bends. However, for a number of sensing and hybrid photonic systems, it would be beneficial to enhance the evanescent field of the guided mode so that it will interact with the surrounding environment.
Devices based on thin shallow ridge waveguide structure , when operated in the TM polarized mode, exhibit strong evanescent fields. Using a thermal oxidization process , thin-ridge SOI waveguides with ultra smooth sidewalls have been practically realized providing very low scattering losses . However, when operating in the TM mode, these waveguides can experience severe inherent lateral radiation leakage losses [5, 6]. We have experimentally observed the lateral leakage loss in thin-ridge SOI waveguides . We have also demonstrated that these inherent losses can be effectively mitigated at so-called “magic widths”  where radiation loss components experience resonant lateral leakage effect . However, in order to use these waveguides to form practical subsystems such as resonators, add-drop filters, and active components such as modulators, switches, and possibly emitters, they must be configured into architectures including rings, disks, bends, and directional couplers. Because the resonant effect phenomenon relies on precision destructive interference of wavefronts in longitudinally invariant waveguides, it is likely that the effectiveness of this cancellation will be impacted by perturbations. The impact of directional coupling and disk architecture on the resonant lateral leakage was analyzed in  and , respectively. We have also recently predicted the dependence of the propagation loss in thin-ridge SOI ring resonators; when operating in TM-polarization; on both the ring radius and waveguide width .
In this paper, we show the impact of waveguide curvature on the lateral leakage loss mechanism. A simple phenomenological model is presented which predicts that both the ring radii and waveguide width play a crucial role in the loss mechanism. The model is rigorously tested by simulating the structure using two independent techniques - analytic mode matching and the finite element method. The analysis reveals that the propagation loss is sensitive to both ring radius and waveguide width and that the interplay between these parameters is not trivial. Opportunities for new devices exploiting this behavior are also identified.
2. Lateral leakage of TM-like mode in thin-ridge SOI ring
Figure 1 shows the cross-sectional [Fig. 1(a)] and plan view [Fig.1(b)] of a section of a ring resonator formed by a thin-ridge SOI waveguide. Consider a simplified ray-tracing view of wave propagation. The rays of the guided TM-like mode of the ring are shown in Fig. 1(b). In general, when the rays of a guided mode are incident on the waveguide boundary, they are totally internally reflected. However, it has been shown previously [7, 9, 10] that when a TM ray of a thin-ridge SOI waveguide is incident on a waveguide boundary, transmitted and reflected TE rays are generated in addition to the reflected TM ray due to strong TM-TE mode coupling at the waveguide ridge boundary. On the outer waveguide boundary, the transmitted TE ray (T TE1) propagates away from the ring, while the reflected TE ray (R TE1) propagates across the waveguide, intersecting the inner waveguide boundary where it is largely transmitted. The reflected TE ray (R TE1) then traverses a secant across the ring, intersecting with the inner and outer waveguide boundaries again, and then propagates away from the waveguide mostly unaltered. On the inner waveguide boundary, the incident TM ray generates reflected TE (R TE2) and transmitted TE (T TE2) rays. The reflected TE ray (R TE2) propagates across the waveguide to radiate away from the ring. The transmitted TE ray (T TE2) traverses a secant across the ring, intersecting the inner and outer waveguide boundaries and propagates away unaltered. The TM guided mode suffers from high leakage loss due to power coupling to TE radiation at the two waveguide boundaries.
Based on the above ray model, at any point outside the ring, there are four different TE rays: transmitted (T TE1) and reflected (R TE1) TE rays generated from the TM ray incident on the outer waveguide boundary, and the transmitted (T TE2) and reflected (R TE2) TE rays generated from TM-TE mode coupling on the inner waveguide boundary. As the relative phases of all these TE rays depend on both the ring radius and waveguide width, it is possible that there might exist some combinations of waveguide width and ring radius for which all four of these TE waves interfere destructively outside the ring, resulting in cancellation of the lateral leakage radiation.
