Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity

Optics Communications 283 (2010) 2957–2960

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Influence of two-photon absorption on bistable switching in a silicon photonic crystal microcavity Chao Li a,b,⁎, Jun-Fang Wu b, Wen-Cheng Xu a a b

Laboratory of Photonic Information Technology, School for Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China Department of Physics, School of Science, South China University of Technology, Guangzhou 510640, China

a r t i c l e

i n f o

Article history: Received 11 November 2009 Received in revised form 16 March 2010 Accepted 22 March 2010 Keywords: Photonic crystal Bistable switching Two-photon absorption Coupled mode theory

a b s t r a c t By employing a simplified nonlinear coupled mode theory, we discuss the influence of two-photon absorption (TPA) on the characteristics of bistable switching. It is revealed that the critical value of frequency detuning for bistability rises linearly with increasing TPA coefficient k (when k is less than 30), and eventually access to a saturated value. It is also found that TPA effect will be enhanced for a greater frequency detuning, especially when transmission reaches its peak value. As a result, the peak transmission will decrease monotonously with the increasing frequency detuning. Based on this simplified model, the TPAinduced temperature rise in microcavity is also estimated. The theoretical predictions show good agreement with the simultaneous results, as well as the proposed experimental phenomena. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Optical bistable switching is one of the key components in alloptical signal processing. For actual application, small size and low switching energy are demanded. Recently, photonic crystals (PCs) with nonlinear microcavities have shown great advantages in optical bistable switching fabrication due to their good ability to confine light in a small size and strong light–matter interaction [1–3]. To design such a PC switching, several nonlinear effects, e.g., Kerr effect, twophoton absorption (TPA) and free-carrier absorption (FCA), should be taken into account [4,5]. Obviously, these nonlinear factors will influence the characteristics of a PC switching, such as the critical frequency detuning for bistability (i.e., hysteresis loop occurs), the nonlinear transmission, etc., which are key parameters for bistable switching design. By far, however, very few theoretical discussions have been made on this question, mainly due to the complexity of the formula when introducing all of these nonlinear factors into coupled mode theory (CMT) [6]. To make a further study on this question, a simplified model is desired. Actually, the relative weights of these nonlinear terms are material related [7], and the PC structures in most of the proposed experiments [2,4,5] are based on a Si system, where only TPA and Kerr effect are dominant [2,7–9]. Consequently, we can

⁎ Corresponding author. Department of Physics, School of Science, South China University of Technology, Guangzhou 510640, China. E-mail address: [email protected] (C. Li). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.053

simplify the nonlinear CMT by only introducing TPA and Kerr effect into it. In this paper, we use a simplified nonlinear CMT to study the influence of TPA on the characteristics of bistable switching in the Si system. As a comparison, nonlinear finite-difference time-domain (FDTD) method is employed to make simulations. It is found that the theoretical predictions not only show good agreements with the proposed experimental phenomena, but also with the simultaneous results, and are helpful for the design and operation of PC-based switching. The paper is organized as follows. In Section 2, we derive the nonlinear transmission, reflection and loss formulas including TPA effect, and the corresponding influences of TPA on the characteristics of bistable switching are discussed, including the critical frequency detuning. Then, in Section 3, we simulate the transmission, reflection and loss of bistable switching, and make a comparison between the theoretical predictions and the simulation results. Finally, a brief summary is given in Section 4.

2. TPA effects on nonlinear transmission, reflection and total loss To describe conveniently, we depict a typical structure of bistable switching based on PC in Fig. 1, which is consist of one nonlinear microcavity with two waveguides (WG) connected to the two ends. Suppose the decay rates of the input WG (WG1) and output WG (WG2) are γ1 and γ2, respectively, and the intrinsic decay rate of the nonlinear cavity is γC. γC can be phenomenologically written as γC = γ0 + γTPA, where γ0 is the linear intrinsic decay rate, and γTPA is

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Introducing Eqs. (5) and (7) into Eqs. (2) and (4), we obtain

Fig. 1. Sketch of a typical bistable switching composed of a resonator cavity coupled to both input and output WGs. γ1 andγ2 are the decay rate of WG1 and WG2, respectively, and γC is the intrinsic decay rate of the cavity.

T=  1+

1

 kηpout 2 2

η ⋅ ;  pout 2 δ− p +1

ð8Þ

0

ðγ + γTPA Þ 2ð1−ηÞT ⋅T = + kpout T: α= 0 γ1 η

ð9Þ

and R can be calculated by employing Eqs. (8) and (9) the decay rate correlated with TPA, therefore, the total decay rate γ = γ1 + γ2 + γ0 + γTPA. When a continuous wave (CW) with frequency ω is incident from WG1, according to CMT, the energy amplitude A reads " ! # 8 pffiffiffiffiffiffiffiffiffi > dA jsout j2 > > −γ A + 2γ1 sin ; = j ω0 −γ > > p0 < dt pffiffiffiffiffiffiffiffiffi > > sout = 2γ2 A > > > pffiffiffiffiffiffiffiffiffi : sin + sR = 2γ1 A

