1. Introduction

A resonator can serve as a channel-dropping filter [1]. In this device, input light containing several wavelengths will normally bypass the resonator and exit in the same waveguide. Only wavelengths that satisfy the resonance condition in the cavity will be “dropped” through the resonator to the other output waveguide. In analogy with a Fabry-Perot interferometer, the straight-through output is denoted the reflection port, and the “drop” output the transmission port. The free-space resonance wavelengths (λ_o) are given by the resonance condition, L _c=mλ_o/n_e (where m is an integer, L_c is the cavity length, and n_e is the effective index), and the free spectral range (FSR) of the filter is determined by the cavity round trip length, FSR=${\mathrm{\lambda }}_{\mathrm{o}}^2$/(n _e L _c) (in nm) [2].

In order to pick out a unique resonant wavelength, the FSR must be greater than the WDM signal bandwidth, which is typically of the order of 10 nm. For an FSR of 10 nm, the resonator round-trip length is of the order of 50λ_o, assuming λ_o is around 1550nm. Such a small-radius ring can only be realized by using strongly-confined waveguides. These waveguides have a deeply etched ridge surrounded by air resulting in a very high refractive index contrast, typically of 3:1 for InP or GaAs semiconductors. The strong lateral confinement allows the waveguides to be bent at small radius without significant leakage loss. These waveguides have unique properties, unlike conventional, weakly guided waveguides. The fundamental properties of these waveguides pose unique challenges for device applications that go beyond fabrication issues. One dominant issue for micro-resonators has been the polarization sensitivity of these waveguides. Polarization sensitivity limits the applications of optical devices in fiber-optic communication systems, in which the polarization states of input optical signals to the photonic devices may change randomly as a function of time.

The micro-resonator filter exhibits quite different spectral characteristics for TE and TM polarizations. This sensitivity arises from the polarization dependence of the coupling factor, which affects the resonance linewidth, as well as the propagation constant and loss in the waveguide, which affect the resonance wavelength and magnitude (or transmittance).

2. Directional coupling

Figure 1(a) shows the schematic of a race-track resonator, and Fig. 1(b) shows the SEM image of an actual device fabricated by electron-beam lithography and inductive-coupled plasma reactive-ion etching [3]. The resonator is defined by the 2.5-µm deep trenches etched into a GaAs/AlGaAs waveguide structure. The single-mode waveguide has a width of 0.4 µm, and the gap is 0.2 µm.

 

Fig. 1. (a) Schematic, and (b) SEM image, of a race-track shaped resonator.

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In a “race-track” resonator, the coupling sections form directional couplers with a coupling factor determined only by the length l, while the gap width g is fixed. The coupling factor is given by [4]:

where l _c is the coupling length, i.e., the distance required to couple 100% of the power from one waveguide to the other. For the case of a waveguide-coupled resonator with a cavity length of 31 µm and l=6 µm, the transmission spectrum from the crossover output waveguide for the TM polarization is shown in Fig. 2. On the other hand, for the TE case, the resonances are much weaker and indistinct. A switching efficiency (or maximum transmission) of 80% and a finesse of 20 were achieved in the TM case. From these values, the analysis discussed in [3] shows that L _rt=0.03 and P _c=0.13, where L _rt=1-exp(-αL_c ) is the round-trip loss and α is the loss coefficient. From the determined value of P _c, and l=6 µm, we obtain l _c~25 µm according to Eq. (1).

 

Fig. 2. The output spectrum for the TM polarization, measured at the transmission port of a race-track resonator. The FSR is 20 nm. The output is normalized by the input power. The variation in peak height is primarily due to the variation in power coupling with wavelength.

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Theoretical simulation using a full-vectorial finite difference mode solver [5] shows strong polarization dependence for directional couplers using sub-wavelength waveguides and gap sizes. Fig. 3 shows the coupling lengths as a function of gap size and waveguide ridge width. Note two unique features: (1) the coupling length can be very short, as small as 10 microns. (2) The coupler can be used as a polarization filter: being TM-selective for w>400nm and TE-selective for w<250nm.

 

Fig. 3. Calculated coupling length as a function of the gap size and the waveguide width (w), for both TE (dotted) and TM (solid) polarizations

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This means that, for the case w=0.4µm, the coupling length is much shorter for TM than for TE. At g=0.2µm, we note that l _c=25 µm, which is in agreement with the experimental results. On the other hand, for TE, l _c=70 µm, which implies that P _c is much smaller. The difference in resonance behavior between TE and TM is attributed to the fact that P _c>L _rt for TM whereas P _c<L _rt for TE. For the resonances to be pronounced, the fractional power coupled into the resonator must be greater than the cavity round-trip loss [2].

To further verify this result we fabricated directional couplers (DC) of various lengths, starting with as small as 3 µm, which is shown in Fig. 4(a). The fractional power transfer from one waveguide to another was measured for each DC and plotted as a function of coupler length, with w=0.4 µm and g=0.2 µm, in Fig. 4(b). Note that full power transfer occurs at approximately l _c=25 µm, again consistent with the calculated value.

 

Fig. 4. (a) SEM image of a 3-µm coupler, (b) measured results of power coupling fraction as a function of the coupler’s length.

