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We propose a new scheme for buffering optical signals in two-ring resonator system that has a larger delay-bandwidth product than those achievable in single-ring and two-ring configurations of the optical analog of electromagnetically-induced transparency (EIT).
©2008 Optical Society of America
1. Introduction
An optical buffer stores and holds an optical data stream for a fixed duration without conversion to electrical format, an important role in all-optical information processing [ 1]. In order to perform buffering on chip in a small footprint, a great reduction in group velocity is necessary. Slow light schemes have been proposed using both optical [ 2–13] and electronic resonances [ 14–19]. Optically, the forward propagation of light is delayed by circulating it through many round trips in a resonator, which may be in the form of a ring resonator or as a defect mode in photonic crystal. This delay can be extended by simply cascading many resonators together, as in coupled resonator optical waveguides (CROW) and side-coupled ring resonator structures that generally employ more than 10 cavities. Electronically, the group velocity is greatly reduced by means of electromagnetically induced transparency (EIT), which is the result of destructive quantum interference between two coherently coupled atomic energy levels [ 14]. A very large delay, but associated with a very narrow bandwidth, has been demonstrated experimentally at very low temperatures [ 15]. Such a remarkable phenomenon causes a growing interest in optically mimicking EIT using two coupled resonators [ 4, 12], whose coupled resonances resemble the two energy levels in an atom, but without the limitation of low temperature which has been a major hurdle for electronic resonance. This method is interesting because it is a relatively simple configuration that can produce a large delay.
Ideally, an optical buffer should have not only a large delay, but the delay should be constant over a broad bandwidth with low insertion loss. However, causality dictates that there is a constant delay-bandwidth product determined by the physical mechanism underlying the delay. The delay-bandwidth product is a measure of the number of bits that can be stored in the buffer ( N ST). For buffers based on resonators the delay-bandwidth product is typically less than one. For example, in the simplest configuration of an all-pass filter (APF) which consists simply of one ring coupled to one bus, the delay-bandwidth product is given by
where Δ f is the normalized FWHM, τ is the maximum delay at the resonance, and r is the coupling coefficient between the waveguide and the ring. Note that here FWHM is used only for convenience, and the usable bandwidth is actually less than that due to the presence of higher order dispersion reducing the N ST. In the case of 56 APF, for example, the system should be able to buffer 56×2/π=35 bits based on FWHM, but in reality it only buffers undistorted 10 bits which means the usable bandwidth is actually smaller [ 1].
Simply cascading the resonators does not necessarily increase N ST. For example, the delay-bandwidth product of a CROW structure theoretically is given by N ST≅ N/2 π, where N is the number of resonators [ 2]. However, experimentally a CROW with N=100 has only achieved undistorted buffering for one bit [ 1]. This is because while the delay is increased N-fold by using N resonators, the delay-bandwidth product remains about the same as that for a single ring since the passband is also saddled with a ripple profile with N peaks which inevitably imposes severe distortions in the signal and effectively diminishes the operating bandwidth by 1/ N. The ripple, however, can be removed when the waveguide loss is high or when the coupling between the bus waveguide and the ring is different from that between the rings [ 2]. This is demonstrated in [ 3] where a delay of ~110ps combined with ~17GHz bandwidth is achieved with 12 coupled rings and a waveguide loss of ~17dB/cm. This corresponds to N ST of 1.87, which is close to the value of (12/2π)=1.9 given by the theoretical estimate. This larger N ST, however, is compromised by a high insertion loss of about 30dB.
In this paper, we propose a different scheme based on a simple two-ring structure, as shown in Fig. 1, that can exhibit a theoretical delay-bandwidth product of 4/π (greater than 1) and a low insertion loss at the same time, that is, a two-ring structure that has a comparable performance with a 10-ring CROW structure. In section 2 we derive the general transmission properties of the two-ring structure and its optimization as a buffer, and in sections 3 and 4, its performance relative to other schemes will be discussed in detail, including time-domain simulations of signal propagation in cascaded structures in the presence of higher order dispersion. It particular, we show that this scheme is different and 4 times better than the optical EIT schemes [ 12] based on similar two-ring structures.
