1. Introduction

Optical nonlinearity is a material property, which is defined as the nonlinear response of the polarization P to the electric field E of light when light is propagating in the material. It is used for high speed optical signal processing [1, 2], broadband electro-optic modulation [3], mass detection [4] and Raman lasing [5]. Due to the optical nonlinearity of materials, a variation of the refractive index is induced, which is proportional to the local intensity of light and can be expressed as $n=n_0+n_{2}I,$ where $n_0$is the refractive index, and $n_2$is a constant. This phenomenon is well known as the optical Kerr effect and $n_2$is defined as the Kerr coefficient. The optical Kerr effect is one of the important nonlinear optical properties of silicon, which attracts much attention due to the well-developed semiconductor technologies. However, the silicon’s Kerr coefficient is only in the range from 2.8 × 10⁻¹⁸ to 14.5 × 10⁻¹⁸ m²W⁻¹ [6], which does not have the superior nonlinear characteristics for many potential applications. It is desirable to improve the optical Kerr effect of the silicon material, for example, to decrease the threshold power for lasing or increase the high sensitivity for sensing. There are few methods used to achieve high optical nonlinearity in the silicon material. For example, ion doping is used to change the optical nonlinearity of the silicon [7], but it induces a large free-carrier absorption, which decreases the capability of light intensity enhancement in the silicon–based devices. Therefore, methods to improve the optical nonlinearity effect without inducing a large absorption loss are desired. Optical ring resonator is an efficient method to enhance the optical nonlinearity of the silicon through the thermal-optic effect or the free-carrier absorption effect [8–10]. However, the intrinsic optical nonlinearity of silicon is not improved. In the contrary, an extremely strong optical nonlinearity is generated due to the opto-mechanical effect in nanoscale mechanical structures, such as two coupled silicon waveguides [11] or waveguide substrates [12]. On the other hand, the optical ring resonator enlarges the opto-mechanical effect via the increase in light intensity, which is widely applied in mesoscale back action [13], sideband cooling [14], thermal nonlinearity [15–17] and photonic structure controlling [18].

In this paper, an opto-mechanical ring resonator system is proposed to produce an ultra-high optical nonlinearity via the opto-mechanical effect. The mechanical deflection of the free-hanging arc of the ring resonator caused by the optical gradient force changes the effective refractive index (ERI) [19–21], which induces a mechanical optical nonlinearity into the system and a Kerr-like coefficient is derived to describe this phenomenon. The theoretical analysis, the numerical simulations and the experimental results are presented and discussed.

2. Theoretical analysis

A ring resonator system is designed to enhance the optical nonlinearity induced by the optical gradient force as shown in Fig. 1(a) . It consists of a bus waveguide and two ring resonators, which has a free-hanging arc with radius R in each ring resonator. The propagating light in the bus waveguide is coupled into the two ring resonators and the free-hanging arc is perpendicularly deformable by the optical gradient force. The free-hanging arc has a length of L_i = 4R arcsin (D_i/X), where$X=\sqrt{2R{D}_i}$ and D_i is the height of the arc in ring i (i = 1, 2) as shown in Fig. 1(b).

 

Fig. 1 (a) Schematic illustration of the opto-mechanical ring resonator system; and (b) demonstration of the deformation of the free-hanging arcs caused by the optical gradient forces.

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The input light intensity I₀ (measured in Wm⁻²) is expressed as $I_0={\epsilon }_0{n}_{eff}c{\left|E\right|}^2/2,$ and the guiding power P_g in the waveguide equal to$P_g=I_{0}A$, where ${\epsilon }_0$ is the permittivity of vacuum, $n_{eff}$is the ERI of this guiding mode, c is the speed of light in vacuum and A is the mode area. Therefore, the electric field in the waveguide can be written as $\left|E\right|=\sqrt{2P_g/Ac{n}_{eff}{\epsilon }_0}.$ When the optical fields pass through the coupling region, the electric field due to the evanescent coupling can be expressed as [22]

In considering the deformation of the free-hanging arcs, the uniform load (F_m) can be expressed as $F_m=K_{eff}\Delta g/L,$ where $\Delta g$ is the deflection and $K_{eff}$is the effective stiffness, which depends on radius of ring, height of the free-hanging arc and the Young’s modulus of the material [24]. Based on the effective index method, the optical gradient force (F_o) can be expressed as [19]

