1. Introduction

In recent years, bistable switching in nonlinear optical microcavities has attracted considerable attention for its promising application in all-optical information processing. [1–8] To describe the nonlinear optical feedback that governs bistability, Soljačič et al. related the shift of the cavity modes with the called nonlinear characteristic power and the output power of the cavity by using the perturbation theory. [3] In this way, the analysis of bistability becomes very easy. It revealed that 100% peak transmission could be achieved for the two-dimensional photonic crystal (PC) defects. [3] Lee and Agrawal studied bistablity in the phase-shift gratings by solving the nonlinear coupled-mode equations. [5] But their results showed that the maximum transmission was smaller than unity, especially when the input frequency was far from the cavity mode (Figs. 4 and 6 in Ref [5]). It is well known that refractive index is time-dependent under Kerr nonlinearity, and the dependence can be negligible for the nonlinear optical fibers. [9] For the nonlinear optical microcavities, however, it causes the oscillating shift of the cavity modes. Based on this fact, Lan et al. explained the ultrafast response of PC defects to the excitation of ultrashort pulses. [10] By merely considering this dynamic characteristic, they also indicated that the maximum transmission did not reach unity. [10] Thus, it seems that the nonlinear optical feedback should be corrected by including the dynamic shift effect. Of course, it is expected that the effect should be reflected by the experimental results. Actually, two recent experiments on the ring resonator structures showed that the minimum pass-transmission increased (Fig. 6 in Ref. [7]) or the maximum drop-transmission decreased (Figs. 4 in Ref. [8]) monotonously when the wavelength detuning increased. These are the typical behaviors of dynamic shift effect, but all of them were attributed to other cause. [7,8] Therefore, to clarify theoretically the effect of the dynamic shift on bistability becomes very important.

In this paper, we introduce the dynamic shift of the cavity modes into the transmission formula, trying to describe bistability in the nonlinear optical microcavies properly. The paper is organized as follows. In Section 2, we derive the dynamic shift of the cavity modes by using the perturbation theory. Then, in Section 3, a time-average approach is utilized to modify the relation between the transmission and the input power. We simulate the nonlinear transmission of a PC defect based on the finite-difference time-domain (FDTD) method in Section 4. Finally, a brief summary is given in the conclusion.

2. Dynamic shift of cavity modes under Kerr nonlinearity

As the response of a launch continue wave (CW) whose frequency is near the cavity mode of an optical microcavity, the stable electric field can be written as E(r,t) = [E(r)exp(iωt)+E*(r)exp(-iωt)]/2 = E ₀(r)cos(ωt+φ), where E(r) and E ₀(r) are the complex and real amplitudes related by E(r) = E ₀(r)exp(iφ>), with φ the position-dependent phase. To survive inside a cavity, E(r,t) yields a kind of stationary wave. Thus, it can be written as E(r,t) = E ₀(r)cos(ωt), i.e., the harmonic form as indicated in Ref. [11]. For simplicity, here we focus on the transverse-magnetic (TM) mode whose magnetic field is in the launch plane, the cavity mode under the linear condition satisfies

where n ₀(r) and c are the linear refractive index distribution and the speed of light in vacuum. Under the instantaneous Kerr nonlinearity, the refractive index distribution becomes time-dependent, which can be written as n(r,t) = n ₀(r)[1 + cε ₀ n ₂(r)${\mathbf{E}}_0^2$ (r)cos²(ωt)], where n ₂(r) and ε ₀ are the Kerr coefficient and the permittivity of vacuum, respectively. It shifts the cavity mode from ω to (ω + δω) as expressed below

where δω denotes the shift of the cavity mode. E ₀(r) is assumed invariable when the refractive index distribution changes from n ₀(r) to n(r, t). Based on Eqs. (1) and (2), we obtain

Similar to Ref. [3], a parameter p ₀, known as the nonlinear characteristic power of the optical microcavity, is defined as

where Q is the quality factor of the cavity mode, (2γ _out) is the decay rate of the cavity mode energy to the output port of the cavity, i.e., (2γ _out) = p _out/∫ε ₀ $n_0^2$(r)${\mathbf{E}}_0^2$(r)d r, with p _out the time-averaged output power. Then, Eq. (3) can be simplified as

where (2γ) = ω/Q is the total decay rate of the cavity mode energy. According to Eq. (4), p ₀ describes the Kerr nonlinearity and the spatial confinement of the field in the cavity, with the bracket term very similar to the effective core area of an optical fiber. For the asymmetrically confined nonlinear cavity, p ₀ is launch-direction-dependent as indicated in Ref. [12]. For the symmetrically confined one, p ₀ can be considered as a constant. Eq. (5) shows that a time-varying refractive index leads to a time-varying cavity mode, oscillating in the frequency of (2ω). It is easy to find that the time-averaged value of δω is (- γp _out/P ₀), the same as in the literature. [3] However, to obtain the time-averaged transmission, we cannot treat the shift of the cavity mode like that as will be discussed.

3. Dynamic shift effect on bistability

When a CW with a frequency ω is launched into a linear cavity, the instantaneous transmission can be expressed as T = 2T¯ cos² (ωt), where T¯ = η γ ²/[γ ²+(ω-ω ₀)²] is the well-known Lorentzian lineshape, η is the linear peak transmission and is unity for a lossless cavity. The time-averaged T does not change the shape of spectrum at all since the cavity mode frequency ω ₀ does not move. That is

where p _in is the input power, and δ = (ω ₀-ω)/γ is the frequency detuning of the input CW with respect to the cavity mode. Under Kerr nonlinearity, if we ignore the dynamic shift of the cavity mode and just fix the shift at its time-averaged value, i.e., δω = (-γP _out/P ₀), the time-averaged approach also changes nothing.

