We propose a simple iterative method for calculating the dynamic behavior of ring resonators with fast and slow (cumulative) optical nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. In the case of a slow nonlinearity, the nonlinear phase shift and nonlinear absorption are temporally-integrated over the incident pulse. In this paper, we consider two types of single-ring resonators made out of As2Se3 chalcogenide glass with high nonlinearity and investigate the dynamic properties (especially the effect of the cumulative nonlinearity on optical bistability) using known nonlinear material parameters. It is found that the cumulative nonlinearity suppresses overshoot and ringing after switching, decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The obtained results are useful in developing chalcogenide-based bistable optical devices and the proposed approach is applicable to modeling of a variety of nonlinear optical devices.
© 2011 OSA
Optical ring resonators made of optical waveguides including optical fibers play an important role in integrated optics and fiber optics . Especially, the integrated microring resonators have recently received considerable attention because of their compactness and suitability for integration with other components [2,3]. They have a variety of applications in optical filters, channel drop filters, delay lines, sensors, lasers, modulators, dispersion compensators and so on [1,3]. In addition to these applications, intensity-dependent nonlinear functions such as optical bistability, all-optical switching, time-division multiplexing, four-wave mixing, and photonic logic have been experimentally demonstrated by making use of a ring resonator to enhance their effects [4–8]. Before the demonstration of these nonlinear functions, the steady-state and dynamic transmission properties of nonlinear ring resonators have been extensively investigated using an iterative method, multiple-beam interference method, and so on [9–13]. All these analyses were done under the assumption of an ideal Kerr nonlinearity in which the refractive index change varies as the optical intensity instantaneously. However the nonlinear microrings developed to date employ not the ultrafast Kerr effect, but the free-carrier effect caused by the two-photon absorption or thermal effect [4–8]. Therefore the switching speed of these devises is not fast since the response time of these two effects is slow, especially that of the thermal effect is of the order of milliseconds. Although optical bistability is not yet observed, we are developing faster nonlinear microring resonators using chalcogenide glasses. These glasses (especially, As-Se system and related glass system such as Ag-As-Se) have been attracting attention in recent years because they possess a high Kerr nonlinearity with an ultrafast time response [14–16].
In fact, there are several physical mechanisms that contribute the nonlinear refractive index and nonlinear absorption of materials, depending on the wavelength, pulse width, and peak intensity of an incident light. Moreover ultrafast nonlinearities accompany more or less slow (cumulative) nonlinearities (i.e., thermal), in which case the refractive index change and the absorption change are accumulated over the incident pulse since the decay time is longer than the pulse width [17–19]. It has recently been found that a significant slow nonlinearity with a response time of 15-20 ms is present in chalcogenide glasses, which is presumably attributed not to free-carrier effects or thermal effects, but to photostructural changes inherent in these glasses [20–22]. Although it is very important to investigate the effect of such cumulative nonlinearities on the performance of nonlinear optical devices, few quantitative studies have been done to date.
In this paper we develop a simple iterative method for calculating the dynamic transmission properties of ring resonators with both instantaneous and cumulative nonlinearities. Moreover we investigate the effect of the cumulative nonlinearity on optical bistability in ring resonators made of As2Se3 chalcogenide glass for its realization. Two types of single-ring resonators with a single coupler and double couplers are investigated in this work. Although the existing iterative method under the assumption of an ideal Kerr nonlinearity is described by a set of simple difference equations [13,23,24], we have to solve an integro-differential equation in the presence case. It is found that optical bistability can occur even if such a cumulative nonlinearity is present in the devices. However the hysteresis loop showing the relationship between the input and output optical powers is affected by it.
2. Numerical analysis of nonlinear ring resonators
2.1 Single-coupler ring resonator
A. Iterative method considering n2 and α only
Figure 1 (a) shows a schematic diagram of a single-coupler nonlinear ring resonator, which consists of a bus waveguide and a ring of length L(=2πr, r being the ring radius). In order to facilitate understanding of a novel approach proposed in this work, we briefly summarize the conventional iterative method for this configuration [13,23,24]. For this purpose, the nonlinearity of the ring waveguide is assumed to be of the Kerr type, i.e., the refractive index is given byFig. 1(a). At the coupling point between the bus waveguide and the ring waveguide, the field amplitudes of incident wave E in(t), transmitted wave E out(t), and circulating cavity wave E c(z,t) satisfy the following equations:
Assuming the instantaneous Kerr effect and linear loss of the ring waveguide, the cavity field E c(L,t) at the end of the ring can be expressed by the prior cavity field E c(0,t-τ R) at the entrance as follows:Eq. (4) into Eqs. (2) and (3), we have
As seen from Eq. (6), the output field E out(t) is simply an iteration of the cavity field E c(0,t), with respect to the cavity round-trip time τ R . Thus, we can calculate the dynamic properties when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using Eqs. (6) and (7).
