The effect of initial frequency chirp is investigated numerically to obtain efficient supercontinuum radiation in photonic crystal fibers (PCFs) with two closely spaced zero-dispersion wavelengths. The positive chirps,instead of zero or negative chirps, are recommended because self phase modulation and four-wave mixing can be facilitated by employing positive chirps. In contrast with the complicated and irregular spectrum generated by negative-chirped pulse, the spectrums generated by positive-chirped pulses are wider and much more regular. Moreover, the saturated length of the PCF,corresponding to the maximal spectrum width, can be shortened greatly and the efficiency of frequency conversion is also improved because of initial positive chirps. Nearly all the energy between the zero-dispersion wavelengths can be transferred to the normal dispersion region from the region within the two zero-dispersion wavelengths provided that the initial positive chirp is large enough.
© 2007 Optical Society of America
Much attention has been paid to Supercontinuum generation (SCG) in optical fibers more than two decades. Photonic crystal fiber (PCF) is a promising method to generate the supercontinuum (SC) spectrum because of its high nonlinearities and flexible dispersion profiles compared with standard single mode fiber [1–5]. An interesting example of the possibilities offered by this flexible dispersion design is the fabrication of PCFs with two zero-dispersion wavelengths (ZDWs) . It is well known that the gain bandwidth for four-wave mixing (FWM) is widest in the vicinity of the ZDW due to phase matching conditions , and the solitons in the vicinity of the ZDW can amplify dispersive waves in the normal dispersion region . Therefore, PCFs with two ZDWs could afford advantages for efficient SCG. There have been some experimental and theoretical studies about SCG in PCFs with two ZDWs. Literature  indicated that a stable and intense SCG is demonstrated in PCF with two ZDWs through self-phase modulation (SPM) and phase-matched four-wave mixing (FWM). In literature , 15 femtosecond pulses with milliwatt average power were launched into a PCF with two ZDWs tapered to normal dispersion, and two distinct smooth spectral parts are generated. The bandwidth, the center wavelength, and the power of the two parts depend on the taper parameters and the pump power. The dependence of the pump wavelength on SCG in PCFs with two ZDWs is investigated experimentally and numerically in literature . However, these investigations only concern the pulses without chirp. To the best of our knowledge, the influence of frequency chirp on SCG in PCFs with two ZDWs has not been studied in detail. In the case of single ZDW, it has been turned out that suitable chirp parameter is a key factor to promote the bandwidth and average power of the SCG [11, 12].Since the initial frequency chirp is an important parameter of the pulse, it is necessary to take the initial frequency chirp into accounts adequately in the PCF with two ZDWs.
This paper aims to investigate the influence of initial linear chirp on SCG in PCFs with two ZDWs closely spaced. The propagation equation and the parameters of PCFs with two ZDWs are proposed in Section 2. The influence of positive chirp and negative chirp is compared numerically in Section 3. It shows that the positive chirp prefers to the zero or negative chirp because the former gives birth to more regular and flat SCG spectrum. Moreover, by choosing appropriate positive chirp, the efficiency of FWM can be enhanced sufficiently to transfer the energy from anomalous dispersion region to normal dispersion region thoroughly. Optimal value of the chirp and length of the PCF are also discussed to maximize the output bandwidth. In section 4, the influence of the peak power, type of the chirps and span of the two ZDWs are discussed in brief. The conclusion is drawn in section 5.
2. Propagation equation
The numerical model of the PCSs with two ZDWs is the well-known generalized nonlinear Schrödinger equation with which the evolution of femtosecond pulses in PCFs can be described accurately even when its bandwidth is of the same magnitude as its central frequency .
where u(ξ,τ) is the normalized amplitude, ω0 is the central frequency, L (n) D=τn P/βn is the nth-order dispersion length, βn(n≳2) is the nth-order dispersion coefficient. L D2=∣L D (2)∣,ξ=z/L D2 is the normalized distance, τ=(t - z/vg)/τp is the normalized retarded time for the pulse traveling with group velocity vg, and N 2=L D2/L NL=L NL is the order of the soliton without chirp, L NL=(γP 0)-1 = is the nonlinear length, γ=2πn 2/λA eff,A eff is the effective cross-sectional area of the guided mode, and n 2 ≈ 2.2×10-20m2 W-1 is the nonlinear refractive index of fused-silica glass. The right-hand side of Eq. (1) models chromatic dispersion, self-phase modulation (SPM), self-steepening, and stimulated Raman scattering. The response function R(T)=(1-f R)δ(T)+f R h R(T) includes both instantaneous electronic and delayed Raman contributions.f R is the fraction of the Raman contribution to the nonlinear polarization and is assumed to be 0.18. h R(T) is the Raman response function of silica fiber and can be approximated by the expression
where T 1=12.2fs and T 2=32fs.