ð1Þ

where sin, sout and sR are the field amplitudes of the incident, output and reflective waves, ω0 is the resonant frequency of the microcavity, and p0 is the characteristic power [10]. Denoting Pout = |sout|2 and Pin = |sin|2, then transmission coefficient T and reflection coefficient R can be derived from Eq. (1) 4γ1 γ2 2 jsout j ðγ1 + γ2 + γ0 + γTPA Þ2 T= = ;  2 jsin j2 δ− ppout + 1

ð2Þ

0

js j2 R= R 2 = jsin j

ðγ1 −γ2 −γ0 −γTPA Þ2 ðγ1 + γ2 + γ0 + γTPA Þ2  2 p

δ−

out

p0

 2 + δ− ppout 0

;

ð3Þ

+1

where δ = (ω0 − ω) / (γ1 + γ2 + γ0) is the frequency detuning of the input CW with respect to the cavity mode. One can see that Eq. (2) is very similar to the nonlinear transmission formula as shown in Refs. [10,11], except for a term of TPA effect included in our case, which will lead to quite different results on bistable switching, as will be seen in the following. Using Eqs. (2) and (3), the total loss rate α can be calculated as α = 1−T−R =

ðγ0 + γTPA ÞT : γ2

ð4Þ

Noticing that γTPA is proportional to the energy stored in the cavity [4,5], while γ0, γ1 and γ2 keep invariant for a certain PC structure, therefore, γTPA can be expressed as 2

γTPA = 2kjAj γ1 γ2 = kγ1 pout ;

ð5Þ

where k is defined as TPA coefficient which denotes the strength of the TPA process, and can be obtained by experiments or simulations, as will be shown in Section 3. Now, for simplicity, let us consider a symmetric PC structure, i.e., γ1 = γ2. In this case, when the incident power is very weak, the linear peak transmission coefficient η can be obtained from Eq. (2) η=

4γ1 γ2 4γ21 = : 2 ðγ1 + γ2 + γ0 Þ ð2γ1 + γ0 Þ2

ð6Þ

Thus, we immediately have γ0 2ð1−ηÞ = : γ1 η

ð7Þ

R = 1−T−α:

ð10Þ

Since k, η and p0 can be obtained by experiments or simulations, to compare better with the simulation results shown in Section 3, k, η and p0 are selected as 7.2 W−1, 0.978 and 0.017 W respectively, which are the same as the values calculated from simulations. Based on Eqs. (8)–(10), the theoretical predictions of the TPA effects on the characteristics of a PC switching can be obtained, as depicted in Fig. 2. In Fig. 2(a), we describe how TPA effect impacts the critical frequency detuning for bistability. One can see readily that when k = 0 (i.e., no nonlinear loss in PC cavity), the critical value of δ is exactly pffiffiffi equal to 3, which coincided with the one proposed in [10], also known to all. Then, the critical value rises with increasing TPA coefficient k, and a good linearity is shown when k is less than 30. Henceforth, the increase of the critical value of δ slows down, and eventually accesses to a saturated value (3.875, by theoretical calculation). This point can help select an appropriate frequency detuning for bistable switching design. In Fig. 2(b)–(d), the nonlinear transmission, reflection and loss of a PC microcavity are depicted in solid lines with respect to several different δ. From Fig. 2(b) and (c), one can see that the peak transmission will decrease monotonously with the increasing frequency detuning, while the corresponding valley reflection will rise gradually, which are coincident with the experimental phenomena proposed by Notomi et al. [2] and Priem et al. [12]. As comparison, the transmission for the case without TPA (say, only with Kerr effect) is also plotted in dash lines in (b), according to Soljačić's theory [10] (i.e., let k = 0 in Eq. (8)). In this case, one can see readily that the peak transmission keeps invariant (equal to η) for different δ, as well as the valley reflection, and the switching power is lower than the one when TPA is taken into account. To get an intuitive sense of the strength of TPA, the variance of the total loss in PC cavity is shown in Fig. 2(d). We can see clearly that the peak loss increases for a greater frequency detuning, because a greater δ leads to more energy stored in the PC microcavity, and TPA effect is accordingly strengthened. This theoretical prediction has been verified by the proposed experiments (Fig. 5(b) in [12]). From Fig. 2(d), it is also found that the enhancement of TPA is gradually weakened with the increasing frequency detuning, and the total loss will be eventually saturated. In addition, by carefully comparing Fig. 2(b) with (d), one can find that TPA-induced loss is strengthened sharply at the exact point of incident power when T switches to its peak value. Therefore, one can see that TPA effect makes important impacts on the characteristics of bistable switching, and it is reasonable and effective to treat TPA and Kerr effect as dominant factors in nonlinear process of the Si system. Essentially, the abovementioned results originated from the variance of the refractive index in the microcavity, which can be mainly attributed to Kerr effect and thermal-optic effect [4,5], say, the energy of the light absorbed by TPA transfers into thermal energy, and the increase of temperature ΔTt leads to a rise of refractive index. Taking these factors into account and using an equivalent method based on Eq. (8), we obtain an approximate formula of ΔTt in a bistable region rffiffiffiffiffiffiffiffiffiffiffiffi   n0 γ p η δ− out + ΔT t ∼ −1 ; ð11Þ p0 T ω0 ∂n = ∂T t