Fig. 1. The schematic of the 2R1B structure, with two mutually coupled rings R1 and R2, where R1 is coupled to the waveguide bus The excited optical pathways are shown in the right inset.
2. General formulation of the proposed two-ring buffer
The two-ring structure is schematically shown in Fig. 1. The transmittance of the two-ring system is similar to that of one ring coupled to one bus, and can be expressed as [ 24–25],
where
is the embedded one-ring-one-bus transmission that incorporates the loading effect of Ring 2 on Ring 1. The r 1 and r 2 are, respectively, the coupling between ring 1 and the bus waveguide and that between the rings, δ j= ωT j= ωn eff L j/ c is the round trip phase of ring j with effective index n eff, cavity length L j, and round trip time T j, and finally is the round trip amplitude loss in ring j with absorption coefficient α. Eq. ( 2) can be reduced to the single-ring form
where
is the phase response of the complex transmission,
is the modified round trip phase, and a= a 1| τ 21| is the effective loss. Note that δ is the phase difference between two optical pathways, one of which is resonant in ring 1 (P1) and the other in ring 2 (P2), as illustrated in Fig. 1. When these two pathways destructively interfere, which is at odd multiple of δ/π, there is no light localization in both rings and the transmission is at maximum. On the other hand, at constructive interference or even multiples of δ/π, the transmission is minimum and the light is localized within the two rings in a ratio depending on the relative size between the two rings γ=δ 2/δ 1. The group delay τ D=(∂ θ/∂ ω)=(∂ θ/∂ δ 1) T 1, in the lossless case is given by
Note that the delay is proportional to the intensity buildup factor in ring 1, B 1, and the relative buildup factor in ring 2 with respect to ring 1, B 21, which are defined in Eq. ( 7).
Fig. 2. (a). The transmission spectra for γ values varying from 1 to 2. (b). Close-up of the boxed area near δ 1=0. (c). The FDTD-simulated field distributions for γ=1 and (d). γ=2, where S, AS, and NR denote the symmetric, anti-symmetric and narrow resonance, respectively.
Generally, the transmission characteristic is similar to that shown in our previous work, in which the two-ring system is coupled to two waveguide buses [ 25]. Figure 2(a) shows the transmission spectra for various γ values. When the rings are identical (γ=1), the light is equally distributed in both rings and the resonances evenly split. The field distributions for the resonances, shown in Fig. 2(c), show a symmetric (S) and an anti-symmetric (AS) field profiles at the coupling point. As γ departs from 1, the resonance splitting becomes uneven and further apart, with the broader resonance associated with stronger light confinement in ring 1 and the narrower resonance associated with ring 2. With increasing γ, the transmission also gets progressively lower because of longer latency in the ring and therefore higher effective loss.
A particular case of interest is where γ=2, when the two resonances are the farthest apart and the narrow resonance is located at the anti-resonance of ring 1, which is at odd values of δ 1/π. Because of this, light is mostly localized in ring 2 which is completely isolated from the bus waveguide, and the narrow resonance has the highest possible finesse. On the other hand, when ring 1 is resonant, so is ring 2, hence there is a symmetric splitting at even values of δ1/π, similar to that for γ=1, but with the light intensity approximately twice larger in ring 1 than in ring 2. The field distributions for each of the resonances are shown in Fig. 2(d). The finesse enhancement is proportional to the intensity build-up factor in ring 2, and implies that the delay is also increased correspondingly. However, the delay-bandwidth is not enhanced by such simplistic increase of finesse. Indeed, the simple γ=2 two-ring scheme based on the narrow resonance has worse buffer performance compared to the APF scheme because it has the same N ST (<1) but lower transparency.
