Open Access
Add to BibSonomyAbstract
We propose and experimentally demonstrate an energy-efficient optical diode based on the optomechanical effect. The optical signals could transmit during forward propagation while be blocked during backward propagation. When launching optical signal with a low power of 4.0 mW, the maximum resonance red-shift of the asymmetric silicon microring resonator (MRR) could be up to 0.74 nm, in this case, a forward-backward nonreciprocal transmission ratio (NTR) of 12.7 dB has been achieved. The 10-dB and 5-dB operation bandwidths are 0.08 nm and 0.24 nm, respectively. The operating bandwidth could be continuously tuned theoretically by changing the input power.
© 2017 Optical Society of America
1. Introduction
Optical nonreciprocal devices such as optical diode [1], optical isolator [2, 3] and optical circulator [4–6] are fundamental components for information processing in optical communication systems, which allows light to pass in a direction while blocking it in the opposite direction. In the past decade, optical non-reciprocal devices have been attracting great interests due to their significant applications in protecting lasers from back reflections, and improving the stability of the optical transmission system [7]. Of these devices, most successful commercial products are based on magneto-optical effect consisting of some discrete devices including Faraday rotators, half-wave plates and so on with a bulky system [8, 9]. In order to minimize the device size, many schemes for better integration have been proposed utilizing different mechanisms. One method is to deposit a layer of magneto-optical materials on the waveguide, which can break time-reversal symmetry and achieve non-reciprocal transmission. Such schemes include a layer of (Ce1Y2)Fe5O12(80 nm)/Y3Fe5O12(20 nm) deposited on a silicon race ring resonator [10], a Ce: YIG/SGGG layer grown on a silicon microring resonator [11]. An alternative method is to modulate the refractive index in the time domain, for instance, by means of electric driving, then the indirect photon migration in silicon waveguide is realized to achieve non-reciprocal effect [12]. The other method is to utilize the nonlinear effects in waveguides such as thermo-optic effect to achieve the asymmetric spectral shift [13].
The fabrication of non-reciprocal devices based on the magneto-optic effect or temporal refractive index modulation are difficult because of their complex fabrication process. Therefore it is still a challenge to realize optical non-reciprocal devices with low-cost fabrication and simple manipulation [14]. With the huge application prospects and superiorities of silicon photonic technology, such as compact footprint and low-cost fabrication [15–17], silicon integrated platform is promising to be the mainstream fabrication of photonic devices. Several schemes with silicon photonic crystal [18], silicon asymmetric microring resonator (MRR) have been demonstrated in pure silicon platform. In 2012, Qi et al demonstrated a passive optical diode on silicon-on-insulator (SOI) platform [19], this device has a large nonreciprocal transmission ratio (NTR) in a low input power (20 dB at 0.085 mW) while the operating bandwidth is narrow and the response is slow due to the usage of high-Q resonators. In 2015, we demonstrated a pure silicon optical circulator on the basis of two asymmetric MRR [6]. It has a larger NTR and a tunable operating bandwidth but needs a high input power (maximum 33 dB NTR at 32 mW). The red-shift caused by thermo-optic effect is 0.23 nm. More recently, to pursue better tunability, we have optimized the parameters and realized the scalability and bandwidth tunability of the device [20, 21].
In order to pursue an energy-efficient optical diode in pure silicon platform, the optomechanical effect of asymmetric silicon MRR is utilized to demonstrate an on-chip optical diode with lower power consumption. This device is energy efficient, and the red-shift of MRR is as large as 0.74 nm with input power as low as 4 mW, the maximum NTR is 12.7 dB. The operating bandwidth with NTR larger than 10 dB (10-dB BW) is 0.08 nm, and with NTR larger than 5 dB (5-dB BW) is 0.24 nm. The operating bandwidth could be continuously tuned by optimizing the input power.
