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We report the first experimental study of the effects of coupling phase dispersion on the spectral response of a two-dimensionally coupled quadrupole micro-racetrack resonator. Negative coupling in the system is observed to manifest itself in the sharp stop band transition and deep extinction in the pseudo-elliptic filter response of the quadrupole. The results demonstrate the feasibility of realizing advanced silicon microring devices based on the 2D coupling topology with general complex coupling coefficients.
© 2015 Optical Society of America
1. Introduction
In integrated photonic circuits, power transfer between two adjacent waveguides is typically accomplished by evanescent field coupling. For short coupling lengths the field coupling coefficient between the two waveguides, as defined in the conventional coupled mode theory [1], is positive. For such a coupler, the coupled electric and magnetic fields have a π/2 phase lag with respect to the uncoupled fields. All integrated photonic circuits, including coupled optical microring filters [2,3], are typically designed using positive coupling elements.
An emerging application that requires negative field coupling is the realization of two-dimensionally (2D) coupled microring resonators for synthesizing the spectral responses of a class of optical filters known as pseudo-elliptic filters [4,5]. These types of filters are characterized by transfer functions with transmission nulls in the stop band, resulting in frequency responses with very sharp stop band transitions and high frequency discrimination. It is well known that conventional microring filters based on the serial coupling configuration (also known as CROW filters) have transfer functions containing only poles and no transmission zeros. These filters have slow stop band transitions and poor frequency discrimination since the spectral response never reaches perfect extinction in the stop band. By contrast, in a 2D microring coupling topology, light follows multiple paths through the device which can interfere destructively with each other at the output to produce transmission nulls at certain frequencies. The use of 2D resonator coupling topologies can greatly expand our ability to engineer the spectral responses of optical devices [4,5]. However, a major barrier to the practical implementation of 2D coupled microring filters is that to realize transmission nulls, the topology requires negative coupling elements between adjacent microrings, which are difficult to realize in integrated optics.
In a directional coupler with negative coupling, the coupled fields acquire a π/2 phase lead with respect to the uncoupled fields. The concept of negative coupling is quite well known in microwave engineering where a common approach using standing-wave resonators is to engineer the stationary electric and magnetic field patterns in the vicinity of the coupling junction to obtain negative coupling [6]. Since microring resonators are travelling-wave resonators, the fields inside the resonators are not spatially stationary so it is not possible to engineer the field pattern at the coupling junction between two microrings to obtain negative coupling in this manner. An exception to this is that in a 2D microring coupling topology, the fields in the resonators have a fixed phase relationship with each other, making it possible to engineer the relative phases of the coupled fields so that they acquire a π/2 phase lead with respect to the uncoupled fields, thereby resulting in an effective negative coupling. In fact, in general it is possible to engineer the relative phases of the fields of the coupled resonators to obtain an effective complex coupling coefficient , where δ is the coupling phase. This technique has been employed to realize negative coupling in a quadrupole microring resonator by shearing the topology by 1/8 of a wavelength [5]. An experimental demonstration of the structure has also been attempted in [7] but due to fabrication errors and other experimental uncertainties, the effect of negative coupling was not clearly evidenced in the device spectral response, so that ambiguity remains as to whether negative coupling was achieved in the system.
In this paper we experimentally investigated the effects of frequency dispersion of the coupling phase on the spectral response of a quadrupole resonator. Clear evidence of complex and negative couplings is observed as the coupling phase changes from 0 to π. The quadrupole resonator used consists of four coupled micro-racetrack resonators in which the positions of the coupling junctions can be adjusted to achieve a wide range of coupling phases in the system. In order to experimentally observe the effect of the coupling phase, we designed the quadrupole to have the spectral response of a 4th-order pseudo-elliptic filter and investigated how the transmission null response of the device varies with frequency. A crucial advantage of the proposed racetrack coupling topology over the sheared microrings in [7] is that it enables a wide range of coupling phase values to be realized so that the effects of frequency dispersion of the coupling phase on the device response could be experimentally observed. Measurements of the fabricated quadrupole resonator on silicon-on-insulator (SOI) showed that the stop band transition of the spectral response became increasingly steepened as the coupling phase approaches π, thereby providing for the first time unambiguous evidence of the effect of negative coupling on the device response.
