## Abstract

In this paper, optical bistability effect in an ultracompact plasmonic racetrack resonator with nonlinear optical Kerr medium is investigated both analytically and numerically. The properties of optical bistability and pump threshold are studied at 1.55µm with various detuning parameters by an analytical model. The transmission switch from the upper branch to the lower branch with a pulse is also demonstrated by a finite-difference time-domain method. An extinction ratio of 97.8% and a switching time of 0.38ps can be achieved with proper detuning parameter. Such a plasmonic resonator design provides a promising realization for highly effective optical modulators and switch.

© 2011 OSA

## 1. Introduction

Highly integrated optical communication circuits require practical realization of ultracompact optical resonant cavities [1, 2]. Traditional optical micro-cavities made of dielectric or semiconducting materials have large footprints due to the rapidly increasing radiated loss with the decreasing cavities dimensions. Plasmonics is a promising field that can merge photonics and electronics at nanoscale dimensions [3–6]. Plasmonic cavities that integrate surface plasmon polariton (SPP) with optical cavities can realize ultracompact optical resonators [7–12]. Plasmonic cavity not only is beyond the diffraction limit, but also achieve large local field enhancement within it. This strong field enhancement at nanoscale volume can be exploited to manipulate light-matter interactions [13, 14] and improve the performance of photonic devices based on optical nonlinear effect, such as optical bistability and all-optical plasmonic modulators and switching. Owing to its many potential applications, various plasmonic cavity structures have been investigated recently [9, 11, 14–19]. In those various plasmonic structures, circular cavity has relatively low loss [16, 18] owing to its smooth surface, while rectangle cavity can offer large coupling efficiency [11, 19] because of its long coupling region. Plasmonic racetrack resonator that combine plasmonic cirular resonator with plasmonic rectangle resonator can realize more excellent plasmonic device [20]. Among many characteristics of nonlinear plasmonic modulators and switching, the extinction ratio is a very important figure of merit [21, 22] that should be investigated in details to make these devices suitable for practical applications.

In this work, we propose and investigate a bistable switching with high extinction ratio based on plasmonic racetrack resonator with nonlinear optical Kerr material. Taking into account perturbations caused by the Kerr nonlinearity, the properties of optical bistability and pump threshold are studied at resonant mode at 1.55µm with various detuning parameters by coupling mode theory. A numerical model is developed by the finite-difference time-domain (FDTD) method to confirm the theoretical results. The bistable switching from the upper branch to the lower branch with an extinction ratio of 97.8% and a switching time of 0.38ps is also demonstrated by the FDTD method.

## 2. Analytical model

The proposed configuration consists of a single mode metal-dielectric-metal (MDM) plasmonic gap waveguide side-coupled to a plasmonic racetrack cavity with nonlinear optical Kerr material, which is illustrated in Fig. 1 . Silver is chosen for its low absorption. The Drude model is used to describe the dispersion of Ag that is used in the device.

where*ε*=3.7,

_{∞}*ω*=1.3826×10

_{p}^{16}rad/s,

*ν*=−1.3674×10

_{c}^{13}s

^{−1}[12]. The core index of the coupling MDM plasmonic waveguide is chosen as 1.52 (e.g. fused silica). The gap of racetrack cavity is filled with nonlinear organic polymer, the linear refractive index

*n*= 1.52, and the third-order susceptibility is ${\chi}^{(3)}=1.4\times {10}^{-7}\text{esu}\approx {10}^{-15}{\text{m}}^{2}{\text{/V}}^{\text{2}}$. The width of the coupling plasmonic waveguide and the racetrack resonator is

_{l}*w*, the distance between the boundaries of the racetrack and waveguide is

*g*, and the outer radius of curve section is

*R*while the length of straight section is

*L*considered as the coupling length. The amplitudes of incident light and transmitted light are denoted by S+ and S-, which are normalized such that

_{c}*p*

_{in}= |S + |

^{2}and

*p*

_{out}= |S-|

^{2}are the incident and transmitted power. Power exchange dominantly takes place in the coupling area (the dashed box). For the resonant system, the coupling of modes formalism [23] can be expressed as:

^{2}gives the energy in the cavity mode. The frequency

*ω*

_{0}is the interest eignfrequency of the plasmonic resonator, and 1/τ

_{0}denotes the decay rate due to the internal loss in the resonator, while 1/τ

_{e}denotes the decay rate due to the escaping power. If the incident light is at frequency

*ω*($S+\sim \mathrm{exp}(i\omega t)$), according to Eq. (2), a can be expressed as:

Applying the property of time reversibility and energy conservation to Eqs. (2) and (4), we can obtain

Then the transmission efficiency of the resonator in the steady state is derived from Eqs. (3) and (5):

Acorrding to Eq. (6), when 1/τ_{e}=1/τ_{0} and the system is at resonance (*ω*=*ω*
_{0}), the transmission power vanishes. This condition, known as the critical coupling, is due to the perfect destructive interference in the outgoing waveguide between the transmission field and the internal field coupled back into the output waveguide. When the system is at critical coupling, the transmission efficiency can be simplified as:

*γ=*1/τ

_{0}+ 1/τ

_{e}is decay rate related to the resonator quality factor

*Q*by

*γ*=

*ω*

_{0}/(2

*Q*).

Taking into account perturbations caused by the Kerr nonlinearity, the eigenfreqency of the plasmonic resonator *ω*
_{0} will change into *ω*
_{0}-Δ*ω*, and the transmission efficiency Eq. (6) can be written as:

*ω*-

*ω*

_{0})/

*γ*is a detuning parameter, which is the detuning of the incident excited frequency

*ω*from the plasmonic resonator

*ω*

_{0}.According to first-order perturbation theory of the eignmode in the plasmonic resonator and the characteristics of the system [24], the change in the resonant frequency of the mode can be given by

*κ*is the nonlinear feedback parameter determined by the overlap of the plasmonic cavity mode with the nonlinear material.

*n*

_{2}(

*r*) and c are the local Kerr coefficient and the speed of light in vacuum respectively. We define${p}_{{}_{in}}^{\text{'}}={p}_{in}/{p}_{0}$and${p}_{{}_{out}}^{\text{'}}={p}_{out}/{p}_{0}$, then applying Eq. (9) to Eq. (8), the transmission efficiency becomes

*p*

_{0}(

*p*

_{0}=c/(

*ω*

_{0}

*κQ*

^{2}

*n*

_{2}(

*r*)|

_{max})) is the characteristic power of the resonator. The Eq. (10) shows that the resonant system has bistable phenomenon or not, depending on the value of the detuning parameter δ. According to Eq. (10) and the assumption of

*p*

_{0}=0.73MW/cm

^{2}which can be achieved with a particular set of parameters to be detailed later, the transmission efficiency

*p*

_{out}/

*p*

_{in}as a function of incident power

*p*

_{in}with different detuning parameter is shown in Fig. (2) . It shows that $\delta =\sqrt{3}$ is the minimum detuning requirement for the presence of bistability, and the bistable loop becomes more obvious with increasing the detuning parameter, while the threshold of the incident light power increase. We also note that one state of every bistable loop can possess a near zero transmission efficiency, so the extinction ratio can be extremely high. It can be used to design bistable switching with high extinction ratio.

## 3. Numerical simulation results and discussions

FDTD simulation is used to verify feasibility of the theoretical model for the nonlinear plasmonic racetrack resonator. In the FDTD simulation, the grid size is 4nm for Δ*x*, 5nm for Δ*y* and 3nm/c for the time step, which are found to be sufficient for the convergence of numerical results. Firstly, when nonlinear plasmonic racetrack resonator doesn’t present nonlinear effect (optical intensity is very weak in racetrack cavity), we choose appropriate structure parameters which result in critical coupling resonance at 1.55*µm* in the nonlinear resonator system. Using a similar procedure to achieve critical coupling resonance which outlined in Ref [20], the optimal structure parameters can be obtained, which are *w=*180*nm, R*=*L _{c}=*500

*nm,*and

*g=*20

*nm.*To verify the correctness of chosen parameter, a 5.2fs broadband Gaussian pulse with a center wavelength of 1.55µm is used to scan the transmission efficiency spectrum of the structure with the chosen structure parameters. The frequency scope of a broadband pulse should contain several resonant wavelengths of the device. Figure 3 shows that 1.55

*µm*is one of the resonant wavelengths, and the structure achieves the critical coupling condition.

