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We propose a new device configuration that incorporates a nested ring with a Mach-Zehnder interferometer. The nested ring is analogous to a dual-bus coupled ring resonator, with the ends of the two buses connected to form a semi-closed loop. With proper design of the length of the U-shaped loop, as well as the coupling coefficient between the ring and the waveguide, the device is capable of generating a box-shaped spectral response. This is shown to be mainly due to the double-Fano resonances that arise from constructive interference between the nested ring and the outer loop. The device is simple in that it requires only one ring, and unique in that it harnesses a pair of Fano resonances to generate a reasonably box-like filter response. The analysis is based on the transfer matrix formalism, and compared and verified with the FDTD simulations.
©2007 Optical Society of America
1. Introduction
Ring resonator (RR) is a versatile photonic building block with applications for filtering [1, 2], sensing [3] and switching [4, 5]. Two basic configurations of ring resonator are shown in Fig. 1(a), showing a single ring coupled to one and two bus waveguides, respectively. The fundamental design parameters are the ring cavity length (L c) and the coupling coefficient (κ) between the ring and the bus. In this paper, we introduce a modified single-ring building block that has interesting phase and transmission properties and applications. We shall call it the nested ring resonator (NRR) as it looks like a ring nested by a U-shaped waveguide. It is equivalent to the dual-bus ring with the two buses connected by a loop with an arbitrary length Lv, which is expressed as a multiple of the inner-ring circumference Lv =v(2πR). A key result of this paper is that the NRMZI can give a box-like transmission when v is an integer or half-integer. The loop feedback path provides an alternative route for the propagation of light to that through the ring. The interference between two pathways is required to generate Fano resonances as have been demonstrated in other ring-based structures [6, 7]. Thus, the nested ring is expected to exhibit a periodic phase response with sharp nonlinear phase changes. This phase response can be translated to a power spectrum with sharp filter-like response when the nested ring is coupled to one arm of a Mach-Zehnder interferometer. We study the dependence of the power spectrum on the length of the feedback loop and the coupling coefficient between the ring and the bus waveguide, and show that the nested ring MZI (NRMZI) can exhibit a box-like transmission spectrum with sharp transition edges, and thus is highly desirable as a “digital” sensor or switch as well as a filter.
Fig. 1. (a). Single-bus and dual-bus coupled ring resonator; (b) The NRR. The dashed line refers to the outer feedback arm with a length of Lv = v(2πR) where R is the radius of the inner-ring.
In section 2 we present a theoretical formulation of NRR based on the transfer matrix formalism. In section 3 we discuss the unique properties of NRMZI and some potential limitations. In section 4, we compare the transfer matrix formalism with the finite-difference time-domain (FDTD) simulation. Lastly, we consider the effects of loss and sensitivity to fabrication variations in Section 5.
2. Theory of the nested ring resonator (NRR)
For a ring resonator coupled to one bus waveguide, the transfer function is given by the reflectivity ρ =∣ρ∣ exp(iφ) , which is defined as ρ = b 1/a 1, where b 1 is the output field and a 1 is the input field [8, 9]:
Here, r is the reflectivity of the coupler, which is related to the coupling coefficient between the ring and the bus waveguide by κ=(1-r 2)1/2; a ≡ exp(-αLc/2) is the round-trip amplitude transmission factor, and δ=ωneffLc/c is the round-trip phase, Lc is the round-trip length of the ring, α is the power loss coefficient, and neff is the waveguide effective index. As a filter, δ contains the frequency dependence, and is also known as the normalized frequency as it can be written in the form δ = 2πf/FSR = 2π(m + Δf /FSR), with Δf = f- fo is the frequency detuning, fo is the resonance frequency, and FSR = c /(neffLc) is defined as the local free spectral range. Most of the interesting features of the resonator occur near resonance where δ = 2πm. For switching and sensing applications δ shifts in response to a change in neff, hence a transmission function exhibiting a sharp response with δ is desirable.