2. Operation principle
The physical mechanism of this optical diode is the optomechanical effect of MRR. When the gap of two waveguides is designed small enough, the evanescent wave from one waveguide would be coupled into another waveguide or a dielectric substrate and an optical gradient force is produced [22]. If light power is high enough and the waveguide is free-standing, the gradient force could cause a deflection of the free-standing structure. In an MRR, owing to its nanoscale and high energy density, the optical gradient force can be enhanced with a relatively low input power [23]. The MRR structure is shown in Figs. 1(a) and 1(b), where half of MRR (the racetrack region) is designed to be suspended from SiO2 substrate and leaves a vertical air-gap (labeled as g0). When we control g0 small enough, light injected into MRR will be coupled into the SiO2 substrate in the form of the evanescent wave, then optical gradient force will drive the free-standing waveguide to bend down toward the SiO2 substrate until the mechanical force and the optical force has a balance. The central deflection of the arc is defined as x. Consequently, as the waveguide bends down, its effective refractive index (neff) increases and the resonant wavelength of MRR is red-shifted.
Fig. 1 Theoretically analysis of optical diode based on optomechanical effect. (a) Layout of the proposed optical diode; (b) Cross-sectional view of the proposed diode; (c) Top view of the proposed diode; (d) Theoretical forward and backward propagation spectrum.
It should be emphasized that the nonreciprocal device is realized owing to the nonlinearity enabled by the optomechanical effect. The time reversal symmetry has not been broken in our experiment. Figure 1(c) shows the top view of the optical diode, where the basic unit of the device is an asymmetric add-drop racetrack MRR with different coupling gaps named as GS (the forward route) and GL (the backward route) between bus waveguide and MRR, respectively. GL is much larger than GS in order to realize a weaker coupling during backward propagation. Lr is the length of the racetrack, the radius of MRR is defined as R, the green arrows represent forward propagation and the red arrows correspond to backward propagation. The upper bus waveguide is also suspended from SiO2 substrate in order to prevent refractive index mismatching in the coupling region. Figure 1(d) shows the red-shifts of resonant wavelength in both forward and backward transmission. The initial resonant wavelength of the unbent waveguide is λ0 (black solid line), the monochromatic input light wavelength is λS (blue dash line). When input light is injected from Port 1 and propagates forward via MRR with a gap of GS, a strong coupling occurs between the lower bus waveguide and MRR, energy stored in both MRR and SiO2 substrate is large so as to produce a strong optical force and a significant red-shift appears (green dash line). At the same time, the shifted resonant wavelength is almost aligned to the input wavelength (λS), thus the light can transmit to Port 2 with low loss, which is the “on” state of forward propagation. On the contrary, when input light is injected from Port 2 and propagates backward with a gap of GL, a weak coupling happens, the energy stored in MRR and SiO2 substrate is small, producing a tiny red-shift (red dash line). The new resonant wavelength is not aligned to λS, thus light cannot be detected at Port 1. This means it is the “off” state of backward propagation.
3. Device fabrication
We fabricate the device on a commercial silicon-on-insulator (SOI) wafer which is from Silicon on Insulator Technology (Soitec) company, France. The thickness of top silicon and SiO2 layer are 220 nm and 2 μm, respectively [Fig. 2(a)]. The whole fabrication of the proposed device consists of three dry etching procedures (for Si etching) and one wet release procedure (for SiO2 etching). The order of the three Si etching procedures are etching the grating couplers, the silicon waveguide, and the “window” (the released area). After the grating couplers are etched, we begin fabricating the key structure of the released MRR. Firstly, we use E-beam lithography to transfer the pattern on the E-beam resist, then the bus waveguide and MRR are fabricated by inductively coupled plasma (ICP) etching with an actual width of 440 nm and a height of 185 nm, leaving 35 nm-thickness silicon layer to protect the substrate from corrosion in the solution of hydrofluoric acid, as shown in Fig. 2(b). Secondly, the remaining 35 nm-thickness silicon layer in the released area is then etched off for the second time to form a ’window’ [Fig. 2(c)], and the SiO2 substrate is exposed to the air. Lastly, the wafer is dipped into the hydrofluoric acid solution, a chemical reaction occurred between the SiO2 and the HF solution, then a thin SiO2 layer of the substrate in the released area is etched by the hydrofluoric acid solution [Fig. 2(d)]. Thus half of the MRR is suspended from the SiO2 substrate. In the experiment, we choose Buffered-HF to precise control the wet release depth g0 to achieve a large optical gradient force. Since the cantilever is relatively short which is within 100 μm, it will not collapse under our careful wet release process.
