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Coupled microresonators exhibit great potential for nonlinear applications. In the present work, we explore the nonlinear performance of an embedded ring resonator analogous to an electromagnetically induced transparency (EIT) medium, also known as coupled resonator induced transparency (CRIT). Interestingly, an EIT-like amplitude response can have a remarkably different power enhancement factor that varies by more than one order of magnitude, which is attributed to the different phase regimes of the embedded micro-ring resonators. In addition to the non-monotonic phase profile reported in atomic EIT systems, the phase responses featuring 2π and 4π monotonic transitions are identified and analyzed. We also present an interesting phenomenon, in which the power enhancement changes greatly, even with the same transfer function (both intensity and phase responses). This reveals that wisely choosing the operating regime is critical to optimize nonlinear performance of the embedded double resonator system, without adding to design or fabrication difficulty.
©2013 Optical Society of America
Corrections
Xiaoyan Zhou, Lin Zhang, Andrea M. Armani, Raymond G. Beausoleil, Alan E. Willner, and Wei Pang, "Power enhancement and phase regimes in embedded microring resonators in analogy with electromagnetically induced transparency: erratum," Opt. Express 21, 28414-28414 (2013)https://opg.optica.org/oe/abstract.cfm?uri=oe-21-23-28414
1. Introduction
Integrated microring resonators are promising optical devices to realize various photonic functions [1, 2], such as on-chip lasers, modulators, filters, logic gates, and biological sensors [3–8]. An intriguing effect in coupled resonators has recently been identified, which is analogous to electromagnetically induced transparency (EIT) in atomic systems [9] and is referred to as coupled resonator induced transparency (CRIT). The EIT-like effect is found in various coupled ring structures, such as parallel rings [10–12], serial rings [13, 14], 3 × 3 coupler-connected rings [15], and embedded rings [16, 17]. The on-chip realization of this effect provides a simplified system configuration and increases design flexibility as compared to atomic systems, enabling new opportunities for optical delay lines [11, 14, 16], switches and modulators [12, 16], and sensing platforms [17]. For nonlinear applications, the large power enhancement in the optical cavity is critical. The embedded rings [16, 17] can offer a cascaded power enhancement and small footprint (Fig. 1) as compared to other types of double-ring structures.
Fig. 1 (a) Schematic of the embedded rings. (b) and (c) Equivalent split-models of power distribution in Case I (m1 - m2 = odd number) with t2 = t4.
In previous theoretical and experimental investigations into the EIT-like effect, the majority of the research effort has focused on studying the intensity response of the coupled resonator system. To gain a thorough understanding of this behavior, studying only the intensity response is not sufficient. Moreover, many applications strongly depend on the phase characteristics, such as reducing the speed of light [11, 14, 16], encoding multi-level phase information [18, 19], and enhancing optical nonlinearity [20, 21]. However, the EIT-like effect is typically not characterized in terms of the phase response.
Here, we present a detailed study of both intensity and phase responses of the EIT-like effect in embedded ring resonators shown in Fig. 1(a). It is found that the EIT-like intensity response features several distinctly different phase profiles. Only one of them is observed in atomic EIT phenomena. The newly identified phase regimes exhibit different power enhancement factors. By judicious selection of phase, the power enhancement factor can be tuned by more than one order of magnitude. Intriguingly, some embedded configurations have the same intensity and phase responses but significantly different power distributions. This reveals that examining both intensity and phase responses is not sufficient to predict the nonlinear characteristics of the coupled resonator devices.
