Microwave generation on an optical carrier in micro-resonator chains
Abstract
We consider self-pulsing regimes in chains of Kerr non-linear optical micro-resonators. By means of a super-modal diagonalization procedure of the conventional coupled-mode theory in time, we theoretically and numerically study the bifurcation diagrams of a singly-pumped 3-cavity and a doubly-pumped 4-cavity systems: the latter allows us to predict thresholdless frequency tripling of a GHz modulation. These self-pulsing regimes are proven robust and will find applications in generation and conversion of microwaves on an optical carrier.
presently at ]GAP-Nonlinearity and Climate, Institut of Environmental Sciences, Université de Genève, Bd. Carl Vogt 66, 1205 Genève, Switzerland
I Introduction
Optical microresonators confine light in a small volume on many optical cycles and thus provide a fundamental building block for high speed all-optical signal processing Ilchenko and Matsko (2006). Applications include frequency conversion Turner et al. (2008); Azzini et al. (2013); Pu et al. (2015); Combrié et al. (2017), switching Ibanescu et al. (2002); Sylvain et al. (2013), signal regeneration in communications Ghisa et al. (2007) and optical generation of microwaves Matsko et al. (2005); Del’Haye et al. (2008); Razzari et al. (2010).
One of the topics that recently attracted the researchers’ attention most is the generation of oscillations at microwave frequency by optical means. The underlying mechanism is simply the beating of optical oscillations separated by about 10 GHz to 200 GHz. This enabling technology is necessary not only in high-speed communications (e.g. in aerospace industry), but also in metrology, optical clocks and sensing. Provided that the resonator has a large enough size [so that its free spectral range (FSR) is in the target range], a set of adjacent resonances of an optical microresonator can be used as an optical rule Matsko et al. (2005). Starting from a synchronous quasi-continuous input at high enough power, the ubiquitous four-wave mixing (FWM) leads to the formation of an optical frequency comb (OFC) Del’Haye et al. (2009); Chembo and Yu (2010); Soltani et al. (2012); Cazier et al. (2013); Pasquazi et al. (2017). This can also form train of pulses and solitary waves inside the cavity Herr et al. (2014). The threshold power to observe such phenomena depends on the nonlinearity of the medium and on the cavity lifetime, or equivalently the quality factor . This poses strong technological constraints and octave-spanning frequency combs are generally observed in large diameter () glass or crystalline microresonators. Thus, due to the smallness of instantaneous optical nonlinearities, the required power levels cause thermal dissipation concerns and thus competition with thermal nonlinearities: is required. Finally in order to obtain oscillations at e.g. GHz, a microresonator of mm radius or more is needed to obtain the desired FSR. This clearly poses a serious limit to integration.
In order to overcome these constraints, one may rely on larger nonlinearities, such as those found in semiconductors (III-V as well as Silicon or Diamond Hausmann et al. (2014)), or on atom-like resonant effects (in bulk or low-dimensional structures, like quantum dots Grinberg et al. (2012) or nitrogen-vacancy Faraon et al. (2010)), where, nevertheless, the control of light-matter interactions requires cryogenic apparatuses to preserve the coherence among quantum states Combrié et al. (2017).
Nevertheless, an integrated nano-cavity hardly achieves , due to the unavoidable disorder and fabrication tolerances and the FSR is in the THz range (because of its size).
Let us consider optical micro-cavities supporting a single resonant mode around the telecommunication wavelgength (m). In order to trigger the system to oscillate in the GHz range, we can couple the optical degree of freedom to a degree-of-freedom of the material, such as a time-delayed nonlinear response Malaguti et al. (2011); Armaroli et al. (2011); Vaerenbergh et al. (2012). Beyond theoretical speculations, this proved effective for the dynamic control of photon lifetime at room temperature Huet et al. (2016). The coupled degrees of freedom can be also purely optical: several cavities mutually exhanging energy. It was thought that, owing to excessive structural complexity, such a solution would be limited to two cavities Maes et al. (2009); Grigoriev and Biancalana (2011); Dumeige and Féron (2015). The disadvantage is that stable self-pulsation is limited to a period of the same order of the cavity lifetime. A short lifetime, e.g. ns, to obtain a GHz oscillation imposes an upper limit on , thus requiring very large optical injection. Moreover large injection leads to a period-doubling bifurcation cascade to chaos. In Armaroli et al. (2015), we showed that a system of three evanescently-coupled optical micro-race-track resonators with instantaneous Kerr response can be tuned to oscillate at a frequency that depends only on the coupling between the cavities (supposed large). This solutions is compatible with a , so it cuts down the injection energy requirements; our approach is more robust and flexible.