Fig. 2 Fabrication process of the device. (a) E-beam lithography of the device; (b) ICP etching the bus waveguide and MRR; (c) ICP etching the released area to form a ‘window’; (d) Etching the released area by the hydrofluoric acid solution.
Figure 3 shows the scanning electron microscope (SEM) images of the device. The whole device is shown in Fig. 3(a). The radius of MRR (R) is 30 μm, the length of the race (Lr) is 12 μm, the actual width of the upper gap (GL) and the lower gap (GS) are 255 nm and 100 nm which are shown in Figs. 3(d) and 3(e), respectively. The actual width of both the bus waveguide and the MRR is 440 nm [Fig. 3(e)]. The zoom in images of the grating coupler is shown in Fig. 3(f), which is about 19 μm long. The zoom in images of the free-standing arc and bus waveguide is shown in Figs. 3(b) and 3(c), respectively. The width of the air gap (g0) is measured as 325 nm by the profilometer.
Fig. 3 SEM images. (a) Whole scanning of the device; (GL); (b) Free-standing arc; (c) Free-standing bus waveguide; (d) The upper gap (GL); (e) The lower gap (GS); (f) Vertical grating coupler.
4. Experimental results and discussions
4.1 Experimental results
The experimental configuration is shown in Fig. 4(a). A tunable laser source (TLS) emits a continuous wave (CW) light with an output power of 6 dBm, then the monochromatic light is amplified to ~19 dBm by an erbium-doped fiber amplifier (EDFA), the attenuator (ATT) after EDFA is used to make the output power tunable. In order to characterize the transmission spectrum of the device, an amplified spontaneous emission (ASE, Amonics ALS-CL-15-B-FA) source whose power density is around 0.1 mW/nm is coupled with the monochromatic light by an optical coupler (OC) to measure the resonant red-shift and the forward-backward NTR. The polarization beam splitter (PBS) and the polarization controller (PC) are used to ensure light to be TE mode before the vertical grating coupler. Lastly, an optical spectrum analyzer (OSA, YOKOGAWA AQ6370C) is used to record and analyze the optical spectrum. The resolution of the OSA is 0.02 nm. During the experiment, we exchange the input and output ports of the single device and get the forward and backward optical spectrum. As is shown in Fig. 4(b), we choose two resonant wavelengths of MRR to characterize red-shift and NTR of the device. The wavelength of input CW light is set around the second resonant wavelength (i.e., 1548.77 nm) to investigate the NTR. Assisted by the ASE source, we can measure red-shift conveniently around the first resonant wavelength, i.e., 1546.25 nm.
Fig. 4 Experimental configuration and non-reciprocal phenomenon. (a) Experimental setup. TLS, tunable laser source; EDFA, erbium-doped fiber amplifier; ATT, attenuator; ASE, amplified spontaneous emission source; OC, optical coupler; PBS, polarization beam splitter; PC, polarization controller; OSA, optical spectrum analyzer; (b) Optical spectrum of forward and backward propagation at the input power as 4 mW.
As Fig. 4(b) shows, the free spectrum range (FSR) of MRR is 2.5 nm, and the Q factor is around 8400. The extinction ratio (ER) of the resonance is 17 dB. Considering an extra 6.5 dB fiber-to-chip coupling loss, the real light power injected into the MRR is about 4 mW, and the wavelength of input light is aligned at 1549.36 nm, longer than the resonant wavelength at 1548.77 nm, thus the detuning wavelength is 0.59 nm. When the light is injected from Port 1 and propagates forward, a distinct red-shift of 0.74 nm is observed (green solid line), and the second resonant wavelength is shifted to 1549.51 nm, well aligned with the input light wavelength. Thus the output power at Port 2 is high. On the contrary, when the light is injected from Port 2 and propagates backward, we observe a very tiny red-shift (i.e., 0.06 nm, red dash-dot line). Thus the output power at Port 1 is much lower since the wavelength of input light is mismatched with the resonant wavelength. The forward-backward NTR is measured to be 12.7 dB.