Since the objective is to increase N ST, our proposed scheme is to make use of the two symmetrically split resonances of the two-ring structure where the transparency is higher, in order to form a reasonably flat delay with a larger delay bandwidth product (although the delay may be smaller). This can be achieved by controlling the splitting and the broadening of the resonances. In the vicinity of the split resonance, the absorbance A=1- T, can be expressed in the form A=(1- a 2) B 1 (see Appendix for the detailed derivation), where
is a split-Lorentzian function similar to that given in [ 12], B 1(0)=(1- r 2 1)/(1- a 1 r 1) 2 is the maximum value of B 1, is the linewidth of ring m, is the total decay rate from both resonators, and Ω γ=δ + 1-δ - 1 is the resonance splitting which depends on γ:
Equation ( 9) shows that the splitting depends only on the coupling factor between the two rings, and is generally smaller for γ=2. On the other hand, as a 2 r 2 is typically close to 1, the total decay rate Γ is mainly dependent on a 1 r 1 in ring 1 and weakly dependent on ring 2. Figure 3(a) shows the absorbance near the Ring 1 resonance for different values of r 1 and a fixed value of r 2 (i.e., fixed Ω). When Ω>Γ, the splitting dominates and there are two distinct resonances. When Ω<Γ, the two resonances merge into one. When Ω/Γ ~ 0.6, a broad resonance with minimum loss is obtained. This offers the optimal condition for the buffer and the criterion Ω/Γ=0.6 can be achieved by various combinations of ( r 1, r 2), as shown in Fig. 3(b). Note that the required r 2 value is generally very high, which is difficult to control as it is highly sensitive to the gap separation between the waveguides. Moreover, the inter-cavity coupling shifts the resonance wavelength of the neighboring resonator and this makes the fabrication optimization more complex. The required r 2 value is slightly smaller for γ=2, hence it may be easier to implement the scheme in this case.
Using Eqs. ( 7) and ( 8), the maximum delay for the proposed scheme is shown to be
where the fact that r 2~1 is assumed for both γ=1 and γ=2. Note the delay is independent of γ. Using the criterion Ω/Γ~0.6, the delay for the two-ring system is almost 3 B 1(0), or equivalent to 3 modules of APF with the same coupling parameters, for which τ D/ T 1= B 1(0). However, the FWHM bandwidth is somewhat smaller than the APF, giving rise to a net increase in the delay-bandwidth product by a factor of about 2.
Fig. 3. (a). The absorbance spectrum depends on the ratio Ω/Γ, displays a flat top shape when Ω/Γ~0.6. (b) The combinations of ( r 1, r 2) required to achieve the criterion Ω/Γ~0.6. In all cases, a 1=0.999.
The result can be verified by the phase response. Figure 4(a) shows the phase responses and transparencies corresponding to three different regimes of the Ω/Γ ratio. In the splitting-dominant regime (blue curve), the phase response is split into two 2π swings. In the broadening-dominant regime (red curve), the phase converges to one 2π swing, similar to that in a one-ring system. Finally, the balanced regime (dotted curve) gives a combined phase swing of 4π, meaning that the delay-bandwidth product is about twice that obtained in the one-ring APF scheme, consistent with the analytical prediction based on Eq. ( 10).
The dependence of transparency and delay response on the loss parameters ( a 1) is shown in Fig. 4(b) for the case Ω/Γ~0.6. Note that the delay is independent of loss while the transparency decreases with increasing loss. The transmittance is reduced to half with a moderate loss of a 1=0.99, but much smaller loss of a 1=0.999 has been demonstrated which suggests that a 90% transmittance is possible.
Fig. 4. (a). Different situations resulting from different values of r 1 or splitting-broadening ratio Ω/Γ. (b). Transparency and delay response for different loss parameters.
Finally, we note that the scheme can be generalized to more rings to further enhance the delay bandwidth product. With N rings it is the same as an apodized N-ring CROW coupled to one waveguide bus, in which the coupling between the waveguide and the first ring is stronger than those between rings. However, there is a limit to how large N can be. As N increases it will create a photonic bandgap mechanism causing the large delay to be concentrated at the band edges, as happens in the CROW scheme, which reduces the delay-bandwidth product.
