Abstract

We propose and demonstrate an all-optical tuning mechanism to tune the response of a microwave photonic filter (MPF) based on a nonlinear silicon microring resonator (MRR). The tuning mechanism relies on the optical nonlinearities induced resonant wavelength shift in the silicon MRR, leading to the change of frequency difference between the optical carrier frequency and resonant frequency of the silicon MRR. A detailed theoretical model is established to describe the operation of the proposed all-optical tunable MPF. Two cases are studied in the experiment, i.e. the optical carrier frequency is located at the left or right side of the MRR resonant frequency. Both forward and backward pumping configurations in each case are demonstrated. Using the fabricated silicon MRR and exploiting light to control light, the central frequency of the notch MPF can be flexibly tuned by adjusting the pump light power. Moreover, the presented all-optical tuning mechanism might also facilitate interesting applications such as microwave switching and microwave modulation.

© 2015 Optical Society of America

1. Introduction

Silicon photonics has become one of the most promising photonic integration platforms owing to its small footprint, reduced power consumption, and availability of complementary metal-oxide-semiconductor (CMOS) fabrication technology [1–6 ]. The attractive small footprint feature of silicon waveguide structures is because of the high refractive index contrast between silicon and its oxide or air, which enables tight light confinement and benefits small bending radius of waveguide. Because of its unprecedented small size for potential large scale integration, silicon microring resonator (MRR), i.e. compact bended silicon waveguide in a small loop configuration, is of great importance to accelerate the success of silicon photonics [7, 8 ]. Owing to its strong light confinement in a small modal volume, MRR allows nonlinear interactions with relatively low power levels, and is of great interest for nonlinear optics. The nonlinear effects in MRRs include Kerr effect, two-photon absorption (TPA), free-carrier effect (FCE), and thermo-optic effect (TOE) [9, 10 ]. These nonlinear effects in MRRs have been widely used to facilitate miscellaneous optical processing functions, such as all-optical switching [11], all-optical signal processing [12, 13 ], optical bistability [14], optical diode [15, 16 ], and modulator [17].

In the recent years, it has been proved that photonic systems can be also used to process microwave radio frequency (RF) signals [18, 19 ]. As a key element of microwave photonics, several schemes of microwave photonic filters (MPFs) have been proposed and demonstrated using fiber-based devices [20–22 ]. Very recently, owning to the great advance on photonic integrated circuits, some microwave photonic devices such as MPFs have been implemented on silicon-on-insulator (SOI) platforms. MPFs based on MRR and Mach–Zehnder interferometer (MZI) have been proposed and demonstrated showing impressive operation performance [23, 24 ]. Remarkably, most of previous works employed linear effects of microwave photonic devices to perform tunable operations (e.g. tunable MPFs). Actually, nonlinear effects might also provide alternative approaches enabling tunable operations of microwave photonic signal processing. To the best of our knowledge, there have been limited research efforts on tunable MPFs exploiting nonlinear effects in silicon waveguide devices.

In this paper, using light to control light, we present an alternative approach to tuning the response of MPF, i.e. optically-controlled tunable MPF [25]. A detailed theoretical model is established to describe the operation of the proposed all-optical tunable MPF. The tunability of the MPF is based on the combined nonlinear effects in SOI MRR. TOE is identified as the dominant nonlinear effect. With optical single sideband (OSSB) modulation, the central frequency of the notch MPF can be flexibly tuned by changing the pump light power. Two cases depending on the relative position between the optical carrier frequency and MRR resonant frequency are discussed. Both forward and backward pumping configurations are demonstrated in the experiment.

2. Concept and operation principle

Figure 1(a) illustrates the typical scheme of the all-optical tuning process of the MPF based on an SOI MRR. The microwave signal is modulated on a signal optical carrier, and then processed by an SOI MRR and detected by a photodiode (PD). Since the optical response of the MRR can be adjusted by a pump light, the electrical response of the link can be tuned. The inset of Fig. 1(a) depicts typical transmission spectrum of an SOI MRR, in which the microwave modulated signal is located close to one notch resonance frequency and pump light positioned at another notch resonance frequency. Figures 1(b)-1(e) summarize the operation principle of the proposed all-optical tuning process of the MPF. An optical carrier is modulated by an RF signal with OSSB modulation. The output field after modulation is then applied to the MRR for microwave photonic signal processing. The frequency of the signal optical carrier is fs. There are two cases depending on the relative position between the signal optical carrier and the notch resonance of the MRR. For the Case 1 with the signal optical carrier fs located at the left side of the notch resonance of the MRR, when the sideband component (fs + f 2) is just aligned to the notch frequency of the MRR spectrum as shown in Fig. 1(b), the output RF response in the absence of the pump vanishes at f 2. Hence, a notch MPF with a central frequency of f 2 is obtained. As shown in Fig. 1(c), when the pump light is on, one would expect a red shift of the notch peak of the MRR owing to the combined nonlinear effects in the MRR, resulting in a notch MPF with a central frequency of f 1. Hence, the central frequency of the notch MPF can be tuned from f 2 to f 1 by increasing the pump light power. For the Case 2 with the signal optical carrier fs located at the right side of the notch resonance of the MRR, as shown in Figs. 1(d) and 1(e), similar tunable notch MPF with its central frequency changed from f 1 to f 2 is also achievable by increasing the pump light power. Both Case 1 and Case 2 shown in Figs. 1(b)-1(e) indicate possible all-optical tunable MPF using nonlinear effects of MRR.

 figure: Fig. 1

Fig. 1 (a) Schematic illustration of the proposed all-optical tuning process of the MPF. (b)-(e) Operation principle of the all-optical tuning process of the MPF. (b)(d) Pump off. (c)(e) Pump on. Case 1: the frequency of the signal optical carrier (fs) is located at the left side of the notch resonance frequency of the MRR. Case 2: the frequency of the signal optical carrier (fs) is located at the right side of the notch resonance frequency of the MRR. The inset of (a) shows typical transmission spectrum of the MRR with relative positions of microwave modulated signal and pump light.

