This paper addresses the polarization sensitivity issue of micro-ring resonators by proposing a novel design of an MMI-coupled resonator with substantially reduced or zero polarization sensitivity, while maintaining singlemode and low-loss conditions. The design is based on polarization-independent, single-mode waveguide obtained by using a judicious combination of critical ridge width and etch depth. The design is limited to relatively large resonators having small FSR (free spectral range). For the first time, it gives designer a handle on the intrinsic polarization-dependent characteristics of waveguide microresonators.
© 2004 Optical Society of America
Waveguide ring resonators are versatile, wavelength-selective elements that can be used to synthesize a wide class of filter functions . Many compact microring resonators have been demonstrated in a number of semiconductor materials [2–4]. The extremely compact size of these micro-resonators is made possible by the use of strongly confined single-mode ridge waveguides with a width of 0.5 µm or less. These narrow ridge waveguides are strongly polarization dependent, resulting in unpredictable and non-overlapping resonance wavelengths and filter transfer functions for TE- and TM-polarized input lights [4,5]. This polarization dependence presents a problem for application in fiber-optic networks in which the optical signal polarization input to the device is random. In this paper we explore the theoretical possibility of designing the resonator as an intrinsically polarization-independent (PI) device, as opposed to extrinsic solutions that require additional devices for polarization management, such as polarization diversity . Although early part of this work is alluded to in , this approach has never been reported in the literature.
The primary parameter of a resonator is the resonance wavelengths λo, given by the equation Lrt =, where L rt is the round trip length of the resonator, are the effective indices for the TE or TM modes, and m is the order of the resonance. Thus, to realize an intrinsic PI resonator the fundamental step is to make the effective index identical for both TE and TM. Furthermore, the waveguide must be single-mode to avoid excitation of multiple-order resonance spectra. The next step is to shape the resonator reflectance (R) and transmittance (T) functions to be identical for TE and TM. This requires a balanced cavity where the coupling factors (i.e., the fractional power transfers between the resonator and the coupled waveguides) are fine-tuned around a given round-trip loss .
2. Type I single-mode and polarization-independent waveguide
We consider a Type I waveguide  which has strong lateral optical confinement due to a very high index contrast, but weak vertical confinement provided by a conventional waveguide layer structure with a small index step. For this waveguide, the effective indices of the fundamental TE and TM modes are plotted as a function of the ridge width in Fig. 1, for a particular wavelength λ=1.55 µm. At the critical width where the mode is circular the effective indices become equal . Similar behavior occurs for curved waveguide, except the critical width is larger, increasing with decreasing radius. Therefore, to design a polarization-independent resonator, the underlying waveguide must be at the critical width, where, however, it will be multi-mode. Moreover, as the critical with depends on wavelength and temperature, polarization independence will hold only over a small wavelength range and if the temperature is stabilized.
A single-mode waveguide is required for the resonator since different modes will have different sets of resonance spectra. The multi-mode waveguide can be made effectively single-mode by using the ridge height (or the etch depth) as a control parameter to eliminate the higher-order modes through differential leakage loss. This is based on the fact that the leakage losses for the higher-order modes increase much faster with decreasing etch depth than those for the fundamental modes. However, if the etch depth is too shallow, then the leakage loss for the fundamental mode will also become too high. For a curved waveguide the differential loss is even greater, and the etch depth will depend on the bend radius as well. The optimum etch depth is determined by the acceptable loss for the fundamental mode and the required differential loss for the first-order modes.
Figure 2 shows leakage losses as a function of etch depth for the fundamental and the higher-order modes of a waveguide at the critical width (i.e., βx=βy) and with a bending radius of 30µm. To satisfy, for example, the criterion that there should be less than 1 dB/mm leakage loss for the fundamental mode and at least 10x larger loss for the first-order EX1 and EY1 modes, an etch depth between 2.0 and 2.1 µm is needed to make the waveguide single-mode. Likewise, if the bending radius is 40 µm, the required etch depth would be between 1.9 and 2.0 µm. In general, the smaller the bending radius, the larger is the required etch depth.
There is, however, a critical radius below which no etch-depth can meet the differential loss criterion, as losses for the fundamental modes also become unacceptably large. For the waveguide structure under consideration this occurs when the bending radius is smaller than 30 µm. As long as the radius is greater than 30 µm, the bending losses for the fundamental modes are less than 1 dB/mm (0.001 dB/µm). The polarization-dependent loss (PDL) is also insignificant.
In summary, a ring waveguide, with the proper combination of critical width, critical etch depth, and ring radius, can be a polarization-independent, low-loss and single-mode waveguide resonator. The only limitation is that the resonator radius must be greater than 30 µm for the waveguide structure considered.
3. Polarization independent coupling
Given a polarization-independent (PI) single-mode ring resonator, the final step in realizing a PI waveguide-coupled resonator filter is to design a PI coupler. This coupler may be realized with either a directional coupler  or a multi-mode interference (MMI) coupler , and it is critical for the coupler to be compact. However, based on the wider ridge waveguides used for the resonator, the directional coupler will require a very small (or zero) gap size in order to yield reasonable coupling over a very short length. It may also not be easy to achieve a polarization-independent coupling length.
