In this paper, we study analytically and numerically the light propagation in microring resonator chains that exhibit distributed loss or gain. We derive the stability conditions for the latter and demonstrate the feasibility of the group index control within a certain amplification range. Possible applications of the discussed effects are proposed.
©2007 Optical Society of America
Microring resonators are the basic building blocks for high-performance photonic devices used to implement various functionalities in optical networks in an efficient way. Among other applications, they can be used for ultra-selective optical filtering in WDM networks , optical add/drop multiplexing , dispersion compensation  or wavelength conversion . Control of the group velocity of the propagating optical signal can be used to enhance the nonlinear response [7, 8] for all-optical switching. Microring resonators are particularly appealing for photonic integration: programmable integrated circuits which perform advanced all-optical signal processing have already been demonstrated .
Most of previous research has focused on the behavior of passive devices where theoretical analyses assume lossless propagation in the waveguides sections. Such assumption is unrealistic, since propagation in bent waveguide sections will always be accompanied by radiation loss. Also, the use of active propagation sections implemented either as semiconductor  or Erbium-doped  optical amplifiers can be fundamental in order to overcome the practical limitations of propagation losses on the achievable physical size of integrated circuits or in the design of novel microring laser sources. Moreover, the work presented here illustrates how the control of the propagation gain or loss by optical or electronic means can be used to develop novel devices which can find application in high-performance optical networks.
In this paper, we first study the propagation properties of optical signals in microring resonator chains in the presence of gain and loss. In Section 2, we present a stability analysis of active microring chains. The results presented are of fundamental nature not only in the design of novel microring-based sources but, most important, the stability thresholds must be known and not crossed when one uses optical signal amplification in order to increase the size of microring photonic integrated circuits . In the final section, numerical simulations of Gaussian pulse propagation are used to illustrate the dynamical behavior of this type of structures. The presented work specially focuses on the active control of the group delay of the structure by changing the waveguide gain and loss. The aspects addressed are the physics of the problem and how it can be used as the basis of a new class of photonic devices with applications such as in-line timing jitter correction in high-performance optical transmission systems or advanced optical signal processing in microwave photonics [10, 11].
2. Analysis of the transmission characteristics of lossy and amplifying chains
We will consider a chain of N +1 identical cells, i.e. N rings plus the two semi-rings at the ends of the structure, as shown in Fig. 1. Calling M the matrix relating the the input and the output of the n-th unit cell, we have , 
Here d = L/2 and , with t the transmission coefficient, which in this case is real and equals the field coupling ratio between resonators. is the propagation constant of the single-mode waveguide, which can in general be complex due to losses or gain along the waveguides. We note that det(M) = (1-r 2)/t 2; thus, provided the coupler has no insertion losses (t 2 +r 2 = 1, as assumed), the transfer matrix is unimodular regardless of whether the waveguide has losses or not. This is not an “accidental” result for this particular simple cell; contrary to what has been frequently implied in the literature ( is an exception, for example), the unimodularity is related to the reciprocity of the structure, which is not necessarily violated because a waveguide is lossy (or amplifying). We will write:
with α = 0 if the waveguide is ideally lossless and α > 0 (< 0) for a lossy (amplifying) waveguide.
Using Sylvester’s formula for a unimodular 2× 2 matrix, the (N + 1)-th power of M can be written as 
where I is the identity matrix and β is given by cos(βd) = (M 11 + M 22)/2. In this case, according to Eq. (1),
For k̅ real, relation (4) turns out to be the dispersion relation of the corresponding infinite-length, β being the propagation constant of the Bloch wave . No such interpretation is possible if k̅ is complex, but (3) and (4) hold in any case.
We will define the field transmissivity through a finite structure of N rings as
Thus for k̅ real, the periodicity of sin(βd) = (1 - sin2(kd)/t 2)1/2 determines transmission “minibands” whose centers lie at kd = mπ, m being an integer. The condition |sin(kd)| < t, determines the range of the minibands, each having N transmission peaks corresponding to the minima of (M N+1) 11 as given by Eq. (3).