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3. Theory

3.1 Pump induced refractive index change

The nonlinear system in silicon waveguide devices can be summarized as follows [9]: when light is launched into a silicon waveguide, it can be absorbed through TPA process and results in associated loss of light power and a change in the refractive index proportional to the light intensity (Kerr effect). Through the TPA process, free carriers are excited which lead to additional absorption (free carrier absorption, FCA) and an associated index change (free carrier dispersion, FCD). FCA and FCD are collectively known as free carrier effect (FCE). By the law of energy conservation, the optical energy absorbed in TPA process and the Joule loss caused by free carriers are eventually converted to thermal energy, resulting in heating of the structure and giving rise to a thermal refractive index change through TOE. Here we use nonlinear coupled mode theory (NCMT) to comprehensively describe the numerous nonlinear effects in SOI MRR and accurately calculate the pump induced refractive index change of the waveguide [16]. NCMT is a semi-analytical dynamic model for nonlinear resonator systems. It takes into account both linear loss (intrinsic loss and coupling between the resonator and bus waveguide) and nonlinear loss in a microresonator system. The nonlinear resonator frequency shift in this model is viewed as a perturbation to the linear resonant mode. The following set of differential equations shows the basic form of NCMT model for SOI MRR,

ddta=j(ωr+δωNL(U))aa(1/τin+1/τi+1/τNL(U))jμisiδωNL(U)=δωKerr(U)+δωTOETPA(U)+δωTOEFCA(U)+δωFCD(U)1/τNL(U)=1/τTPA(U)+1/τFCA(U)
where a is the complex electric amplitude of the field of the MRR, which is normalized so that U(t) = |a(t)|2 denotes the total energy stored in the MRR. s i is the complex electric amplitude of the field at the input port of the MRR, which is also normalized in a way that P i = |s i|2 represents the input power. τin is the intrinsic photon lifetime in the MRR. 1/τi is the photon lifetime reduction associated with the temporal coupling coefficients between MRR and the input waveguide, which is denoted by μi. 1/τi is related to μi through 1/τi = μi2/2. 1/τNL(U) is the nonlinear photon lifetime reduction, including the photon lifetime reduction associated TPA and FCA, which are denoted by 1/τTPA(U) and 1/τFCA(U), respectively. ωr is the resonant angular frequency of the MRR in linear region. δωNL(U) is the resonant frequency shift induced by the pump light. δωKerr(U), δωTOE-TPA(U), δωTOE-FCA(U), and δωFCD(U) represent resonant frequency shift induced by Kerr effect, TOE associated with TPA, TOE associated with FCA, and FCD, respectively. Detailed nonlinear effects formulas and physical parameters are given by [16].

With a steady state monochromatic input s i with time dependency ejωpt, where ωp is the angular frequency of the pump light. ddta can be written asddta=jωpa. Steady-state equation of the nonlinear SOI MRR system can be then derived from Eq. (1) [26],

[(ωpωrδωNL(U))2+(1/τin+1/τi+1/τNL(U))2]U=μi2Pi
The pump induced refractive index change is expressed by
Δn=nδωNL(U)/ωr
where n is the refractive index of waveguide in linear region.

We further evaluate the contribution of each nonlinear effect. The linear resonant wavelength of the SOI MRR is set to be 1591.786 nm. The input pump light wavelength is aligned to the linear resonant wavelength. τi and τin is fitted from the experiment. Figure 2 shows the simulated refractive index changes associated with different nonlinear effects in SOI MRR. It is obvious that TOE is the dominated nonlinear effect throughout the power range. The contribution of FCD is about one order of magnitude lower than TOE. In addition, the refractive index change associated with Kerr effect can be ignored.

 figure: Fig. 2

Fig. 2 Calculated refractive index changes associated with different nonlinearities in SOI MRR.

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The resonant wavelength shift at another resonant mode (near the signal light wavelength in the experiment) induced by the pump light is also examined. Since the resonant frequency shift is directly related to the wavelength shift, the resonant wavelength shift Δλ of the MRR is determined by Δλ = -λδωNL(U)/ωr . Figure 3 shows the calculated resonant wavelength shift as a function of pump light power. The calculated results agree well with the experiment data.

 figure: Fig. 3

Fig. 3 Resonant wavelength shift as a function of pump light power.

