Abstract
Alloptical analogtodigital converters based on the thirdorder nonlinear effects in silicon waveguide are a promising candidate to overcome the limitation of electronic devices and are suitable for photonic integration. In this paper, a 2bit optical spectral quantization scheme for onchip alloptical analogtodigital conversion is proposed. The proposed scheme is realized by filtering the broadened and split spectrum induced by the selfphase modulation effect in a silicon horizontal slot waveguide filled with siliconnanocrystal. Nonlinear coefficient as high as 8708 W^{−1}/m is obtained because of the tight mode confinement of the horizontal slot waveguide and the high nonlinear refractive index of the siliconnanocrystal, which provides the enhanced nonlinear interaction and accordingly low power threshold. The results show that a required input peak power level less than 0.4 W can be achieved, along with the 1.98bit effectivenumberofbit and Gray code output. The proposed scheme can find important applications in onchip alloptical digital signal processing systems.
Introduction
Analogtodigital converters (ADC) as the key frontend digital processing device have extensive applications in ultrawideband systems, such as advanced radar systems, realtime signal monitoring, and ultrahighspeed optical communication systems. Limited by the inherent aperture jitter and the comparator ambiguity, electrical ADCs cannot meet the demands of ultrawideband applications^{1,2,3,4}. Benefited from the ultrafast response time and passive operation, alloptical ADCs (AOADC) based on thirdorder nonlinear effects are a promising candidate to overcome the limitation of electronic devices^{5,6}. In last decade, many quantization schemes have been proposed using the nonlinear effects in highlynonlinear fibers (HNLFs) and photonic crystal fibers (PCFs). For example, the phase quantization schemes use crossphase modulation (XPM)^{5,7,8,9,10}, the spectral quantization schemes use selfphase modulation (SPM)^{11}, XPM^{12}, intrapulse stimulated Raman scattering (ISRS)^{6,13,14,15,16,17,18}, and supercontinuum (SC)^{19}. However, because of the weak nonlinear interaction in these SiO_{2}based materials, the requirements on the fiber length, pulse width, and pulse power are typically very high. These fiberbased AOADCs therefore cannot meet the demands of miniaturization and photonic integration, which are the inevitable development trend of ADCs. Silicon photonic technology is suitable for photonic integration owing to their compatibility with complementary metal oxide semiconductor (CMOS) process. The optical waves can be confined to submicron region by the high refractive index of silicon using the silicononinsulator (SOI) technology^{20}. Some SOIbased nonlinear signal processing schemes have been proposed^{21,22,23,24,25,26,27} but, to the best of our knowledge, there is still no reports on AOADC. In a SOIbased AOADC, the nonlinear interaction is expected to occur only in the specific waveguide but not in other interconnected devices, which requires that the waveguide should provide stronger nonlinear interaction than conventional silicon strip waveguides. The slot waveguide structure allows for very strong mode confinement in a low index nonlinear material sandwiched between two silicon wires^{28,29}. In particular, the horizontal slot structure has the advantages of much thinner slot thickness and minimized scattering loss when compared to the vertical structure^{30}. For the embedded nonlinear materials with high nonlinear refractive index, e.g. the organic material PTS (n_{2} = 2.2 × 10^{−16} m^{2}W^{−1} at 1600 nm), the nonlinear coefficient can reach to more than 2 × 10^{4} W^{−1}/m^{28,30}. However, the organic materials require nonCMOS processes and strict temperature limitation, which render them not suitable for massmanufacturing^{31,32}. The material siliconnanocrystal (Sinc) embedded in silica (SiO_{2}) has a high n_{2} (about one or two orders of magnitude higher than that of silicon) at telecom wavelengths^{33} and is completely CMOS foundry compatible using standard chemical vapor deposition (CVD) methods^{31}. Thus Sinc/SiO_{2} is quite suitable for the proposed horizontal slot waveguide with strong nonlinear interaction.