2. Modeling and classification
The embedded rings are shown in Fig. 1(a), and consist of an outer ring (Ring 1) and an inner ring (Ring 2) coupled with each other in two coupling regions. The outer ring is coupled with two waveguides. The amplitude coupling coefficients in the four coupling regions are labeled t1, t2, t3, and t4, respectively. The azimuthal mode orders of Ring 1 and Ring 2, m1 and m2, are integers so that the two rings have the same resonance wavelength. The transfer functions at the through and drop ports are derived using coupled mode theory [22]. We temporarily disregard optical loss for simplicity without losing generality as it is added in Section 5. The transfer functions are:
where ri is related to ti by ri2 + ti2 = 1 (i = 1, 2, 3, or 4); e1 and e2 are phase factors between two neighboring coupling regions in Ring 1 and Ring 2, respectively. That is, e1 = exp(φ1/4) and e2 = exp(φ2/2), where φj is the round-trip phase shift in the jth ring and is related to mj by φj = 2mjπ (j = 1 or 2).The EIT-like effect is obtained in the through port, so here we only focus on Eq. (1). Due to the complexity of Eq. (1) and the flexibility of tuning the parameters in it, it is critically important to classify the operating regimes of the device. We first look at the shared resonance wavelength. It is noted that there are four combinations of phase factors in Eq. (1): e22, e14, e14e22 and e12e2. The first three terms always equal 1, since m1 and m2 are integers. The last term, e12e2 = exp(φ1/2 + φ2/2) = exp[2π(m1 + m2)/2], is determined by the difference of m1 and m2: (I) when m1–m2 = odd number, (m1 + m2)/2 = M + 0.5 (M is an integer), i.e., e12e2 = −1; (II) when m1–m2 = even number, (m1 + m2)/2 = M, i.e., e12e2 = 1. In this way, embedded ring systems generally degenerate into two cases, depending on (m1–m2).
As a representative example, we set m1 = 101 and m2 = 70 for Case I, and m1 = 100 and m2 = 70 for Case II. A device comprised of a silicon nitride (Si3N4) waveguide with cross-section of 700 nm × 800 nm (height × width) is considered. The corresponding effective refractive index is 1.87 at a resonance wavelength of 1.55 μm, as determined using a finite-element mode solver. Under this configuration, Ring 2 has a radius of approximately 10 μm.
3. EIT-like effect in Case I: m1–m2 = odd number
Although a high-level classification of the device is made by differentiating phase factors (Case I or II), there are still many parameter combinations in terms of coupling coefficients. Here, we first set the same coupling between the two rings, i.e., t2 = t4 = 0.2 in Fig. 1(a).
When the ring-waveguide coupling coefficients are t1 = 0.3 and t3 = 0.4, we obtain an EIT-like intensity response and a monotonic phase profile, as shown in Figs. 2(a) and 2(b). The phase profile features a sharp change of 2π near resonance wavelength. Switching t1 and t3 results in the same intensity response, yet changes the phase profile in Fig. 2(d). This behavior mimics a similar phenomenon in a single ring. When the operating regime is shifted from under-coupling to over-coupling though an exchange of coupling and loss, its intensity response remains the same, while the phase profile changes dramatically [20]. Figures 2(c) and 2(e) show the power distributions in the system on the resonance, corresponding to Figs. 2(b) and 2(d), respectively. In both cases, the optical power accumulates considerably in the inner ring.
Fig. 2 Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case I, with the same ring-ring coupling coefficients, i.e., t2 = t4 = 0.2. (b) and (c) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4. (d) and (e) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3.
To more quantitatively compare the power accumulation between geometries and among different operating conditions, it is useful to define a power enhancement factor as the ratio of the intra-cavity power in Ring 2 to the input power. In both cases described above, the power enhancement factor is more than a few hundred. However, the power enhancement in Fig. 2(c) is nearly twice that in Fig. 2(e), which is highly desirable for enhanced optical nonlinearity.