To investigate the impact of input power on the NTR performance, we then experimentally measure the maximum NTR at different input power levels, as shown in Fig. 5(a). One can see that as the input power increases, the maximum NTR is increasing rapidly at first and incrementally after the input power is larger than 1 mW. Owing to the Lorentzian curve resonance, when the input power is 1 mW, the red-shift has achieved 0.35 nm, which is about double of the 3-dB bandwidth of the resonance, i.e., 0.18 nm. And the detuning wavelength is 0.32 nm. In this case, we can already get a large NTR. After the input power is increasing, the NTR changing rate is smaller because the slope of resonance edge is much smaller than that in the resonance center. To verify whether the red-shift is mainly caused by optomechanical effect or thermo-optical effect, we compare the red-shift performances of two chip samples whose geometric structures are the same except that one is etched in the released area and another is not. Figure 5(b) shows the red-shift amount at different input powers for these two samples. For the sample of optomechanical effect, the experimental red-shift (purple solid line) is as large as 0.74 nm with 4 mW input power. However, for the sample of thermo-optic effect, the experimental red-shift (red solid line) is only 0.05 nm, much smaller. This proves that it is the optomechanical effect rather than thermo-optic effect that generates such large red-shift.
Fig. 5 Simulation and experiment results. (a) Experimental result of maximum NTR-input power relationship; (b) Simulation (sim or fit) & experimental (exp) result of the optomechanical (OM) effect & thermo-optic (TO) Effect causing red-shift-input power relationship; (c) Simulation of wavelength shift-deflection relationship; (d) Simulation of optical force-deflection relationship under different input power respectively.
We further investigate the optomechanical effect in MRR theoretically. We simulate the effective refractive index (neff) for various waveguide deflection (x) and calculate the wavelength shift with different x when g0 is 325 nm in COMSOL software. Figure 5(c) shows the simulated result of the wavelength shift-deflection (neff -x) relationship. By using neff -x relationship, we calculate different optical gradient forces for different input powers respectively. The optical gradient force induced by the optomechanical effect can be expressed as [22]:
Where g is the depth of air gap, U is the energy stored in MRR, τe−1 is the extrinsic decay rate induced by external coupling occurred in GL and GS coupling region of the transmitted wave, it should be noted that when the suspend MRR bending down to the SiO2 substrate, the GL would be larger and the coupling parameter would change. While the change is too small to significant influence the coupling state (e.g. the ER of MRR is still the same before and after the red-shift), so we regard the coupling parameter of GL and τe−1 as the stable ones. And τi−1 is the intrinsic decay rate due to the internal cavity loss, λin is the wavelength of input light, and λr is the resonant wavelength of MRR. As is shown in Fig. 5(d), the optical gradient force is a nonlinear function of the deflection, the black solid line shows the mechanical elastic coefficient of the waveguide which is kmech = 0.14 N/m [22]. Considering that the parameter scale of our device is similar to that in [22], we approximately choose kmech = 0.14 N/m as the real mechanical elastic coefficient of our device and make the simulation. The intersection between the black solid line and other curves is where mechanical force equals to optical gradient force, called the balancing point. By combining Figs. 5(c) and 5(d), we calculate the wavelength shift of different balancing points under different input power injection, which is shown as the blue dash line in Fig. 5(b). The red-shift induced by thermo-optic effect can be expressed by [24]The index change induced by thermal effect can be written as [25, 26]