It is the same as shown in Ref. [3]. Based on Eq. (7), bistability appears when δ > √3 and, T¯ = η when p _in = p ₀δ/η no matter what value is set for δ. This is inconsistent with the fact that the transmission gets η only at some moments, not all the time because of the oscillating behavior of the cavity modes. Thus, the dynamic shift represented by Eq. (5) should be taken into account, and we obtain the time-averaged transmission as

Note that [δ-2(p _out/p ₀)cos²(ωt)]² ≥ 0 is always valid, it is not difficult to derive that T¯ is always less than η. Based on Eq. (8), we no longer get a simple expression. However, for a setting δ, we can first calculate numerically the above integral for a given pout, 1.5p ₀ for example, then obtain p _in by using Eq. (8). In this way, we can still get the relation between T¯ and p _in. Concretely, we plot the results with respect to six values of δ in Fig. 1(a). For comparison, the results based on Eq. (7) are plotted in Fig. 1(b). We see that while the maximum transmission reaches η for any δ value in Fig. 1(b), it never reaches η for any δ > 0 in Fig. 1(a). Actually, it reduces monotonously with the increase of δ. Careful calculation reveals that bistability (or transmission jump) starts when δ is about 1.95 as shown in Fig. 1(a), other than √3 as shown in Fig. 1(b). This can be attributed to the time average effect that depresses the sensitivity of the cavity mode to the CW frequency.

 

Fig. 1. Theoretical transmissions of the nonlinear optical resonator when the dynamic shift character is considered (a); is not considered (b). The values shown in the top right corners are the corresponding frequency detuning.

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4. FDTD simulations

To verify the above analyses, we consider a two-dimensional PC structure depicted in Fig. 2(a). It consists of a square lattice of AlGaAs rods with a linear refractive index n ₀ = 3.4 and a radius of 0.30a, where a = 0.60 μm is the lattice constant. The Kerr coefficient is chosen as 1.5×10^-5 μm²/W. A defect cavity is introduced by reducing the radius of one rod to 0.15a and, along the launch direction, two PC defect waveguides serving as the input /output ports are created by removing three lines of the rods.

 

Fig. 2. (a) PC defect structure used in this paper. (b) Defect mode lineshape obtained by using the FDTD simulations. A Lorentzian lineshape (ηγ ²)/[γ ² +(ω-ω ₀)²] is also given for comparison.

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We focus on the TM mode whose magnetic field is in the launch plane and perform nonlinear FDTD simulations by using the commercial software developed by Rsoft Design Group (www.rsoftdesign.com). Here, the grid sizes in the two directions are set to be a/20, and the perfectly matched-layer boundary of 1.0a width is employed. When the input power is very small, the defect structure exhibits a band gap ranging from 0.2335(2πc/a) to 0.2993(2πc/a). In the band gap, there is a defect mode located at ω ₀ = 0.2546(2πc/a) with quality factor Q = 410 and peak transmission η = 0.96, as shown in Fig. 2(b).

Now we turn to examine the nonlinear transmission behaviour of the PC defect. To do so, CWs with Gaussian spatial shape (FWHM = 0.2a) are launched from the left port into the PC defect. For a given frequency detuning, different stable output powers are measured by a monitor set at the right port while increasing the input power. In this way, we obtain the lower branch of bistability, as the empty circles shown in Fig. 3. To obtain the upper branch of bistability, we boost the PC defect into high transmission state by superposing CW input with a high peak-power Gaussian pulse, as suggested in Ref. [3]. The results are plotted as the solid circles in the same figure. As we have discussed in Section 3, here the maximum transmission reduces monotonously with the increase of δ. Their values are very close to the analytical one shown in Fig. 1(a). We also find that bistability does not start when δ = √3. Instead, it appears when δ = 1.95, with the up- and down- jumps of transmission overlap with each other almost. This is exactly the same δ value as showed in Fig. 1(a). Bistability becomes obvious when δ increases further to 3 and 4, with the shapes of hysteresis loop very similar to those shown in Ref. [5].

 

Fig. 3. Nonlinear transmissions of the PC defect structure based on FDTD simulation. The solid circles denote the stable transmissions when the input is the superposition of CW and Gaussian pulse. The empty circles denote the stable transmissions when the input is the CW only. The values shown in the top right corner are the corresponding frequency detuning.

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Of course, obvious discrepancy between the analytical and simulation curves exists. For example, if the characteristic power p ₀ is “extracted” by using the values of the threshold input power showed in Figs. 1(a) and 3, it is found that p ₀ is a little different when δ are 1.95 and 3, but obvious different when δ is 4. We explain it by considering the validity of the perturbation theory. When the input CW frequency is far from the defect mode, i.e., δ is large, the mode field E ₀(r) in Eqs. (1) and (2) cannot be regarded as the same. Under the circumstances, the resulting Eq. (3) is no longer valid.

4. Conclusion

In summary, the theoretical bistability is very close to the simulation one when we correct the nonlinear feedback mechanism by taking into account the dynamic shift of the cavity modes. Under Kerr nonlinearity, the maximum transmission is always smaller than the linear peak transmission. As the frequency detuning increases, it decreases monotonously. Also, the frequency detuning to start bistability should be corrected as about 2 due to the sensitivity depression of the cavity mode to the input frequency. Therefore, to properly describe the bistability in nonlinear optical microcavities, the dynamic shift character should not be neglected.