An optically bistable device based on a Bragg grating resonator with a nonlinear medium in metal-insulator-metal waveguides is proposed. Its properties are numerically investigated by a finite-difference time-domain method and further qualitatively analyzed by adopting Airy equation. Cavities with different Q factors are compared with respect to bi-stability. Cavities with a small Q factor lead to a high transmission and a narrow hysteresis loop. The response time of such cavities is found to be in the sub-picosecond region. Our nano-scale switching structure is comparatively easy to fabricate and integrate in plasmonic circuits and promises to be useful for future all-optical computing and communication technology.

## I. INTRODUCTION

All-optical signal processing in integrated photonic circuits requires the ability to control light with light. There are two main drawbacks that limit practical applications of traditional all-optical devices: large scale and relatively high operation light intensity.^{1} Surface plasmon polaritons (SPP), i.e. electromagnetic waves coupled to collective oscillations of electron plasma at the interface between metal and dielectric, have been proposed to solve these problems.^{2,3} Confinement and enhancement of the SPP field at the interface make it possible to manipulate light on a nanometer scale with relatively low intensity. Quite a number of SPP based optical devices have been proposed or produced, including couplers,^{4} amplifiers,^{5} interferometers^{6} and switches.^{7–9} On the one hand, SPP functional devices which utilize surface plasmons in metal-insulator-metal (MIM) waveguides could be easily integrated in more complex plasmonic circuits. MIM structure based splitters,^{10} filters,^{11} demultiplexers^{12} and Bragg reflectors^{13–15} have been reported recently. On the other hand, optical bistability is a rapidly expanding field of current research because of its potential application in all-optical transistors, switches, memories and logic gates.^{16,17} Recently, several types of optically bistable devices based on SPPs have been proposed and studied such as nonlinear extraordinary optical transmission structures formed by coating sub-wavelength metallic gratings with nonlinear media^{1,8} and optical Tamm cavities with a Kerr medium sandwiched between a metal layer and a Bragg mirror.^{18} However, such structures are rather large which limits their applicability. Nano-cavities containing nonlinear media,^{9,19,20} are another proposal, they may be integrated efficiently, but are either difficult to adjust for different operational requirements or with low extinction ratio or modulation depth. In this paper, we propose a nano-scale optically bistable switch constructed by Bragg grating resonators in metal-insulator-metal waveguides. This kind of device can easily be adjusted by varying the Bragg grating resonators and readily be integrated in all-optical circuits.

## II. MIM BRAGG RESONATORS

There are two possible plasmon modes in the MIM waveguide, the odd mode (*E*_{x} is odd, *H*_{y} and *E*_{z} are even) and the even mode (*E*_{x} is even, *H*_{y} and *E*_{z} are odd).^{3} The even mode exhibits cut-off when the thickness of the dielectric layer is smaller than a certain thickness.^{21} Here we only consider the odd mode which can be excited by a dipole source. The dispersion relation of the odd mode SPP propagating in a MIM waveguide can be written as^{3}

where *w* is the width of the insulator, ɛ_{m} and ɛ_{d} are the dielectric constants of the metal and the insulator, respectively. *k*_{m} and *k*_{d} denote the transverse propagation constants in the metal and the insulator, which are related to the effective index of refraction *n*_{eff} as

*k*

_{0}= 2π/λ is the propagation constant in vacuum. It follows from Eq. (1) that

*n*

_{eff}is related to ɛ

_{m}, ɛ

_{d}and

*w*for a certain wavelength. Therefore, an SPP waveguide Bragg grating (SPP-WBG) could be obtained through periodical modulation of ɛ

_{m},

^{13}ɛ

_{d}

^{14}or

*w*.

^{15}Among these three approaches, the modulation of the insulator width

*w*is suggested because it can be fabricated using standard top down lithography methods

^{22}or direct etching on bulk metal/metal film as long as the thickness is sufficiently large.

^{23}

The scheme of the optically bistable device is illustrated in Fig. 1. It should be noted that all our study is operated on 2D. We plot a 3D figure of a cavity fabricated via etching a metal film, in which we believe would be convenient for those elaborate MIM waveguide structures.^{19,20} The interesting region does not depend on *z*, and we therefore performed a 2D calculation only. However, one has to justify the plane wave approach. Consider a fundamental mode Gaussian beam. The angle of divergence is defined as θ = λ/(π*w*_{0}), where *w*_{0} is the laser beam waist. For a wavelength of 1550 nm, a waist of several tens of micrometers can keep light traveling several hundreds of micrometers without noticeable divergence. So we believe several tens of micrometers are enough for the depth of the metal film. Moreover, the impedance matching method is suggested to efficiently guide the micro-scaled light into the nano-scaled waveguide.^{24} Field enhancement comes after the compression.