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FCA and TPA processes in SOI MRR may also introduce extra loss, which may cause the degradation of the Q-factor of the cavity. Considering the nonlinear loss, the total Q-factor of a MRR can be written by Qtotal = (Qin −1 + Q i −1 + QNL −1)−1, where Qin and Q i are the intrinsic Q-factor of the cavity in linear region and the Q-factor associated with the coupling between the ring and bus waveguide. Qin and Qi are related to τin and τi through Qin = ωrτin/2 and Qi = ωrτi/2. Here we use QNL to describe intrinsic Q-factor reduction caused by nonlinear loss. QNL can be expressed by QNL = (QTPA −1 + QFCA −1)−1, where QTPA = ωrτTPA/2 and QFCA = ωrτFCA/2 denote the Q-factor associated with TPA and FCA, respectively. A lower QTPA or QTPA means stronger TPA or FCA induced Q-factor degradation. We analyze the Q-factor as a function of input optical power, considering the nonlinear loss associated with TPA and FCA. The calculated results are shown in Fig. 4 . It can be seen that the nonlinear loss caused by FCA and TPA is much smaller than the radiation loss over all the power range. Even with an input power level of about 10 dBm, the total Q-factor reduction is below 1%. In addition, the extinction ratio (ER) variation can also be ignored according to theoretical calculations.

 figure: Fig. 4

Fig. 4 Q-factor variation as a function of pump light power.

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3.2 MPF response

We use a phase modulator (PM) and a tunable bandpass filter (TBF) to generate OSSB signal. The optical field at the output of the PM can be written as

A(t)=ejωctejβsin(ωRFt)
where ωc and ωRF are the angular frequency of the input optical carrier and applied electrical drive signal, respectively. β = πVRF/Vπ is the modulation index. Vπ is the half-wave voltage of the PM. And VRF is the amplitude of the microwave signal. Based on the Jacobi–Anger expansions, and considering small-signal modulation, Eq. (4) can be expanded to be
A(t)=J0(β)ejωct+J1(β)ej(ωc+ωRF)tJ1(β)ej(ωcωRF)t
where Jn is the nth order Bessel function of the first kind. Note that the two sidebands are out of phase. When this phase modulated signal is directly sent to PD, there is no RF signal at the PD output because the beating between the anti-phase sidebands and the carrier exactly cancel. When this signal passes through a TBF, the lower/upper sideband is suppressed in case 1/case 2. Considering case 2, the resulting output optical filed becomes
A(t)=J0(β)ejωctJ1(β)ej(ωcωRF)t
When this signal is further launched to a SOI MRR, assuming the field transmission of the MRR at ω is TMRR (ω), without considering pump induced resonant shift, the signal can be further described as
A(t)=J0(β)ejωctTMRR(ωc)J1(β)ej(ωcωRF)tTMRR(ωcωRF)
In Eq. (7), TMRR (ω) is expressed by
TMRR(ω)=rαejωcneffL1αrejωcneffL
where neff is the effective index of the waveguide. L is the circumference of the MRR. α is the field round-trip loss coefficient of the ring (zero loss: α = 1). r is the field transmission coefficient through the coupling region of MRR. From Eq. (7), we can see that if the lower sideband is aligned at the notch of the MRR, the response of the MPF is a null; otherwise, if the lower sideband is far away from the notch, the amplitude of the detected RF signal is a constant. Therefore, a notch MPF can be obtained.

Taking the nonlinear resonance shift into consideration, the output optical field can be finally written as

A(t)=J0(β)ejωctTNLMRR(ωc)J1(β)ej(ωcωRF)tTNLMRR(ωcωRF)=J0(β)ejωctTMRR(ωc+δωNL)J1(β)ej(ωcωRF)tTMRR(ωc+δωNLωRF)
where TNLMRR (ω) is the field transmission of the MRR at ω when the pump is on. TNLMRR (ω) is related to TMRR (ω) through TNLMRR (ω) = TMRR (ω + δω).

We further calculate the RF responses with different pump light power levels to show the all-optical tuning process of the MPF. The parameters used to calculate MRR field transmission are extracted from the measured MRR transmission spectrum. The MPF responses under different pump powers in Case 1 and Case 2 are shown in Figs. 5(a) and 5(b) , respectively. As shown in Fig. 5(a), the central frequency of the MPF decreases from 15.5 GHz to 9.5 GHz as the pump power increases in Case 1. The central frequency of the MPF increases from 9.5 GHz to 15.5 GHz as the pump power increases in Case 2, as shown in Fig. 5(b).

 figure: Fig. 5

Fig. 5 Calculated microwave responses under different pump power levels in (a) Case 1 and (b) Case 2, respectively. Case 1: the central frequency of the MPF decreases as the pump power increases. Case 2: the central frequency of the MPF increases as the pump power increases.

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4. Fabrication and characterization of the silicon microring resonator

To experimentally demonstrate the all-optical tuning of the MPF, we fabricate the MRR on a commercial SOI wafer with a 340-nm-thick silicon slab on the top of a 2-μm silica buffer layer. The device pattern is transferred to photoresist by E-beam lithography (Vistec EBPG5000 + ES). The upper silicon layer is etched downward for 220 nm to form a ridge waveguide through induced coupled plasma (ICP) etching (Oxford Instruments Plasmalab System100). Figure 6(a) shows the scanning electron microscope (SEM) image of the fabricated device. The waveguide width of both bus waveguide and MRR is about 500 nm. The radius of the MRR is 10 μm. The coupling gap between the bus waveguide and MRR is about 400 nm. Figure 6(b) shows the measured transmission spectrum of the MRR. The pump light wavelength is centered at the right resonant wavelength λp = 1591.786 nm, while the signal light wavelength (optical carrier) is located around the left resonant wavelength λ0 = 1581.606 nm. Figure 6(c) shows the measured zoom in details of the left resonant wavelength with pump light off and on. The resonant wavelength red shifts to 1581.634 nm with a pump light power around 0.8 dBm. We use a vertical coupling system to couple the light in and out of the SOI chip. The pump light power here is the collected power in the SOI waveguide taking into account the coupling loss of the grating coupler.

 figure: Fig. 6

Fig. 6 (a) SEM image and (b) measured transmission spectrum of the fabricated silicon microring resonator. (c) Zoom-in of the measured transmission spectrum with pump light off and on (resonance red shift with pump on).

