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Under the constraint of fixed pulse distortion, we study the bandwidth-delay product and nonlinear phase shift performance of coupled-resonator slow-light waveguides with designs that produce maximally-flat transmission and maximally-flat group delay responses. Even though improvement in bandwidth-delay product can be obtained with increasing number of resonators, the nonlinear response fails to improve beyond a certain number of resonators due to the increased filter bandwidth necessary to maintain a fixed pulse distortion. However, as expected, the nonlinear response improves with resonator finesse and degrades with resonator loss.
©2007 Optical Society of America
1. Introduction
Coupled-resonator optical waveguides, or CROW’s, have been proposed Refs [1, 2] and demonstrated Refs [3, 4] for the reduction of the group velocity of light. Of particular interest is the operation of CROW’s in the nonlinear regime Refs [5, 6, 7], where enhanced light-matter interactions can be utilized for a variety of functions such as optical switching, frequency conversion, and amplification. One of the key performance parameters of these structures is the bandwidth-delay product, which is related to the number of pulse delays that can be achieved. This metric also has relevance in the nonlinear regime, but of greater importance is the pulse energy required to perform a given nonlinear operation. Using the nonlinear phase shift, we show that, as a function of the number of resonators, the pulse energy required to produce a π phase shift has a lower bound due to the nonlinear distortion of an otherwise idealized transfer function.
2. Maximally-flat designs
Figure 1 shows the generic CROW slow-light waveguide geometry which consists of a sequence of directly-coupled microring resonators. Each ring has a physical length, or circumference, L, with free-spectral range FSR=c/nL, where n = 3.0 is the effective refractive index, and a slow-light waveguide consists of N such coupled resonators. The coupling coefficients are denoted by ci, where i indexes the coupling regions, numbering from 1 to N +1. The coupling coefficients are varied in order to produce either a maximally flat transmission Ref [8], which has idealized spectral amplitude cutoff, or maximally-flat group delay Ref [9], which has idealized spectral phase (i.e. minimal dispersion). Typical linear transfer functions are shown in Fig. 2, where δv =v - vm, v is the optical frequency, and vm is a resonance frequency for a wavelength near 1550 nm for these studies.
3. Bandwidth-delay product
Using the flat transmission and flat group-delay designs, the bandwidths of the CROWs are optimized to obtain 20% pulse distortion at the output. Pulse distortion is calculated via the following expression
where the pulse delay tgd represents the average group delay of the CROW and is calculated by the first moment of the output pulse intensity
Fig. 2. Maximally-flat transmission (left) and group delay (right) designs with six resonators (N = 6), plotted versus normalized frequency. These designs were optimized for 20% distortion of a 10 ps input pulse.
assuming that the input pulse is presented at t = 0. At each value of N, tgd is calculated and maximized under the constraint of fixed D for a 10 ps input pulse duration. Note that a similar approach was utilized in Ref [10] to maximize delay based upon the deviation of the complex transfer function from the ideal, distortionless slow-light case. Another approach is to compute a power penalty associated with the distortion [11].
Optimizing the bandwidth of the flat transmission design such that distortion levels D = 20%,10% or 5% in the output pulse shape are produced, Fig. 3 shows the bandwidth-delay product versus the number of resonators. In this calculation, the bandwidth is fixed at 44.1 GHz, corresponding to the FWHM bandwidth required for 10 ps, transform-limited, Gaussian input pulse rather than the FWHM filter bandwidth, which is necessarily greater; the number of pulse delays is a factor of ~2.3 times the bandwidth delay product. As shown, the bandwidth-delay product increases with the amount of allowable distortion and with the number of resonators. Similar results are obtained for flat group delay, but at a slightly lower bandwidth delay product with an offset of about -0.2. This reduction indicates that in the linear regime, the spectral filtering of the flat delay design has a greater degradative effect than the higher-order dispersion of the flat transmission design.
Output pulse profiles for these two designs are shown in Fig. 4 for N = 6 and D = 20%. The output of the flat transmission design exhibits noticeable dispersive pulse broadening (broadening is not due to spectral filtering because the time-integrated intensity at the output is 98% of the input) which is the primary source of the calculated distortion. For the flat group delay design, very little broadening occurs, but the overall pulse energy is reduced (to about 85%) due to the spectral filtering caused by the rounded transmission response.