B. Iterative method considering ultrafast and cumulative nonlinearities
Considering the concept of the iterative scheme, what we have to do is to establish the relation between the two cavity fields E c(0,t-τ R) and E c(L,t). Since these two cavity fields are complex, we decide to express them in polar form as follows:Eq. (3). In the general case where there are a variety of nonlinearities, we must determine the relation between E c(0,t-τ R) and E c(L,t) by directly solving the wave equation that governs the propagation through the nonlinear medium.
In the characterization of third-order nonlinear optical materials, the nonlinear equations of the optical intensity I (∝|E|2) and the phase ϕ N are usually employed for the purpose and the material parameters associated with their optical nonlinearities are defined. Using the slowly-varying envelope approximation, we can fully determine the light propagation within the nonlinear medium by [18–20]Eq. (12) and then have
In this case, the changes in the absorption coefficient and refractive index are accumulated. We must solve Eq. (10) together with Eq. (13) for an initial value I(0,t), which changes with time. However we cannot expect its analytical solutions since that is an integro-differential equation. Therefore we must develop a simple numerical approach for solving Eq. (10).
In this work the ring length is subdivided into a large number of segments. If the segment size is small enough, the term proportional to I(z,t) in Eq. (10) can be assumed to be a constant within it, giving a solution of exponential function. Figure 2 shows a schematic drawing for explaining the concept of a proposed approach, where a computation scheme for the light propagation along the ring over one round-trip is shown using the space (z,t). We subdivide the ring length L into M equally spaced segments so that the step size Δz=L/M and the time step size Δt=τ R/M. The time region from t=t-τ R to t=t is shown in Fig. 2. We can numerically solve Eq. (10) along the straight line t=t-τ R +(τ R/L)z by successively connecting the solutions of exponential function as follows:Eq. (11), the nonlinear phase shift ϕ Ν(t-τ R) is given by
Thus, we can calculate a unknown complex value of E c(L,t) at z=L from the known complex value of E c(0,t-τ R) at z=0. Moreover we can determine the transmitted field E out(t) and the cavity field E c(0,t) for the given incident field E in(t) by substituting the obtained E c(L,t) into Eqs. (2) and (3). The obtained cavity field E c(0,t) will be used as an initial value for the computation after τ R. We can entirely calculate the transmission changes when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using repeating such a process at a fixed time interval of Δt.
2.2 Double-coupler ring resonator
The application of the proposed approach to a nonlinear double-coupler ring resonator is straightforward. Figure 1(b) shows the configuration of the ring resonator and the definition of each electric field for analysis, where a ring of length L is coupled to two identical waveguides and is divided into a right and a left half. The naming of each port is based on the analogy between the ring resonator and the Fabry-Perot resonator. E r(t) and E t(t) are the transmitted fields at the reflection and transmission ports, respectively. E c1(z,t) and E c2(z,t) are the right-half (0<z<L/2) and left-half (L/2<z<L) cavity fields, respectively. The complex electric fields are connected as follows:
3. Numerical results and discussion
We present the numerical results for the transient properties of two kinds of nonlinear ring resonators shown in Fig. 1. Since the aim of this work is to obtain useful information in realizing bistable optical devices using As2Se3 glass, the practical computation should be performed using as realistic device and material parameters as possible. We also assume a Nd:YAG laser as a light source. All numerical results presented here were calculated for the following values: free-space wavelength λ=1.064 μm, ring radius r=100 μm, effective mode area S eff=2.0 μm 2, linear refractive index n 0=2.818, linear absorption coefficient α=0.621 cm−1, nonlinear refractive index n 2=3.0×10−17 m2/W, two-photon absorption coefficient β=5.0×10−11 m/W, refractive index change per unit photo density σ r=0.89×10−22 cm3, absorption change per unit photo density σ ab=4.46×10−18 cm2. These linear and nonlinear material parameters are the experimental data obtained using the Brewster-angle technique  and z-scan technique [20,21]. In this case, the cavity round-trip time τ R given by n 0 L/c is 5.90 ps for r=100 μm and the linear absorption coefficient α=0.621 cm−1 corresponds to a transmission loss of 0.17 dB per round-trip. In order to clarify the effect of the linear loss and the cumulative nonlinearity on optical bistable behavior, the numerical simulations will be performed for the following three cases:
- Case (i): n 2≠0, α=0, and the other nonlinear parameters=0.
- Case (ii): n 2≠0, α≠0, and the other nonlinear parameters=0.
- Case (iii): All linear and nonlinear parameters≠0.
Note that the difference between cases (i) and (ii) is the linear loss of the ring waveguide. Although the nonlinear absorption is caused by the fast two-photon absorption and thecumulative effect, the contribution of the former effect is small in this case.