2. Device design
The quadrupole resonator used in the experiment consists of four micro-racetracks arranged in the coupling configuration shown in Fig. 1(a). Racetracks 1 and 4 are also coupled to an input and an output bus waveguide, respectively, with input signal si applied to the input port and transmitted signals st and sd measured at the through port and drop port, respectively. The field coupling coefficient between microrings i and j is denoted by κi,j. The coupling coefficients to the input and output bus waveguides are assumed to be equal and denoted by κ0.
Fig. 1 (a) Schematic of a quadrupole micro-racetrack resonator. (b) Target spectral responses of the quadrupole pseudo-elliptic filter (black and grey solid lines). The drop port response of the quadrupole filter with positive κ14 value (blue dashed line) and a 4th-order CROW filter (red dashed line) are also plotted for comparison.
The transfer function at the drop port of the quadrupole resonator has the general form [8],
where K is a constant, is the roundtrip delay variable, φ is the roundtrip phase of the racetracks, and pi and zi are the poles and zeros, respectively. The above transfer function can be used to realize the spectral response of a pseudo-elliptic filter with equiripples in both the pass band and stop band [4]. Using the microring filter design method in [8], we designed the quadrupole to have a pseudo-elliptic response with 0.1dB ripple in the pass band, a 3dB bandwidth of 150GHz and a free spectral range (FSR) of 600GHz. The resulting device transfer function has four poles located at and , and two transmission zeros that are placed at ± 120GHz on both sides of the pass band to obtain a stop band rejection of over −30dB. The coupling values obtained for the racetracks in the quadrupole are κ0 = 0.82, κ12 = κ34 = 0.477, κ23 = 0.454, and κ14 = −0.10. The negative coupling value of κ14 results from the placement of the two transmission zeros on the unit circle in the z-plane (or equivalently, the jω-axis in the complex frequency s-plane) in order to achieve sharp transmission nulls.Figure 1(b) shows the spectral responses at the drop port (black line) and through port (grey line) of the designed quadrupole filter. To show the effect of the negative coupling coefficient κ14 on the device response, we also plotted the spectral response at the drop port of the same device but with positive value for κ14 (κ14 = 0.10, blue dashed line). It is apparent that the transmission nulls responsible for the sharp stop band transitions of the pseudo-elliptic filter disappear when the phase of κ14 changes from π to 0. For comparison, we also show in Fig. 1(b) (red dashed line) the drop port response of a 4th-order CROW filter designed to have the same bandwidth and pass band ripple as the pseudo-elliptic filter. It is evident that the CROW filter has a much slower stop band transition than the 2D quadrupole resonator.
To realize the negative coupling element κ14 in the quadrupole system we adjust the positions of the coupling junctions of the racetracks to vary the relative phases of the coupled fields in the resonators. In the design shown in Fig. 2(a), the coupling junctions κ14 and κ23 are located at the midpoints of the straight waveguide sections of racetracks 1 and 3, but the positions of the coupling junctions κ12 and κ34 are deviated from the midpoints of the straight sections of racetracks 2 and 4 by an amount ∆L = (L2 – L1)/2. To obtain an exact relation between the offset ∆L and the phases of the coupling elements, we transform the racetrack quadrupole into an equivalent coupled waveguide array [9]. This is done by “cutting” the micro-racetracks at the positions indicated by the red lines in Fig. 2(a), and unfolding them into straight waveguides while keeping track of the positions of the coupling junctions. The resulting straight waveguide array is shown in Fig. 2(b), where coupling between two waveguides is indicated by the line connecting them. The lines corresponding to κ12 and κ34 are skewed due to the unequal racetrack lengths caused by the coupling junction offsets.
Fig. 2 (a) Implementation of the quadrupole pseudo-elliptic filter using racetrack resonators. The red lines indicate the position where each racetrack is “cut” and unfolded to obtain the equivalent coupled waveguide array in (b). (c) Scanning electron microscope image of the fabricated quadrupole micro-racetrack resonator in silicon.