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5. Experimental setup

Figure 7 depicts the experimental setup for the proposed optically-controlled tunable MPF. A tunable laser diode (TLD) emits a continuous wave (CW) light. An electrical amplifier (EA) is used to amplify the RF signal from a vector network analyzer (VNA). The CW light is modulated by a phase modulator (PM) to produce an optical double sideband signal. A tunable optical filter is used to eliminate one of the first order sideband to obtain an OSSB signal. In the forward pumping configuration, the signal light together with the pump light is coupled through a 3-dB coupler and fed into the MRR by a vertical coupling system, as shown in Fig. 7(a). In the backward pumping configuration, the coupler is placed at the output port, as shown in Fig. 7(b). After passing through the MRR device, the optical signal is converted to an electrical signal by a photodetector (PD) and analyzed by the VNA. By comparing in detail the forward and backward pumping configurations, it is found that an optical filter is required at the output port of the MRR to separate the optical signal from the pump in the forward pumping scheme, while such optical filter can be removed in the backward pumping scheme.

 figure: Fig. 7

Fig. 7 Experimental setup for optically-controlled tunable MPF with (a) forward and (b) backward pumping configurations. Solid lines: optical path. Dash lines: electrical path. TLD: tunable laser diode, PM: phase modulator, EDFA: erbium-doped fiber amplifier, VOA: variable optical attenuator, PC: polarization controller, PD: photodetector, EA: electrical amplifier, VNA: vector network analyzer.

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6. Experiment results and discussions

The measured microwave responses of MPF under different pump power levels are shown in Fig. 8 . The pump light wavelength is fixed at 1591.786 nm. In the Case 1, the optical carrier wavelength is 1581.730 nm in both forward and backward pumping configurations. As shown in Figs. 8(a) and 8(b), the central frequency of the notch MPF is tuned from 15.64 to 8.79 GHz in the forward pumping configuration while from 15.60 to 10.04 GHz in the backward pumping configuration by adjusting the pump light power from −27.2 to 4.8 dBm. In the Case 2, the optical carrier wavelength is 1581.564 nm in the forward pumping configuration and 1581.534 nm in the backward pumping configurations. As shown in Figs. 8(c) and 8(d), the central frequency of the notch MPF is tuned from 5.27 to 12.47 GHz in the forward pumping configuration and from 8.84 to 15.04 GHz in the backward pumping configuration as increasing the pump light power from −27.2 to 4.8 dBm.

 figure: Fig. 8

Fig. 8 Measured microwave responses of all-optical tunable MPF under different pump power levels with (a) Case 1/Forward pump, (b) Case 1/Backward pump, (c) Case 2/ Forward pump, and (d) Case 2/ Backward pump, respectively.

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The degradation of the notch filter rejection ratio in Fig. 8 might comes from the non-ideal filtering characteristics of the TBF (i.e. incomplete suppression to the unwanted sideband) and resultant residual unwanted sideband. Figure 9 shows the simulated MPF response considering the effects of the TBF, taking Case 2 as an example. The theoretical results in Fig. 9 agree well with the experimental results in Fig. 8(d). By employing a more effective optical single sideband modulation scheme, the performance of the MPF is expected be further improved.

 figure: Fig. 9

Fig. 9 Calculated microwave responses considering the imperfect filtering of TBF.

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The all-optical tuning of the MPF might also facilitate interesting applications such as optically-controlled microwave switching, optically-controlled microwave amplitude modulation, optically-controlled microwave frequency modulation, and even optically-controlled multi-level microwave modulation. For the Case 1 with backward pumping configuration corresponding to Fig. 8(b), we focus on two frequencies of 13.55 GHz and 15.55 GHz and analyze their power variation as changing the pump light power. Figures 10(a) and 10(b) show measured output electrical spectra with a 13.55 GHz RF input under two pump light powers. When the pump light power is switched from −27.2 dBm to 0.8 dBm, the output RF power is changed from −53.59 dBm to −69.90 dBm (i.e. switch off) with a switching ratio of 16.31 dB. Figures 10(c) and 10(d) depict measured output electrical spectra with a 15.55 GHz RF input under two pump light powers. The output RF power is switched from −75.49 dBm to −58.46 dBm (i.e. switch on) with a switching ratio of 17.03 dB as increasing the pump light power from −27.2 dBm to 0.8 dBm. The obtained results shown in Figs. 10(a)-10(d) indicate that optically-controlled microwave switching off/on operations are available for 13.55 GHz and 15.55 GHz RF signals. Additionally, microwave frequency modulation between 13.55 GHz and 15.55 GHz (i.e. frequency-shift keying) might be also available by employing data-carrying pump light with its power time varied between −27.2 dBm and 0.8 dBm. In addition to optically-controlled microwave photonic applications, the idea of using light to control light could be also applied to wide interesting all-optical signal processing applications (e.g. all-optical switching, all-optical modulation, all-optical logic gates).

 figure: Fig. 10

Fig. 10 Measured output electrical spectra corresponding to Fig. 8(b) with (a)(b) 13.55 GHz and (c)(d) 15.55 GHz RF input under two pump light powers of (a)(c) −27.2 dBm and (b)(d) 0.8 dBm.