The bistable curves are studied at resonant mode at 1.5*5µm* with different detuning. For comparing the simulated results with theoretical results, the detuning in simulation are the same with theoretical analysis, which are 1.566*µm* ($\delta =\sqrt{3}$), 1.5697*µm* ($\delta =1.33\sqrt{3}$), and 1.571*µm* ($\delta =1.4\sqrt{3}$) respectively. The bistability loops with these detuning are shown in Fig. (4)
, the up and down arrows represent the decreasing and increasing of incident light intensity. According to Fig. (4), the cut-off value of detuning for the presence of bistability and the change of bistable loops with increasing of detuning are the same with theoretical results, so the agreements between the coupled mode equation in time and FDTD simulations are excellent.

Based on the results of the bistable phenomenon of the nonlinear plasmonic racetrack resonator, the bistable switching is studied with FDTD. The bistable loops reveal that the transmission efficiency can be switched to the lower branch from the upper branch with a pulse. The pulse bandwidth should be chosen appropriate, since photon has cavity lifetime in cavity. At the 1.55*µm* resonant wavelength, the nonlinear plasmonic racetrack resonator has a full-width-at-half-maximum (FWHM) of 12.56nm and a quality factor Q=123, the corresponding cavity lifetime can be achieved by

We choose pulse bandwidth of 0.155ps in simulation. The bistable loop with detuning wavelength of 1.5697*µm* is used based on weighing the threshold of incident light intensity and the bistable phenomenon. Figure (5a)
shows switching process with pulse. As the incident continuous wave (CW) is increased to the stable power level of 1.748MW/cm^{2} for some cavity period, the resonator system is at a high transmission state that is at the upper branch of bistable loop. Then the switching occurs after a pulse with a peak power 1.748MW/cm^{2}, the same carrier frequency as that of CW, which is superimposed upon the CW excitation. The pulse make the stored light intensity in cavity exceed the bistable threshold. After the pulse has passed through the plasmonic racetrack resonator, the system switches to the low transmission state. According to Fig. (5a), the system has a switching time of 0.38ps, and a high extinction ratio of 97.8%. In the simulation, the time delay of the nonlinear material is not considered, so the switching time of 0.38ps is only determined by the feedback of the structure, which represents the shortest switching time of the structure. To make the states of switching on and off visible, we give the magnetic field distributions in Fig. (5b) and Fig. (5c). The switching on state shown in Fig. (5b) has most of the power transmits through the waveguide, and Fig. (5c) corresponds to the off state which has most of the power confined in plasmonic racetrack cavity.

## 4. Conclusions

In conclusions, we studied optical bistability phenomenon in an ultracompact nonlinear plasmonic racetrack resonator by theoretical analyses and FDTD method. The analytical model clearly demonstrates that optical bistability and the threshold of incident light intensity have close relationship with detuning parameter δ, and the simulated results are consistent with theoretical analysis. Thus, it shows that the nonlinear dynamics in plasmonic racetrack resonator can be completely accounted for using coupled mode theory, which provides a convenient framework for analyzing complex nonlinear processes. The bistable switch with an extinction ratio of 97.8% and a switching time of 0.38ps can be achieved with proper detuning parameter. Such a nonlinear plasmonic resonator design provides a foundation for realizing low threshold of incident light intensity, fast switching time and highly effective optical modulators and switch.

## Acknowledgments

This work is supported by the Key Program of National Natural Science Foundation of China No. 61036005 and National Natural Science Foundation of China No. 60977019, 11074241.

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