Where λ0 is the resonant wavelength, ng is the group index, and δnTO is the the silicon refractive index variation, Γth is the effective confinement factor of the silicon waveguide, kth = 1.86 × 10−4 K−1 is the silicon thermo-optic coefficient, Rth = 50 K/mW is the thermal resistance of the silicon ring resonator, T(λ) is the transmission coefficient of the MRR at the steady state, and Pin is the input optical power [27]. It indicates that the thermo-induced red-shift is a linear relationship to the input power Pin when a monochromatic wavelength input light is injected. We fit the experimental results with a linear fitting curve, as the green dash line shown in Fig. 5(b). The slope of the fitting curve is 0.014 nm/mW. From the theory of optomechanical effect, one can see that the simulated result accords well with the experimental results.In the experiment, the wavelength of input monochromatic light is aligned to the resonant wavelength, i.e., 1548.77 nm initially. In order to optimize the optical force and red-shift of MRR, we increase the input wavelength by a step of 0.02 nm and measure the red-shift and NTR at each input light wavelength, as shown in Fig. 6(a). We notice that the wavelength to obtain the maximum NTR is not aligned to the resonant wavelength of MRR but has a wavelength detuning (red shift) defined as Δλ. The detuning Δλ has to be optimized, relative to input power. We have also found that when Δλ is small, the NTR is a negative value. Figure 6(b) shows the resonance shift of MRR under both forward and backward propagation when Δλ is very small. The red shift of resonance is much larger in forward propagation than in backward propagation, thus the transmission of input wavelength experience a pass filter in backward propagation while a lossy response in forward propagation. As a result, the NTR is a negative value in this case. Therefore, the optimization of input wavelength is essential in our scheme.
Fig. 6 (a) Experimental results under different power inputs; (b) Transmit spectrum of forward and backward propagation when Δλ is small.
From Fig. 6(a), we can see that when the input power increases from 0.6 mW to 4.0 mW, both the maximum NTR and the operating bandwidth increase. We define 10-dB BW (5-dB BW) as the wavelength range where the NTR is larger than 10 dB (5 dB). Table 1 summarizes several important parameters of the device including measured bandwidths of the device. As is shown, the 10-dB BW is 0.08 nm when the input power is as low as 4.0 mW. At the same time, the operation bandwidth can be tuned by adjusting the input power.
Table 1. Measured parameters of the device
4.2 Discussions
The optical gradient force is a nonlinear phenomenon whose internal operating process is complex. In our experiment, the resonant wavelength is red-shifted under a strong coupling condition, which indicates that the neff increases. To our analysis, in addition to the optical gradient force between the released MRR and the SiO2 substrate contributing to the red-shift, there are other two possibilities that could cause the red-shift too. The first factor is the optical gradient force between the released MRR and the adjacent bus waveguide. If the red-shift is caused by the first one, when the MRR and the bus waveguide are attracted and closed to each other, the neff will increase. If so, the gap between MRR and the bus waveguide would be smaller, which would influence the coupling coefficient and the ER of MRR would change too. However, in our experiments, the ER remains the same when the light is injected. The second factor is the thermo-optic effect. If the red-shift is caused by the thermo-optic effect, the red shift-input power curve should be a linear one according to Eq. (4). Moreover, by increasing the wet release depth, the red-shift should be larger at the same input power because the heat amount dissipated to SiO2 is much more than dissipated to air. But the experiment result is opposite to the above hypothesis. The red-shift to input power curve is a nonlinear one and the red-shift decreases when the wet release depth is increasing. As is shown in Fig. 7, we measured the red-shift of another device whose wet release depth g0 is 365 nm. The maximum red-shift is around 0.2nm which is much smaller than 0.74nm (where g0 is only 325 nm). So, we believe that the dominant operation mechanism is the optical gradient force between the released MRR and the substrate SiO2 waveguide.