In order to compare with previous work [11], Eq. (9) is rewritten as,

In the numerical simulations, the parameters of the ring/waveguide used are as follow: the width b = 450 nm, the height h = 220 nm, the ring radius R = 30 μm, the ring-waveguide gap G = 200 nm, the refractive index of ring/waveguide n_s = 3.5 and the initial ring-substrate gap g_o = 200 nm. The gradient optical force is calculated based on Eqs. (1)- (4) and Eq. (8). First, based on the finite element method (FEM), the ERI ($n_{eff}$) in the gap-wavelength (g-λ) domain is shown in Fig. 2(a) , thus the values of $\partial n_{eff}/\partial g$and $\partial n_{eff}/\partial \lambda$ are determined. Second, according to Eq. (2), a_i is approximately 0.99, r_i and t_i are subsequently simulated by the finite-difference time domain method. Therefore, when the input power $P_g$ is fixed at 1 mW, the normalized optical force $F_o$ is obtained as shown in Fig. 2(b). It can be seen that the maximal values of the optical force is corresponding to the resonant wavelengths (i.e. λ₁₁ = 1545.3 nm and λ₂₁ = 1548.2 nm at g = 180 nm). When the gap g is decreased, the maximal value of the optical force is increased and the corresponding resonant wavelengths are red shifted.

 

Fig. 2 Contour plot of (a) the effective refractive index and (b) the optical force in the gap-wavelength domain.

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Based on the uniform load assumption, the effective stiffness $K_{eff}$of the free-hanging arc is obtained by using the FEM method. For example, $K_{eff}$ is approximately 30 nNμm⁻² when D_i = 5 μm. The mechanical force (F_m, nNμm⁻¹) is linearly proportional to the deflection. However, the optical force (F_o, nNμm⁻¹) is not only varied by the deflection ($\Delta g=g_0-g$), but also affected by the wavelength of the input light as shown in Fig. 2(b). When the wavelength increases from 1548.29 nm to 1548.34 nm at a step of 0.01 nm, the respective curves of optical force versus deflection are shown in Fig. 3 . The cross points between the curves of optical force and the line of mechanical force indicate the net force (F_n = F_o - F_m) acting upon the free-hanging arc is zero since the two forces have opposite directions. When the potential energy of the free-hanging arc is at local minimum, these cross points correspond to the mechanical equilibrium states. The work done by the net force ($W={\displaystyle \int F_n}dg={\displaystyle \int F_o\left(\lambda \right)dg}-{\displaystyle \int F_{m}dg}$) is equal to the change of the potential energy. When the work is positive (same as the direction of optical force), the potential energy is decreased. In the contrary, the potential energy is increased. The wavelength is tuned slowly such that the free hanging arc is assumed to have sufficient time to move from one equilibrium position to the other. When the wavelength increases from 1548.29 nm to 1548.32 nm, the optical force is larger than the mechanical force ($F_o\left({\lambda }_o+d\lambda \right){|}_{\Delta g_o}^{\Delta g_o+dg}>F_m{|}_{\Delta g_o}^{\Delta g_o+dg}$) in a small deflection range from $\Delta g_o$ to $\Delta g_o+dg$, where ${\lambda }_o$is an arbitrary wavelength in the range from 1548.29 nm to 1548.32 nm, $d\lambda$is a small wavelength increase, and $\Delta g_o$is the deflection of the equilibrium state corresponding to ${\lambda }_o$. Therefore, the work in the small deflection range ($\Delta g_o$,$\Delta g_o+dg$) is positive ($W{|}_{\Delta g_o}^{\Delta g_o+dg}>0$), and the total work ($W_{total}={\displaystyle {\int }_{1548.29\text{nm}}^{1548.32\text{nm}}W{|}_{\Delta g_o}^{\Delta g_o+dg}d\lambda }$) is also positive. The cross points corresponding to mechanical equilibrium states are highlighted by the circles in Fig. 3. When the input wavelength is larger than 1548.32 nm, the optical force is smaller than the mechanical force (F_o < F_m), thus the free-hanging arc returns to the original position (i.e. $\Delta g$ = 0). Similarly, when the input wavelength decreases from 1548.34 nm to 1548.32 nm, the free-hanging arc cannot be deflected. When the wavelength is decreased from 1548.32 nm to 1548.31 nm, the free-hanging arc cannot be largely bent from the original position because the bending process requests negative work (${\displaystyle {\int }_0^{\Delta g_o}{F}_{m}dg}>{\displaystyle {\int }_0^{\Delta g_o}{F}_{o}dg}$, $\Delta g$> 27.5 nm). Deflection is only observed when the input wavelength is decreased to 1548.30 nm in which the optical force is always larger than the mechanical force (F_o > F_m) in the deflection interval from 0 nm to 17.5 nm (as indicated by the red circle) when the mechanical oscillation is ignored. These phenomena are experimentally observed and discussed in the next section.