The spectral response of the quadrupole resonator system is obtained by applying the transmission matrix method to the coupled waveguide array. Denoting as the array of fields in the racetrack waveguides, we trace the transmitted fields through the coupled waveguide system to obtain the following equation for a in terms of the roundtrip delay variable z:
In the above equation I is the 4 × 4 identity matrix and , where , is the matrix representing coupling to the input and output waveguides. The array accounts for the input excitations of the resonators and M1 and M2 are the waveguide coupling matrices given by where . In the matrix M1 the phase δ associated with coupling elements κ12 and κ34 is , where nr is the effective index of the racetrack waveguides. Thus the effect of offsetting the coupling junctions κ12 and κ34 by ∆L is to impart a phase shift δ to these coupling elements, effectively making them complex. These coupling phases are relative to the phases of other coupling elements in the quadrupole and are not unique, since it is always possible to apply a similarity transformation to Eq. (2) to obtain new coupling matrices. For example, by applying the similarity transform to Eq. (2), we obtain , where is the matrix M1 in Eq. (3) but with zero coupling phase (δ = 0) for κ12 and κ34, and the new matrix is given byNote that the above transformation does not alter the spectral response of the quadrupole since it preserves both the poles (or eigen-frequencies) and zeros of the system. In the new coupling matrix , the coupling element κ14 now acquires a phase of 2δ. For κ14 to be negative, we require that at the center resonance wavelength λ0 of the quadrupole,where m is an integer. From Eq. (6) we obtain as the offset length which will result in an effective negative value for the coupling element κ14 in the quadrupole.3. Device fabrication and experimental results
We implemented the quadupole filter design on an SOI substrate with a 340nm-thick Si layer and a 1µm-thick SiO2 buffer layer. The waveguide width was 300nm and the device was designed to operate in the TM mode. The racetracks had a bending radius R = 12µm and straight sections of length L = 12µm. The input and output bus coupling value (κ0 = 0.82) was achieved using an input/output coupling gap of 200nm and a coupling length of 1µm. The coupling gaps between racetracks were designed to be g12 = g34 = 240nm (κ12 = κ34 = 0.477), g23 = 250nm (κ23 = 0.454), and g14 = 450nm (κ14 = 0.10). To realize an effective negative coupling value for κ14, we set the coupling junction offset ∆L = 4.9µm, which corresponds to m = 7 at the center wavelength λ0 = 1.55µm in (6). We intentionally chose a fairly long ∆L value so that the phase 2δ varies from 0 to 2π over the 1.5-1.6µm wavelength range, thereby enabling us to observe the effect of coupling phase dispersion on the spectral response of the quadrupole over this wavelength range. The device was fabricated using electron beam lithography and reactive ion etching. Figure 2(c) shows a scanning electron microscope (SEM) image of the fabricated device, which was left air-cladded. The chip was cleaved to expose the input and output waveguide facets for butt-coupling to lensed fibers.
Figure 3(a) (lower panel) shows the measured spectral responses at the drop port (red line) and through port (blue line) of the quadrupole over the C-band. The measurement data have been filtered using a minimum phase technique to remove the Fabry-Pérot ripples in the spectral responses caused by reflections from the chip end facets [10]. To verify that the quadrupole behaves as designed, we performed curve fitting of the measured spectral response using the coupled waveguide model given by Eq. (2). The best curve fits of the drop port and through port responses over the C-band are shown in Figs. 3(a) and (b) (green and black dashed lines), from which excellent agreement between the model and the measured spectral responses can be seen over the broad wavelength range of study. The best curve fits were obtained by setting the coupling coefficients to be 17% larger than designed values to account for the smaller coupling gaps (about 20%) of the fabricated device due to etching bias. To account for loss in the resonators we set the roundtrip amplitude attenuation coefficient of each racetrack to be 0.97, which is similar to the loss value in silicon microring devices fabricated using the same process [11]. The model also indicated that the resonance frequencies of racetracks 2, 3 and 4 were slightly detuned from the first racetrack by ∆ƒ2 = −10GHz, ∆ƒ3 = 35GHz, and ∆ƒ4 = 15GHz, respectively.