A simple qualitative description of the mechanism goes as follows: thanks to large coupling, the three resonant frequencies (that are assumed coincident for isolated resonators) split far apart, much more than the detuning between the central uncoupled resonance and the laser injection. The coupling among the resulting super-modes (defined as the modes of the coupled system) by means of degenerate FWM trasfers energy from the (externall excited) central mode to lateral ones, similarly to frequency combs or a multi-modal nano-cavity Belotti et al. (2010); Combrié et al. (2017), and may result in stable oscillations.
Inspired by similar analyses in the theory of frequency combs Hansson et al. (2013), in the present work we develop a more detailed analysis of the three-cavity oscillator and present its bifurcation diagram. After having assessed the validity of this method, we extend it to a doubly pumped four cavity chain, analogous to bichromatically pumped frequency combs Strekalov and Yu (2009); Hansson and Wabnitz (2014), which are based on non-degenerate FWM. This is a non-autonomous set of ordinary differential equations (ODEs), which is greatly simplified by our method.
We prove that in the large coupling coefficient regime, a microwave frequency can be converted with high efficiency to —up to of the total energy inside the cavity super-modes, i.e. half of the energy is used to excite super-modes at and half at . The bifurcation diagram is similar to that of the three-cavity system, but, owing to the threshold-less nature of non-degenerate FWM, no minimum power is required.
Contrary to bichromatically pumped combs, where FWM is further cascaded over a broad bandwidth and a truncated dynamics behaves appropriately only for quite a specific range of parameters, our treatment is consistent and robust: it can be extended to chains (or molecules) of an arbitrary number of coupled resonators.
In section II we introduce the diagonalization procedure of the nonlinear time-dependent coupled-mode equations (CMT) Haus (1983) to coupled super-mode equations (CSMT). In III we apply this to revisit and improve our understanding of the results of Armaroli et al. (2015). Remarkably, we present a new phase-space representation and a more detailed bifurcation diagram. In section IV we study the four cavity case and contrast its results to that of the previous section. In section V, we discuss the accessibility and technological feasibility of the proposed solutions, by means of a Monte-Carlo approach to explore the parameter space. Finally, in section VI, we conclude.
Ii Coupled-modes and coupled-supermodes
Let us consider a system composed by evanescently coupled optical microcavities (single mode or with a large FSR), the time evolution of which reads, in dimensional units, as Fan et al. (2003); Grigoriev and Biancalana (2011); Abdollahi and Van (2014); Dumeige and Féron (2015)
(1) |
where , is the detuning of the laser excitation from the -th cavity resonance frequency, is the cavity lifetime, quantifies the coupling from the input waveguide to the first cavity. is the Kronecker delta. We further assume, for the sake of simplicity, that the decay into the waveguide is negligible with respect to the intrinsic cavity contribution, i.e. (undercoupling), as opposed to the critically coupled (the escape and decay rates are equal) or overcoupled (the escape in the waveguide is dominant) case.
are normalized such that the square amplitude is the energy stored in the cavity (in J), is the power in W in the external waveguide coupled into the first cavity, are the coupling rate of cavity and and are assumed to be real. The independent variable represents the time expressed in seconds. We let the effective nonlinear coefficient , where is the Kerr coefficient, is the modal effective index and is the modal effective volume.
As in Armaroli et al. (2015), we assume that the lifetimes and modal properties are the same for each cavity , , , for . As for the coupling coefficients, we consider a linear chain of resonators (as in Fig. 1). We thus limit ourselves to for , , and otherwise. We will define below what is the most symmetric choice for the different situations.
By introducing the normalization , , with , we derive from (1) the following adimensional model
(2) | ||||
where the dot denotes the derivative in ; moreover , and is the actual power coupled in the first cavity. is a normalized function which denotes the envelope of the input signal coupled from the waveguide.
If , the conventional approach consists in finding the equilibria (fixed points) of (2) by imposing , for , and characterize their properties.
We proved in Armaroli et al. (2015) that, if the coupling coefficients are large compared to the cavity lifetime, , the dynamics greatly simplifies and regular self-pulsing regimes are found, which can be thought as beating between linear resonances of the whole system, i.e. super-modes Grigoriev and Biancalana (2011). In this case a simplified harmonic balance technique is readily obtained.
However this is not possible if the system is not autonomous, i.e. if has an explicit time-dependence. This is the case, for example, of an harmonically forced system, see below. Alternative approaches for computing bifurcation curves are possible Sracic and Allen (2011), the most common is to include several coupled oscillating degrees of freedom to transform it to an autonomous form.