3. Comparison with the EIT buffer scheme
It is instructive to compare our proposed scheme with the optical analog of EIT which is based on a similar two-ring configuration. The analogy between EIT and the coupled resonators has been discussed extensively by Smith [ 12]. We note that our Eq. ( 8) encompasses the EIT case, reducing to Eq. ( 5) in [ 12] in the limit where a 2=1 and Δ 2 approaches 0. To achieve EIT, three conditions must be satisfied: (i) the splitting is very small ( r 2~1), (ii) Ring 2 is in the over-coupling condition ( r 2< a 2) in order to have strong interaction between the split resonances, and (iii) Ring 1 is in the under-coupling regime ( r 1> a 1) in order to behave like an atom. In other words, the combined condition is a 1< r 1< r 2< a 2, which will ensure the condition Ω/Γ≪1.
This condition can only be achieved when the two rings have different loss coefficients (i.e., a 2≠ a 1 γ). This requirement complicates the design. More importantly, although the EIT version of the coupled resonator has a large delay, it is not the optimum design for an optical buffer because its delay-bandwidth product is small. Figure 5(a) shows the EIT spectrum with the same parameters as in [ 12]: a 1=0.88, r 1=0.9, a 2=0.9999 and r 2=0.999. Note that the swing in the phase response over the EIT resonance is generally less than π, implying that the delay bandwidth product of the EIT scheme is smaller, and approximately half the value of the APF scheme:
Compared with Eq. ( 10), the NST for the EIT scheme is 4 times smaller than our proposed design which is based on the same two-ring configuration. The reason for the smaller N ST lies in the under-coupling condition for ring 1 in the EIT scheme, as opposed to over-coupling in our design. As a result, the phase response in the EIT scheme resembles two closely split under-coupled phase responses from the first ring, connected by a linear and steep phase. The under-coupling phase response is analogous to the index change spectrum associated with a two-level atomic transition. The delay given in this scheme may be large but has a narrow operating bandwidth. Fig. 5(b) shows the delay and transparency for two different EIT cases. The N ST of EIT is numerically calculated to be ~0.3 in agreement with Eq. ( 12). As a reality check, we also numerically calculate the delay-bandwidth product of another EIT configuration based on two rings side-coupled to two waveguide buses [ 10], which has a FWHM bandwidth of ~0.13nm and a delay of 17.9ps. The N ST is again ~ 0.3. This is expected as the delay-bandwidth product for schemes based on the same mechanism should be independent of the specific configuration. Hence, we conclude that the EIT scheme has a smaller delay bandwidth product than even the APF (one ring) scheme. However, it should be noted that this limitation may be broken by using active dynamic tuning, as has been demonstrated in [ 11].
4. Discussion
For conciseness we summarize in Fig. 6 all the one- and two-resonator buffer schemes considered in this paper, with their different signatures in transparency ( T), modified round trip phase (δ), and phase response (φ). The delay-bandwidth product is a fundamental limitation in any buffering system, and has a different value for different mechanism. The value is fundamentally determined by the phase change across the resonance, thus it is clear that our proposed scheme has the largest delay bandwidth product per module.
Fig. 6. Buffer schemes based on ring resonators: (a) the APF schemes based on one ring coupled to one waveguide bus and (b) three other schemes based on two mutually coupled rings coupled to one waveguide bus. Each scheme has its own signatures for transparency (T), modified round trip phase (δ), and phase response (φ). The dashed lines represent the location of the resonances.
In Fig. 7, we compare our proposed two-ring scheme with a 10-ring CROW and two different EIT schemes, all of which have been optimized. For our proposed scheme, the coupling coefficients are r 1=0.95 and r 2=0.999. For the CROW, the coupling between waveguide and ring is r WG=0.95 and that between rings is r=0.999. For both schemes, the loss of 0.999 is assumed [ 21–23]. In the EIT schemes, the losses are those used in Fig. 6(b) with a 2~1. It can be seen that the EIT has the highest transparency but the smallest bandwidth. Our simple proposed scheme has the same delay as the EIT and the 10-ring CROW, and about the same bandwidth as the 10-ring CROW. This is a reflection of the fundamental delay-bandwidth product of each system. For our proposed scheme, the group delay is ~100 T 1 and the flat region of the delay has a bandwidth of Δδ 1/2π~7×10 -3. For a ring with a radius of 5 µm and a group effective index of 4.4, these values imply a group delay of ~46ps and a bandwidth of 15GHz.