4. Nonlinear response
The optimized linear designs are now characterized for their nonlinear responses. The figures of merit (FOMs) used here are the input pulse intensities required to produce π/10 and π phase shifts, or Iπ/10 and Iπ. In the small-signal regime, the nonlinear change in phase is given by
where I in is the input intensity, n 2,eff is the effective nonlinear refractive index which is enhanced due to the average intensity buildup within the slow-light waveguide (which scales as finesse ℱ [12]), while Leff is the effective propagation length of the resonator system as given by the overall group delay (which scales as NQ, with Q the quality factor). By Eq. 3, one would expect that the intensity would drop with increasing number of rings due to increase in Leff.
Fig. 3. Bandwidth-delay product versus number of resonators for maximally-flat transmission and group delay designs with the constraint of 20%, 10%, or 5% output pulse distortion. For each resonator, L = 50 μm.
Fig. 4. Input (dashed linestyle) and output (solid linestyle) pulse profiles for maximally-flat transmission (left) and group-delay (right) designs with N = 6, optimized for 20% output pulse distortion. The input pulse duration is 10 ps, and the pulse has Gaussian shape.
Figure 5 plots the FOMs versus number of resonators using L = 50 μm for each design. The purpose of Iπ/10 is to represent a small-signal nonlinear phase change, which should follow Eq. 3 closely. The roughly inverse scaling of I π/10 with Leff (i.e. N) is apparent in Fig. 5, where some saturation is observed near N = 25. However, achieving a π phase change perturbes the system well beyond the small signal regime, as indicated by the saturation of Iπbeyond N = 5.
This saturation of Iπ can be understood by examining the distortion of the output pulse in the nonlinear regime, as shown in Fig. 6. The distortion in nonlinear propagation at Iπ exceeds the 20% distortion of the initial linear design, and grows rapidly with number of resonators, while the distortion at I π/10 remains close to 20%. Much of the distortion is due to reduction in signal transmission under resonator detuning. Nonlinear detuning of the transfer functions of both designs at their respective values of Iπis shown in Fig. 7 for N = 6. It is clear that strong distortion of the flat transmission transfer function occurs under detuning, with significant narrowing of the magnitude response leading to spectral filtering. The flat group delay design has less distortion, but due to the rounded transmission, detuning results in significant additional spectral filtering.
Fig. 5. Comparison of intensity required to produce π/10 (dashed linestyle) and π (solid linestyle) nonlinear phase change versus number of resonators for maximally-flat transmission (blue) and group delay (red) designs with 20% output pulse distortion in linear propagation.
The distortion at high intensity levels can be reduced by increasing the filter bandwidth, which requires a separate optimization.
5. Optimization for nonlinear response
Filter optimization can be performed for nonlinear pulse propagation, where the filter bandwidth is optimized to produce 20% distortion of the output pulse at the input intensity Iπ, for example. Performing this optimization, the flat group delay design exhibits bistable behavior for large phase shifts and large N, preventing convergence of the optimization routine in these regimes. As a result, optimizations for the flat delay design are performed over a limited range as compared to optimizations for flat transmission. Iπ, I π/4 and I π/10 are plotted in Fig. 8 versus the number of resonators; I π/4 is added to represent an intermediate regime. Note that 20% distortion in nonlinear propagation at Iπand I π/4 is only achievable for N ≥ 6. The total phase change across one FSR is πN; working initially on resonance, the maximum phase change upon complete detuning is therefore πN/2. A minimum of two resonators is needed in order to obtain a π phase change, but this would result in negligible transmission due to complete detuning; with more resonators, a smaller detuning fraction (i.e. resonance shift normalized by resonance width) will produce a π phase change. This condition, combined with the increased filter bandwidth required to maintain a given level of pulse distortion in nonlinear propagation, establishes a minimum value of N.
Nonlinear detuning for the case N = 6 is shown in Fig. 9 for flat transmission, and should be compared to Fig. 7. In the case of optimization for the nonlinear response, the bandwidth of the filter is increased so as to produce less distortion under nonlinear detuning. This is clear from Fig. 9, where flatness of the magnitude response is more closely maintained and less bandwidth narrowing occurs in the detuned transfer function.
Fig. 6. Output pulse distortion at intensity required to produce π/10 (dashed linestyle) and π (solid linestyle) nonlinear phase change versus number of resonators for maximally-flat transmission (blue) and group delay (red) designs with 20%output pulse distortion in linear propagation.
Fig. 7. Nonlinear detuning of maximally flat transmission (left) and group delay (right) designs optimized for 20% linear distortion with N = 6.