In this paper, we propose a CMOScompatible 2bit optical spectral quantization (OSQ) scheme for the first time by filtering the broadened and split spectrum induced by SPM in a silicon horizontal slot waveguide filled with Sinc/SiO_{2}. The dimensions of the slot waveguide are optimized to achieve the smallest effective mode area, thus making strong nonlinear interaction possible. The quantization performances are studied numerically. In addition, high quantization resolution of the proposed OSQ scheme is demonstrated.
Results
Principle of operation
The schematic diagram of the proposed OSQ scheme is shown in Fig. 1. The sampled pulses at different peak powers, illustrated in inset (a), are delivered into a specially designed silicon horizontal slot waveguide. When the input peak power increases, the output spectra of the pulses first broaden and then split from the center frequency owing to the SPM effects inside the waveguide. The degree of broadening and splitting is directly proportional to the peak powers of the pulses, which can be used for the mapping between the variations of the peak powers and the spectrum parameters. As shown in inset (b), the spectrum of the pulse with higher power broadens and splits while the one with lower power remains unchanged. An arrayed waveguide grating (AWG) with two proper filtering windows at wavelengths λ_{1} and λ_{2} is used to slice the spectrum of the output pulses of the waveguide. The power transfer functions of the two filtering channels are shown in inset (c). When the input peak power increases, the output power of the λ_{1}channel first increases owing to the spectral broadening and then decreases owing to the spectral splitting. In contrast, the output power of the λ_{2}channel only increases with the input peak power. This is because the wavelength λ_{2}, when compared to λ_{1}, is farther away from the center wavelength λ_{0} so that the filtering range of λ_{2}channel does not contain the spectral splitting region. The output signals of the two filtering channels are then detected by the photodiodes and binary decisions are made by comparators. The available encoding results are shown at the bottom of inset (c), which exhibits a 2bit Gray code output.
Design of the horizontal slot waveguide
The crosssection of the Sinc/SiO_{2}based horizontal slot waveguide is shown in Fig. 2. The lower cladding is a silicon dioxide layer and the upper cladding is air. The slot is filled with Sinc/SiO_{2}, which is sandwiched between two silicon strip wires and has a thickness s. The upper and lower silicon wires are formed by the amorphous silicon (aSi) and the crystalline silicon (cSi), respectively, and have the same height h. The width of the waveguide is w. Since the electric field discontinuity occurs at the horizontal interface for such a waveguide, the quasi transversemagnetic (TM) polarization should be used for the analysis. The deepUV lithography method, together with the plasmaenhanced chemical vapor deposition (PECVD) or lowpressure chemical vapor deposition (LPCVD), can be used for the fabrication of such horizontal slot waveguides. The detailed fabrication procedure can be found in^{34}.
In order to obtain the strongest nonlinear interaction inside the waveguide, the minimum effective area (A_{eff}) is investigated with different geometrical parameters. The effective areas at the wavelength of 1.55 μm are calculated for different combinations of w and h when the slot thickness s is increased from 5 to 30 nm. For each given slot thickness, the minimum achievable effective area is found and shown in Fig. 3(a). From Fig. 3(a), the effective area does not vary monotonically with the slot thickness. The minimum value of A_{eff} ≈ 0.0268 μm^{2} is obtained around s = 13 nm. Fig. 3(b) shows the optimum geometrical parameters which correspond to the minimum effective area as a function of the slot thickness. We note that the calculated results agree with the results in^{30}.