Fig. 3 (a) Top panel: coupling phase (2δ) of the quadrupole as a function of wavelength. Bottom panel: measured and fitted spectral responses of the quadrupole filter. Blue and red solid lines are the measured through and drop port responses; green and black dashed lines are the fitted responses. (b) Zoomed-in view of the transmission bands at 1551, 1556 and 1561nm showing the effect of coupling phase dispersion on the spectral response. The grey dash-dotted line is the response when the coupling phase is zero. (c) Comparison of the measured pseudo-elliptic filter response to simulated responses of a 4th-order CROW filter for the ideal case (pink line) and when the device has the same resonance mismatches as the fabricated quadrupole (green line).
From the plots in Figs. 3(a) and (b), it is seen that near 1.55μm, the drop port response of the quadrupole exhibits the steepest stop band transition with a deep extinction of more than −30dB. Away from the 1.55μm wavelength the filter shape degrades gradually, exhibiting slower roll-off and shallower stop band extinction. We emphasize that such a rapid change in the spectral response over a short wavelength range is typically not observed in a CROW microring filter and is here attributed to the dispersion of the coupling phase in the quadrupole. As shown in the top panel of Fig. 3(a), around 1.55μm the phase 2δ of the coupling element κ14 is close to π, leading to the formation of the transmission null which is responsible for the observed sharp band transition and deep extinction. Away from the 1.55μm wavelength, the coupling phase approaches 0 or 2π causing the positions of the transmission zeros to move away from the jω-axis in the complex frequency domain (or equivalently, away from the unit circle in the z-plane) so that the transmission nulls become less sharp and eventually disappear. As a result, the filter roll-off becomes increasingly slower and the device response approaches that of a quadrupole resonator with positive coupling coefficients as shown by the grey dash-dotted line in Fig. 3(b).
In Fig. 3(c) we compare the experimental performance of the 2D quadrupole resonator to the simulated response of a 4th-order CROW filter subject to the same conditions of loss and overcoupling as the fabricated quadrupole. The pink line shows the CROW response for the ideal case of no resonance mismatch while the green line is the response when the device has the same resonance mismatches as the fabricated quadrupole. It is seen that in both cases the CROW filter has a slower skirt roll-off than the pseudo-elliptic filter. More specifically, the slope roll-off rates for the pseudo-elliptic filter and the CROW filter at their respective 10dB-bandwidth points are 0.39dB/GHz and 0.23dB/GHz, respectively. In addition, we find that the CROW filter exhibits much more sensitivity to resonance mismatch than the 2D quadrupole device. Specifically, in the presence of the same resonance mismatches, the CROW filter suffers from severe degradation in the pass band, while the 2D quadrupole still retains its relatively flat-top transmission and sharp roll-offs. The robustness of the 2D coupling topology can be attributed to the fact that light can travel multiple pathways through the device so the transmission is less susceptible to individual resonance detunings.
4. Conclusion
In conclusion, the effects of coupling phase dispersion on the spectral response of a quadrupole micro-racetrack resonator were experimentally investigated for the first time. Clear evidence of negative coupling was observed through the sharp band transitions and deep extinction ratio in the resonator spectral response near the 1.55μm wavelength. This work also provides the first demonstration of a pseudo-elliptic optical filter based on the 2D resonator coupling topology on the SOI platform. Both the proposed racetrack coupling topology and the experimental performance achieved help lay the groundwork for realizing a new class of microring devices based on 2D coupling topologies with complex coupling coefficients.
In the filter design presented in this paper we chose a fairly long ΔL length in order to capture the effect of coupling phase dispersion over the measured wavelength range. In practical applications, however, a smaller ΔL value should be used so that the coupling phase does not vary much over the wavelength range of interest to ensure that the filter response maintains its shape and sharp roll-off across many FSRs. Finally, since silicon microrings are known to be susceptible to fabrication induced phase errors and thermal sensitivity, mismatch in the microring resonances can be corrected for and thermally stabilized by using a post-fabrication tuning method, such as thermo-optic tuning using microheaters placed on individual microring resonators.
Acknowledgments
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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