Optical multi-stability in a nonlinear high-order microring resonator filter

Optical bistability, addressed initially by the pioneering work of Szöke et al. in 1969,¹ is a phenomenon that is still of significant interest due to its many applications in ultrafast communications and signal processing.^2–9 Photonic crystals and micro-cavities are two popular geometries for the implementation of all-optical devices such as switches,¹⁰ logic gates,¹¹ and memories.¹² These devices utilize the free carrier or Kerr induced nonlinear resonance shift to generate their bi-stable behavior.¹³ In general, low switching threshold power, high on/off contrast, and multiple operation states are desired in these devices.

Recently, integrated optical ring micro-cavities fabricated with CMOS compatible platforms such as silicon,¹⁴ silicon nitride,^15,16 and high-index-contrast doped silica glass^17–19 have demonstrated excellent nonlinear optical performance. Bistability in these micro-cavities has been investigated both theoretically^20–22 and experimentally,^23–26 based on the Kerr and/or thermo-optical effects, which yield intensity dependent nonlinear responses. In particular, bistability can be triggered at very low power levels in these devices, due to the strong mode confinement, long interaction lengths, and high enhancement of the resonant field.^27,28

In this framework, high-order resonator filters consisting of multiple cavities have been explored for optical functions such as slow light²⁹ and optical buffers.³⁰ Unlike the linear Lorentzian response of a single ring resonator, in high-order filters, both the bandwidth and shape of the linear high-order filter response can be adjusted by controlling the coupling between the cavities, as well as their relative resonances, thus providing additional degrees of freedom^31,32 in designing these devices. For example, Vlasov et al. demonstrated optical switching in a high-order ring resonator filter by thermal tuning of the coupling between the cavities achieved with an external laser.³³ The Kerr nonlinearity has been studied in these systems showing slow-light enhanced four-wave mixing,^34,35 self-pulsation,^36,37 and multistability.^38,39

Although theoretical models that consider both the optical nonlinear Kerr and thermal effects have previously been proposed and investigated,^24,37 the bistable and, especially, the multi-stable behavior in high-order microresonator filters have not been experimentally demonstrated. In this work, we present both theoretical and experimental studies of the multi-stable response of a fifth-order microring resonator filter. The characterization of the intracavity power distribution in each of the coupled resonators, performed by observing the spectra over different nonlinear regimes, provides a further understanding of the nonlinear dynamics in coupled nonlinear systems. Here, we use a theoretical model based on coupled mode theory³¹ to describe a nonlinear high-order microring resonator filter and experimentally verify its multi-stable behavior. We determine the magnitude of the thermally induced resonant frequency shift by comparing experiments with simulations. Finally, we investigate the effects of input frequency detuning on the nonlinear response and derive a method to measure the intracavity power distribution at various stages of the nonlinear operation.

Figure 1 shows a schematic of a device consisting of a series of N coupled resonators with the first and last cavities coupled to the input and output waveguides, respectively. The individual resonators are characterized by the energy wave amplitudes for the rings and are represented by the energy amplitude array $a=a1,a2,…,aN−1,aNT$⁠, where |a_q|² is the energy in the q^th ring defined in units of joules (J). We assume that the power P_in, defined in units of watts (W), is excited at the input waveguide, with the input array $s=−jμiPin1/2,0,…,0,0T$⁠. Here, μ_i is the input coupling coefficient, in dimensionless units. The time evolution of the energy amplitude a(t) has the following form:^31,37

where M is the coupling matrix.

To account for the thermal contribution, an array $ΔωT$ is introduced with $ΔωT=Δω1T,Δω2T,…,ΔωN−1T,ΔωNTT$⁠, where $ΔωqTt$ is the time-dependent temperature induced resonance shift of the q^th ring in unit of rad/s. The amount of thermally induced frequency shift for the individual ring is a function of the instantaneous energy stored in the ring so that the time evolution of $ΔωT$ is expressed as follows:

Here, $τT$ is the thermal relaxation time and γ_T is the thermal coefficient defined in J⁻¹ s⁻¹ and characterizing the thermally induced frequency shift, while the array $Q=a12,a22,…,aN−12,aN2T$ takes into account the energies in each ring. When only the first terms of the right-hand side of Eq. (2) are considered, the equation exhibits an exponential decay type solution, showing that the angular frequency deviation approaches zero with time. In other words, the system gradually becomes stable, and there is no angular frequency deviation with increasing time if no external effect is introduced. The second term in the right-hand side of Eq. (2) describes the angular frequency deviation due to the power dependent thermal effect. It is noted that both the nonlinear Kerr effect and the thermal effect lead to a red shift of the angular frequencies in our simulations and experiments. The cancellation between the nonlinear Kerr effect and the thermal effect is not considered in this model. Other effects including the plasma effect are also not considered.^40–42 