Fig. 7. Comparison of transparency and delay, between the proposed scheme with r 1=0.95 and r 2=0.999 (1), and the 10-ring CROW (2) with r WG=0.95 and r=0.999, and the EIT schemes with (3) r1=0. 9, a 1=0.88 and (4) r 1=0.96, a 1=0.95. The second ring for the EIT scheme is assumed lossless, for other cases a=0.999.
Table. 1. The parameter comparison between three schemes.
Except for the proposed scheme, the delay-bandwidth products for buffers involving resonators are typically less than one, thus a cascade configuration is necessary to buffer more than 1 bit. In Table I we summarize the relevant parameters and the performance of three different buffer schemes with N number of modules, where N is limited to around 10 so that the insertion loss is not excessive. In the first scheme, the CROW configurations have been discussed in the Introduction, where it is mentioned that the apodized configuration ( r WG=0.95, r=0.999) can have a flat spectrum with a small insertion loss of ~2dB for 24 modules. The N ST for this case is about 3.51, consistent with the value given by N ST= N/2π [ 2]. Although N ST tends to be lower for CROW, the advantage of CROW is the low higher order dispersion, as long as the operating bandwidth is far from the edges of the CROW transmission band where higher order dispersion is dominant. In the EIT scheme, a cascade of 4 modules is necessary to buffer 1 bit. It should also be noted that the second ring in reality is not lossless, thus there is insertion loss in the cascade EIT configuration. If a 2=0.999, the transparency per module would be ~0.8 for a 1=0.88 and r 1=0.9, which means a cascade of 24 modules would have an insertion loss (IL) of 10log(0.8) 24~23dB. Similarly, the case of a 1=0.95 and r 1=0.96 would correspond to a transparency of ~0.9 and an insertion loss of ~ 11dB for 24 modules. Finally, for our proposed scheme, two cases are presented. In the case with very weak inter-resonator coupling ( r 2=0.999) we have an insertion loss of 8dB for an 8- module structure, and in the case with stronger inter-resonator coupling ( r 2=0.99) we have IL =2.9dB for a 10-module structure. The N ST for both cases are 10.8 and 14. The insertion loss is reasonably low for such a large delay-bandwidth product.
Note that we have left out the APF in Table 1 because the delay and 3dB bandwidth are of little difference from the 2R1B scheme for the 2-APF-cascade. Cascading two identical APF will double the delay and hence in principle can have comparable N ST as the 2R1B scheme. However, we show below that there is a practical difference in how they respond to signal propagation in long cascaded structures in the presence of higher-order dispersions, which have been shown to limit the buffer characteristics [ 20]. Figure 8(a) shows a side-by-side comparison of the signal propagation between a single-ring and a two-ring cascade structures, with the same coupling parameters and for different number of rings in each case. In order to show the inter-symbol interference, we insert 8 return-to-zero (RZ) bits of logic 1 and plot the time domain results with the time normalized to the bit length. Note that the APF starts to show symbol ambiguity when N>40 with ~6 buffered bits, whereas the proposed scheme starts to have inter-symbol interference only when there are more than 60 rings in the structure ( N>30) with ~14 buffered bits. Clearly this shows that the proposed scheme, due to its flatter delay spectrum, is more immune to high-order dispersion and can store twice the number of undistorted bits compared to the APF. Another possible two-ring APF configuration is a cascade of two APF with “detuned resonances” to double the 3-dB bandwidth while keeping the same delay. If the resultant spectrum is flat, then this configuration should be equally immune to inter-symbol interference while having the same N ST as the proposed scheme. However, the main challenge for this configuration is the practical difficulty in achieving the desired detuning in a controlled manner. Simulations show that the resonance shift is not linear but rather random for very small size detunings of less than 1% and high resonance order of the order of 100 (corresponding to a ring radius of 10 µm) because of the Vernier effect. In contrast, it is easier to achieve the Ω/Γ=0.6 condition required for the proposed 2R1B scheme, as illustrated by Fig. 3(b).