Though the device is energy-efficient and the input power is as low as 4.0 mW, it can be further decreased. The optomechanical effect is mainly determined by four factors, including the input power, the length of suspended waveguide, the detuning wavelength Δλ and the gap between MRR and SiO2 substrate (g0). For a well-fabricated chip, when the light is injected with a certain input power, the former three factors are fixed theoretically. The only one factor we could control is the g0. The minimum g0 we could achieve theoretically is half of the waveguide width under the isotropic wet corrosion. The actual width of the gap g0 is 325 nm, which is much larger than the half of the waveguide, i.e., 220 nm. For instance, Liu et al demonstrated the similar device whose g0 was controlled as 160 nm and the red-shift could achieve 1.62 nm at input power is 2.8 mW [22]. Therefore the input power can be further decreased by controlling g0 smaller.
There also has much potential improvement on the nonreciprocal transmission ratio (NTR) and the operating bandwidth of the device. Besides the red-shift which is limited by the optomechanical effect, the other limiting factor of NTR is the extinction ratio (ER) of the resonant wavelength, which is mainly decided by the actual device design and fabrication process. The maximum NTR we could achieve theoretically is equal to the actual ER which is relatively low as 17 dB. By optimizing the parameter of MRR such as the length of race track Lr and the gap between bus waveguide and MRR at the drop port, the ER of MRR could be increased, so as to enhance the NTR of the device when the red-shift and input power are fixed. The operating bandwidth of the device is mainly limited by the difference of GL and GS, the larger the difference is, the larger the bandwidth is. As the waveguide-ring coupling of GL is in the under-coupling state, when GL increases actually, the coupling would be weaker, and the operating bandwidth of the device would be larger. The actual GL and GS are 255nm and 100 nm, respectively. By optimizing GL and GS we can further increase the bandwidth.
There are some other novel applications with the optomechanical effect. On the basis of the optical diode, a further design of an optical circulator [6] with two and even more MRR units is valuable to be tried. And we can also further use the optomechanical effect in all-optical signal processing devices which need a distinct red-shift [27].
In conclusion, by optimizing this non-reciprocal device, we can improve the performance of optical diode with more energy-efficiency, higher Q [28, 29], larger NTR and larger bandwidth. And based on this device, further research on other applications such as circulator and optical logical code generation in all-optical communication networks is available to be tried.
5. Conclusion
In this paper, we have experimentally demonstrated an energy-efficient optical diode based on the optomechanical effect on a pure silicon platform. The maximum forward-backward nonreciprocal transmission ratio (NTR) is 12.7 dB under the input power is as low as 4.0 mW, the obvious large red-shift is 0.74 nm. The operating bandwidth with NTR larger than 10dB (10-dB BW) and 5dB (5-dB BW) are 0.08 nm and 0.24 nm, respectively. The operating bandwidth could be tuned theoretically by changing the input power. This kind of integrated device without any extra assistance such as an epitaxial layer, magnetic fields or strong pumping signal has a promising appliance in all-optical communication systems.
Funding
The research is supported under National Natural Science Foundation of China (61622502, and 61475052).