In details, let the vector of complex modal amplitudes, we write in compact form
(3) |
where the definitions of , , and are obvious, by comparing to Eq. (2).
We suppose that the non-linear part is small and the solution is a perturbation of the linear solution.
We start from the homogeneous linear system obtained for , , and , which provides the super-modes of the system. The complex resonant frequencies (eigenvalues) , with (i.e. the lifetime is the same as for isolated cavities, provided that every cavity has the same resonant frequency). Notice that we diagonalize only the coupling part, i.e. , with the diagonal matrix of eigenvalues and the matrix of eigenvectors.
By defining the super-mode complex amplitude as , we obtain a system of coupled non-linear equations in the form
(4) | ||||
No approximations with respect to Eq. (3). Further simplifications can be made by neglecting all terms in Eq. (4) but those oscillating at , . This is possible if is a sum of sinusoids with frequencies equal to linear combinations of with integer coefficients, i.e. in the rotating wave approximation (RWA).
In this work we consider a three-cavity and a four-cavity system, respectively in secs. III and IV, as prototypes of degenerate- and non-degenerate four-wave mixing (FWM) among super-modes. Our approach allows us to derive an autonomous system, the fixed points of which are easily characterized.
We will highlight their analogies and differences by analyzing their full nonlinear evolution.
Iii Three-cavities revisited
iii.1 Derivation of coupled-super-modes theory
We start by revisiting the results of Ref. (Armaroli et al., 2015). Consider the system depicted in Fig. 1(a). The coupling coefficients are supposed identical, , and . This highly symmetric configuration supports three super-modes at and , with unit normalized lifetime. The eigenvector matrix reads as
Then, in the RWA, the amplitudes of the super-modes evolve according to Eq. (4), which specifically reads as
(5) | ||||
Each super-mode undergoes equal dephasing and losses, owing to the equality of lifetimes and resonant frequencies. The central mode is coherently pumped by the external waveguide. The nonlinear response is composed of a self-phase modulation (SPM), a cross-phase modulation (XPM), and a coherent transfer of energy from one mode to the others. While this form is easily predictable by virtue of the cubic nonlinearity, the coefficients differ from similar systems Chembo and Yu (2010). Notice that does not appear in Eq. (5).
The bistable response of the system is calculated by letting . We can write , where . The system is thus bistable if , and the two saddle-node bifurcations occur for
The self-pulsing threshold is derived by assuming so that no appreciable conversion of energy from sidebands to carrier is possible. The linearized system reads as
and predicts growing sidebands (thus the initiation of self-pulsing) if the gain overcomes losses (), i.e. if , with
A more detailed analysis of the limit cycles is also possible: Eq. (5) has the same form of the coupled-mode theory derived in Hansson et al. (2013), apart from the different wights of SPM and XPM. On the lines of that work, in order to precisely estimate the equilibria of Eq. (5), we transform it to real variables. We define , for , and notice from Eq. (5) (and confirm numerically below, in Figs. 2 and 5) that the steady-state sideband imbalance is a constant of motion. It is verified that . Thus, only four real variables are required, i.e. the relative phase , the relative pump-central-mode phase, (Eq. (5) is not phase invariant), the total intensity in the modes, , and the fraction of the intensity in the lateral modes, . We derive, after some simple algebra,
(6) | ||||
The bistable response mentioned above corresponds to ; solutions with are always unstable. The non-trivial fixed points (, corresponding respectively to the bistable curve and to a homogeneously zero solution) can be expressed in implicit form, by means of some cumbersome algebra. represents directly the energy in the sidebands, i.e. the generated microwave signal.
iii.2 Bifurcation diagram and numerical results
In Fig. 2, we compare the results of the present approach a case qualitatively similar to what we presented in Armaroli et al. (2015) [Fig. 5(b)]. We let , , and study the evolution in time from noisy initial conditions of the cavity towards its self-pulsing state (the slight increase in (from 30 in our previous study to 40) allows us to show more clearly the different bifurcations). We plot the intensities of the super-modes and observe that the time at which the steady state is achieved is different from Eq. (2) to (5), but the values at which and stabilize are close and the behavior is qualitatively similar: importantly the energy transferred to sidebands, i.e. the microwave output, is particularly well-predicted. The imbalance is oscillating about zero, thus reassuring us on the soundness of the present approximation.