Inside the ring, there are reflected (R TE1) TE rays generated from the outer waveguide boundary and transmitted (T TE2) TE rays generated from the inner waveguide boundary. The relative phase between these TE rays depends primarily on the waveguide width. For some waveguide widths, these TE rays can interfere constructively to generate a strong TE field inside the ring. On the other hand, if the waveguide is at a right width, the TE rays inside the ring will be out of phase, resulting in minimization of the TE field inside the ring.
3. Rigorous simulation approaches
To rigorously model the TM-like modes of thin-ridge bent waveguides, a full vectorial mode matching technique  in cylindrical coordinates  was employed. In the mode matching technique, the waveguide cross-section is divided into a number of radially uniform sections. Each section corresponds to a multi-layer slab. In each section, the waveguide mode field was expanded into a superposition of the TE and TM normal modes of the corresponding slab waveguide. The amplitudes of slab normal modes in each section are the solutions of Bessel equations. By matching the fields of these sections at the vertical interfaces between two adjacent regions, the modes of the waveguide can be determined. The propagation loss of each mode was then calculated from the imaginary part of the complex azimuthal propagation constant. The calculated loss includes both the lateral leakage and conventional bending loss.
In the mode matching simulation implementation, to avoid treatment of the continuum of the radiation modes of each section, two perfectly conducting planes were introduced above and below the waveguide [6, 13]. The positions of these conducting planes were chosen to be sufficiently far away from the Si core so that they do not affect the waveguide loss. In order to achieve adequate accuracy, a sufficiently large number of normal modes including guided and radiation modes in both the silica and air claddings must be included in the field expansion. In this paper, 50 pairs of TM and TE normal modes were used in each waveguide section. There is no boundary imposed on the lateral direction and therefore, the mode matching simulation can accurately model the lateral leakage in thin-ridge SOI waveguides.
To validate the mode matching results, the structure was also simulated using a finite element model (FEM)  with cylindrical-perfectly matched layer (C-PML) boundary conditions; developed for full vectorial analysis of an axi-symmetric structure. The FEM model utilized the COMSOL  FEM engine along with its geometry definition, meshing and post-processing features.
4. Numerical results and discussion
This section presents the numerical results of the analysis of the lateral leakage of the TM-like mode in thin-ridge SOI bent waveguides and ring resonators. The refractive indices and waveguide dimensions are presented in Fig. 1(a).
4.1. The impact of waveguide curvature on the leakage cancellation without re-entry
It has been shown in [5, 7, 9] that for straight waveguides, there exist resonant widths at which the generated TE waves from two waveguide boundaries coherently cancel resulting in low propagation losses for the TM-like guided mode. The effectiveness of this leakage cancellation is limited by the imperfect balance of the generated reflected and transmitted TE waves  at the waveguide boundaries. It is expected that this imbalance in the amplitude of the generated TE waves is further enhanced by the waveguide curvature of bent waveguides. The impact of the waveguide curvature on the leakage cancellation at the resonant widths is now investigated.
To isolate the effect of the waveguide width, the secant TE waves inside the ring were absorbed so that they did not re-enter the waveguide region. For the mode matching simulation, this was done by forcing the amplitudes of the TE waves inside the ring to take the form of Hankel functions instead of Bessel function of first kind as in . In the FEM simulation, a C-PML layer was placed inside the ring to absorb the secant TE waves. It should be noted that by preventing the TE waves from re-entering the waveguide region, we are effectively simulating the loss characteristics of a section of a ring resonator i.e. a bent waveguide structure. Figure 2(a) shows the simulated propagation loss of a thin-ridge bent waveguide as a function of waveguide width for different bend radii without considering re-entrant TE waves. The results for a straight waveguide  are also shown in Fig. 2(a) which has a resonant width of 1.43 μm. The results obtained from the mode matching and FEM simulations are in good agreement. The two simulation approaches make significantly different approximations. The FEM simulation assumes a limited simulation domain and a finite, numerically implemented PML. The mode matching simulation considered only a limited number of normal modes of the slab regions. Both simulation approaches produced the same solution, proving that the solution is not a result of either of these approximations and hence is likely to be a valid representation of the actual waveguide behavior. It is clear that all waveguides with different bend radii show the same resonant width which is identical to that of a straight waveguide at 1.43 μm. However, the propagation loss increases as the bend radius is decreased. Figure 2(b) shows the propagation loss calculated for different bend radii with the waveguide width fixed at the resonant width of 1.43 μm. Also shown is the conventional bending loss for these curved waveguide sections. It is evident that the conventional bending losses are too small to account for the simulated total propagation losses in these curved waveguides. Even at a bend radius of 100 μm the conventional bending loss of a bent waveguide with equivalent refractive index contrast is only 0.13 dB/cm which is much smaller than the loss of ~2 dB/cm obtained from the simulations. It is thus evident that the radius of a bent waveguide has a significant impact on the effectiveness of the leakage cancellation at the resonant width. As shown in , the leakage cancellation is effected by the imbalance of the reflected and transmitted TE waves generated from TM-TE mode coupling. For a bent waveguide, the inner and outer waveguide boundaries will have different radii and thus the reflections from these boundaries will differ significantly. This will exacerbate the imperfect cancellation at the resonant width. The amplitude difference between the radiated TE waves increases as the bend radius reduces. Therefore, the leakage cancellation is less effective with smaller radius as seen in Fig.2.