The observed cascaded power enhancement in the embedded rings can be explained as follows. The cause of the cascaded power enhancement can be determined by evaluating the optical power evolution at four places, labeled A, B, C, and D in Fig. 1(a), near the coupling regions between the two rings. We assume the initial phase of light at A is 0. The optical wave partially enters into Ring 2, travels from B to C, and eventually is coupled out to D, with a phase shift of π + 35⋅2π. The other portion of light travels in Ring 1 from A to D and obtains a phase shift of π + 50⋅2π. The two optical waves are in phase and interfere constructively. Then we consider the light travelling back from Ring 2 to A after one circulation. With a phase shift of π + 70⋅2π, it is out of phase with the initial light at A, resulting in a destructive interference. Since t2 = t4, eventually optical power builds up only in the lower part of Ring 1 at steady state, and there is almost no light in the upper part, as shown in Figs. 2(c) and 2(e). Noting this fact, we describe the power distribution in a more straightforward way by regarding the power distribution as a summation of two parts, as shown in Figs. 1(b) and 1(c). The inner ring can be viewed as a single ring critically coupled with two waveguides since loss is ignored, while the double-ring system can be treated as a side-coupled single resonator. Power accumulation at resonance is described as follows:
From Eq. (4), the power enhancement in the lower part of Ring 1 is directly obtained, while the power enhancement factor defined as PRing2-lower/Pin can be derived by multiplying Eq. (3) and Eq. (4). When t1 = 0.3 and t2 = t4 = 0.2, the power enhancement factor is calculated to be 1061, in excellent agreement with the result in Fig. 2(c). When t1 is switched with t3 as in Fig. 2(e), the power enhancement is decreased to 551. With loss ignored, reducing t2 (or t4) or t1 will result in increased power enhancement. However, in practice, the power enhancement is limited due to optical loss. It is noted that, although the double-ring system seems to be under-coupled in Figs. 2(b) and 2(c) with in-coupling t1 less than out-coupling t3, it is indeed over-coupled because very little light leaks out from the upper part of Ring 1 in the complicated power distribution in the embedded rings.Next, we study the EIT-like effect with asymmetric coupling between rings by setting t2 = 0.2 and t4 = 0.17, while keeping the ring-waveguide coupling coefficients unchanged: t1 = 0.3 and t3 = 0.4. An EIT-like intensity response is obtained in Fig. 3(a), which has a normalized peak value less than unity even in the lossless case. Featuring an anomaly, the according phase profile in Fig. 3(b) is dramatically different from the 2π phase transitions above. The power enhancement in the system also drops greatly, as shown in Fig. 3(c), as compared with those in Fig. 2. By setting different t2 and t4 values, we note that the EIT-like resonance in Fig. 3(a) has a wider peak than that in Fig. 2(a). This increase is because the inner ring is no longer critically coupled, and optical power flows through the upper part of Ring 1 and leaks out from the waveguide. Exchanging t1 and t3 leads to the same intensity profile with a marked change in phase response, as shown in Fig. 3(e), which has a 4π shift in the center. The phase profiles in Figs. 3(b) and 3(e) are in direct analogy with the phase responses in under- and over-coupled single resonator. This can be explained by viewing the two rings as a single resonator. That is, a double-ring resonator is coupled with two waveguides, with in-coupling t1 and out-coupling t3. Figure 3(b) and 3(e) are corresponding to under- and over-coupled double-ring resonator. The power distribution for Fig. 3(e) is shown in Fig. 3(f). The power enhancement factor in Fig. 3(f) is 232, which is higher than that in Fig. 3(c), i.e., 176, because of the over-coupling of the double-ring resonator.
Fig. 3 Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case I, with different ring-ring coupling coefficients, i.e., t2 ≠ t4. (b), (c), and (d) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4, where t2 and t4 are switched in (c) and (d). (e), (f), and (g) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3, where t2 and t4 are switched in (f) and (g).
So far, from Figs. 2 and 3, one can see that an EIT-like intensity profile obtained from embedded rings exhibits dramatically different power enhancement factors, which are accompanied by four different phase responses. Thus, identifying the phase characteristics is critically important to determine the optimal operation of the device for nonlinear applications.