Fig. 3 shows the bifurcation diagram for the trivial (bistable curve of fixed points, in black) and non-trivial solutions (limit-cycles, in red) of Eq. (6) obtained by means of a standard numerical continuation package Dhooge et al. (2003). This is equivalent to Fig. 4 of our previous work, we simply change the observables that we consider: instead of the energy stored into the third cavity. The inset shows the fraction in the lateral modes and the relative phases and , which were not straightforward to obtain from our previous calculations. The results of the main panel are indistinguishable from the results of the bifurcation analysis presented in Armaroli et al. (2015), once adapting them to the new variables, in the main panel the comparison is irrelevant. As we know well two types of instability coexist: the absolute instability which leads to the conventional bistable response and the Andropov-Hopf bifurcation which destabilizes the upper stable state and corresponds to the initiation of self pulsing. The bifurcation diagram of limit-cycles (in red) exhibits in turn a saddle-node bifurcation, i.e. a bistability. While the branch of limit-cycles for corresponds to a larger energy stored in the cavity than in the unstable equilibria, for lower , decreases and stabilizes after a sudden surge (red line in the inset). For the system is attracted to the lower fixed point. Another branch of stable limit-cycles (separated by a short branch of unstable solutions, dashed lines in the inset and red dashed line in the main panel) exists for , but it can be reached only from a hot cavity state, i.e. non-zero initial mode amplitudes. Limit-cycles in this branch undergo themselves a bifurcation, namely a Neimark-Sacker (NS, i.e. secondary Andropov-Hopf) bifurcation, from a cycle to a torus, as discussed below. Notice that the conversion efficiency for , the central mode contains less energy than the sidebands.
Finally, the family of limit cycles reconnects to the family of fixed points via a branch of unstable solutions. No other states separated from the curve of fixed points where found, in contrast to Hansson et al. (2013). If the interval between the two NS points is small enough the most-detuned segment of the second branch can be reached adiabatically from the first one. We can thus state that our system is a soft-excitable self-pulsating system.
The advantage of the present approach is to obtain the relative phases and : in Fig. 4, we show the phase space evolution, in terms of polar representation in the two planes and . (a-b) correspond to Fig. 2, (c-d) is the homologue of Fig. 5(c) in Armaroli et al. (2015) (with and , instead). In (a), it is apparent how the cold cavity is first excited into a state corresponding to an unstable non-oscillating solution, making a turn around it. Once the central mode gains enough energy [in (b), is attracted initially to 0, its initial value being random], it loses stability and converts abruptly —about of its energy—to the lateral modes. The phases suddenly lock, to the values shown in the inset of Fig. 3. In panel 4(b), the random initial condition passes through , then abruptly switches to a finite value with a locked .
Panels (c-d) show the two possible scenarios, obtained by changing the initial condition from cold (small random intensity) or hot cavity (the central mode is already strongly excited, as shown by a red cross, ). The former, solid lines, just stabilizes to the fixed point lying on the lower branch (c), the rate of conversion drops to zero (d) (the apparent large initial is again an artifact of small random amplitudes). The latter is first attracted to the upper equilibrium branch, which is in turn unstable but allows the mode to enter in the oscillating regime. As it was shown in (Armaroli et al., 2015), the two conditions are connected in the bifurcation diagram and we can switch adiabatically from one to another, by changing .
The region below is quite richer, because a stable and unstable branches of limit cycles coexist with a stable and an unstable equilibria. The observation of these limit cycles is harder and harder, because, as we approach the limit point where the two cycles merge at , their basin of attraction shrinks. Moreover a bifurcation to tori exists, see the dotted interval in Fig. 3.
For the sake of completeness, we picture this bifurcation by a means of a numerical example: we let and an excited central super-mode (). The system settles on a limit-cycle (oscillating at ) modulated at a persistent low frequency , see Fig. 5. We did not discuss this bifurcation before, albeit it was observable in the adiabatic transition in Fig. 5(d) of Armaroli et al. (2015), because it represents a source of noise and instability in view of the generation of microwaves on the optical carrier and should thus be avoided; however, this concerns a small region of the parameter space. The modal and super-modal approaches agree quite well even in the present case, except for a slightly shorter period in the secondary oscillation predicted by the latter. does not significantly deviate from over all the considered temporal range.
Iv Four cavities
iv.1 Derivation of coupled-super-modes theory
We now consider the system, depicted in Fig. 1(b). The most general symmetric system is and .
The super-modes are located at frequencies and , with and . We focus here on the specific case , which is the most symmetric and efficient, as far as conversion is concerned, Trillo et al. (1994); Marhic (2007); Ott et al. (2013). This condition is satisfied for . This allows us to obtain a simple eigenvector matrix
with , . We suppose the central modes are excited at the same time by a modulated input and study the energy conversion to .