As shown in Fig. 8, the intensity for the optimized nonlinear designs are greater than the previous results (Fig. 5) and also appears to have a lower bound; since the actual bandwidth of nonlinear operation is determined by the input pulse bandwidth, this result also shows that there is a lower bound in energy. The floor in Iπ and I π/4 is due to the increase in filter bandwidth (and therefore reduction in ℱ and Q) required to maintain a 20% distortion level. Since the effective n 2 scales with ℱ, the increase in bandwidth serves to reduce the nonlinear sensitivity; however, the group delay (i.e. Leff) continues to increase with N which compensates for the reduction in intensity buildup. For N > 20, I π/10 also begins to saturate, demonstrating that distortion of the transfer function under nonlinear detuning can occur even for small phase shifts.
Fig. 8. Intensities required to produce π, π/4 and π/10 nonlinear phase changes versus number of resonators for maximally-flat transmission and group delay designs with 20% output pulse distortion in nonlinear propagation.
Fig. 9. Nonlinear detuning of maximally flat transmission design optimized for 20% nonlinear distortion with N = 6.
The increase in group delay with N in the nonlinear case is evident in Fig. 10, which shows an increase in bandwidth delay product, where the bandwidth is fixed at 44.1 GHz, corresponding to the pulse bandwidth. Again, even though delay increases, there reaches a point at which no further improvement in nonlinear response in obtained.
6. Scaling of nonlinear response with resonator finesse
Given that there is a lower bound on Iπ, the question arises if improved nonlinear response can be obtained through manipulation of the resonantor properties. Since n 2,eff scales linearly with ℱ, increase in resonator FSR (through decrease in L) can improve nonlinear performance without affecting bandwidth or group delay; optimization of delay in the linear regime results in a constant bandwidth-delay product. In the nonlinear regime, however, the results are different. Even with the optimization constraint of 20% output pulse distortion at input intensities I π/10 and Iπ, nonlinear performance is improved with increase in resonator finesse, as shown in Fig. 11 for ring diameters of 20, 30, 50 and 75 μm. The reason for the improvement is that the average internal intensity scales with resonator finesse; even though the bandwidth of the slow-light waveguide increases to maintain 20% distortion, the finesse is greater for systems with smaller resonators than for systems with larger resonators. The improvement is more pronounced for I π/10, which essentially remains in the small-signal regime and follows Eq. 3 more closely.
Fig. 10. Bandwidth-delay product versus number of resonators for maximally-flat transmission design with the constraint of 20% output pulse distortion in nonlinear propagation atIπ. For each resonator, L = 50 μm.
Fig. 11. Intensity required to produce π and π/10 nonlinear phase shifts versus number of resonators for maximally-flat transmission design of different rings lengths, with the constraint of 20% output pulse distortion in nonlinear propagation.
7. Effects of ring loss
Finally, the effects of resonator loss are considered, with attenuation of 1 cm-1 assumed. The difference between the lossy and lossless cases is shown in Fig. 12, where the calculations are based upon the optimized designs for 20% nonlinear distortion in the lossless case. It is apparent that loss degrades nonlinear performance in both the small-signal and large-detuning regimes, as expected, resulting in greater values for I π/10 and Iπ.
Fig. 12. Intensity required to produce π and π/10 nonlinear phase shifts for maximally-flat transmission design with ring loss, with the constraint of 20% output pulse distortion in nonlinear propagation. For each resonator, L = 50 μm, and loss is included at 1 cm-1.
It is interesting to optimize directly for the lossy case to determine if any improvement in nonlinear performance can be obtained. In this optimization for 20% nonlinear distortion, the output pulse amplitude is corrected for attenuation, so that loss does not factor in to the distortion calculation. This result is also shown in Fig. 12, which shows that the improvement is negligible.
8. Conclusion
In summary, we have shown that energy limitations exist in the nonlinear performance of coupled-resonator slow-light waveguides. After the first few resonators beyond the minimum, the intensity required to produce a π phase shift achieves a lower bound due to distortion under strong resonator detuning, although I π/10 continues to decrease over a greater range. Given the lower bound n 2 Iπ ~ 10-5 assuming no loss, and using a 10 ps pulse duration, we arrive at a lower bound in energy density of E π ~ 10-16/n 2 J/cm2, where the area is the effective mode area of the waveguide. Using estimates of n 2 ~ 10-14 cm2/W and effective mode area of 1 μm2, the energy Eπ ~ 100 pJ; pJ energies would be sufficient for π/10 phase shifts.
This research was supported in part by contract number HQ 00604C0010 from the Missile Defense Agency.
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