4.2. Leakage cancellation in waveguide rings with re-entry
Referring back to Fig. 1(b), when the reflected TE wave from the outer boundary and transmitted TE wave from the inner boundary traverse a secant across the ring and intersect with the waveguide again, it is possible for these TE waves to interfere destructively with the transmitted TE wave from outer boundary and reflected TE wave from inner boundary resulting in low loss propagation for the TM-like mode. The effect of the secant TE waves inside the ring was simulated by allowing them to re-enter the waveguide region. For the mode matching simulation, this was done by setting the form of the TE slab modes of the inner section to be Bessel functions of the first kind. For the FEM simulation, the C-PML inside the ring was removed and a perfect electric conductor (PEC) boundary was placed significantly far away from the inner boundary of the waveguide so that this PEC boundary does not interfere with the TE waves inside the ring as illustrated in Fig. 3. The waveguide width was fixed at the resonant width of 1.43 μm. Simulations were conducted for rings with radii in a range of ±3 μm around nominal radii R 0=100, 200, and 400 μm. In the FEM, due to the necessity of a large simulation domain and hence proportionately high computational resources and time requirements for accurate modeling of the loss characteristics, simulations were performed only for the R 0 = 100 μm case.
Figure 4 shows the propagation loss of the TM-like mode as a function of the ring radius. Both mode matching and FEM simulations produced very similar results. It can be seen that although the waveguide width is at the resonant width of the straight waveguide, perfect leakage cancellation and hence low loss does not occur at all ring radii. Complete leakage cancellation only occurs at specific “magic radii” for which the resonance of lateral leakage occurs. The resonant effect originates from the destructive interference between the secant TE waves and the TE waves outside the ring. The loss of the TM-like mode shows a cyclic dependence on the ring radius. This behavior has previously been predicted in disks .
The propagation loss shown in Fig. 4 includes both lateral leakage loss due to TM-TE conversion and conventional bending loss. For large rings, at resonant radii, the propagation loss approaches zero due to near-perfect leakage cancellation and negligible bending loss. However, for smaller rings e.g. R 0 = 100 μm, the losses at the resonant radii are limited by conventional bending losses. This effect can be seen clearly in Fig. 4 by comparing the propagation losses at the resonant radii for the R 0 = 100 μm ring case with the conventional bending loss for R 0 = 100 μm. The small difference between the bending loss and the total loss at resonant radii can be attributed to the imperfect leakage cancellation resulting from imbalance in the amplitudes of the reflected and transmitted TE waves for small rings as discussed above.
We now consider the dependence of the lateral leakage loss on both the ring radius and waveguide widths other than the resonant width of the straight waveguide. Figure 5 shows the variation of the propagation loss with waveguide width for a number of ring radii R. Also shown is the loss for a straight waveguide corresponding to the case of infinite radius. Again, good agreement is achieved between the results obtained from the mode matching and FEM simulations. It can be seen that the loss depends strongly on both the waveguide width and ring radius. Consider the loss at a waveguide width of 1.43 μm. This is the optimum width to achieve lowest loss for a straight waveguide (infinite radius). For bent waveguides, the loss has a local minimum at 1.43 μm, but significantly lower losses can be achieved at other waveguide widths.