For a RR coupled to two bus waveguides, the two outputs are defined as the reflection R and the transmission T, which are given by
where t = ∣t∣ exp(iφt), ρ =∣ρ∣ exp(iφρ). The corresponding phase terms are given by
One effective approach to analyze the NRR is to consider the dual-bus coupled ring as a “black box” with reflection and transmission coefficients, ρ and t given by Eqs. (2) and (3), respectively. The U-loop connecting the two buses has a length of vLc. Hence, by summation, the total transmission is given by,
To focus on the phase behavior, we first consider the lossless case where a = 1 and the nested ring behaves as an all-pass filter with phase-only response. In this case it can be shown [9] that ∣ρ∣2+∣t∣2=1 and(ρ 2 - t 2)/ ρ = 1/ ρ*, and with φρ - φt = π/2 from Eq. (4), the transmission of the NRR can be simplified to,
Hence the phase response of the NRR may be written as φNRR = φt + φload, where
The effect of feedback is embodied in ϕload (denoted as the loading term), which is similar to the phase response of a single-bus RR, Eq. (1b), with the modifications δ → vδ+φt and r →∣t∣. This implies that the outer feedback loop behaves like a single-bus RR with respect to the inner dual-bus RR (DBRR). The resonance in the outer loop depends on the transmission ∣t∣ of the DBRR, and increases with the coupling factor κ2. The inner ring resonance, on the other hand, increases with the reflectivity (r). Since r and κ are complementary, it is impossible to have resonance in both loops simultaneously. As shown in Fig. 1, when r is low, the phase response is dominated by the loading phase, which is highly nonlinear with a step-like feature, where each step represents a 2π phase shift caused by resonance in the outer loop. On the other hand, when r is large, the phase response is dominated by the resonance in the inner ring.
The phase term (vδ+φt) in φload contains primarily the factor (v + 1/2)δ, which implies: (a) if v is an integer (v = m), then the period of φload is 4π, and the number of possible resonances in the outer loop (corresponding to the number of steps in the phase response) is 2m in each period; (b) if v is a half-integer (v = m-1/2), then the period of φload is 2π, and the number of resonances per period is m. Only these two cases are of interest, and two examples are shown in Fig. 3. For other values of v, the power spectrum would be irregular with relatively large periodicities.
Fig. 4. The NRR transmission showing the critical coupling (a = 0.95) and (b) the oscillation (a = 1.1) conditions for the case v = 5.5, and for various values of r. The corresponding pole-zero diagrams and the contour plots of TNRR on the r-δ plane are shown on the right. The zeros are red, the poles are blue, and the arrows indicate the direction of movement as r is increased from 0 to 1. The dashed lines in the contour plots correspond to the transmission curves.
In the more general case where a ≠ 1, the transmission is no longer all-pass and depends critically on the value of a. In particular, for a < 1, specific combinations of r and a can result in the numerator of Eq. (5) becoming zero at specific values of δ, leading to the critical coupling condition similar to that in a single-bus ring [10]. Likewise, for a > 1 (which is possible if an optical amplifier is incorporated in the ring cavity [11]), specific combinations of r and a can result in the denominator of Eq. (5) becoming zero at specific values of δ, leading to the oscillation condition [8]. As an illustration, Fig. 4 shows the transmission for an NRR with v = 5.5 for (a) a < 1 and (b) a > 1. The zeros and poles corresponding to critical coupling and oscillation, respectively, occur in pairs and move closer to δ= 0 as r increases. The resonances at δ = 2πm correspond to the inner-loop, whereas other resonances correspond to the outer-loop. The inner-loop resonance requires a very high r value as compared to the outer-loop resonances, which is consistent with the phase response shown in Fig. 2. The zeros and poles can be displayed on a pole-zero plot as a function of r, as discussed in detail elsewhere [8] and shown on the right of Fig. 4. There are 6 pole-zero pairs for v = 5.5. As r increases the zeros (poles) move outward (inward) from the unit circle, except for the pair nearest to the zero angle which give rise to the inner-ring resonance at δ = 0 when r is sufficiently large. Critical coupling (oscillation) occurs when the zeros (poles) cross the unit circle. For stability, all poles must lie within the unit circle [12]. The resonances indicated as “unstable” correspond to poles which fall outside the unit circle.