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Remarkably, for the proposed optically-controlled flexible microwave photonic applications such as tunable microwave photonic filter, microwave switch, and microwave modulation, the operation speed is in principle dependent on the modulation speed of the pump light. However, the nonlinear effects in the silicon MRR could also impact on the operation speed of microwave photonic applications.

  • 1)When using a low-speed pump light (~Mbit/s), the TOE with typical thermal dissipation time of ~1 μs in an SOI waveguide [14] can respond to the time-varying pump light. Moreover, the TOE is the dominant nonlinear effect (Fig. 2) in an SOI MRR causing significant nonlinear resonant wavelength shift under a relatively low power of pump light. Consequently, the operation speed of microwave photonic applications is mainly affected by the TOE. For example, the typical switching time of microwave switch is expected to be ~1 μs.
  • 2)When increasing the modulation speed of pump light (~Mbit/s to ~100Mbit/s), the TOE might not respond to the pump light. Considering that the typical free carrier recombination time is about 0.5 ns [27], it is believed that the FCD effect could respond to the time-varying pump light. As a result, the operation speed of microwave photonic applications is limited by the FCD effect. Note that FCD-induced nonlinear resonant wavelength shift is much smaller than TOE (Fig. 2). To improve the operation performance, one might employ silicon photonic crystal nanocavity with greatly reduced mode volume and enhanced nonlinearity [28–30 ] to achieve a large FCD-induced nonlinear resonant wavelength shift.
  • 3)When using a relatively high-speed pump light (~Gbit/s), the bandwidth of the time-varying pump light could be beyond the ability of both TOE and carriers to respond. The FCD related carrier lifetime might be considerably reduced using several mechanisms, including the use of ion implantation [31] and reverse bias p-i-n diode [32, 33 ]. The smallest reported experimental value of carrier lifetime is 12.2 ps [33], which indicates the potential fast operation speed of microwave photonic applications up to ~Gbit/s.

7. Conclusion

In summary, we have reported an all-optical tunable MPF based on a nonlinear silicon MRR. The working mechanism is based on the optical nonlinear effects in silicon MRR. The dominant nonlinear effect in the experiment is TOE in MRR. Both forward and backward pumping configurations are demonstrated in two cases. For the Case 1 with the optical carrier located at the left side of the notch resonance of the MRR, the central frequency of the notch MPF is tuned from 15.64 to 8.79 GHz in the forward pumping configuration while from 15.60 to 10.04 GHz in the backward pumping configuration by adjusting the pump light power from −27.2 to 4.8 dBm. For the Case 2 with the optical carrier located at the right side of the notch resonance of the MRR, the central frequency of the notch MPF is tuned from 5.27 to 12.47 GHz in the forward pumping configuration and from 8.84 to 15.04 GHz in the backward pumping configuration as increasing the pump light power from −27.2 to 4.8 dBm. Additionally, the proposed all-optical tuning mechanism might also find potential applications in microwave switching and microwave modulation. With future improvement, more interesting flexible integrated microwave photonic signal processing applications might be developed by employing compact silicon photonic devices and exploiting light to control light.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under grant 61222502, the Program for New Century Excellent Talents in University (NCET-11-0182), the Wuhan Science and Technology Plan Project under grant 2014070404010201, the Fundamental Research Funds for the Central Universities (HUST) under grants 2012YQ008 and 2013ZZGH003, and the seed project of Wuhan National Laboratory for Optoelectronics (WNLO). The authors would like to thank Han Zhang, Chao Li, Chengcheng Gui, Qi Yang and Shuhui Li for their valuable technical supports and helpful discussions.

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15. 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]  

16. J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013). [CrossRef]  

17. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef]   [PubMed]  

18. J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013). [CrossRef]  

19. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009). [CrossRef]  

20. J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol. 24(7), 2500–2509 (2006). [CrossRef]  

21. W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011). [CrossRef]  

22. L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013). [CrossRef]  

23. D. Zhang, X. Feng, X. Li, K. Cui, F. Liu, and Y. Huang, “Tunable and Reconfigurable Bandstop Microwave Photonic Filter Based on Integrated Microrings and Mach–Zehnder Interferometer,” J. Lightwave Technol. 31(23), 3668–3675 (2013). [CrossRef]  

24. J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010). [CrossRef]  

25. Y. Long, H. Zhang, C. Li, C. Gui, Q. Yang, and J. Wang, “Ultracompact optically-controlled tunable microwave photonic filter based on a nonlinear silicon microring resonator,” in Asia Communications and Photonics Conference (Optical Society of America, 2014), paper AF1B.7. [CrossRef]  

26. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]  

27. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). [CrossRef]   [PubMed]  

28. 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), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-13-3-801&id=82543. [CrossRef]   [PubMed]  

29. T. Uesugi, B.-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14(1), 377–386 (2006), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-1-377&id=86921. [CrossRef]   [PubMed]  

30. K. Nozaki, E. Kuramochi, A. Shinya, and M. Notomi, “25-channel all-optical gate switches realized by integrating silicon photonic crystal nanocavities,” Opt. Express 22(12), 14263–14274 (2014), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-12-14263&id=289558. [CrossRef]   [PubMed]  

31. M. Waldow, T. Plötzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express 16(11), 7693–7702 (2008), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-11-7693&id=159698. [CrossRef]   [PubMed]  