References and links
1. Y. Zhang, D. Li, C. Zeng, Z. Huang, Y. Wang, Q. Huang, Y. Wu, J. Yu, and J. Xia, “Silicon optical diode based on cascaded photonic crystal cavities,” Opt. Lett. 39(6), 1370–1373 (2014). [CrossRef] [PubMed]
2. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009). [CrossRef]
3. M. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5(9), 549–553 (2011). [CrossRef]
4. W. Śmigaj, J. Romero-Vivas, B. Gralak, L. Magdenko, B. Dagens, and M. Vanwolleghem, “Magneto-optical circulator designed for operation in a uniform external magnetic field,” Opt. Lett. 35(4), 568–570 (2010). [CrossRef] [PubMed]
5. K. Mitsuya, Y. Shoji, and T. Mizumoto, “Demonstration of a silicon waveguide optical circulator,” IEEE Photonics Technol. Lett. 25(8), 721–723 (2013). [CrossRef]
6. L. Liu, J. Dong, D. Gao, A. Zheng, and X. Zhang, “On-chip passive three-port circuit of all-optical ordered-route transmission,” Sci. Rep. 5(1), 10190 (2015). [CrossRef] [PubMed]
7. D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7(8), 579–582 (2013). [CrossRef]
8. M. Shirasaki, H. Kuwahara, and T. Obokata, “Compact polarization-independent optical circulator,” Appl. Opt. 20(15), 2683–2687 (1981). [CrossRef] [PubMed]
9. Y. Fujii, “High-isolation polarization-independent optical circulator,” J. Lightwave Technol. 9(10), 1238–1243 (1991). [CrossRef]
10. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011). [CrossRef]
11. D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. Bowers, “Silicon microring isolator with large optical isolation and low loss,” in Optical Fiber Communication Conference (Optical Society of America, 2016), Th1K. 2. [CrossRef]
12. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009). [CrossRef] [PubMed]
13. L. Fan, L. T. Varghese, J. Wang, Y. Xuan, A. M. Weiner, and M. Qi, “Silicon optical diode with 40 dB nonreciprocal transmission,” Opt. Lett. 38(8), 1259–1261 (2013). [CrossRef] [PubMed]
14. M. Soljacić and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3(4), 211–219 (2004). [CrossRef] [PubMed]
15. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431(7012), 1081–1084 (2004). [CrossRef] [PubMed]
16. Q. Xu and M. Lipson, “All-optical logic based on silicon micro-ring resonators,” Opt. Express 15(3), 924–929 (2007). [CrossRef] [PubMed]
17. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon–organic hybrid slot waveguides,” Nat. Photonics 3(4), 216–219 (2009). [CrossRef]
18. C. Wang, X.-L. Zhong, and Z.-Y. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 674 (2012). [CrossRef] [PubMed]
19. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335(6067), 447–450 (2012). [CrossRef] [PubMed]
20. L. Liu, J. Dong, H. Qiu, F. Jiang, M. He, H. Zhou, and X. Zhang, “An all-silicon passive six-port circuit of all-optical ordered-route transmission,” in CLEO: Science and Innovations, (Optical Society of America, 2016), paper SM3G. 4.
21. H. Qiu, “Scalability and tunability of the silicon circuit supporting on-chip ordered-route light transmission,” in Asia Communications and Photonics Conference, (Optical Society of America, 2016), paper AF3G. 7. [CrossRef]
22. M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A.-Q. Liu, “Nano-optomechanical actuator and pull-back instability,” ACS Nano 7(2), 1676–1681 (2013). [CrossRef] [PubMed]
23. M. He, S. Liao, L. Liu, and J. Dong, “Theoretical analysis for optomechanical all-optical transistor,” Frontiers Optoelectron. 9(3), 406–411 (2016). [CrossRef]
24. M. Xu, J. Wu, T. Wang, X. Hu, X. Jiang, and Y. Su, “Push–pull optical nonreciprocal transmission in cascaded silicon microring resonators,” IEEE Photonics J. 5(1), 2200307 (2013). [CrossRef]
25. P. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13(3), 801–820 (2005). [CrossRef] [PubMed]
26. H. Chen, X. Luo, and A. W. Poon, “Cavity-enhanced photocurrent generation by 1.55 mum wavelengths linear absorption in a pin diode embedded silicon microring resonator,” Appl. Phys. Lett. 95(17), 1111 (2009). [CrossRef]
27. L. Liu, J. Dong, and X. Zhang, “Chip-integrated all-optical 4-bit Gray code generation based on silicon microring resonators,” Opt. Express 23(16), 21414–21423 (2015). [CrossRef] [PubMed]
28. K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, “Fabrication of ultralow-loss Si/SiO(2) waveguides by roughness reduction,” Opt. Lett. 26(23), 1888–1890 (2001). [CrossRef] [PubMed]
29. M. Soltani, S. Yegnanarayanan, and A. Adibi, “Ultra-high Q planar silicon microdisk resonators for chip-scale silicon photonics,” Opt. Express 15(8), 4694–4704 (2007). [CrossRef] [PubMed]