Besides the usual linear M_L and nonlinear M_NL elements, the coupling matrix M in the model also consists of an additional thermal coupling matrix element M_T to account for the thermally induced frequency detuning, such that M = M_L + M_NL + M_T, where

Here, ∆ω_q = 2π∆f_q is the angular frequency detuning from the resonance angular frequency ω_q of the q^th ring. The coupling coefficients are expressed in dimensionless units, where μ_L²/2 is the decay rate due to intrinsic loss, while μ_i²/2 and μ_o²/2 represent coupling to the input and output waveguides, respectively. The relationships between the power coupling coefficient k_z and the normalized coefficient μ_z are $ki=μiTr$⁠, $ko=μoTr$⁠, $k1=μ1Tr$⁠, $k2=μ2Tr$⁠, $k3=μ3Tr$⁠, and $k4=μ4Tr$⁠, where T_r = 2πR/v_g is the roundtrip time, R is the ring radius, and v_g the group velocity.²⁴ Also, $μL=αc/n0$⁠, where α represents loss in units of nepers/m. Here, n₀ is the effective index. In Eq. (4), we have expressed the Kerr effect in J⁻¹ s⁻¹, γ_K = γc/(n₀T_r), so that it can be directly compared to the thermal coefficient γ_T. The parameter γ is the nonlinear coefficient due to the Kerr effect in W⁻¹ m⁻¹, defined as γ = n₂ω₀/(cA_eff). Here, n₂ is the Kerr coefficient at the frequency ω₀, c is the speed of light in vacuum, and A_eff is the effective area of the cross section of the ring waveguide.

Equations (1)–(5) can be used to obtain a linear relationship between the frequency detuning ∆f_q and the instantaneous ring power P_q = |a_q|²/T_r as follows:

To study the instability property of the system described by Eqs. (1)–(6), we implemented a linear stability analysis. We introduced a perturbation vector δa to the energy amplitude stationary state a_f and a perturbation vector $δΔωT$ to the thermal detuning stationary state $ΔωfT$⁠, where $δa=δa1,$$δa2,…,δaN−1,δaNT$⁠, $af=af,1,af,2,…,af,N−1,af,NT$⁠, and $δΔωT=δΔω1T,δΔω2T,…,δΔωN−1T,δΔωNTT$ and $ΔωfT$ is referenced to the fixed point $ΔωfT=Δωf,1T,Δωf,2T,…,Δωf,N−1T,Δωf,NTT$⁠. By solving the complex eigenvalue problem of the first order perturbation equations, one can determine the different stability regions of the system.³⁷ 

We now consider the multi-stable behavior of the fifth-order microring resonator filter for theoretical verification and experimental demonstration, starting with its theoretical analysis. Our device is the so-called Chebyshev filter, and it has been designed with a configuration presenting the maximum flat passband.³¹ Such a configuration is obtained by setting the six coupling coefficients as μ_i = μ_o, μ₁ = μ₄ = 0.309µ_i², and μ₂ = μ₃ = 0.178µ_i². A value for μ_i of 1.7665 × 10⁵ is used in the simulation, which corresponds to the power coupling coefficients k_i = 0.235. μ_L is assumed to be zero. The relaxation time τ_T for the thermal effect is assumed to be 1 ms in the simulations, which is reasonable and beyond the estimated value of about 1 μs for the thermal relaxation time used in Ref. 24. The static properties of the fifth-order microring resonator filter (given an input laser detuning ∆f_p = −2.3 GHz) calculated and shown in Fig. 2. As an example, the left panel of Fig. 2(a) reports the output power P_drop of the steady state as a function of the input power P_in for γ_T/γ_K = 100. Here, the blue parts of the multi-stable curve mark the unstable state, while the red parts represent the stable state, where the maximum number of unstable states is proportional to the filter order. In the right subgraph, the real and imaginary parts of the dominant eigenvalue corresponding to such a stationary state are also calculated.⁴⁰ The multi-stable regions with positive real parts of the eigenvalues and zero imaginary parts are shaded in blue, matching the threshold boundaries of the multi-stable regions depicted in the left subgraph. Figure 2(b) presents a map of the stable and unstable regions of the filter as a function of both the output power P_drop = |a_N|²/T_r and the ratio γ_T/γ_K. The expected decrease in the output power threshold boundaries of the bi-stable and multi-stable regions is observed when increasing the ratio γ_T/γ_K. The theoretically calculated map for a specific high-order microring resonator filter can be used to fit the experimental results and obtain an estimate of the thermal coefficient γ_T, using the measurement of the Kerr coefficient γ ∼ 233 W⁻¹ km⁻¹.¹⁶ The linear coupling coefficients can be obtained by fitting the output filter shape.