We now derive, along the same lines of the previous section, the complex ODEs that govern the amplitudes (the subscripts referring to oscillations at multiples of frequency ) of the super-modes—only terms rotating at frequencies are retained,
(7) | ||||
with .
The SPM, XPM and coherent interactions among modes are similar to those found in Eq. (5). Moreover, the terms originate from the particular choice of the FWM process, and act as a forcing at . Thus the steady-state solution for the pumps
As explained above, for large spurious bifurcations and chaotic regimes are largely suppressed and, if , the pump couples evenly to both pump super-modes. Thus the imbalance of each harmonic pair, and , can be thus safely assumed to be zero. We will discuss deviations below.
We derive the simplest real form for the system (7), by assuming and and defining the positively detuned half energy , the sideband fraction and the relative phase . Together with the relative injection to -mode phase , these variables evolve according to the following system,
(8) | ||||
For the sake of keeping the notation at a minimum, we use the variable names of the previous section. Whenever we need to distinguish between them, we will add a superscript . It is straightforward to verify that the trivial fixed points of Eq. (8) are always unstable, thus we study numerically the other equilibria, which correspond necessarily to a small generated signal at .
iv.2 Bifurcation diagram and numerical results
In Fig. 6 we show the bifurcation diagram of obtained from Eq. (8) as a function of , for different values of (the actual value of is not important, as it does not appear in Eq. (7)). Do not forget that is the detuning of the laser frequency (the center of the two pumps) with respect to the uncoupled linear cavity resonance, i.e. the midpoint between . Far from resonance, at , the line connects smoothly with the undepleted pump solutions (). This is the precise meaning of thresholdless excitation of the lateral resonances: contrary to the previous section, we do not need a specific combination of detuning and power levels in order to observe self-pulsing. As in the previous section, bistabiliy of limit-cycles (two stable and one unstable branches coexist) occurs for , as can be approximately predicted by looking for the steady state for . By increasing , we observe that a second hump appears (at , blue line, is already noticeable), then the hump folds towards negative and the branch of larger splits in two stable and one unstable limit cycles. We now have three stable and two unstable branches: we will denote the stable limit-cycles as higher, intermediate (the one extending towards extreme ), and lower. The saddle-node bifurcation, where the lower stable and unstable branches merge is well approximated by the solution in the limit. The two stable solutions split farther and farther apart as we increase . In Fig. 7 we show the bifurcation of , , and . Notice that, on the intermediate branch, the conversion , i.e. more energy is trasnferred to than that in . In contrast to frequency combs Hansson and Wabnitz (2014), where the cascaded FWM distributes energy to higher and higher frequencies, depending on dispersion, and the spectral shaping demands complicated contrivances (feedback control of injection, dispersion engineering…), here an efficient energy conversion occurs spontaneously. The upper branch is characterized by , while the intermediate and lower ones require . The bifurcation analysis of Eq. (8) predicts other bifurcations, notably a supercritical Neimark-Sacker bifurcation from cycles to tori (for ).
We present below some numerical examples of evolution in time in order to validate the super-modal approach and to understand the different permitted behaviors. We let , , (as in Fig. 7(d)).
Fig. 8 shows the evolution towards a stable limit-cycle, for , i.e. where only the upper branch exists, starting from random noise-like initial perturbations (cold cavity). Notice that the pump and signal imbalances, and oscillate around 0 during the whole time window. A steady-state is quickly attained, apart from small oscillation at a frequency , stemming from the non-resonant contributions that are neglected in Eq. (7). The comparison of the numerical integration of Eq. (7) and of the conventional CMT shows a much better agreement than in the previous section, for . This is due to the threshold-less nature of non-degenerate FWM, which provides an active forcing for . The phase-space representation [defined, as above, by the two planes and ] is provided in Fig. 10(a-b). They show how the energy starts soon converting to the side-modes, and does not need to heat the cavity before starting oscillations. This is evident particularly in the absence of sharp phase jump away from a fixed point in (a) and the smooth growth of in (b); compare the present behavior to Fig. 4.
Next, we study the more complex case, where the lower branch (where ) coexists with the most energetic intermediate one. Let . Obviously, starting from a cold cavity leads the system to decay to the lower branch. We thus impose the hot cavity conditions and , i.e. the pump pair is already intense enough inside the cavity, in order to lie in the basin of attraction of the intermediate branch. The agreement between the two models is still satisfactory: we cannot neglect that is excited more strongly than its mirror , the imbalance turns indeed negative. stays quite steadily around 0. are only slightly overestimated, while is a bit underestimated. In Fig. 10(c-d) we map this solution in the pair of phase-planes. Notice that initial conditions are very close to the final state, the basin of attraction of which is quite narrow (verified numerically, not shown). The system spirals around its steady state. Panel (c) is quite similar to the dashed line in Fig. 4(c), while (d) differs owing to the thresholdless conversion mechanism: starts to grow at the very beginning.