32. S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, “Ultrafast all-optical modulation on a silicon chip,” Opt. Lett. 30(21), 2891–2893 (2005). [CrossRef]   [PubMed]  

33. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18(4), 3582–3591 (2010), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-4-3582&id=195454. [PubMed]  

References

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  1. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23(1), 401–412 (2005).
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  3. B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
    [Crossref]
  4. B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
    [Crossref]
  5. R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006).
    [Crossref]
  6. B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
    [Crossref]
  7. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
    [Crossref]
  8. Y. Long and J. Wang, “Optically-controlled extinction ratio and Q-factor tunable silicon microring resonators based on optical forces,” Sci . Rep. 4, 5409 (2014).
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  9. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, “Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures,” Opt. Express 13(23), 9623–9628 (2005), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-13-23-9623&id=86249 .
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  10. S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express 20(7), 7454–7468 (2012), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-7-7454&id=230426 .
    [Crossref] [PubMed]
  11. V. R. Almeida, C. A. Barrios, R. R. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, and A. L. Gaeta, “All-optical switching on a silicon chip,” Opt. Lett. 29(24), 2867–2869 (2004).
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  12. Y. Ding, C. Peucheret, M. Pu, B. Zsigri, J. Seoane, L. Liu, J. Xu, H. Ou, X. Zhang, and D. Huang, “Multi-channel WDM RZ-to-NRZ format conversion at 50 Gbit/s based on single silicon microring resonator,” Opt. Express 18(20), 21121–21130 (2010), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-20-21121&id=205874 .
    [Crossref] [PubMed]
  13. J. Wang, C. Gui, C. Li, Q. Yang, J. Sun, and X. Zhang, “On-Chip Demultiplexing of Polarization and Wavelength Multiplexed OFDM/OQAM 64/128-QAM Signals using Silicon 2D Grating Coupler and Microring Resonators,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A. 48.
    [Crossref]
  14. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004).
    [Crossref] [PubMed]
  15. 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]
  16. J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
    [Crossref]
  17. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
    [Crossref] [PubMed]
  18. J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave photonic signal processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
    [Crossref]
  19. J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  20. J. Mora, B. Ortega, A. Díez, J. L. Cruz, M. V. Andrés, J. Capmany, and D. Pastor, “Photonic microwave tunable single-bandpass filter based on a Mach-Zehnder interferometer,” J. Lightwave Technol. 24(7), 2500–2509 (2006).
    [Crossref]
  21. W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011).
    [Crossref]
  22. L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013).
    [Crossref]
  23. D. Zhang, X. Feng, X. Li, K. Cui, F. Liu, and Y. Huang, “Tunable and Reconfigurable Bandstop Microwave Photonic Filter Based on Integrated Microrings and Mach–Zehnder Interferometer,” J. Lightwave Technol. 31(23), 3668–3675 (2013).
    [Crossref]
  24. J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
    [Crossref]
  25. Y. Long, H. Zhang, C. Li, C. Gui, Q. Yang, and J. Wang, “Ultracompact optically-controlled tunable microwave photonic filter based on a nonlinear silicon microring resonator,” in Asia Communications and Photonics Conference (Optical Society of America, 2014), paper AF1B.7.
    [Crossref]
  26. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
    [Crossref]
  27. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006).
    [Crossref] [PubMed]
  28. 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), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-13-3-801&id=82543 .
    [Crossref] [PubMed]
  29. T. Uesugi, B.-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14(1), 377–386 (2006), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-1-377&id=86921 .
    [Crossref] [PubMed]
  30. K. Nozaki, E. Kuramochi, A. Shinya, and M. Notomi, “25-channel all-optical gate switches realized by integrating silicon photonic crystal nanocavities,” Opt. Express 22(12), 14263–14274 (2014), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-12-14263&id=289558 .
    [Crossref] [PubMed]
  31. M. Waldow, T. Plötzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express 16(11), 7693–7702 (2008), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-11-7693&id=159698 .
    [Crossref] [PubMed]
  32. S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, “Ultrafast all-optical modulation on a silicon chip,” Opt. Lett. 30(21), 2891–2893 (2005).
    [Crossref] [PubMed]
  33. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18(4), 3582–3591 (2010), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-4-3582&id=195454 .
    [PubMed]

2014 (2)

2013 (4)

2012 (3)

S. Chen, L. Zhang, Y. Fei, and T. Cao, “Bistability and self-pulsation phenomena in silicon microring resonators based on nonlinear optical effects,” Opt. Express 20(7), 7454–7468 (2012), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-7-7454&id=230426 .
[Crossref] [PubMed]

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

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]

2011 (1)

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

2010 (3)

2009 (1)

2008 (2)

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

M. Waldow, T. Plötzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express 16(11), 7693–7702 (2008), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-11-7693&id=159698 .
[Crossref] [PubMed]

2006 (5)

2005 (5)

2004 (2)

2000 (1)

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

1998 (1)

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Almeida, V. R.

Andrés, M. V.

Asano, T.

Baets, R.

Barclay, P.

Barrios, C. A.

Beckx, S.

Bergman, K.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Biberman, A.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. Van Campenhout, P. Bienstman, and D. Van Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23(1), 401–412 (2005).
[Crossref]

Bogaerts, W.

Bolten, J.

Buchwald, W. R.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006).
[Crossref]

Cao, T.

Capmany, J.

Chen, S.

Chen, X.