Based on the results obtained above, the slot thickness of s = 15 nm is chosen, which corresponds to A_{eff} = 0.027 μm^{2} at 1.55 μm. Accordingly, the width (w) and height (h) are set at 220 and 210 nm respectively. In order to obtain the accurate dispersion properties of the designed horizontal slot waveguide, the material dispersions of aSi^{35}, cSi^{36}, Sinc/SiO_{2}^{37,38}, and SiO_{2}^{39} are considered. The calculated transverse profiles of the electric field for quasiTM polarization at three different wavelengths are shown in Fig. 4. We observed that the electric field is tightly confined inside the slot even for wavelength as long as 1.8 μm. The effective refractive index n_{eff} is calculated and shown in Fig. 5(a). Using the results of n_{eff}, we further determined the group index, second and thirdorder dispersion parameters, as shown in Figs. 5(a) and 5(b). The designed waveguide has two zero dispersion wavelengths around 1.38 and 1.51 μm, which results in about 130 nm anomalous dispersion regime. The second (β_{2}) and thirdorder (β_{3}) dispersion coefficients at 1.55 μm are 1.418 ps^{2}/m and −0.064 ps^{3}/m, respectively.
Quantization performance
The dynamic of an optical pulse inside a silicon waveguide is governed by the modified generalized nonlinear Schrödinger equation (GNLSE), which includes the effects of twophoto absorption (TPA), freecarrier absorption (FCA), and freecarrier dispersion (FCD). The GNLSE is given by where A(z, t) is the slowly varying envelope of the electric field along the propagation direction z, β_{m} is the mth dispersion coefficient, ω_{0} is the center angular frequency of the optical field, γ_{0} and γ_{TPA} are the real and imaginary part of the complex nonlinear coefficient, α_{l} is the linear loss of the waveguide, α_{FCA} and n_{FCD} are the coefficients of FCAinduced absorption and FCDinduced index change, which are given by α_{FCA} = 1.45 × 10^{−21}(λ_{0}/λ_{ref})^{2}N_{c} and n_{FCD} = −5.3 × 10^{−27}(λ_{0}/λ_{ref})^{2}N_{c}^{20}. Here, λ_{0} is the input center wavelength, λ_{ref} = 1.55 μm is the reference wavelength, and N_{c}(z, t) is the freecarrier density defined by where β_{TPA} is the TPA parameter, h is the Planck constant, f_{0} = ω_{0}/2π is the center frequency, τ is the effective carrier lifetime that estimated to be about 100 μs^{33}.
The Sinc/SiO_{2} used in the simulation is assumed to be with 8% silicon excess, deposited by PECVD method and annealed at 800°C. The Sinc/SiO2 has the achievable α_{l} = 4.6 dB/cm^{40}, n_{2} = 5.8 × 10^{−4} cm^{2}/GW, and β_{TPA} = 5 cm/GW^{33}. Using the available A_{eff} of 0.027 μm^{2}, the parameter values of γ_{0} = 2πn_{2}/λ_{0}A_{eff} = 8708 W^{−1}/m and γ_{TPA} = β_{TPA}/2A_{eff} = 926 W^{−1}/m can be obtained.
The nonlinear response function R(t) in Eq. (1) is related to the Raman contribution and is given by R(t) = (1 − f_{R})δ(t) + f_{R}h_{R}(t). The Raman response function h_{R}(t) can be deduced from the Raman response spectrum. The Fourier transform H_{R}(Ω) of h_{R}(t) with a Lorentzian shape is defined by^{20} where Ω_{R}/2π = 15.6 THz is the Raman shift, Γ_{R}/π ≈ 105 GHz is the bandwidth of Raman gain spectrum at room temperature. The parameter f_{R} is the fractional contribution of the nuclei to the total nonlinear polarization, and f_{R} = 0.043 is obtained by using the normalization condition of .