Figure 3 shows a schematic of the bi-stability and multi-stability experiments, which also allow for the measurement of the high-order ring resonator filter response. The fifth-order ring resonator filter device consists of a cascade of five microring resonators comprised of high-index (n = 1.7) doped silica glass core waveguides embedded in the silica cladding layer. The five rings have the same radius of 50 µm, where the dimension of the waveguide cross section is 1.45 × 1.45 µm². The high-index-contrast waveguide has negligible linear (<0.06 dB/cm) and nonlinear losses with nonlinear parameter γ as high as ∼233 W⁻¹ km⁻¹.¹⁶ The packaged device was under temperature control with a resolution of 0.01 °C, which corresponded to a shift of 0.2 pm in wavelength. In the bi-stability and multi-stability experiment, the input signal was supplied by a quasi-CW tunable laser with frequency ω_input subsequently amplified by an erbium doped fiber amplifier (EDFA). Here, the amplified spontaneous emission noise was suppressed via a 1 nm tunable filter with an extinction ratio greater than 55 dB. A polarization controller was used to adjust the polarization of the input signal into the device. To determine the intracavity power distribution, the evolution of the filter response during the operation was measured with an Agilent insertion/polarization dependent loss (IL/PDL) fast scanning system. Such a system consisted of a quasi-CW laser acting as the probe beam and a polarization synthesizer. To minimize the effect of the scanning signal on the instability behavior, the input frequency ω_input and the probe frequency ω_probe were kept to within a few free spectral ranges (FSRs) apart, with the power of the probe beam kept at least 30 dB below the input beam. The output signal from the drop port was frequency separated into two paths, with the input and probe signals detected by using two separate detectors.

In the experiment, the linear response of the fifth-order device was measured with the input signal switched off and with the output at the drop port directly connected to the Agilent N7744A optical power detector. Figure 4(a) shows the measured drop response of the device across a span of two FSRs, with FSR = 577 GHz at 1565 nm. The full width at half maximum (FWHM) of the filter pass band is 3.5 GHz. Since the individual rings in the fabricated device are not identical due to fabrication process variations, there is a slight difference in the spectral shape at different resonances, as shown in the enlarged insets. The measured through port loss is 2.75 dB, indicating that the coupling loss between the fiber and the waveguide is <1.5 dB per facet, while the pass band loss is about 4 dB. The pass band near 1560 nm is fitted against the linear model in Ref. 31 to determine the relative detuning between the individual rings of the filter and the corresponding coupling coefficients. The extracted individual detuning is zero, and the coupling coefficients between the ring and waveguide are μ_i = 0.13 and μ_o = 0.10, while between the individual rings are μ₁ = 0.015, μ₂ = 0.01, μ₃ = 0.0115, and μ₄ = 0.012. The loss is 0.22 dB/cm, which corresponds to a coefficient μ_L of about 94.5. The simulated linear response with the above-mentioned parameters is shown in Fig. 4(b).

Figure 5(a) shows the simulated nonlinear response at a pump detuning of ∆f_p = −3.0 GHz. Using the extracted filter parameters and according to the steady state analysis in Fig. 2(b), γ_T/γ_K is set to 180 in the simulations to obtain the best agreement with experiments, which tends to apply to all frequencies. In other words, thermal effects form the bulk of the contribution to the observed bi-stable behavior. Based on this thermal coefficient value, we investigated the influence of the input detuning ∆f_p on the instability response both experimentally and with the nonlinear model. Figures 5(b) and 5(c) show the simulated and measured responses when the input frequency was tuned to ∆f_p = −2.7 GHz and −2.4 GHz, respectively. It can be seen from the experimental results that a slight detuning of 0.3 GHz, from −3.0 GHz to −2.7 GHz, induced a drastic change to the response, shifting the behavior from bi-stable to multi-stable, in good agreement with the model. At ∆f_p = −2.7 GHz, both the experimental and simulated results show that the variation of the multi-stable response at larger powers creates an additional hysteresis loop. When the detuning is further reduced by an amount of 0.3 GHz at ∆f_p = −2.4 GHz, the additional hysteresis loop observed at ∆f_p = −2.7 GHz becomes narrower with the main bi-stable behavior having a much smaller hysteresis loop compared to that at ∆f_p = −3.0 GHz. The large change in the observed nonlinear behavior with only a slight change of input frequency offers an interesting route for creating tailored pulse shapes by controlling the power, frequency, and spectral width of the input pulse.