3.1 Single-coupler ring resonator
Figure 3 shows the temporal power profile of the output pulse when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident into the nonlinear ring resonator with κ=0.1and Δϕ 0=−0.1π for three cases (i) to (iii). In this numerical example, two values of the coupler loss γ=0 (Fig. (a)) and γ=0.1(Fig. (b)) are assumed. Note that the nonlinear ring resonator is completely lossless (γ=0 and α=0) for the case (i) in Fig. 3(a)). For the cases of (i) and (ii), we can calculate the nonlinear pulse response using Eqs. (6) and (7), as described in the preceding section. In these two cases, the numerical results obtained by the proposed approach coincide with those obtained using Eqs. (6) and (7). The number of segments used for these numerical simulations is M=500 and the time required for computing one pulse response is within a few seconds on a standard personal computer. For a proper understanding of these pulse responses, we show the corresponding input-output characteristics in Fig. 4 . In the simulations for the cases (i) and (ii), optical bistability never occurs although clear switch-off, i.e., switching from high to low transmissions and subsequent overshoot and ringing take place, as already pointed out in . Some discussion of such an oscillation can be found in . If the cumulative nonlinearity is added to the linear loss α (case (iii)), we can have optical bistability in which the output power describes a clockwise hysteresis loop as the input power is increased and then decreased. As seen from these examples, the losses in the ring resonator are absolutely essential for the appearance of optical bistability. However the extinction ratio, i.e., the ratio of the high and low power levels is not high and the switch-off response is not sharp. Hence, let us calculate the cavity response (build-up or decay) time and discuss the steady-state operation of the ring resonator. From the analogy between the ring resonator and the Fabry-Perot resonator , the cavity decay time τ c (of the intensity) is given byFig. 6 ). According to our previous work , an about 10 times longer pulse is necessary to attain near-stationary conditions.
Figure 5 shows the dependence of the input-output characteristics on the initial detuning Δϕ 0. In this numerical example, a Gaussian pulse with τ P=500 ps is assumed to be incident into the ring resonator with γ=0.1 and κ=0.1. It is found that the width of the hysteresis loop and the switch-off power increase as the magnitude of initial detuning is increased. If the input power does not reach the switching threshold, the ring resonator exhibits linear behavior.
Figure 6 shows the dependence of the input-output characteristics on the pulse width τ P of an incident Gaussian pulse. The ring resonator with γ=0.1, κ=0.1, and Δϕ 0=−0.1π is assumed in this numerical example. For purpose of comparison, the numerical result for case (ii) and τ P=5.0 ns is also presented in the figure. It is confirmed that the width of the hysteresis loop decreases with increasing pulse width. In the absence of the cumulative nonlinearity, the transient hysteresis loop approaches steady-state solution as the pulse width is increased. However such a situation does not occur in the presence of the cumulative nonlinearity. Cumulative nonlinear refraction mainly shifts the operating point (initial detuning) of the device, moving the center of the hysteresis loop on the higher input-power side. Moreover cumulative nonlinear absorption increases the switch-on threshold, decreasing the width of the hysteresis loop. It can be expected that the width of the hysteresis loop decreases with increasing incident pulse width.
3.2 Double-coupler ring resonator
In the double-coupler ring resonator, we can use two types of optical bistability, i.e., transmission bistability (of an anticlockwise hysteresis loop) and reflection bistability (of a clockwise hysteresis loop). However there is essentially no difference between the single-coupler and double-coupler ring resonators. The sole difference between the two resonators is that the double-coupler ring resonator has a loss because of the power extraction through the second coupler even if the two couplers and the ring waveguide are lossless (γ=0 and α=0). Therefore we pay attention to optical bistability in transmission mode, i.e., the power coming out from the transmission port shown in Fig. 1(b). In such case, it should be noted that the output power P t(t) at the transmission port is related to the circulating power P c1(L/2,t) inside the ring through Eq. (18).
Figure 7 shows the nonlinear pulse response at the two output ports when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident in the nonlinear double-coupler ring resonators with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b). In order to clarify thecontribution of the cumulative nonlinearity, the output powers, P t and P r, were calculated for two cases (ii) and (iii). Regarding transmission bistability, the switch-on and switch-off occur at the leading edge and the trailing edge of the incident pulse, respectively. The cumulative nonlinearity can suppress overshoot and ringing after switching, but it brings an additional loss. The temporal variation of P t shows that the induced loss increases with time since the nonlinear absorption continues to accumulate. Figure 8 shows the input-output characteristics at the transmission port of the double-coupler ring resonators with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1(b). The numerical results were computed for two cases (ii) and (iii), and two pulse widths τ P=500 ps and 5 ns. We can clearly confirm that the effect of the cumulative nonlinearity on the switch-off is greater than that on the switch-on. It is also confirmed that the width of the hysteresis loop decreases with increasing incident pulse width. Moreover it is worth pointing out that we cannot assume an infinite pulse width since the accumulated loss becomes infinite.
We proposed a novel approach for calculating the dynamic transmission properties of single-ring resonators with both ultrafast and cumulative nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. Single-coupler and double-coupler ring resonators made of As2Se3 chalcogenide glass with high nonlinearity were considered and the effect of the cumulative nonlinearity on optical bistability was investigated using the proposed numerical method and known nonlinear material parameters. It has been found that we can obtain optical bistability under suitable conditions although the cumulative nonlinearity is considerable intense. Moreover the cumulative nonlinearity can suppress overshoot and ringing after switching. But it decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The proposed approach is easily applicable to higher-order nonlinearities, finite carrier decay time, i.e., the case where the loss term Ν(z,t)/τ in Eq. (12) is not negligible, and so on.
This work was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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