 

Fig. 3 Optical force and mechanical force versus the deflection. The circles indicate the different deflections corresponding to different wavelengths.

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3. Experimental results and discussions

The opto-mechanical ring resonator system, as shown in Fig. 4(a) , is fabricated on the silicon-on-insulator wafer by using the nano-fabrication processes. The waveguide-ring gap is 200 nm, the thickness of the silicon structure layer and the buried oxide layer are 220 nm and 2 µm, respectively. Except for the free-hanging arcs, the remaining parts of the system are deposited with a 2-µm thick SiO₂ cladding layer. The free-hanging arcs are released by the selective etching process and the gap g between the free-hanging arc and the substrate is controlled to be approximately 200 nm.

 

Fig. 4 (a) SEM image of the opto-mechanical ring resonator system with a released bus waveguide and two partially released ring resonators, and (b) the schematic of experimental setup.

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The experimental setup is shown in Fig. 4(b). In the experiments, a pair of tapered fiber is used to couple the input light into the nano-waveguide and detect the output light, respectively. The probe light is a broadband light source with a central wavelength of 1550 nm and a bandwidth of 70 nm. With the probe light, each ring has two absorption modes within the spectrum range, i.e. ring resonator 1 with λ₁₁ = 1545.4 nm and λ₂₁ = 1548.2 nm while ring resonator 2 with λ₁₂ = 1545.6 nm and λ₂₂ = 1548.4 nm as shown in Fig. 5 . By combining with the numerical results of$n_{eff}$, r_i, and t_i, the simulated transmission spectrum and the experimental results are well matched with each other via the optimization of the attenuation factor a_i and the height of the free-hanging arcs D_i. Subsequently, a control light with an output power of 160 μW and a tunable wavelength ranging from 1500 nm to 1600 nm with a tuning step of 0.01 nm are coupled into the system. When the control light is adjusted to different wavelengths, the transmission spectrum of the ring resonator is affected as shown in Fig. 6 . It can be seen that the absorption peaks are not shifted as compared to those peaks in the transmission without the control light when the control wavelength is 1547.97 nm. When the control wavelength increases from 1548.22 nm to 1548.26 nm, the resonant peak λ₁₁ is shifted from 1545.4 nm to 1545.6 nm (melted with λ₁₂). This occurs because the effective input power $P_{eff}$ of the ring 1 is increased when the overlapped range between the control laser peak and the absorption peak (λ₂₁ = 1548.2 nm) increases. The effective input power ($P_{eff}$, a part of guided power in waveguide, which is absorbed by ring 1) is calculated by the integration of the product of the laser power $p_L(\lambda )$ and the absorption ratio (1- T) in the wavelength domain ($P_{eff}={\displaystyle \underset{{\lambda }_{\min }}{\overset{{\lambda }_{\max }}{\int }}{p}_L(\lambda )\left(1-T\right)d\lambda }$), in which the laser power $p_L(\lambda )$can be directly determined by the transmission spectrum. The absorption ratio (1- T) is estimated based on the absorption peak with the central wavelength λ₂₁. The normalized effective input power is associated with the control wavelength as shown in Fig. 7(a) . It is observed that the effective input power is increased from 0 μW to 51.9 μW when the control wavelength increases from 1548.18 nm to 1548.76 nm. Thereafter, the free-hanging arc cannot be kept at the position with the maximum deflection and returns to the initial position by the mechanical force. On the contrary, when the control wavelength decreases from 1548.76 nm to 1548.45 nm, the effective input power is zero in the beginning position, and jumps to 0.98 μW at the wavelength of 1548.5 nm. Thereafter, the effective input power is gradually decreased to zero. It is shown that one control wavelength corresponds to two possible effective input powers in this particular range, which means an opto-mechanical bistability is induced in this system.