To obtain a better understanding of the characteristics of the observed bi-stability and multi-stability behaviors at different stages, we plot in Fig. 6 the simulated intracavity power distribution of the filter stemming from the nonlinear behavior of the individual rings. Figures 6(a)–6(c) show the simulated individual ring powers as a function of increasing P_in for different input detuning values. The first three rings near the input waveguide have very different behavior compared to the remaining two rings. The last two rings exhibit classical instability behavior, while the response of the first three rings is more erratic. The detuning variation ∆f_q, of the individual rings with P_in at different stages of the bi-stable and multi-stable curves, given by Eq. (6), is shown in Figs. 6(d)–6(f).

In the initial stable region where P_in is below the lower transition threshold of the bi-stability or multi-stability, the values of ∆f_q are small and they all shifted together, nearly “in unison” with each other. Here, the filter pass band slightly shifts in the negative frequency direction with an increase in P_in while maintaining its original shape, as marked in red in Figs. 6(g)–6(i). When P_in is further increased, the middle rings 2 and 4 encounter larger nonlinear shifts than the other rings because the optical power is larger in these rings. Therefore, the bi-stable and multi-stable curves are determined by the distribution of the individual ring detuning. Figures 6(g)–6(i) show that once the detuning of the individual rings changes, the original near flat-top filter shape becomes distorted and the ripples of the responses become more complex, as marked in magenta and green. This corresponds to the redistribution of the power between the individual rings. Generally, as the shape of the filter response evolves with P_in, the spectral power tracks the changes in the bistable and multi-stable responses, as deduced when comparing the bistable and multi-stability curves in blue with the evolution of the spectral shapes in Figs. 6(g)–6(i). Furthermore, it can be seen from Fig. 6 that the frequency of the input power can be used to select the bistable or multi-stable behavior with different intracavity power distributions at different stages. From this analysis, it is clear that the main difference between the instability behaviors in the single-order and high-order ring resonator filters lies in the high-order filter’s ability to redistribute its intracavity power, so as to drastically alter the filter response and create much more complex instability dynamics.

Using the scanning probe beam and the Agilent N7744A detector in Fig. 2, the output filter response two FSRs away from the input at various P_in was measured to experimentally investigate the evolution of the filter responses. The individual ring detunings ∆f_q were extracted by assuming that the coupling coefficients μ_z were fixed at their initial values. The extracted ∆f_q as a function of input power P_in at three input frequencies are presented in Figs. 7(d)–7(f), showing the distinct distribution of the individual ring detuning in the simulations discussed above. However, it is the evolution of the filter response in Figs. 7(a)–7(c) that clearly demonstrates the basis of the bi-stable and multi-stable behaviors in the experiment.

We theoretically analyze and experimentally demonstrate optical bi-stability and multi-stability in a high-order integrated nonlinear microring resonator filter. We present a nonlinear model for the analysis of instability in the N^th order resonator filter, which includes both thermal and Kerr effects. The model is used to provide insight into the measured bistable and multi-stable behavior in a fifth-order microring resonator filter. By comparing the simulated and measured bi-stable responses, the thermal effect has been found to dominate the response, being two orders of magnitude larger than the Kerr effect. Using an additional scanning probe beam and a separate detector, we measured the evolution of the filter response for different input powers. We observed that the detunings of the individual rings all shifted uniformly at low-power, while they had different distributions at different stages of the bi-stable and multi-stable curves at high-power. We have shown that the different nonlinear filter responses in the high-order ring resonator filter are due to the redistribution of the optical power within the single resonators. The complex instability behavior achieved in the high-order ring resonator filter can be potentially exploited to produce advanced switching devices.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors acknowledge the support of the EPSRC, Industrial Innovation Fellowship Programme, under Grant No. EP/S001018/1, from INNOVATE UK, project “IOTA” Grant Agreement No. EP/R043566/1 and from the University of Sussex RDF program. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, Grant Agreement No. 725046. A.P. acknowledges the support of the EPSRC, Industrial Innovation Fellowship Programme, under Grant No. EP/S001018/1, from INNOVATE UK, project “IOTA” Grant Agreement No. EP/R043566/1 and from the University of Sussex RDF program. MP has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, Grant Agreement No. 725046. S.T.C. acknowledges support from the Research Grant Council of Hong Kong (GRF No. 9042663). B.E.L. acknowledges support from the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB24030300). R.M. is affiliated to 5 as an adjoint faculty and acknowledges funding by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Strategic, Discovery, and Acceleration Grants Schemes, by the MESI PSR-SIIRI Initiative in Quebec and by the Canada Research Chair Program.