3. Nested ring Mach-Zehnder interferometer (NRMZI): Theory and results
To convert the phase response of the nested ring to a power spectrum the nested ring may be incorporated into a balanced Mach-Zehnder interferometer, as shown in Fig. 5. The bar and cross outputs of a lossless NRMZI are given by,
When r = 1, φNRR = vδ since no light is coupled into the ring as if the ring is not present, so both arms of the MZI are identical.
The bar transmission is shown in Fig. 6 for various values of r. A smaller r results in steeper phase nonlinearity, which gives rise to the shar p asymmetric Fano resonances that define the sharp edges of the passband. However, the sharp Fano resonances are inevitably accompanied by ripples in and outside the passband which are undesirable for a bandpass filter. Increasing the value of r can flatten the ripples but at the expense of the band edge roll-off. This trade-off is an inherent limitation of this device, which fundamentally relies on the double Fano resonances to generate the box-like output.
To investigate the origin of the Fano resonances, we analyze the behavior of the device in terms of the field build-up in each resonant loop. In the notations of Fig. 1(b), the inner-loop and outer-loop build-up factors are defined and given, respectively, by
The extra term +κ 2 a v+1/2 e i(v+1/2)δ in the denominator indicates the effect of the outer-loop ‘coupling’ based resonance. For the inner-loop we have defined the build-up factor as B 31 instead of the conventional circular intensity B 21 since the latter is still a part of the outer-loop branch.
Fig. 6. The bar transmission of NRMZI for various r values, showing the double Fano resonances giving rise to a box-like transmission profile (a = 1; v = 1.5).
Fig. 7. Left: The build-up factor of the inner (B 31) and the outer loop (B 61) of NRR. Right: The corresponding bar transmission of NRMZI (a = 1; v = 1.5). The r value is varied showing 3 different modes of operation: (i) MZI-like mode; (ii) Box-like NRMZI mode; (iii) Lorentzian DBRR mode.
Figure 7(a) shows the build-up spectra and the corresponding MZI output spectra for different values of r. In general, when r is sufficiently large, B 61 is flat and greater than B 31 except at the ring resonance, where B 31 dominates and the NRR behaves like the drop output of a dual-bus RR which has a Lorentzian profile. On the other hand, when r is low, B 31 is flat and greater than B 61 except at the Fano resonances where B61 rises sharply, giving rise to the sharp asymmetric spikes in the MZI output. This shows that the double-Fano resonances in the MZI output are related to the resonant build-up in the outer-loop. For an intermediate value of r (e.g., 0.65), the peaks in B 61 are broadened and hence the sharpness of the Fano resonances is reduced, while B 31 remains quite flat in between, thereby resulting in a flatter and rounder MZI output profile which is considered as a better approximation to a “box-like” filter. A more optimized r value corresponds to the situation where the power is more evenly distributed between the outer and the inner rings. In Fig. 8, we show that only for v = m or v = m-1/2 is the build-up factor uniformly distributed over the round-trip phase δ, and for other values there is a chirping effect in the distribution of the build-up factor.
Fig. 8. The build-up factor for two cases of v values (a = 1; r = 0.65), showing a more evenly distributed build-up factor when v = m or v = m-1/2, and chirped build-up factor when v ≠ m or v ≠ m-1/2.