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Fig. 3 also shows a point where the TE and TM curves cross, at w=0.3 and g=0.2 µm, and so the corresponding directional coupler is polarization balanced (i.e., no preference for TE or TM). Indeed, polarization independent DC’s can be designed with the right combinations of w, g, the refractive index of the waveguide, and that of the surrounding medium. Figure 5 shows a beam propagation simulation for an example of this DC with l _c<20 µm. However, even if the directional coupler is polarization independent, the waveguides themselves may still be polarization sensitive.

 

Fig. 5. Polarization-independent coupler: beam propagation simulation of an ultra-small directional coupler with l _c=18 µm for TE and TM modes. The numbers shown are the powers at the two output waveguides. About 2.5 dB is lost to radiation.

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3. Propagation constants

The polarization dependence of the propagation constant is a geometrical birefringence due to the asymmetric shape of the waveguide and the fact that the index difference between the guiding and cladding regions along the vertical direction is much smaller than that along the lateral direction. As the waveguide width shrinks below a wavelength, the waveguide modes are hybrid, combining features of both TE and TM, but can be described as quasi-TE or quasi-TM [6]. Figure 6 shows the calculated effective indices for the fundamental (quasi-)TE and TM modes as a function of waveguide width (w), based on the InP/InGaAsP/InP waveguide structures. Note that below some critical width (in this case w=1.5 µm), TE has a lower effective index than TM, while above that critical width, the TE effective index is higher, as is normally the case for planar waveguides. In the single-mode region (w<0.5 µm) the waveguides are inevitably strongly polarization dependent, with n_TM significantly larger than n_TE. This behavior applies also to AlGaAs/GaAs/AlGaAs waveguide structures.

 

Fig. 6. Calculated effective indices for quasi-TE and TM modes as a function of ridge waveguide width. The insets show mode profiles at various ridge widths, and the waveguide structure. D, the waveguide core thickness, is 0.5 µm.

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This behavior can be understood from the changing mode profiles as the width is varied, as shown in the insets of Fig. 6. For example, when the waveguide is very narrow, the mode is more elongated in the Y direction. This means the propagation constant of the TE mode (with dominant electric field along X-direction) will be lower than that of the TM mode (with dominant electric field along Y-direction). The opposite applies for sufficiently large widths, which resemble slab waveguides. The critical width, then, corresponds to the point where the mode profile is nearly symmetrical to both TE and TM polarizations, so that the propagation constants of the TE and TM modes will be equal, thereby leading to a polarization independent waveguide in the absence of polarization-dependent loss. Similar behavior is true in curved waveguides, as in the case of micro-resonators, except that the critical width is also dependent on the bending radius, increasing as the radius decreases and the optical field shifts more to the edge of the ridge waveguide.

4. Towards a polarization-independent resonator

Several methods have been invented to reduce or eliminate the polarization sensitivity for guided-wave optical devices in general. Such methods as polarization control, polarization scrambling, and polarization diversity involve additional components and increase the complexity of the devices. The ideal solution is therefore a polarization independent waveguide design.

At the critical width where the birefringence is smallest, the waveguide may be polarization-independent but it may also be multi-mode, and thus its use for the ring resonator is not feasible as high-order modes can be excited. One potential solution is to make the multi-mode waveguide effectively single-mode by shedding the higher-order modes through waveguide leakage loss while maintaining negligible loss for the fundamental mode. A suitable design criterion may be that there should be less than 1dB/mm loss for the fundamental mode and at least 10x larger loss for the first-order mode. This can be achieved by a judicious choice of the etching depth for the ridge, based on the fact that the leakage losses for the higher-order modes increase much faster with decreasing etch depth than that for the fundamental mode (somewhat similar to [7]). Thus, for a given bending radius, it may be possible, by proper choice of the ridge width and the etching depth, to design a polarization-independent, single-mode, and low-loss waveguide.

 

Fig. 7. Polarization independent single-mode ridge waveguides for different bending radii. The waveguides are defined by the critical width and the etching depth, and satisfy the criteria β_X=β_Y for fundamental Ex mode and Ey modes, and less than 1dB/mm loss for the fundamental mode and at least 10x larger loss for the first-order mode.

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Figure 7 shows the combinations of critical width and etch-depth that meet the polarization independent criterion for the case of a straight waveguide and several bending waveguides, based on the same InP/InGaAsP waveguide structure. Some general observations based on these simulations are summarized below:

For the typical waveguide structure used in our micro-resonators, the critical ring radius is on the order of 30 µm, much larger than the radii of micro-resonators used thus far that are designed for large FSR. In other words, polarization-independent waveguides based on this approach seem to work only for relatively large resonators with radius greater than 30 µm. The implication for small resonators (i.e., radius<30 µm), therefore, is that they may not simultaneously satisfy the requirements of polarization independence and large FSR, and thus may have limited applications. Practical applications of micro-resonators may require architectures that do not require the devices to have large FSR, so that the resonators may be relatively large in size. Larger resonators have other advantages as well, such as being less sensitive to dimensional variations and easier to control.

Finally, even if the resonator waveguide is single-mode and polarization independent, the waveguide-coupled resonator filters are polarization insensitive only if the coupling between the resonator and the input and output waveguides is also polarization independent. This may be achieved by using multi-mode interferometers (MMI) as couplers instead of waveguide coupling [8].

In conclusion, we have characterized the polarization dependence of a waveguide-coupled ring micro-resonator, explained it in terms of the coupling and waveguide properties, and discussed a possible direction for the design of single-mode polarization-independent micro-resonators based on a polarization-independent waveguide. Further work in this direction is needed.