As shown in Fig. 1, the Fabry-Perot (F-P) cavity has cavity mirrors formed by SPP-WBG at each side. The insulator is air, the optical response of metal silver which can be characterized by the well-known Drude model

_{∞}= 3.7, ω

_{p}= 9.1 eV and γ = 0.018 eV.

^{15}InGaAsP is chosen as the Kerr-type nonlinear material, the third order nonlinear susceptibility of which is χ

^{(3)}= 1 × 10

^{−18}

_{l}= 12 at the glass fiber telecommunication wavelength 1550 nm.

^{25,26}The dielectric constant ɛ

_{d}depending on the electric field intensity |

*E*|

^{2}is expressed as ɛ

_{d}= ɛ

_{l}+ χ

^{(3)}|

*E*|

^{2}.

The SPP-WBG (as shown in the inset of Fig. 1) could be adjusted to reflect light at certain wavelength according to the Bragg conditions

where *m* is an integer, *n*_{neff1} and *n*_{neff2} are, respectively, effective refractive indexes for insulator widths *w*_{1} and *w*_{2}, *L*_{1} + *L*_{2} is the period of the grating. If the light wavelength is 1550 nm, we can set *m* = 1, *w*_{1} = 90 nm and *w*_{2}=160 nm to get *n*_{neff1} = 1.21 and *n*_{neff2} = 1.12 from Eq. (1). Then, we set *L*_{1} = *L*_{2}. *L*_{1} and *L*_{2} are calculated to be 332 nm according to Eq. (2). The length of the nonlinear medium *L* can be determined by a 2D finite-difference time-domain (FDTD) method computation. In this simulation, the spatial and temporal steps are set as Δ*x* = Δ*y* = 5 nm, Δ*t* = Δ*x*/2*c*. The fundamental TM mode (magnetic field parallel to *z* axis) of the waveguide is excited by a dipole source. *L* is 395 nm in our simulation. The low intensity transmission spectra of the MIM resonator with Bragg mirrors of period number *N* = 2, 3 and 4 at each side are shown in Fig. 2(a), for which Q factors are calculated to be 70, 123 and 163, respectively.

## III. OPTICAL BISTABILITY

It is expected that the nonlinear behavior of output intensity will couple by intrinsic feedback to the outbound light if the input intensity is varied. This will result in hysteresis. For *N* = 3 as an example, we plot the spectra of the F-P cavity depending on different input intensities as shown in Fig 2(b). The transmission peak drops and red shifts as *I*_{input} increasing. The transmission appears a sudden jump at wavelength 1550 nm when *I*_{input} increases to 1.20 × 10^{18}, which turns out to be the high jump threshold value. The effective refractive index of the nonlinear cavity at low intensity is calculated to be *n*_{effNL} = 4.33 + 0.01*i* according to Eq. (1). While at the high jump threshold *I*_{input} = 1.20 × 10^{18}

*E*|

^{2}inside the cavity is found to be 2.28 × 10

^{18}

*i*.

To verify this bistable behavior, we calculated the output intensity by smoothly increasing the incident intensity and then decreasing it back to zero. The input-output characteristics of the F-P cavities with period numbers *N* = 2, 3 and 4 are compared. Consider e.g. the bistability loop of period number N = 2 (solid line) in Fig. 3(a). The arrows identify the input power increasing or decreasing. Initially, *I*_{output} increases slowly on the lower branch as *I*_{input} increases. When *I*_{input} reaches the threshold value (∼6 × 10^{17}

*I*

_{output}jumps to a much higher value (A → B). If then

*I*

_{input}is slowly decreased (B → C),

*I*

_{output}will decrease slowly as well but remains on the upper branch due to the strong localized field in the structure until

*I*

_{input}reaches another threshold value (∼1 × 10

^{17}

*I*

_{output}strongly decreases (C → D). It is possible to have two stable

*I*

_{output}if

*I*

_{input}is between the two threshold values. The pronounced overshooting and damped oscillations at the jump up point result from the transient buildup of the cavity field from its initial value to the final value at switching.