Finally we verify that two branches can be connected by an adiabatic variation of detuning. We start from a cold cavity system and , let it stabilize up to , then adiabatically decrease it up to at . In Fig. 11 we show the results of our simulation. First we notice that the predicted existence and stability of the intermediate branch ends at for Eq. (2), while at for Eq. (7), as predicted by the bifurcation diagram of Fig. 6. This represents the main limitation of the present approximation. Then we observe that after the upper branch disappears at , strong oscillations set in, similar to Fig. 5. This confirms the presence of the torus bifurcation shown in Figs. 6 and 7.
For larger (not shown), it becomes harder to observe the most detuned part of the intermediate branch. The saddle-node bifurcation where the higher branch disappears occurs at nearly the same of the NS bifurcation. Moreover, larger region of existence for tori between the two NS points imply larger secondary oscillations. The system is preferably attracted to the small conversion branch.
V Concrete implementation and limits
v.1 Estimate of design parameters
We finally comment on the physical accessibility of this approach. We assume to operate at and take ns, so that the cavity has a . Consider a racetrack cavity with minimum curvature radius and mode area (upper bound): the modal volume is . and the effective index of the mode is . A semiconductor of refractive index and Kerr index is considered Wagner et al. (2009), thus ; we finally get fJ. We also assume weak waveguide coupling , so that the first cavity is undercoupled to the waveguide and the quality factor does not vary considerably from a cavity to the other.
With these values, for , corresponds to a power in the waveguide mW These power levels are feasible despite the undercoupling regime. As far as the coupling is concerned, a basic modal calculation, Little et al. (1997); Haus (1983), permits to estimate that two waveguides of cross section nm with a gap of nm require only about a coupling length to achieve the normalized value of . This value was chosen in order to obtain an oscillation frequency of about 9 GHz. The secondary frequency, where cycles bifurcate to tori is around 318 MHz.
The use of the same parameters for leads to a forcing (generated) frequency GHz ( GHz). corresponds, for the considered sinusoidal forcing, to an average input power mW. Still these values are attainable in current technological platforms.
v.2 Robustness to fabrication tolerances
An important question is whether the fabrication tolerances with respect to nominal values inhibits to observe the phenomena we are studying.
We perform some Monte-Carlo simulations, by letting , , and randomly vary according to a Gaussian distribution around their nominal values; is kept constant. As in the previous paragraph the coupling time fluctuates around . We simulate different realizations of Eq. (2) and look for the maximum standard deviation at which the system behaves as expected.
In the previous sections, we slightly abused of notation by using the same symbols for the autonomous and sinusoidally forced oscillators. Here we use superscripts to distinguish between them.
For the system is very robust: an independent choice of every parameter with a standard deviation is still tolerable.
In Fig. 12, for and , the energy coupled in the cavity (a) and its fraction in the limit cycles (b) are distributed around the predicted solution (see Fig. 2) with a standard deviation of about and respectively. The systematic inconsistency of the average (lower than expected) is explained by the inclusion of a finite external coupling . The imbalance (not shown) is distributed around 0 and does not represent a major problem.
In the case of , at each realization we assume to tune the input frequency in order to match the eigenvalue of a pair of supermodes. This situation is more sensitive to small deviations, owing to the higher complexity of the system. Anyway a standard deviation is well tolerated.
This is shown in Fig. 13, for and : the energy stored in the cavity (a) is in most cases around the expected value (compare to Fig. 8), but with a difference limited to at most, on top of the systematic error mentioned above. The generated microwave energy (b) is peaked around the expected steady state and exhibits a long tail of smaller values, too. Their distribution is much broader on account of the inefficient mechanism which occurs if . This latter is shown in panel (c) to be distributed according to a Gaussian curve with a standard deviation of about . The main limitation on conversion efficiency is thus the spacing of the super-mode resonances compared to their lifetime, but FWM is effective in more than half of the cases. Reducing the uncertainty on parameters improves the results.
Vi Conclusions
Based on a diagonalization procedure of the coupled-mode theory in time, which allows us to write the nonlinear equations which rule the coupling between super-modes in a chain of Kerr-nonlinear optical micro-resonators, we present a thorough bifurcation analysis of (i) degenerate four-wave mixing in a three cavity chain (ii) non-degenerate four-wave mixing in a four cavity chain. Their bifurcation diagrams and behavior in phase space are similar in many aspects (bistability of limit-cycles, Neimark-Sacker bifurcations, phase locking between injection and pump and between pump and sidebands). The main difference relies on the thresholdless microwave generation in the four-cavity system.