The splitstep Fourier method is used to numerically solve Eqs. (1) and (2). A hyperbolic secant pulse source centered at 1.55 μm with 1.2 ps pulse width is utilized. The length of the designed waveguide is 8 mm. The peak power of the pulse is adjusted to simulate the sampled pulses. Fig. 6(a) shows the spectral dynamic of the output pulse of the waveguide when the peak power is increased from 0 to 0.4 W. The spectrum begins to broaden around 0.1 W and splits at 0.2 W. In order to investigate the influences of TPA and FCA, the spectral dynamic without the FCA or TPA effects are shown in Figs. 6(b) and 6(c). We observe that FCA has little effect on the spectral broadening, but TPA significantly limits the spectral broadening. The effect of FCD is negligible because of its low magnitude of 10^{−4}. The spectral profile shows two splitting gaps when TPA is neglected (Fig. 6(c)), but is limited to only one splitting gap with TPA (Figs. 6(a) and 6(b)). Fig. 6(d) shows the 2D spectral profiles with peak power of 0.4 W at different nonlinear conditions. The FCA effect affects the spectral broadening slightly but reduces the depth of the spectral splitting by about 15 dB. Owing to the moderate peak power used in our simulation, the FCAinduced spectrum asymmetry is not observed, but the TPAinduced decrease in spectral broadening is about 4 nm.
To simultaneously show the temporal and spectral dynamic, the crosscorrelation frequencyresolved optical gating (XFROG) traces of the output pulses with peak powers of 0.1, 0.2, 0.3, and 0.4 W are illustrated in Fig. 7. Unlike the obvious broadening and splitting of the spectrum, the temporal profiles of the pulses are only slightly broadened even at peak power of 0.4 W. This is because the dispersion length L_{D} = T_{0}^{2}/β_{2} = 0.33 m is about 40 times larger than the waveguide length of 8 mm, which means chromatic dispersion basically does not affect the pulse dynamic. By contrast, the nonlinear length is L_{NL} = 1/γ_{0}P_{0} = 0.3 mm, thus the pulse evolution dynamic is dominated by the nonlinear interaction.
The output signal of the waveguide is delivered into an AWG with two filtering channels, Channel1 and Channel2. The frequency transfer functions of the two filtering channels are both 6order Chebyshev type, which have the stopband extinction of 34 dB and the passband amplitude ripple of 0.5 dB. In order to find the optimum filtering windows, we investigate the variation of the power transfer function of the filter by increasing the center wavelength from 1550.8 to 1553.2 nm with 0.2 nm filtering bandwidth. As shown in Fig. 8(a), the peak of the transfer function shifts towards higher input power when the center wavelength increases. It should be noted that two requirements must be met to obtain an uniform quantization step length. For Channel1, since the maximum input peak power is 0.4 W, the filtered powers at the input peak powers of 0.1 and 0.3 W must be identical to ensure that the lengths of the first and fourth quantization steps are both 0.1. For Channel2, the filtered power at 0.2 W input should be lower than the case of the 0.4 W input. Thus, the filtered power at 0.2 W input can be used as the decision threshold to obtain two quantization steps with uniform length of 0.2. When both requirements are satisfied, four quantization steps with same length of 0.1 can be obtained. The requirement of the Channel2 can be satisfied with the center wavelength larger than 1551.6 nm. For Channel1, the optimum center wavelength is found to be 1551.2 nm. It is desired that the thresholds of the two channels are the same in order to reduce the complexity of the ADC. This can be achieved by setting the center wavelength of the Channel2 at 1552.9 nm.
We then investigate the influence of the filtering bandwidth by varying the bandwidth from 0.2 to 1.2 nm at the two optimum center wavelengths of 1551.2 and 1552.9 nm, as shown in Fig. 8(b). It is evident that the power transfer function of the Channel1 changes slightly with the bandwidth, and the transfer function of the Channel2 remains almost unchanged.