It is instructive to combine the results of Figs. 4 and 5 into a single three dimensional plot showing the loss as a function of both ring radius and waveguide width as shown in Figs. 6(a) and 6(b) for rings with radius around 200 μm and 400 μm, respectively. The cyclic dependence of propagation loss with ring radii at a particular waveguide width is clearly visible and so is the presence of multiple low loss waveguide widths for a ring with particular radius. For any waveguide width, there exist multiple resonant radii at which the propagation losses are low. It can be seen from Fig. 6 that it is best to design a ring with waveguide width equal to the resonant width of a straight waveguide and then select the ring radius to be one of the resonant radii at this waveguide width. The sensitivity of the propagation loss to waveguide width and radius variations at these resonant radius - resonant width combinations is lowest.
4.3. Mode field distribution
Using the mode matching simulation, the mode profile of the TM-like guided mode was also calculated. Figures 7(a) and 7(b) show the vectorial components of the electric field distributions of the TM-like modes of ring resonators with a waveguide width of 1.35 μm and radius of 199.62 μm (high loss) and 198.95 μm (low loss), respectively. These field distributions of ring resonators are very similar to the field distributions of the TM-like whispering gallery modes of disks presented in . Similar field distributions were also obtained from FEM simulation. Comparing the vertical component (Ex), it can be seen that the TM-like component is almost identical for both ring radii. Similar to straight waveguides and disks, there is a significant radial component (Er) of TE-like mode despite the mode being TM-like. Inside the ring, the radial field components are very similar for both ring radii. At an anti-resonant width (w = 1.35μm), the generated TE-waves inside the ring interfere constructively resulting in a strong radial field inside the ring regardless of whether the mode is high or low loss. Outside the ring, however, there is a significant radial field component only when the mode has high loss. For a ring radius of 198.95 μm which is the resonant radius for 1.35 μm waveguide width, destructive interference of TE waves outside the ring results in the cancellation of the radial field component outside the ring. The azimuthal field component (Eφ) also shows similar radiation behavior as the radial field component.
Next, we consider the field distribution when the waveguide width is at the resonant width of 1.43 μm. Fig. 8(a) and 8(b) show the field distributions of the TM-like modes with radius of 199.84 μm (anti-resonant radius) and 200.46 μm (resonant radius), respectively. Comparing the field amplitudes of the radial and azimuthal components inside the rings in Fig. 7 and Fig. 8, it is clear that at the resonant width, these field components inside the ring are significantly reduced due to the destructive interference of TE waves generated from two waveguide boundaries. However, unlike the case of a straight waveguide, the cancellation is not perfect due to the impact of the waveguide curvature as discussed in Section 4.1. As a result, there are still small TE components inside the ring. At the resonant radius, the radiation field outside the ring is further suppressed resulting in low loss propagation.
4.4. Wavelength dependence of the lateral leakage loss
We have shown that the resonant widths in straight waveguides [7, 9] and resonant radii in disk resonators  have a strong dependence on the wavelength. Therefore, it is expected that the lateral leakage loss of the TM-like mode of a thin-ridge SOI bent waveguide which combines both the phenomenon will also show a strong dependence on the wavelength. Using the mode matching simulator, we calculated the wavelength dependence of the propagation loss for rings with a waveguide width of 1.43 μm (resonant width) and radii of 400 μm and 200.5 μm, and a waveguide width of 1.35 μm (anti-resonant width) and radii of 399.02 μm and 198.9 μm. The chosen radii are the resonant radii for the corresponding waveguide widths at λ= 1.55μm. Figures 9(a) and 9(b) shows the propagation loss as a function of wavelength for waveguides with widths of 1.43 μm and 1.35 μm, respectively. It is evident that the propagation loss depends strongly on the wavelength. The resonant width/resonant radius at a given wavelength may become an anti-resonant width/anti-resonant radius when the operating wavelength changes. Thus, when designing a ring resonator, care should be taken to engineer a ring which exhibits low loss at the required resonant wavelength. The wavelength dependent behavior of the propagation loss also presents an opportunity to eliminate unwanted out of band resonances.