Fig. 8. (a). Side by side comparison of propagation of bits in different structure lengths. Note that APF suffer more inter-symbol interference as N>40. (b) The comparison between the proposed scheme, APF, and CROW in a fixed 4(8) buffered RZ (NRZ) bits. The smaller the required number of modules, the larger is the delay-bandwidth product of the scheme
Finally, we compare the proposed scheme with APF and CROW in their ability to buffer a fixed number of bits. Here, the operating bandwidth is chosen to be smaller than the 3dB bandwidth to avoid inter-symbol interference and the coupling coefficients for each scheme are freely chosen to achieve the same undistorted buffering of 4 RZ bits. To minimize distortion, the usable bandwidth is defined as half the FWHM bandwidth and therefore N ST for the undistorted bit is half the values previously defined. In the case of CROW, the usable bandwidth is defined as half the frequency band between the two band edges [ 2]. The time domain results for the APF, CROW, and the proposed scheme are shown in Fig. 8(b). It is seen that in each scheme, a different number of modules ( N) is required to buffer the same 4 bits, and the ratio of modules, 18:43:78, is in agreement with the ratio of their delay-bandwidth products, 1:2.38:4.33. In other words, both frequency and time domain analyses show that our configuration is 2 and 4 times more compact than the APF and the CROW structure, respectively. Similarly, it should also be 4 times more compact compared with the EIT scheme.
5. Conclusion
We have proposed a buffer scheme based on indistinguishable split resonances in a two-ring structure. The delay bandwidth product, which is the fundamental parameter for optical buffer, is shown to be higher than other schemes such as the cascaded side-coupled ring structure (APF), the coupled resonator optical waveguide (CROW), and the optical analog of EIT. The delay and transparency spectra are reasonably flat and the insertion loss for realistic parameters is quite low, which are all sought-after features in optical buffering. Time-domain simulations of signal propagation through long cascaded modular structures also show superior performance in buffering higher number of bits for the proposed scheme compared with APF and other schemes.
Appendix A: Derivation of the resonance splitting in Eq. ( 9)
Based on Fig. 1, it can be easily shown that the resonance condition for the bare two-ring structure is
where the rings are assumed to be lossless and τ mn represents the loading factor of ring m on ring n. By matching the real and imaginary components of ( A1), we can come to the characteristics equation for arbitrary γ
For γ=1, the two resonances are located at δ ± 1=±cos -1( r 2), thus the splitting is given by
For γ=2, by using the trigonometric identities cos3 x=4cos 3 x-3cos x and cos 2 x=2cos 2 x-1, ( A2) can be transformed to which has three roots:
Two of these corresponds to the symmetrically split resonances around the δ 1=2 mπ, giving the splitting as
while the third root corresponds to a much narrower resonance situated in δ 1=(2m+1)π. The field distribution at the split resonances is proportional to the relative intensity buildup factor of ring 2 with respect to ring 1, B 21( δ ± 1). For γ=1, the B 21( δ ± 1)=1, this means the light is equally confined in both rings. For γ=2, B 21( δ ± 1)~1/2 for very weak inter-resonator coupling ( r 2~1), implying that the light is twice more localized in ring 1 than in ring 2. These facts are verified by the FDTD simulations in Fig. 2.
Appendix B: Derivation of Eq. ( 8)
The intensity buildup factor in ring 1, as defined in Eq. ( 7), is in the presence of loss given by
where δ= π+ δ 1+ δ 2+2 tan -1[ r 2sin δ 2/(1- r 2cos δ 2)]. The ring 2 is assumed lossless for a moment, because the shape of θ 21 is almost unaffected in the presence of low loss. Using trigonometric rules, it can be shown that
By inserting ( B2) in ( B1) and with the high-finesse approximation, cos δ 2~1- δ 2 2/2, B 1 can be expressed as
The effect of loss in ring 2 is then reintroduced by substituting r 2→ a 2 r 2 in ( B3), and by approximating cos δ 1~1- δ 2 1/2 and using the identity (for γ=2), the case of γ=1 and γ=2 in ( B3) becomes
where both Ω 1 and Ω 2 are resonance splitting as derived in Appendix A, is the total cavity decay rate, and is the linewidth of the m th ring.
Acknowledgment
We would like to acknowledge a reviewer’s comment with regard to the comparison with the APF scheme.
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