Fig. 9(a) shows the power transfer functions of the two quantization channels at the optimum center wavelengths with the filtering bandwidth of 0.2 nm. The transmitted powers of the Channel1 at 0.1 and 0.3 W input peak power are both 3.66 μW, which is only 0.02 μW larger than the required decision power of the Channel2. This small difference allows the same decision threshold to be used with only slight degradation of quantization resolution. Fig. 9(b) shows the quantization transfer function of the proposed 2bit quantizer for the decision threshold of 3.66 μW. The simulated transfer function is found to be very similar to that of the ideal ADC. The inset of Fig. 9(b) shows the tiny difference between the two transfer functions. Assuming the analog input is a sinusoidal wave, the effectivenumberofbit (ENOB) can be calculated by^{1,9} where SNR is the signaltonoise ratio, RMS_power and RMS_noise are the rootmeansquare of the power and the quantization errors, P_{FS} is the fullscale of power range, Δ = P_{FS}/2^{B} is the power of the least significant bit (LSB), B is the ideal number of bit, and Δ_i is the nonlinear error of the ith quantization step. The SNR is calculated to be 13.68 dB, thus the ENOB is 1.98bit.
The AWG can also be realized by silicon devices, e.g. silicon ring resonators^{3,41}, which will ensure CMOScompatibility of the proposed quantizer. The required bandwidth of 0.2 nm and center wavelengths of 1551.2 and 1552.9 nm are achievable by appropriate design of the ring resonators^{3}. It is expected that optical pulses with less than 0.4 W peak power do not have nonlinear interaction inside the AWG so that the quantization performances would not be degraded. The silicon ring resonators based filters are typically formed by silicon strip nanowires. For such silicon strip waveguide, the achievable minimum effective area is found to be about 0.05 μm^{2} with both the length and height of the strip waveguide within the range of 200 ~ 400 nm^{28}. Using the same parameters as the above except that n_{2} = 6 × 10^{−5} cm^{2}/GW, β_{TPA} = 0.5 cm/GW^{42}, and A_{eff} = 0.05 μm^{2}, the spectral dynamic of the optical pulses with different input peak powers are simulated and shown in Fig. 10. It is evident the spectrum remains almost unchanged when the peak power is less than 0.4 W and only slightly broadened even at 4 W. This is because the nonlinear coefficient of such strip nanowires is only 486 W^{−1}/m, which is about 18 times less than that of the proposed slot waveguide. The nonlinear interaction length at 0.4 W peak power for such silicon nanowires is found to be 5.1 mm, which is very close to the waveguide length 8 mm, thus the nonlinear accumulation is small. The nonlinear effects in such silicon nanowires are negligible when the input peak power is less than 0.4 W. It is therefore feasible to combine the proposed quantizer with other silicon nanowires based functional devices.
Discussion
Owing to the inherent limitation of spectral broadening, it is difficult for the proposed OSQ to realize more than 2bit quantization. However, as discussed above, since the proposed OSQ requires low power threshold, it is possible to combine the proposed OSQ with other existing quantization schemes that use linear modulation, e.g. phaseshifted optical quantization (PSOQ). The combination can be realized via a cascaded optical quantization (COQ) structure, as shown in Fig. 11.
The COQ structure is composed of two cascaded quantization stages. The firststage quantization, which can be realized by the unbalanced MachZehnder modulator (UMZM) proposed in^{43,44}, implements the Nchannel PSOQ and generates N different sinusoidal power transfer functions corresponding to the different channels. Such a PSOQ scheme can provide a resolution of log_{2}(2N)bit^{44,45}. The inset (a) shows the power transfer functions and the coding results of the firststage quantization in the case of two channels (N = 2). A 2bit resolution is obtained. Instead of direct detection and binary decision, the proposed 2bit OSQ is used to further quantize the output power of the firststage quantization module. With the secondstage quantization, the number of quantization level increases from 4 to 12 without adding additional channels, as shown in the inset (b). The total resolution is then enhanced to log_{2}[2N(2^{2} − 1)] = 3.59bit, which is 1.59bit higher than individual PSOQ and OSQ. In the case of N = 16, a competitive 6.59bit resolution can be achieved. Besides resolution enhancement, such a COQADC can also provide high analog bandwidth because of the alloptical process in both quantization stages. Since the UMZM can also be fabricated by silicon photonics technology^{46}, the entire components of the COQADC is CMOScompatible.