The bandwidth of the filter is determined by the separation between the Fano resonances which, according to the phase response of Fig. 4, is primarily dependent on v; increasing v reduces the bandwidth. The higher the v value, the narrower and sharper would be the Fano resonances if the r was fixed at a small value. Hence, to give a flatter box-like response, the optimal value of r would have to be higher for larger values of v. This is verified in Fig. 9(a) which shows some examples of optimized box-shaped response and the corresponding parameters. The higher the v value, the narrower and sharper are the transmission bands. To minimize the sidelobes while preserving the box-shaped response, a cascaded configuration of identical NRMZI can be adopted. Two stages are sufficient to suppress the sidelobes reasonably, as illustrated in Fig. 9(b).
The box shaped response of NRMZI is distinct from that given by the ring-enhanced MZI (REMZI), which consists simply of a ring resonator coupled with the MZI. In the case of REMZI, the response is Lorentzian at resonance when the MZI is balanced, while a single asymmetric Fano resonance can be generated by applying an appropriate bias Δϕbias on the other arm of the MZI [6]. In the ideal case of NRMZI where the bare MZI is balanced, no bias is necessary and the nested ring generates a mirror pair of Fano resonances that define the box-like passband. Although the NRMZI filter is far from ideal compared with synthesized high-order filters using an array of ring resonators, the latter are far more complex than the NRMZI, and their designs require complex synthesis and apodization techniques [2, 10-12]. The novelty of NRMZI lies in its simplicity. To our knowledge, this is the first time that Fano resonances generated from a single ring are shown to be able to tailor a box-like filter with reasonable performance.
Fig. 9. (a). The optimized “box-shaped” response for various combinations of r and v for a single-stage NRMZI, assuming a =1. (b) The bar transmission of a cascaded NRMZI showing suppressed sidelobes (v = 1.5; r = 0.65). The inset shows the double-stage NRMZI configuration.
4. Comparison with FDTD Simulations
To verify the accuracy of the transfer matrix formalism we compare the results with the one obtained using the finite-difference time-domain method (FDTD), a method which is able to calculate the fields everywhere in a device. For the FDTD simulations, the radius of the microring-resonator is taken to be 1.7 μm, the waveguide width is assumed to be 0.4 μm, the core-cladding index contrast is 3:1, and the gap between the ring and the bus is designed to give a nominal coupling of r = 0.6. The fields in the inner and outer rings are ‘measured’ at each wavelength and normalized by the input spectrum to give B 31 and B 61, respectively. A Gaussian light pulse of 15fs pulse width with near-resonant input wavelength is launched from one of the waveguides. In our simulation, we chose a grid size less than λ/20n and set v = 1 to save computation time. As shown in Fig. 10(a), the build-up spectra are similar to those shown in Fig. 7(a) obtained by the transfer matrix formalism. The insets show the FDTD field distributions at the wavelengths corresponding to the two peaks and the one dip in B 61. The dashed curves show the results of the transfer matrix formalism, which are in good agreement with FDTD. To obtain these results, we have taken into account the wavelength dispersion using the expression n eff(λ) = 3.1334 – 0.2874λ, for 1.7μm < λ < 1.8μm, which is obtained by numerically solving the n eff for various λ. Furthermore, to achieve good agreement the ring radius is increased to 1.717μm, consistent with the fact that the mode in the ring is slightly skewed outward from the waveguide axis.
Fig. 10. Comparison between the 2-D FDTD simulation and the analytic transfer matrix formalism (for the case v = 1) in terms of: (a) The outer-loop build-up factor B 61 and the inner-loop build-up factor B 31. The insets show the field distributions at three different points of the spectrum as indicated. (b) The bar transmission of NRMZI. The inset show the field distribution where the inner ring is on resonance (B 31 is maximum).