A sensible set of parameters is presented to show the accessibility of these oscillatory regimes. Moreover the exploration of the parameter space by means of Monte-Carlo simulations allows us to estimate the robustness of the present solutions to technological inaccuracies. The result indicates that the four cavity solution is less robust (tolerating quite a smaller uncertainty level), but still achievable in current technological platform.
Acknowledgments
Y. D. acknowledges the support of the Institut Universitaire de France (IUF).
References
- Ilchenko and Matsko (2006) V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—Part II: Applications,” IEEE J. Sel.Top. Quant .Electron. 12, 15–32 (2006).
- Turner et al. (2008) A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator.” Opt. Express 16, 4881–4887 (2008).
- Azzini et al. (2013) S. Azzini, D. Grassani, M. Galli, D. Gerace, M. Patrini, M. Liscidini, P. Velha, and D. Bajoni, “Stimulated and spontaneous four-wave mixing in silicon-on-insulator coupled photonic wire nano-cavities,” Appl. Phys. Lett. 103, 10–14 (2013), .
- Pu et al. (2015) M. Pu, H. Hu, L. Ottaviano, E. Semenova, D. Vukovic, L. K. Oxenlowe, and K. Yvind, “AlGaAs-On-Insulator Nanowire with 750 nm FWM Bandwidth, -9 dB CW Conversion Efficiency, and Ultrafast Operation Enabling Record Tbaud Wavelength Conversion,” in Optical Fiber Communication Conference Post Deadline Papers, September (2015) paper Th5A.3.
- Combrié et al. (2017) S. Combrié, A. Martin, and A. de Rossi, “Comb of high-Q Resonances in a Compact Photonic Cavity,” Laser Photonics Rev. 11, 1700099 (2017).
- Ibanescu et al. (2002) M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and John D Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
- Sylvain et al. (2013) S. Combrié, G. Lehoucq, A. Junay, S. Malaguti, G. Bellanca, S. Trillo, L. Ménager, J. P. Reithmaier, and A. De Rossi, “All-optical signal processing at 10 GHz using a photonic crystal molecule,” Appl. Phys. Lett. 103, 193510 (2013).
- Ghisa et al. (2007) L. Ghisa, Y. Dumeige, N. Nguyên Thi Kim, Y. G. Boucher, and P. Feron, “Performances of a Fully Integrated All-Optical Pulse Reshaper Based on Cascaded Coupled Nonlinear Microring Resonators,” J. Lightwave Tech. 25, 2417–2426 (2007).
- Matsko et al. (2005) A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005).
- Del’Haye et al. (2008) P. Del’Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J. Kippenberg, “Full stabilization of a microresonator-based optical frequency comb,” Phys. Rev. Lett. 101, 053903 (2008), .
- Razzari et al. (2010) L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, ‘‘CMOS-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2010).
- Del’Haye et al. (2009) P. Del’Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics 3, 529–533 (2009), .
- Chembo and Yu (2010) Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82, 033801 (2010).
- Soltani et al. (2012) M. Soltani, S. Yegnanarayanan, Q. Li, A. A. Eftekhar, and A. Adibi, “Self-sustained gigahertz electronic oscillations in ultrahigh-Q photonic microresonators,” Phys. Rev. A 85, 053819 (2012).
- Cazier et al. (2013) N. Cazier, X. Checoury, L.-D. Haret, and P. Boucaud, ‘‘High-frequency self-induced oscillations in a silicon nanocavity,” Opt. Express 21, 13626 (2013).
- Pasquazi et al. (2017) A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, ‘‘Micro-combs: A novel generation of optical sources,” Phys. Rep. (2017).
- Herr et al. (2014) T. Herr, V Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L . Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014), .
- Hausmann et al. (2014) B. J. M. Hausmann, I. Bulu, V. Venkataraman, P. Deotare, M. Loncar, and M. Lončar, “Diamond nonlinear photonics,” Nat. Photonics 8, 369–374 (2014).
- Grinberg et al. (2012) P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, ‘‘Nanocavity Linewidth Narrowing and Group Delay Enhancement by Slow Light Propagation and Nonlinear Effects,” Phys. Rev. Lett. 109, 113903 (2012).
- Faraon et al. (2010) A. Faraon, P. E. Barclay, C. Santori, K.-M. C. Fu, and Raymond G. Beausoleil, “Resonant enhancement of the zero-phonon emission from a color center in a diamond cavity,” Nat. Photonics 5, 5 (2010), .