In this paper, the lateral leakage loss behavior of TM-like modes in thin-ridge SOI bent waveguides and ring resonators has been thoroughly analyzed. A simple geometric model has been used to predict new resonant leakage cancellation behavior. These predictions have been verified using rigorous full vectorial mode matching and FEM simulation techniques which gave consistent results. The results of both the simple phenomenological model and rigorous analysis have shown that both the waveguide width and bending radius have significant impact on the propagation loss of the TM-like modes in these structures. The analysis results have also indicated the existence of resonant widths and resonant radii at which the lateral leakage is cancelled resulting in low loss propagation. The optimum propagation loss is predicted for specific, non-trivial combinations of these parameters. Wavelength-dependent propagation loss has also been analyzed.
This work was supported by the Australian Department of Innovation Industry, Science and Research through its International Science Linkage (ISL) program under ISL grant CG130142 and by the Australian Research Council through its Discovery Project scheme under grant DP1096153. RST acknowledges support from the AFOSR MURI on “Electrically-Pumped Silicon-Based Lasers for Chip-Scale Interconnects” under Dr. Gernot Pomrenke and grants from Pennsylvania BFTDA and the Army Research Laboratory Cooperative Agreement.
References and links
1. B. Jalali and S. Fathpour, “Silicon Photonics,” J. Lightwave Technol. 24(12), 4600–4615 (2006). [CrossRef]
2. M. A. Webster, R. M. Pafchek, G. Sukumaran, and T. L. Koch, “Low-loss quasi-planar ridge waveguides formed on thin silicon-on-insulator,” Appl. Phys. Lett. 87(23), 231108–231110 (2005). [CrossRef]
3. T. L. Koch, R. M. Pafchek, and M. A. Webster, “Fabrication of Optical Waveguides,” US Patent App. 20060098928, (2006).
4. R. Pafchek, R. Tummidi, J. Li, M. A. Webster, E. Chen, and T. L. Koch, “Low-loss silicon-on-insulator shallow-ridge TE and TM waveguides formed using thermal oxidation,” Appl. Opt. 48(5), 958–963 (2009). URL http://ao.osa.org/abstract.cfm?URI=ao-48-5-958. [CrossRef] [PubMed]
5. A. A. Oliner, S.-T. Peng, T.-I. Hsu, and A. Sanchez, “Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part II-New Physical Effects,” IEEE Trans. Microwave Theory Tech. 29(9), 855–869 (1981). [CrossRef]
6. K. Ogusu, “Optical strip waveguide-A detailed analysis including leaky modes,” J. Opt. Soc. Am. 73, 353–357 (1983). [CrossRef]
7. M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photon. Technol. Lett. 19(6), 429–431 (2007). [CrossRef]
8. R. S. Tummidi, T. Nguyen, A. Mitchell, and T. L. Koch, “Anomalous Losses in Curved Waveguides and Directional Couplers at “Magic Widths”,” in 21st Annual Meeting of the IEEE Lasers and Electro-Optics Society, pp.521–522, (2008).
9. T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous Modeling of Lateral Leakage Loss in SOI Thin-Ridge Waveguides and Couplers,” IEEE Photon. Technol. Lett. 21(7), 486 (2009). [CrossRef]
10. T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Lateral leakage in TM-like whispering gallery mode of thin-ridge silicon-on-insulator disk resonators,” Opt. Lett. 34(7), 980–982 (2009). URL http://ol.osa.org/abstract.cfm?URI=ol-34-7-980. [CrossRef] [PubMed]
11. T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Thin-Ridge SOI Disk and Ring Resonators with “Magic Radius” and “Magic Width” Phenomena,” in IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2009 (Baltimore, MD, 2009).
12. A. S. Sudbo, “Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides,” J. Opt. A, Pure Appl. Opt. 3, 381–388 (1994).
13. L. Prkna, M. Hubálek, and J. Ctyroký, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11(1), 217–223 (2005). [CrossRef]
14. J. Jin, The finite element method in electromagnetics, 2nd ed. (Wiley-IEEE press, 2002).
15. Comsol, Inc. URL http://www.comsol.com.