In Fig. 10(b) the bar transmission of the NRMZI obtained by both FDTD and transfer matrix formalism are compared. The inset shows the field distribution where B 31 dominates the NRMZI operation. The Tbar spectrum is very similar to the B 31 spectrum. Indeed, the flattop feature in Tbar is closely related to the flatness in B 31 between the two peaks in B 61. The slight sidelobe imbalance in the FDTD simulation is caused by the phase imbalance between the two arms of the MZI, as well as the phase imbalance between the inner ring and the outer loop, due to the perturbation of the ring in the upper arm. The field distribution at the NRR coupler shows that there is a slight phase mismatch between the two propagating bending modes in the inner and the outer loops. This implies that the effective index is not exactly the same for the inner and outer loops, and hence v is not exactly 1, and that there is effectively a slight imbalance between the bare MZI arms. Indeed, a very good match with the transfer matrix formalism is obtained when slight deviations in the scale factor (v) and in the ratio of the two arms [denoted γ in Fig. 10(b)] are taken into account. The excellent agreement with FDTD gives us confidence that the transfer matrix formalism provides a reliable analytical approach to this device.
5. The effect of loss and fabrication variations
In this section we discuss briefly the effect of loss and sensitivity to fabrication variance. The effects of loss on the NRMZI are shown in Fig. 11. It can be seen that even a small loss has a significant effect on the device performance. Therefore, the NRMZI designs are practical only if low loss waveguides can be realized. Recently, a very high order multi-ring filters with very high Q resonators have been realized using low-loss Hydex material [13]. For polymer microring devices a thermal-reflow technique has been used to greatly reduce scattering loss, with an experimental loss reduction of ∼74 dB/cm reported [14]. An ultra-high quality silicon-on-insulator (SOI) microring resonator with a Q of 139,000 has been reported [15]. All these imply that it is possible to achieve a round-trip amplitude transmission factor of a > 0.99, and hence feasible to realize such devices.
The previous section shows that a small deviation in the scale factor and an imbalance in the MZI arms can have significant effects on the transmission. Hence, we discuss the sensitivity to fabrication fluctuation by considering the effect on transmission of a variation of the order of 50 nm in the outer loop length as well as in the lower arm of the MZI. Figure 12 shows the bar transmissions in the worst case, corresponding to (a) maximum deviations in v given by v ± = (Lv ± ΔL)/Lc, where ΔL = 50 nm, and (b) maximum deviations in the lower MZI arm length. We see that such deviations shift the resonance frequency and distort the box-like transmission by giving a slant to the passband and an imbalance in the sidelobe. The sidelobe imbalance gives rise to a crosstalk penalty, which is 3.2 dB due to the imbalance between the MZI arms and 1.6 dB due to the imbalance in the NRR. This shows that the performance is more sensitive to the MZI arm imbalance and that the fabrication tolerance for this device is less than 50 nm. One remedy is to add external bias (i.e., thermal film heater or electrical bias) at the NRR and the lower MZI arm to tune the NRMZI into the desired performance.
Fig. 12. The NRMZI sensitivity to a 50-nm length deviations in: (a) the length Lv variations of NRMZI around v = 1 assuming balanced bare MZI; (b) the length variations in the lower arm of MZI assuming v = 1. All parameters not mentioned in the legends are extracted from Fig. 10(b).
6. Conclusion
We have proposed for the first time a nested ring-loaded MZI which is capable of producing a box-like spectral response, and is potentially useful as a filter, modulator or sensor. The boxlike output is generated by the double Fano resonances caused by the constructive interference between the nested ring and the outer loop. The harnessing of a pair of Fano resonances to generate a reasonably box-like filter response makes the NRMZI unique, and distinct from other similar devices such as the ring-enhanced MZI (REMZI) [6, 7]. The sharpness of these Fano resonances depends on the length of the outer loop (which is an integer or half-integer multiple of the inner loop), the reflectivity of the coupler, and the round-trip loss in the cavity. Although its performance is still far from an ideal box-shaped filter, it is simple to design and requires only one ring, in contrast to other high-order filters that require an array of resonators and complex synthesis and apodization techniques [2, 12-13]. We have presented a transfer matrix based analytical approach to the design which is in excellent agreement with FDTD simulations.
Acknowledgment
This work is supported in part by a Singapore Ministry of Education grant (RG12/03). One of the authors (S.D.) would like to thank Singapore Millennium Foundation (SMF) for granting a scholarship.
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