In our present case k is complex but, taking α as a parameter (approximately constant within the frequency range of operation), we will write tN = tN(k). We next define the real-valued functions TN(k), power transmissivity, and ΨN(k), phase, as follows:
For a total cavity length (N + 1)L corresponding to N rings, the group velocity is
The results for k̅ real can be taken as a guide when α is nonzero but small. To avoid unessential complications in this presentation, we make the customary assumption that the couplers can be treated as lumped gainless devices (Mookherjea has recently considered the effect of gain and loss in the coupling region, which can lead to dispersion tunning ). Thus, Figs. 2 and 3 show T 3(k) and ϕ 3(k), respectively, for a lossless (α = 0), slightly lossy (αd = 0.015), and slightly amplifying 3-ring structure, with several negative values of αd in the latter case. The other parameters are d = L/2 = 78.5 μm (ring radius of 25 μm), a modal index n = 3.2772, and t = 0.5. The central peak in the lossless case lies at k 0 d = 332π, which corresponds to the wavelength λ 0 = 1.5505 μm in this example. It should be noted that tN is periodical with period π in the normalized frequency kd if N is odd. (Of course such periodicity if formal as the physical parameters n̅, α, and t are remarkably wavelength-dependent for large frequency spans.) For example, the analytical expression of t 3, that we will use, is:
As seen in Figs. 2 and 3, the losses significantly decrease the amplitude peaks of the transmissivity function, as is obviously expected, while not any remarkable variation in the shape of the phase function occurs (it actually tends to be somewhat less rippled). Setting αd to somewhat higher positive values confirms this behavior.
Similar but inverse results are, in principle, observed when moderate gain instead of loss is applied. We will call this gain region the “stable operation range.” If gain is further increased, an extremely sudden change occurs when a certain critical value -(αd)c is reached. In the example under consideration, when -αd > -(αd) c ≃ 0.037206, the slope of the phase becomes strongly negative within two narrow frequency bands around the side transmissivity peaks. This terminates the stable operation range, as we explain in the next section.
This behavior is illustrated in the curves of Figs. 2 and 3. The fourth transmissivity curve in Fig. 2 (-αd = 0.037206) is virtually indistinguishable from the third (-αd = 0.0372055), but the corresponding fourth phase curve does change dramatically, showing a large negative slope around the two transmissivity side peaks, as seen in Fig. 3. In general, the presence of gain makes the slope of the phase curve steeper near the side peaks, as compared with the no-gain case (see the curves αd = 0 and -αd = 0.0372055). Thus, further increase of -αd ends up yielding almost step-like changes of magnitude π. At this point, any slight increase of the gain will drive the phase to a value slightly greater than +π, hence a negative angle, as shown in the -αd = 0.037206 curve in Fig. 3.
The amplitude of the central peak of the transmissivity increases with -αd (Fig. 2). On the contrary, the side peaks reach their maximum (which is actually ∞) at the critical value of - αd, then decreasing for higher gains; this behavior is best explained through a zero-pole diagram, which we will do in the next Section. As for the phase, Fig. 3 shows that the effect of increasing the gain is essentially smoothing the curve at the transitions: the spectral ranges where Ψ3(k) has negative slope broaden, while the magnitude of the slope decreases.
A closed expression for the group velocity was derived in  for the lossless (α = 0) case. When there is gain, it follows from the results above that the group velocity in the structure may become very large and even negative in the spectral ranges around the transmissivity peaks. Taking ng ≈ n for the waveguide and using formula (7), we obtain the graphics displayed in Fig. 4.
Figs. 2, 3 and 4 illustrate the expected overall behavior of the structure where both the inverse of the net gain through the coupled resonators chain and the optical bandwidth scale as the absolute value of the group velocity.
Thus, by varying the optical gain within a certain range, the propagation delay can be controlled around the spectral peaks, reaching very high positive values. For even higher gains, the propagation delay becomes “negative,” in the sense of . However, the stability of the structure turns out to be an issue in this case, as we shall see in detail in Section 3.
If the gain were further increased, a “second threshold” would be eventually reached; namely, -αd ≃ 0.06943 in the present case. At this point, the group index also becomes negative around the central peak and all three peaks display negative group index within a narrow frequency range around them. Further increase of the gain would cause no more qualitative changes. However, note that this extended linear analysis beyond the critical value -(αd) c, although potentially useful to study the oscillation properties of the structure, for example, is not appropriate to describe the operation of the resonator as a controlled wave-retarder, since the latter is restricted to the stable, non-saturated regime 0 < -αd < -(αd)c.