Intriguingly, when t2 and t4 are switched, i.e., t2 = 0.17 and t4 = 0.2, while both the EIT-like intensity profile and phase profile are unchanged, the power distribution is different, with the enhancement factor reduced to 18 and 73, as shown in Figs. 3(d) and 3(g). Conventionally, it is believed that once the intensity and phase responses are determined, the linear properties of a device, such as filtering or waveform manipulation of signals, are fully characterized. However, the present analysis shows that the device may not be completely described. For example, the optical power enhancement in the device could change by more than one order of magnitude, which is of great interest to nonlinear operation of the device and provides important design criteria for resonator-based nonlinear devices.
We examine the spectrum evolution as t2−t4 increases from 0 to 0.012 and 0.03, as is illustrated in Fig. 4, with t1 = 0.3, t3 = 0.4. Note, to accurately reflect experimentally realizable values, we set loss = 0.5 dB/cm here. The EIT-like profile is obtained at t2−t4 = 0. When t2−t4 = 0.012, the transmission changes to a profile featuring electromagnetically induced absorption. As t2−t4 further increases to 0.03, the spectrum changes back to an EIT-like spectrum with a larger linewidth. The transmission on resonance varies with t2−t4, as shown in Fig. 4(b), where the transmission first drops quickly from 0.47 to almost 0, and then increases to 0.8. Figure 4(b) also illustrates that the largest power enhancement is obtained when t2 is slightly larger than t4 to compensate for optical loss in Ring 2.
Fig. 4 (a) Transmission spectrums with different value of t2−t4. (b) Changes of transmission and power enhancement with t2−t4 at 1.55 μm.
4. EIT-like effect in Case II: where m1–m2 = even number
In Case II, mode splitting has been reported in [16] when the coupling strength between the embedded rings is strong. We find that one can still observe the EIT-like effect with very small t2 and t4. Here, we set t2 = 0.015 and t4 = 0.005, and to be consistent with Case I, the ring-waveguide coupling coefficients are: t1 = 0.3 and t3 = 0.4. In this configuration, the EIT-like intensity profile and non-monotonic phase profile are obtained, as shown in Figs. 5(a) and 5(b). The corresponding power distribution is depicted in Fig. 5(c). Similar to the results in Fig. 3, when t1 and t3 are switched, the EIT-like intensity response remains the same, while the phase profile is shifted to a monotonic one with a 4π change in the center, as shown in Fig. 5(e). The corresponding power enhancement in Fig. 5(f) is almost twice that in Fig. 5(c). Then we exchange t2 and t4, and different power distributions with unchanged transfer characteristics are obtained in Figs. 5(d) and 5(g), respectively. This phenomenon is similar to that in Fig. 3, though the changes of power enhancement are much smaller. Unlike Case I, the regime of the transfer function and the power enhancement is unchanged whether t2 equals t4
Fig. 5 Transfer characteristics of EIT-like effect and power distributions at resonance wavelength in Case II, with different ring-ring coupling coefficients, i.e., t2 ≠ t4. (b), (c), and (d) have the same ring-waveguide coupling coefficients: t1 = 0.3 and t3 = 0.4, where t2 and t4 are switched in (c) and (d). (e), (f), and (g) also have the same ring-waveguide coupling coefficients: t1 = 0.4 and t3 = 0.3, where t2 and t4 are switched in (f) and (g).
When the coupling strength between the rings is strong, the EIT-like effect evolves into the mode splitting effect, as illustrated in Fig. 6(a). Here, we set loss = 0.5 dB/cm, and t2 = t4 for simplicity. When t2 (or t4) increases, the two notches in the transmission spectrum move apart from each other, which eventually results in a profile featuring the mode splitting effect [16]. Changes in the peak transmission and power enhancement with t2 (or t4) are shown in Fig. 6(b). As t2 (or t4) decreases, the power enhancement first increases to the maximum value and then drops down, indicating that the peak of the EIT-like effect cannot be arbitrarily narrow due to loss. Note that, in the case of mode splitting, although the transmission in the center wavelength almost reaches unity, there is very little power accumulation. That is, the embedded rings are not resonant [23]. This is also a significant difference between the EIT-like effect and mode splitting.