Coupling loss is a key factor to be considered for practical implementation of siliconbased functional devices. There are two types of coupling loss in the proposed device: fibertowaveguide and waveguidetowaveguide. The former only occurs between the transmitter part and the functional waveguide. High coupling efficiency of 93% (0.3 dB loss) at 1550 nm has been numerically obtained using the inverted taper approach and lensed fibers^{47}. To ensure the required power threshold of 0.4 W is attained, the peak power of the transmitter part should reach to at least 0.43 W. Considering a pulse width of 1.2 ps at pulse repetition rate of 10 GHz, the average power is only 5.2 mW. Such a power level is moderate and achievable. Another main fibertowaveguide coupling approach is based on grating couplers. A coupling efficiency of 20% (7 dB loss) at 1550 nm has been achieved experimentally in^{48}, which requires that the peak power of the transmitter part reaches 2 W, i.e. an average power of 24 mW. We note that pulsed sources at peak power of several Watts and pulse width of several picoseconds are achievable even at high repetition rates of tens of GHz^{49}. For higher peak powers, the pulsed sources can be integrated on a separate chip^{3} together with semiconductor optical amplifiers. The discussion above aims at the coupling between fiber and silicon slot waveguide. For the case of fiber to silicon strip waveguide, the low coupling loss of 0.36 dB is obtained experimentally^{50}. The impact of fibertowaveguide coupling loss is therefore negligible.
In contrast, the waveguidetowaveguide coupling loss that exists in optical interconnections will determine the feasibility of the proposed device. This is because the proposed 2bit quantizer may be joint indirectly with the transmitter part through interconnecting with other linear functional devices which can be realized by silicon strip waveguides. If the interconnection loss is very large, the high power inside the linear devices can induce undesired nonlinear effects, which could severely degrade the quantization performance. However, we note that ultralow loss of <0.1 dB striptoslot mode converter at the telecom waveband has been reported in experiment^{51}, which is based on the logarithmically tapering approach. Considering the propagation loss of a strip waveguide to be typically around 2 dB/cm^{51}, the peak power inside a strip waveguide several millimeters long can be less than 0.6 W. It is unlikely that such a moderate peak power will lead to any undesired nonlinear effects in the linear devices from the results shown in Fig. 10. Taking the proposed COQADC as an example, an UMZM could interconnect between the 2bit quantizer and the transmitter. A similar siliconbased structure reported in^{46} can be used to realize such an UMZM, which exhibits a total loss of 6 dB where the coupling loss is 4 dB and the propagation loss is 2 dB. Combined with the 0.1 dB striptoslot coupling loss, the peak power inside the UMZM and at the transmitter part are expected to be about 0.6 and 1.6 W, respectively, which correspond to average power of only 7.2 and 19.2 mW for repetition rate at 10 GHz and pulse width at 1.2 ps. Both of the power requirements can easily be satisfied. We therefore conclude that using the existing efficient waveguide coupling techniques and the moderate peak power threshold we obtained, the problem of coupling loss is unlikely to affect negatively the feasibility of the proposed scheme.
In summary, we propose a CMOScompatible 2bit OSQ scheme by filtering the broadened and split spectrum induced by SPM in a silicon horizontal slot waveguide filled with Sinc/SiO_{2}. By optimizing the dimensions of the slot waveguide, the nonlinear coefficient of 8708 W^{−1}/m can be obtained, which gives a strong Kerr nonlinear interaction and low power threshold. Using simulation, we show that the proposed 2bit quantization can be realized with an ENOB of 1.98bit and the required peak power is less than 0.4 W. Thus the requirement on the optical sources is reduced, and the interconnection between the proposed nonlinear OSQ and other siliconbased functional devices is possible. We also propose to combine the 2bit OSQ with a conventional PSOQ via COQ structure in order to achieve higher quantization resolution. We show that resolution up to 6bit can be obtained with such a COQADC. Because of the moderate power threshold and achievable ultralow loss striptoslot mode converters, the proposed COQADC is feasible and can find important applications in the onchip alloptical digital signal processing systems.