L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013).
[Crossref]

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Chou, C.-Y.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Chu, S.

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Cruz, J. L.

Cui, K.

Dadap, J. I.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Díez, A.

Ding, Y.

Dumon, P.

Emelett, S. J.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006).
[Crossref]

Fan, L.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Fei, Y.

Feng, X.

Foresi, J.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Först, M.

Foster, M. A.

Gaeta, A. L.

Galán, J. V.

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Gao, L.

L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013).
[Crossref]

Gasulla, I.

Gottheil, M.

Green, W. M.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Greene, W.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Haus, H.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Hsieh, I.-W.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Huang, D.

Huang, Y.

Ippen, E.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Kimerling, L.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Kokubun, Y.

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

Kumar Selvaraja, S.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Kuramochi, E.

Kurz, H.

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Lee, B. G.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Levy, J. S.

Li, X.

Lipson, M.

Little, B.

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

Little, B. E.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[Crossref]

Liu, F.

Liu, L.

Liu, X.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Lloret, J.

Long, Y.

Y. Long and J. Wang, “Optically-controlled extinction ratio and Q-factor tunable silicon microring resonators based on optical forces,” Sci . Rep. 4, 5409 (2014).
[Crossref] [PubMed]

Luyssaert, B.

Marti, J.

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Minasian, R. A.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

Mora, J.

Morthier, G.

Niu, B.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Noda, S.

Notomi, M.

Nozaki, K.

Ortega, B.

Osgood, R. M.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Ou, H.

Ouzounov, D. G.

Painter, O.

Palací, J.

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Pan, W.

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

Panepucci, R. R.

Pastor, D.

Peucheret, C.

Plötzing, T.

Poitras, C. B.

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Preble, S. F.

Priem, G.

Pu, M.

Qi, M.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Salem, R.

Sales, S.

Sancho, J.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

Schmidt, B. S.

Sekaric, L.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Seoane, J.

Shen, H.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Shinya, A.

Song, B.-S.

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R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
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Soref, R. A.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006).
[Crossref]

Srinivasan, K.

Steinmeyer, G.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Taillaert, D.

Thoen, E.

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

Turner-Foster, A. C.

Uesugi, T.

Van Campenhout, J.

Van Thourhout, D.

Van Vaerenbergh, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
[Crossref]

Varghese, L. T.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Vidal, B.

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Villanueva, G. E.

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

Vlasov, Y. A.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Wahlbrink, T.

Waldow, M.

Wang, J.

Y. Long and J. Wang, “Optically-controlled extinction ratio and Q-factor tunable silicon microring resonators based on optical forces,” Sci . Rep. 4, 5409 (2014).
[Crossref] [PubMed]

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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]

Weiner, A. M.

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]

Wiaux, V.

Xia, F.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
[Crossref]

Xu, J.

Xu, Q.

Xuan, Y.

J. Wang, L. Fan, L. T. Varghese, H. Shen, Y. Xuan, B. Niu, and M. Qi, “A theoretical model for an optical diode built with nonlinear silicon microrings,” J. Lightwave Technol. 31(2), 313–321 (2013).
[Crossref]

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).
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Yao, J.

L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013).
[Crossref]

J. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
[Crossref]

Zhang, D.

Zhang, L.

Zhang, W.

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

Zhang, X.

Zsigri, B.

IEEE J. Sel. Top. Quantum Electron. (1)

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1678–1687 (2006).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

L. Gao, X. Chen, and J. Yao, “Tunable microwave photonic filter with a narrow and flat-top passband,” IEEE Microw. Wirel. Compon. Lett. 23(7), 362–364 (2013).
[Crossref]

IEEE Photon. Technol. Lett. (5)

W. Zhang and R. A. Minasian, “Widely tunable single-passband microwave photonic filter based on stimulated Brillouin scattering,” IEEE Photon. Technol. Lett. 23(23), 1775–1777 (2011).
[Crossref]

J. Palací, G. E. Villanueva, J. V. Galán, J. Marti, and B. Vidal, “Single bandpass photonic microwave filter based on a notch ring resonator,” IEEE Photon. Technol. Lett. 22(17), 1276–1278 (2010).
[Crossref]

B. Little, S. Chu, W. Pan, and Y. Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12(3), 323–325 (2000).
[Crossref]

B. E. Little, J. Foresi, G. Steinmeyer, E. Thoen, S. Chu, H. Haus, E. Ippen, L. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998).
[Crossref]

B. G. Lee, X. Chen, A. Biberman, X. Liu, I.-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photon. Technol. Lett. 20(6), 398–400 (2008).
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J. Lightwave Technol. (7)

J. Opt. A, Pure Appl. Opt. (1)

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840–848 (2006).
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W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photonics Rev. 6(1), 47–73 (2012).
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Nature (1)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[Crossref] [PubMed]

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Y. Ding, C. Peucheret, M. Pu, B. Zsigri, J. Seoane, L. Liu, J. Xu, H. Ou, X. Zhang, and D. Huang, “Multi-channel WDM RZ-to-NRZ format conversion at 50 Gbit/s based on single silicon microring resonator,” Opt. Express 18(20), 21121–21130 (2010), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-18-20-21121&id=205874 .
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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), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-13-3-801&id=82543 .
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T. Uesugi, B.-S. Song, T. Asano, and S. Noda, “Investigation of optical nonlinearities in an ultra-high-Q Si nanocavity in a two-dimensional photonic crystal slab,” Opt. Express 14(1), 377–386 (2006), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-14-1-377&id=86921 .
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M. Waldow, T. Plötzing, M. Gottheil, M. Först, J. Bolten, T. Wahlbrink, and H. Kurz, “25ps all-optical switching in oxygen implanted silicon-on-insulator microring resonator,” Opt. Express 16(11), 7693–7702 (2008), https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-11-7693&id=159698 .
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Opt. Lett. (4)