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where Tt denotes temperature in microcavity, and ∂n/∂Tt is a coefficient to show the sensitivity of the refractive index to the change of temperature. Using Eqs. (8) and (11), we can estimate the value of ΔTt. For example, when δ = 4.3478, ∂n / ∂Tt = 0.01 K−1, and the other parameters are the same as the values taken in the following simulations in Section 3, then ΔTt is estimated to range from 0.14 K to 0.2 K in the bistable region. Commonly, the intrinsic process of thermal-optic effect is very complicated [4,5], however, we can still include this loss mechanism by simply introducing a TPA-induced loss rate γTPA into CMT, as we have done in this section. While for simulation, we can do it by introducing an imaginary part into the nonlinear coefficient, as will be shown in Section 3.

3. Nonlinear FDTD simulations and comparisons with theoretical predictions To further compare the theoretical predictions mentioned above with the simulation ones, we choose to study a symmetric twodimensional (2D) PC structure. It consists of a square lattice (15 × 15) of dielectric rods with a refractive index of n0 = 3.4 and a radius of 0.2a, where a = 1 μm is the lattice constant. The nonlinear PC cavity is created by reducing the radius of the central rod to half, and, for convenience, the real part of nonlinear coefficient is supposed to be 0.01 μm2/W, and the imaginary part is set to be 0.0025 μm2/W, denoting TPA effect. Two PC WGs are fabricated by cutting off three lines of rods and connecting symmetrically to the two ends of the nonlinear cavity respectively. To measure transmission and reflection, two monitors are set at the input and output ports of the WGs. Thus, we can begin numerical simulations and focus on the TM modes by employing nonlinear FDTD technique [13]. A defect mode is found at resonant frequency ω0 = 0.3074(2πc / a), with line width (γ1 + γ2 + γ0) = 2.3 × 10−4(2πc / a) and linear peak transmission coefficient η = 0.978, where c is the velocity of light in vacuum. To study how TPA effect varies with increasing δ and the corresponding influences on the characteristics of bistable switching, five continuous waves (CWs) with different frequency detuning are respectively employed to launch from the entrance port of the input WG. To obtain the upper branches of the loops in Fig. 2(b)–(d), each incident CW is superposed with a high peak-power Gaussian pulse, as suggested in Ref. [10]. In this way, we obtain the simulation results of T and R, and the total loss can be readily obtained by “α = 1 − T − R”, as shown in scattering dots in Fig. 2(b)–(d). By utilizing these simulation data and Eqs. (8)–(10), k and p0 can be calculated as k = 7.2 W−1 and p0 = 0.017 W, that is why k and p0 take these values for theoretical calculation in Section 2. By carefully comparing the solid lines and scattering dots in Fig. 2 (b)–(d), one can see clearly that the theoretical predictions consist with the simultaneous results quite well, no matter for a little δ (e.g., δ = 0) or a greater one (e.g., δ = 17.39). Of course, one might find that in Fig. 2(d), there are some slight discrepancies between theoretical curves and simulation dots when the input power is very weak, that is because in this case, linear loss is major, while in our simplified theory, TPA effect is dominant, especially when the energy stored in PC cavity is strong enough, that is why with the increasing of input power, those discrepancies disappear soon. Actually, what we really care about is in the bistable region, where the simplified theory works quite successfully.

Fig. 2. Influence of TPA effects on the characteristics of bistable switching. In (a), the impact of TPA on the critical frequency detuning for bistability (i.e., hysteresis loop occurs) is plotted in scattering dots. In (b)–(d), the theoretical predictions of TPA effects on the transmission, reflection and loss of the PC microcavity are depicted in solid lines, while the corresponding simulation results are shown in scattering dots, with respect to five different values of δ. As comparison, the transmission for the case when without TPA and only with Kerr effect is also plotted in dash lines in (b).

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4. Conclusion

References

In summary, by treating Kerr effect and TPA as the major nonlinear process in PC microcavity for the Si system, a simplified nonlinear CMT was acquired. Base on this simplified theory, we discussed the impacts of TPA effects on the characteristics of a PC bistable switching, and the theoretical predictions not only coincide with the proposed experimental phenomena, but also agree with the simulation results quite well. It was revealed that the critical value of δ for bistability rises linearly with increasing TPA coefficient k when k is less than 30, and eventually accesses to a saturated value. It was also revealed that the TPA induced loss will be enhanced for a greater frequency detuning, especially when transmission reaches its peak value, which leads to the monotonous decrease of the peak transmission with the increasing frequency detuning. Based on this simplified model, the TPA-induced temperature rise in microcavity for δ = 4.3478 was estimated to range from 0.14 K to 0.2 K in bistable region.

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