Methods
Search of the minimum effective area and the optimum geometrical parameters
For a high refractive index contrast waveguide, the A_{eff} is defined by^{28,30} where Z_{0} = 377 Ω is the free space wave impedance, n_{NL} the refractive index of the nonlinear material filled in the slot, →E (x, y) the vector electric, and →H (x, y) the vector magnetic field profiles of the TMlike mode. The upper integral covers the whole crosssection D_{total}, whereas the lower integral considers only within the slot region D_{NL}.
A fullvector finite element method (FEM) based mode solver is used for the eigenmode calculations. The simulation domain is 4 × 4 μm^{2} with a 0.4 μm thick perfectly matched layer. An adaptive mesh refinement is used to ensure the accuracy of the calculations. In order to determine the optimum geometrical parameters for the minimum effective area, we adopt the following procedure. Firstly, a cross grid of width (w) and height (h) is defined with w, h ∈ {200 nm, 205 nm,···, 230 nm}. The region is chosen with reference to the results in^{30}. Secondly, when the slot thickness is increased from 5 to 30 nm, the effective areas are calculated at wavelength 1.55 μm for different combinations of w and h. Finally, at each given slot thickness, if the statistics of the calculated effective areas has an inflection point which corresponds to the minimum value, the value is recorded as the minimum effective area and the corresponding w and h are the optimum geometrical parameters. If such an inflection point is not found, we extend the boundary values of w and h, and repeat the above procedure until the minimum effective area is found.
Model of the XFROG trace
The XFROG trace obtained in our simulation is based on the differencefrequency generation XFROG algorithm, which is given by^{52} Where E_{sig}(t) is the electric field of the measured signal, E_{ref}*(t − τ) is the conjugate electric field of the reference signal delayed by time τ. The reference signal is characterized by the initial input signal.
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Acknowledgements
This work was supported in part by the National Basic Research Program under Grant 2010CB327601, the National Natural Science Foundation of China under Grants 61475023, 61307109, and 61475131, the Natural Science Foundation of Beijing under Grant 4152037, the National HighTechnology Research and Development Program of China under Grant 2013AA031501, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20120005120021, the Fundamental Research Funds for the Central Universities under Grant 2013RC1202, the Program for New Century Excellent Talents in University under Grant NECT110596, the Beijing Nova Program under Grant 2011066, the Fund of State Key Laboratory of Information Photonics and Optical Communications (BUPT) P. R. China, the Hong Kong Scholars Program 2013 under Grant PolyU GYZ45, and the Science Foundation Ireland (SFI) under Grants SFI/12/ISCA/2496 and SFI/13/ISCA/2845.
Author information
Affiliations
State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, P.O. Box72 (BUPT), Beijing, China
 Zhe Kang
 , Jinhui Yuan
 , Xianting Zhang
 , Qiang Wu
 , Xinzhu Sang
 & Chongxiu Yu
Photonics Research Centre, Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
 Jinhui Yuan
 , Feng Li
 & P. K. A. Wai
Photonics Research Centre, Department of Electrical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
 Hwa Yaw Tam
Photonics Research Centre, Dublin Institute of Technology, Kevin Street, Dublin, Ireland
 Qiang Wu
 & Gerald Farrell
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Contributions
K.Z. and Y.J.H. designed the research. Z.X.T. and L.F. prepared figures 1–3, W.Q. carried out the mode analysis and prepared figure 4. K.Z., Y.J.H. and S.X.Z. carried out the simulation of nonlinear pulse propagation under the supervision of F.G., C.X.Y. and W.P.K.A. T.H.Y. prepared figure 7. K.Z., Y.J.H. and W.P.K.A. wrote the main manuscript. All authors contributed to revision of the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Zhe Kang or Jinhui Yuan.
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