By comparing these hysteresis cycles, it is clear that the cavity with a small Q factor shows a large transmission and a low switch-on threshold value whereas the cavity with large Q factor has a wider hysteresis cycle. Extinction ratio (ER, defined as the ratio between the logic “1” and “0”) and modulation depth (MD, defined as the difference between the logic “1” and “0” normalized by the incident power) of each cavity are compared as shown in Table I. a wider hysteresis loop allows the cavity to withstand more external disturbance on *I*_{input} to keep the *I*_{output} stable. There is a trade-off between transmission and stability, and these optically bistable switches can be adjusted to operational requirements of real applications by changing the cavity bistability.

. | ER(dB) . | MD . |
---|---|---|

N = 2 | 16.08 | 0.56 |

N = 3 | 21.38 | 0.30 |

N = 4 | 24.89 | 0.10 |

. | ER(dB) . | MD . |
---|---|---|

N = 2 | 16.08 | 0.56 |

N = 3 | 21.38 | 0.30 |

N = 4 | 24.89 | 0.10 |

and the transmission equation at the output side of the cavity

where

*A*

_{1}and

*A*

_{3}are amplitudes of the input and output fields.

*A*

_{2}denotes the amplitude of the forward-going wave within the cavity. ε

_{1}and ε

_{3}are equal to 1 in our structure. ε

_{2}is the permittivity of the nonlinear medium ε

_{d}, δ the total phase shift in a round trip through the cavity.

*R*and

*T*are the intensity reflectance and transmittance of the cavity mirror with

*R*+

*T*= 1, where loss is ignored for simplicity. It can be well understood that cavities with large Q factors also have large

*R*values. According to Eq. (3), there are maxima of

*I*

_{1}when δ = (2

*m*+ 1)π and minima of

*I*

_{1}when δ = 2

*m*π, where

*m*is an integer. Threshold values of the high jump and the low jump points are those maximum and minimum values. We let

*I*

_{2}at the first high jump and low jump points. Threshold value at the high jump point could be deduced from Eq. (3) as

in which a large *R* leads to a large high jump threshold value. Threshold value at the low jump point could be deduced from Eq. (3) as

in which a large *R* leads to a small low jump threshold value. Due to damping, the actual *I*_{3} is smaller than *I*_{1} at the low jump point. This explains why all low jump points are close to the line *I*_{3} = *I*_{1} (dashed line in Fig. 3). The cavity with a large Q factor has a large high jump and a small low jump threshold value which makes the hysteresis cycle wider. According to Eq. (4), large *R* results in small *T* and, consequently, in small *I*_{3}. Therefore the cavity with a large Q factor has a small transmission factor.

Finally, the response time of the optically bistable switches based on SPP-WBG is estimated and shown in Fig. 4. We first set the intensity of the signal light to be 2.5 × 10^{17}

^{18}

*N*= 2, 3, 4, respectively. The field in the resonant cavity cannot make sudden finite change. It has to approach the final value through interference of multiple reflections. It is easily understood that the cavity with a large Q factor needs more time to become resonant. The switching-off time is determined by blocking the signal light for 100 fs and detecting the output intensity. The switching-off time, defined similarly, is estimated to be 0.66 ps, 0.97 ps and 1.05 ps for

*N*= 2, 3, 4, respectively. As to cavities with large Q factors, cavity mirrors at each side have large reflectance and small transmittance. When the switch is turned on, the light wave needs more time to achieve a stable state via accumulating and traveling inside the cavity. In this case, the switching-on time of the cavity with a large Q factor would be longer, and vice versa. The actual switching time depends on the response time of the structure and the nonlinear material. In spite of this, our structure has the potential to shorten the switching time to sub-picosecond level.

## IV. CONCLUSIONS

We propose a new nano-scale optically bistable switch based on a Bragg grating resonator in an MIM structure and demonstrate its properties by FDTD simulations. The main features of the hysteresis loops could be qualitatively analyzed by using Airy equation. By comparing the bistability loops of cavities with different Q factors, one can see that low Q factor lead to high transmission and narrow hysteresis loop. There is a trade-off between transmission and stability. The switching time is found to be in the sub-picosecond region. Our device may be comparatively easy to be fabricated and is flexible to be adjusted to a large span of operational requirements. This structure has potential applications in a future all-optical computing and communication technology.

## ACKNOWLEDGMENTS

We thank P. Hertel for useful discussions. This work was financially supported by the National Basic Research Program of China (2010CB934101, 2013CB328702), the National Natural Science Foundation of China (11004112, 11174161), the 111 Project (B07013), International S&T Cooperation Program of China (2011DFA52870), Oversea Famous Teacher Project (MS2010NKDX023) and the Fundamental Research Funds for the Central Universities.