 

Fig. 5 Transmission spectrum of the opto-mechanical ring resonator system.

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Fig. 6 Shift of absorption peak of ring 1 under different control wavelengths.

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Fig. 7 (a) Effective input power for ring 1 versus the control wavelength; and (b) Kerr-like coefficient versus the effective input power.

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Based on Eq. (9) and the experimental results, the Kerr-like coefficient is calculated. First, the mode area A is calculated based on the optical mode in the nano-waveguide by using the numerical method. Then, $\partial n_{eff}/\partial P_{eff}$ for ring 1 is obtained by$\frac{\partial n_{eff}}{\partial P_{eff}}=\frac{\partial n_{eff}}{\partial {\lambda }_{11}}\frac{\partial {\lambda }_{11}}{\partial {\lambda }_c}\frac{\partial {\lambda }_c}{\partial P_{eff}},$ where $\partial n_{eff}/\partial {\lambda }_{11}$ is determined based on the resonant conditions (i.e.$\partial n_{eff}/\partial {\lambda }_{11}=m/L_i$), $\partial {\lambda }_{11}/\partial {\lambda }_c$ is determined based on the peak shift as shown in Fig. 6, $\partial {\lambda }_c/\partial P_{eff}$is determined from the curve shown in Fig. 7(a), and the resonant mode order m is determined from the simulation results (i. e., m = 309). Therefore, the Kerr-like coefficient $n_2$ for ring 1 is depending on the effective input power as shown in Fig. 7(b). It can be seen that when the effective input power decreases from 51.9 µW to 0.998 µW, the Kerr-like coefficient is increased from 7.64 × 10⁻¹² to 2.01 × 10⁻¹⁰ m²W⁻¹. In other words, the optical force-induced Kerr-like coefficient in the opto-mechanical ring resonator system is at least 6-order higher than the Kerr coefficient in the silicon material (2.8 × 10⁻¹⁸ to 14.5 × 10⁻¹⁸ m²W⁻¹). When the effective input power is lower than 0.998 µW, the calculated Kerr-like coefficient is further increased. However, the calculation value becomes unreliable because a larger fluctuation is observed at lower effective input power due to the noises in the experiments.

In order to verify the Kerr effect is induced by the optical force but not the thermal optic effect, devices with different gaps between the free hanging arc and the substrate are experimented. The gap is controlled by the etching time in fabrication process, which is ranging from 15 to 30 minutes (etching rate: 10 nm/min). The free-hanging arc in each device is completely released and can be deformed by the optical force, in which the Kerr effect is observed in all cases. On the contrary, when the etching time is shorter than 15 minutes, the free-hanging arc is not fully released from the SiO₂ substrate and the maximum observed peak shift is less than 0.1 nm. When the etching time is longer than 30 minutes, the gap between the free-hanging arc and the substrate is larger than 300 nm. The maximum observed peak shift is also less than 0.1 nm because the weak optical force is insufficient to deform the free-hanging arc. Based on these observations, it is verified that the Kerr effect is indeed induced by the optical force. This can also be verified by the measurement of the time-domain response [21, 26].

4. Conclusions

In summary, the theoretical analysis, the numerical simulation and the experimental results of the optical nonlinearity induced by the optical gradient force in the opto-mechanical ring resonator system are presented and discussed. The Kerr-like coefficient is determined, which is in the range from 7.64 × 10⁻¹² to 2.01 × 10⁻¹⁰ m²W⁻¹, and it is at least 6-order higher than the silicon’s Kerr coefficient. The dramatically improved optical nonlinearity in the opto-mechanical ring resonator promises potential applications in low power optical signal processing, modulation and bio-sensing.