A better alternative to directional coupler is the 2×2 multi-mode interference (MMI) coupler which has better fabrication tolerance and is polarization insensitive . It has been used for several race-track resonators , and an analysis has been presented in . There are two possible types of MMI based on different self-imaging mechanisms, as shown in Fig. 3(a). The Type I MMI is based on general interference and is characterized by the smallest possible MMI width, W mmi=(D+w), where w is the given access waveguide width, and D the center-to-center separation between the access waveguides. For this MMI, cross coupling occurs at p(3Lπ ), and 3-dB coupling occurs at , where p is an integer, and Lπ is the beat length given by 
where nr is the effective index of the slab waveguide, λo is the operating wavelength, and Weq is the equivalent MMI width. Type II MMI is based on restricted interference and is characterized by the configuration W mmi=3D. The coupling length is given by pLπ , while the 3-dB coupling length is given by . Using Eq. (1), the ratio of the coupling lengths for Type I and Type II MMI is then approximately given by . Therefore, if we assume w=1.5 µm and D=2.0 µm, then both types of MMI will have about the same Lc even though they have quite different widths. Other important split ratios, such as 85:15 and 15:85, are found to occur only when W mmi=2D. We call this Type III. The MMI lengths required to give the various split ratios are summarized in Table 1.
Of these, only the 50:50 and 85:15 split ratios are of interest to the resonator. It can be seen that the latter ratio requires much longer length, and so will not be considered as it will make the resonator round-trip length very large. For the 3-dB coupler, Type II is somewhat better than Type I as the difference in L π between TE and TM is smaller. Figure 3(b) shows the power split ratios for both TE and TM as a function of wavelength, for a Type II 3-dB MMI with a width of 6 µm and a length of 52 µm. For this particular MMI, the coupling factors (i.e., the cross-over fraction) for TE and TM are nearly the same over the wavelength range of interest. The insertion loss is about 0.3 dB for TE and 0.2 dB for TM.
4. Polarization independent resonator filter
The final design of the PI resonator consists of two half-circles, with a radius of 40 µm, connected to two 3-dB MMI couplers via short straight sections. The overall cavity length (L c) is 360 µm. Due to mode mismatch, the transition between the half-circle and the straight connector section leads to additional loss, resulting in a total loss of 0.5 dB for TE and 0.4 dB for TM for each half-circle (along with the two connectors). These transition losses are more severe the wider the ridge waveguide, and may be reduced by using offsets . Including the losses of 0.3 dB for TE and 0.2 dB for TM per MMI, the total losses per round trip are therefore 1.6 dB (~30%) for TE and 1.2 dB (~24%) for TM, giving a PDL of 0.4 dB. Note that these losses are quite significant, hence the use of 3-dB couplers giving 50% coupling factor is appropriate. These losses can be reduced but no attempt is made to do so as the focus is on the polarization dependence.
The reflectance (through port) and transmittance (drop port) spectra (in dB) for this resonator are shown in Fig. 4. The results are obtained using Apollo APSS2 circuit simulator , in which the individual components are simulated and their input-output relations (S-parameters) used to calculate the overall transfer functions, R and T. The main features of these spectra are discussed below.
First, the resonance wavelengths for TE and TM modes are nearly aligned, with a discrepancy of only 0.05nm which is 10% of the 3-dB resonance bandwidth. This small mismatch is caused by the residual polarization dependence in the whole device. The wavelength shift due to birefringence is given by Δλ=λo(Δn/n eff), so only Δn=10-4 is needed to give a Δλ of 0.05 nm. Second, because of the relatively large cavity length, round-trip loss and coupling factor, the free spectral range is only about 1.8 nm, the FWHM linewidth is about 0.5 nm, and the on-off ratio for T is about 10. The contrast ratio, the finesse, and other features such as the minimum value of R (R min), and the maximum value of T (T max), are determined by the coupling factor and the round-trip loss . All the simulation results are consistent with the coupling factor of about 0.5 and the round-trip loss values given above.
In summary, we have shown that it is possible to use polarization-independent, single-mode and low-loss waveguides, together with polarization-independent MMI 3-dB couplers, to construct micro-resonators that have substantially reduced polarization sensitivity, while maintaining single-mode and low-loss conditions. This theoretical result represents a large improvement over what can normally be achieved with the usual non-PI resonator . It also represents a significant departure from the conventional designs based on single-mode sub-wavelength waveguides. The design concept suggests that, in principle, even the small residual birefringence can be eliminated, and perfect TE and TM resonance matching may be achieved, by more careful design such as balancing the birefringence in the access waveguides with that in the MMI, and minimizing the transition losses at the curved-straight interfaces using offsets .
Although the design is applicable only to relatively large resonators, which will have relatively small FSR and finesse, the spectral tuning is not limited by the FSR. It has been demonstrated that very wide spectral tuning may be achieved by using the vernier architecture where two filters having slightly different FSR are cascaded back to back . In addition, the frequency linewidth can be reduced by using a higher-order filter consisting of a number of mutually coupled rings . In this regard, rings with relatively large radius are advantageous as they are easier to fabricate and control. Finally, post-fabrication waveguide trimming or active wavelength tuning may be required to fine-tune the resonance wavelengths . Such tuning may be inevitable for all types of waveguide resonators, but it will be easier for the larger resonators, especially if there is much less birefringence to begin with.
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