Sci . Rep. (1)

Y. Long and J. Wang, “Optically-controlled extinction ratio and Q-factor tunable silicon microring resonators based on optical forces,” Sci . Rep. 4, 5409 (2014).
[Crossref] [PubMed]

Science (1)

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]

Other (2)

J. Wang, C. Gui, C. Li, Q. Yang, J. Sun, and X. Zhang, “On-Chip Demultiplexing of Polarization and Wavelength Multiplexed OFDM/OQAM 64/128-QAM Signals using Silicon 2D Grating Coupler and Microring Resonators,” in Optical Fiber Communication Conference (Optical Society of America, 2014), paper Th2A. 48.
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Y. Long, H. Zhang, C. Li, C. Gui, Q. Yang, and J. Wang, “Ultracompact optically-controlled tunable microwave photonic filter based on a nonlinear silicon microring resonator,” in Asia Communications and Photonics Conference (Optical Society of America, 2014), paper AF1B.7.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 (a) Schematic illustration of the proposed all-optical tuning process of the MPF. (b)-(e) Operation principle of the all-optical tuning process of the MPF. (b)(d) Pump off. (c)(e) Pump on. Case 1: the frequency of the signal optical carrier (fs ) is located at the left side of the notch resonance frequency of the MRR. Case 2: the frequency of the signal optical carrier (fs ) is located at the right side of the notch resonance frequency of the MRR. The inset of (a) shows typical transmission spectrum of the MRR with relative positions of microwave modulated signal and pump light.
Fig. 2
Fig. 2 Calculated refractive index changes associated with different nonlinearities in SOI MRR.
Fig. 3
Fig. 3 Resonant wavelength shift as a function of pump light power.
Fig. 4
Fig. 4 Q-factor variation as a function of pump light power.
Fig. 5
Fig. 5 Calculated microwave responses under different pump power levels in (a) Case 1 and (b) Case 2, respectively. Case 1: the central frequency of the MPF decreases as the pump power increases. Case 2: the central frequency of the MPF increases as the pump power increases.
Fig. 6
Fig. 6 (a) SEM image and (b) measured transmission spectrum of the fabricated silicon microring resonator. (c) Zoom-in of the measured transmission spectrum with pump light off and on (resonance red shift with pump on).
Fig. 7
Fig. 7 Experimental setup for optically-controlled tunable MPF with (a) forward and (b) backward pumping configurations. Solid lines: optical path. Dash lines: electrical path. TLD: tunable laser diode, PM: phase modulator, EDFA: erbium-doped fiber amplifier, VOA: variable optical attenuator, PC: polarization controller, PD: photodetector, EA: electrical amplifier, VNA: vector network analyzer.
Fig. 8
Fig. 8 Measured microwave responses of all-optical tunable MPF under different pump power levels with (a) Case 1/Forward pump, (b) Case 1/Backward pump, (c) Case 2/ Forward pump, and (d) Case 2/ Backward pump, respectively.
Fig. 9
Fig. 9 Calculated microwave responses considering the imperfect filtering of TBF.
Fig. 10
Fig. 10 Measured output electrical spectra corresponding to Fig. 8(b) with (a)(b) 13.55 GHz and (c)(d) 15.55 GHz RF input under two pump light powers of (a)(c) −27.2 dBm and (b)(d) 0.8 dBm.

Equations (9)

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d d t a = j ( ω r + δ ω N L ( U ) ) a a ( 1 / τ i n + 1 / τ i + 1 / τ N L ( U ) ) j μ i s i δ ω N L ( U ) = δ ω K e r r ( U ) + δ ω T O E T P A ( U ) + δ ω T O E F C A ( U ) + δ ω F C D ( U ) 1 / τ N L ( U ) = 1 / τ T P A ( U ) + 1 / τ F C A ( U )
[ ( ω p ω r δ ω N L ( U ) ) 2 + ( 1 / τ i n + 1 / τ i + 1 / τ N L ( U ) ) 2 ] U = μ i 2 P i
Δ n = n δ ω N L ( U ) / ω r
A ( t ) = e j ω c t e j β sin ( ω R F t )
A ( t ) = J 0 ( β ) e j ω c t + J 1 ( β ) e j ( ω c + ω R F ) t J 1 ( β ) e j ( ω c ω R F ) t
A ( t ) = J 0 ( β ) e j ω c t J 1 ( β ) e j ( ω c ω R F ) t
A ( t ) = J 0 ( β ) e j ω c t T M R R ( ω c ) J 1 ( β ) e j ( ω c ω R F ) t T M R R ( ω c ω R F )
T M R R ( ω ) = r α e j ω c n e f f L 1 α r e j ω c n e f f L
A ( t ) = J 0 ( β ) e j ω c t T N L M R R ( ω c ) J 1 ( β ) e j ( ω c ω R F ) t T N L M R R ( ω c ω R F ) = J 0 ( β ) e j ω c t T M R R ( ω c + δ ω N L ) J 1 ( β ) e j ( ω c ω R F ) t T M R R ( ω c + δ ω N L ω R F )

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