Fig. 6 (a) Transmission spectrums evolving from the EIT-like effect to mode splitting as t2 (t4) increases. (b) Changes of transmission and power enhancement with t2 (t4) at 1.55 μm.
5. Discussion on the role of loss
Here, we examine the influence of optical loss on EIT-like peak value of the intensity response and power enhancement in the inner ring, as shown in Fig. 7. As examples, we choose Case IA and IB corresponding to Figs. 2(c) and 2(e) with t1 and t3 switched. For Case II, we have a wider range of examples, i.e., Case IIA and IIC corresponding to Figs. 5(c) and 5(f) with t1 and t3 switched, and Case IIB and IID corresponding to Figs. 5(d) and 5(g) with t1 and t3 switched. Note that Case IIA and IIB, Case IIC and IID are also related, with t2 and t4 switched.
Fig. 7 Changes of transmission and power enhancement with loss for Case I and Case II. (a) and (c) Case I: parameters in Case IA and Case IB correspond with configurations in Figs. 2(c) and 2(e), respectively. (b) and (d) Case II: parameters in Case IIA, Case IIB, Case IIC, and Case IID correspond with configurations in Figs. 5(c), 5(d), 5(f), and 5(g), respectively.
Figure 7 shows that both the transmission at the EIT-like peaks and the power enhancement for all types decreases monotonically with loss. Comparing Figs. 7(a) and 7(c), we note that the quicker decrease of transmission for Case IA is accompanied by its stronger power enhancement in contrast to Case IB. This relationship between transmission and power enhancement holds true for Case II as well, as shown in Figs. 7(b) and 7(d), which can be explained as follows. When more power accumulates in the cavity, the output power, i.e., summation of optical powers at the drop port and through port, is less due to the intra-cavity optical loss. For a given loss, when t1 and t3 are switched as in Figs. 7(a) and 7(c), one can see from Eq. (2) that the transfer function at the drop port is unchanged, and thus a high power enhancement in Case IA is corresponding to large power reduction at the through port. Similarly, the transmission in Case IIA and IIB (or Case IIC and IID) at the through port remains exactly the same, resulting from unchanged transfer function, as indicated by Eq. (1), when t2 and t4 are switched. The reducing rates of power enhancement with loss in Case IA and IB are also different, as shown in Fig. 7(c), and Case IA, the one with higher enhancement factor, decreases faster with loss than Case IB. Similar trend applies with sub-cases in Case II, too.
The embedded ring system could offer desirable performance for nonlinear applications in terms of power enhancement factor even with strong coupling. When loss is 0.5 dB/cm, which is achievable with current fabrication technology, the power enhancement in Case IA can still be as high as 600, as shown in Fig. 7(c). In addition, the power enhancement in Case IA is twice that in Case IB as the loss of the system varies. Similar trends can be observed in Case II, as shown in Fig. 7(d). These results emphasize the necessity to wisely choose an operation regime in order to enhance the optical nonlinearity without additional design or fabrication difficulties.
6. Conclusion
We have presented a thorough study on the power enhancement in embedded resonators, enabled by EIT-like effect. It is found that power enhancement factors with the same intensity response can vary by more than one order of magnitude, resulting from the different operating phase regimes. In addition to the extensively reported phase profile, new phase regimes (i.e., 2π and 4π phases) of the EIT-like effect are identified and analyzed in detail. Intriguingly, we show that, even with the same intensity and phase responses, the power enhancement can be significantly different. These results not only greatly deepen our understanding but also have practical meaning for achieving enhanced group delay and optical nonlinearity without adding to design or fabrication complexities.
Acknowledgments
This paper is supported by the Natural Science Foundation of China (NSFC No.61006074 and No.61176106).
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