3. Stability of the active structures
As anticipated in the previous section, since the active microring chain resonator constitutes a gain device with feedback, its operation may be unstable. We will here develop a simple procedure to asses its stability. As it is well known, this kind of structures can be regarded as IIR digital filters for computational purposes  (even if the input and output signals are truly analog, therefore not limited to a periodic spectrum in the π range). The stability depends on the situation of the poles of the transfer function in the complex frequency plane. Replacing the “frequency” k by a complex frequency k̃ = k + iσ (not to be confused with k̅), so that ik → ik̃= -σ + ik, and calling
the transfer function of a microring chain with any number of rings can be written in form of a quotient of two polynomial in the complex variable z. We define QN(z) such that QN(z) = QN(ρexp[ikd]) = tN(k) when ρ= 1. For example, for N = 3, Eq. (8) yields:
with B = t 4 and
The gain and the feedback of an active microring chain are indeed distributed along the whole structure. Nonetheless, Q 4(z), or any QN(z) in general, could be formally rewritten as the closed -loop transfer function of an open-loop amplifier HN(z) with a lumped feedback KGN(z) directly from the output to the output, i.e. Q 3(z) = HN(z)/[1+KHN(z)GN(z)], as can be easily deduced by simple algebra. In this form, Q 3(z) looks like the usual expression employed in control systems theory to study the stability of feedback systems . However, its usefulness lies on the fact that the adjustable gain is completely determined by K, there being suitable graphical methods (such as the Nyquist criterion) that help to determine the range of K for which the system remains stable. This is by no means our case since all the bn coefficients of the polynomial in the denominator are affected by the gain coefficient α, so the Nyquist criterion or any other such standard method is useless.
Nevertheless a remarkable simplification occurs if we note that the coefficients bn contain α always within a factor exp(nαd). Call RN(z) the denominator of QN(z). Let rj(0) be the roots of RN(z), i.e. the poles of QN(z), when α= 0. For (10), for example, R 3(z) can be written:
To find the roots for any α ≠ 0, let us call aj = bj(α= 0), so that (11) can be written:
The roots uj of the polynomial (15) in u are those of (14) in z with α= 0, which are rj(0). So, for α≠ 0,
We conclude that the roots of R 3(z) for any α are
Therefore, the numerical computation of the poles of Q 3(z) need be performed only once, for α = 0, and the results for lossy or amplifying guides with any value of α follow immediately from the trivial formula (18). Obviously, these results generalize to a chain with any number N of cells. Furthermore, it is easy to see that a result similar to Eq. (18) will be obtained if only a certain fraction of the length of each ring waveguide is amplifying (provided it is the same in all cells). The result (18) also explains why the complex poles will always have the same phase for any value of α.
The stability can now readily assessed by checking that all rj(α) lie within the unit circle in the complex z plane . A very definite behavior is then found for the gain (α< 0): Whenever the magnitude of α reaches the threshold for which the group index becomes negative anywhere in the k spectrum,
one or more poles rj(α) move outside of the unit circle, making the system unstable. This is discussed next.
Fig. 5 shows the position of the 6 poles of Eq. (10) for three values of the gain, with the rest of parameters as in Section 2. Up to the critical gain, -(αd)c ≃ 0.037, all the poles lie inside the unit circle. Just above this value, four of the six poles pull out of the unit circle, making the system unstable, which corresponds exactly to the appearance of the two narrow spectral regions with negative group index around the two side peaks. Note that, for a certain value of α, the poles lie exactly on the unit circumference, which is the oscillation condition at the corresponding frequencies; so the structure could be designed to lase in its unstable region, which is certainly not a surprising result.
When the “second threshold” (-αd ≃ 0.069) is reached, the two remaining poles move out of the circle, corresponding to the appearance of negative group velocity around the central peak as well. Although we have focused on the the N = 3 case, the key results are similar for any number of rings. For example, Fig. 6 shows the group index of a 5-ring chain for several values of the normalized gain parameter - αd, all other parameters of the structure being the same as in the 3-ring case previously considered. There appear four, rather than three, distinct operational regimes, the last one corresponding to all five peaks having negative group index. In any event, it must be stressed again that only in the first (stable) zone, - αd < - (αd) c, when the group index is positive at all frequencies, is our analysis relevant and meaningful for the purpose of group velocity control.