- Malaguti et al. (2011) S. Malaguti, G. Bellanca, A. De Rossi, S. Combrié, and S. Trillo, “Self-pulsing driven by two-photon absorption in semiconductor nanocavities,” Phys. Rev. A 83, 051802 (2011).
- Armaroli et al. (2011) A. Armaroli, S. Malaguti, G. Bellanca, Stefano Trillo, A. de Rossi, and S. Combrié, “Oscillatory dynamics in nanocavities with noninstantaneous Kerr response,” Phys. Rev. A 84, 053816 (2011).
- Vaerenbergh et al. (2012) T. Van Vaerenbergh, M. Fiers, J. Dambre, and P. Bienstman, “Simplified description of self-pulsation and excitability by thermal and free-carrier effects in semiconductor microcavities,” Phys. Rev. A 86, 063808 (2012).
- Huet et al. (2016) V. Huet, A. Rasoloniaina, P. Guillemé, P. Rochard, P. Féron, M. Mortier, A. Levenson, K. Bencheikh, A. Yacomotti, and Y. Dumeige, “Millisecond photon lifetime in a slow-light microcavity,” Phys. Rev. Lett. 116, 133902 (2016).
- Maes et al. (2009) B. Maes, M. Fiers, and P. Bienstman, “Self-pulsing and chaos in short chains of coupled nonlinear microcavities,” Phys. Rev. A 80, 033805 (2009).
- Grigoriev and Biancalana (2011) V. Grigoriev and F. Biancalana, “Resonant self-pulsations in coupled nonlinear microcavities,” Phys. Rev. A 83, 043816 (2011), .
- Dumeige and Féron (2015) Y. Dumeige and P. Féron, “Coupled optical microresonators for microwave all-optical generation and processing,” Opt. Lett. 40, 3237–3240 (2015).
- Armaroli et al. (2015) A. Armaroli, P. Féron, and Y. Dumeige, “Stable integrated hyper-parametric oscillator based on coupled optical microcavities,” Opt. Lett. 40, 5622–5625 (2015).
- Belotti et al. (2010) M. bibnamefont Belotti, M. Galli, D. Gerace, L. C. Andreani, G. Guizzetti, A. R. Md Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “All-optical switching in silicon-on-insulator photonic wire nano-cavities,” Opt. Express 18, 1450–1461 (2010).
- Hansson et al. (2013) T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulational instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013), .
- Strekalov and Yu (2009) D. V. Strekalov and N. Yu, “Generation of optical combs in a whispering gallery mode resonator from a bichromatic pump,” Phys. Rev. A 79, 041805(R) (2009), .
- Hansson and Wabnitz (2014) T. Hansson and S. Wabnitz, ‘‘Bichromatically pumped microresonator frequency combs,” Phys. Rev. A 90, 013811 (2014), .
- Haus (1983) H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall series in solid state physical electronics, 1983) p. 402.
- Fan et al. (2003) S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
- Abdollahi and Van (2014) S. Abdollahi and V. Van, “Analysis of optical instability in coupled microring resonators,” J.Opt. Soc. Am. B 31, 3081–3087 (2014).
- Sracic and Allen (2011) M. W. Sracic and M. S. Allen, “Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems,” in Civil Engineering Topics, Volume 4 Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY, edited by Proulx T. (Spinger, New York, 2011) 51–69.
- Dhooge et al. (2003) A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov, “MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs,” ACM Transactions on Mathematical Software 29, 141–164 (2003).
- Trillo et al. (1994) S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A 50, 1732–1747 (1994).
- Marhic (2007) M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University Press, 2007).
- Ott et al. (2013) J. R. Ott, H. Steffensen, K. Rottwitt, and C. J. McKinstrie, “Geometric interpretation of four-wave mixing,” Phys. Rev. A 88, 043805 (2013).
- Wagner et al. (2009) S. J. Wagner, B. M. Holmes, U. Younis, A. S. Helmy, David C. Hutchings, and J. S. Aitchison, “Controlling third-order nonlinearities by ion-implantation quantum-well intermixing,” IEEE Photonics Tech. Lett. 21, 85–87 (2009).
- Little et al. (1997) B. E. Little, S T Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Tech. 15, 998–1005 (1997).
- Okawachi et al. (2015) Y. Okawachi, M. Yu, K.Luke, D. O. Carvalho, S. Ramelow, A. Farsi, M. Lipson, and A. L. Gaeta, “Dual-pumped degenerate Kerr oscillator in a silicon nitride microresonator,” Opt. Lett. 40, 5267–5270 (2015), .