The transmission characteristics of passive (lossless, non-amplifying) microring chains have indeed been studied before in the literature, with special consideration to the role of the number of rings [8,21]. For the proposed amplifying structures, the most relevant additional features are those described in the paragraph above, together with a general result that can be easily verified: the longer the chains are, the smaller the values of -(αd)c become. This can be compared with the laser threshold problem, wherein the distributed mirror losses (in our case due to the input/output coupling) are reduced for a longer laser cavity. Table 1 shows the corresponding values for 3, 4 and 5 rings as an illustration.
We will now use the conclusions obtained from the present analysis as a useful guideline for the numerical study of pulse propagation, developed in the next Section.
4. Numerical simulation of optical pulse propagation and discussion
In order to study the dynamics of optical pulses propagating along lossy and amplifying mi-croring chains, we have launched an unit peak amplitude Gaussian pulse with complex optical field
at the input of a three ring structure. The optical carrier frequency ω 0 is tuned to that of either the center resonance or one of the side resonances within the three peak miniband. T 0 is the pulse width and its relative value to one ring round trip propagation delay can be used to set the optical signal spectral width relative to that of the system response at resonance. For the results presented in this section, we use a fixed value of T 0 = 20 τ where τ is the propagation delay for one unit cell of length d.
In the numerical simulations, we take into account gain or attenuation associated to propagation in the optical guides but waveguide or chromatic dispersion is neglected for the short transmission distances, so only the stronger dispersive effects associated to the periodic ring structure affects the pulse propagation in our analysis. Also, couplers linking the ring sections are assumed ideal as in Eq. (1).
Figure 7 shows the envelope of the optical output field when the input carrier frequency (ω 0 is tuned to one of the side resonances of the miniband. The results for various values of αd ranging from 0.1, in the presence of attenuation, to -0.1, in the unstable region of the response, are shown in these figures. The system dynamics agrees with the predictions from the stationary analysis. In Fig. 7, the response turns unstable when the -(αd) c threshold is crossed and the input signal is tuned to one of the side resonances.
The control of the propagation delay of the structure using the waveguide gain or attenuation can be used to perform various signal processing functionalities using linear active and passive microring chain devices. The potential applications will be subject of further research, whereas in this paper we focus on the fundamental propagation properties in this type of structures.
As seen in the previous sections, the spectral ranges where the group velocity can be very strongly varied by controlling the waveguide gain are those around the peaks and, particularly, the side peaks. These ranges are rather narrow. Fig. 8 illustrates this point by explicitly displaying the group index as a function of the frequency for the specific example quoted in Section 2 (λ 0 = 1.5505 μm). Indeed, we have focused on the most elementary chain structure to present and illustrate the key results; the use of more complex active structures, based on new schemes already proposed for passive resonators (for example, ), can broaden the design possibilities largely, which will be the object of a future work.
Figure 9 display the normalized output signal envelope when the input pulse is tuned to one of the side resonances and illustrates the control on the group delay by acting on the signal gain or attenuation. Only values of αd falling into stable regimes are considered. The results show how the propagation delay dependence on propagation gain or loss can be used to perform a controllable shift in the pulse position. This could be used to perform a correction of timing jitter in signal regeneration schemes for long haul optical systems. In any practical device, such control should be accompanied by a corresponding adjustment of the pulse amplitude level.
As we approach the instability thresholds given by (αd)c in Fig. 9, the effect of the system transmission bandwidth narrowing resulting in pulse distortion (and strong broadening) becomes more evident. This effect will limit the practical realization of the largest attainable group delay in applications of active microring structures other than optical signal generation or amplification even though pulse reshaping using saturable absorbers or nonlinear optical loop mirrors could be used up to increase the group delay tuning range.
We have studied the effect of distributed losses and, particularly, gain in a microring chain resonator. We have shown that, by slightly varying the gain parameter of the amplifying waveguide, the group velocity can be modified dramatically within narrow spectral ranges around the structure transmission peaks. The control of the group velocity could be achieved by using waveguide structures with SOAs or, more likely, with the promising EDWAs (because of their distributed nature).
We have presented a straightforward method to predict the stability of the active structures and establish a limit to the maximum amplification that can be applied. This article has focused on developing the novel key concepts, not intending to carry out an exhaustive survey, but some possible applications and improvements have been also suggested that will be the object of future work.
The authors thank Prof. José Capmany for his valuable assistance in the preparation of this manuscript. This work has been funded by MCyT and FEDER, grant number TIC2003-07020, and Junta de Castilla y León, project number VA083A05.
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