The input pulses are assumed to be of the form
where u(0,τ)=A(0,τ)/√P 0 and A(0,τ) is the amplitude envelope. P 0 is the peak power,τp is the half width at 1/e maximum intensity of the input pulse width, and C is the initial linear frequency chirp.
The temporal and spectral width were defined as 
where Ωc=〈ω〉 are the centre of the pulse and spectrum, respectively. The angle bracket denotes the average over the propagation pulse and spectrum for time or frequency. The intensity modulation, which develops as the pulse energy increasing, is taken into account in the two expressions.
3. Numerical simulation
We consider a highly nonlinear triangular PCF with pitch Λ=1.0μm , hole diameter d=0.57μm , and core diameter 1.4μm . The dispersion profile of the sample PCF is shown in Fig. 1. The two ZDWs are located on 778nm and 950nm, respectively. The dispersion between the two ZDWs is anomalous and the dispersion outside the region is normal. Numerical Simulations are carried out by setting λ0 = 808nm, γ=100W-1km-1, β2 = -2.4739 ps2 km , β3 = 2.3863×10-2 ps3 km , β4 = 9.5116×10-5 ps4 km , β5 = -4.6886×10-7 ps5 km , β6 = 2.2151×10-9 ps6 km , β7 = -7.4373×10-12 ps7 km , P 0 = 5kW and τp = 60 ps. The propagation loss is neglected due to the very short fiber length.Figure 2 shows the evolution of spectrum width along with propagation distance for different chirps.
As shown in Fig. 2 (a), there are three stages in spectrum width for positive-chirped and zero-chirped pulse, similar to the case in literature . In the first stage, the initial positive chirp reinforces the positive chirp induced by SPM , and the spectrum width broadens symmetrically and gradually, which can be observed in Fig. 3 (a). In the second stage, more frequency components of the broadened spectrum fall into the normal dispersion region. Therefore, the seed wavelengths for phase-matched FWM are provided. Then, the spectrum broadens dramatically in a short distance. In the final stage, as most energy has been transferred to the normal dispersion region, the spectrum width begins to saturate and approaches its maximum. There is a saturated length of the PCF corresponding to the maximal spectrum width. The larger the frequency chirp, the wider the spectrum width and the shorter the saturated length of the PCF. As shown in Fig. 2 (b), the spectrum width of negative chirps is compressed at first because the initial negative chirp weakens the positive chirp induced by SPM . During the further propagation, the chirp induced by SPM exceeds the initial negative chirp and the spectrum starts to expand. Eventually, the saturated spectrums of the negative-chirped pulses are a little narrower than those of positive-chirped pulses. Moreover, their saturated lengths are much longer than those of positive-chirped pulses. Compared between Fig. 2 (a) and (b), it can be concluded that wider spectrums and much shorter saturated length of PCFs can be reached by employing positive chirps instead of negative chirps.
Figure 3 (a) and (b) illustrate the spectrums varying along with the distance for the case of C =10 and C = -10, respectively. In Fig. 3 (a), the initial positive chirp reacts with the nonlinear effect of SPM, the temporal width is compressed  and more frequency components exceed the boundary of the two ZDWs during the propagation. The phase-matching conditions for FWM begin to be satisfied and the energy is therefore transferred to the normal dispersion region gradually. Moreover, because the temporal width is compressed, the intensity of the pulse is high and the nonlinear effects are enhanced. Eventually, all the energy between the two ZDWs is depleted and there are two distinct parts located in visible region (less than ZDW 778nm) and near-infrared region (greater than ZDW 950nm), respectively. The output spectrum is comparatively regular.
While in Fig. 3 (b), the initial negative chirp partly suppresses the positive chirp induced by SPM . The spectrum width is compressed and the temporal width is expanded at first.The intensity of the pulse is reduced and the nonlinear effects are thus weakened. During the propagation, the spectrum broadens slowly and thus, less energy is extended to the normal dispersion region as the seed wavelengths of FWM. So the efficiency of phase-matched FWM is reduced. Finally, much energy is still reserved between the two ZDWs and the spectrum is not divided into two distinct parts as in Fig. 3 (a). Furthermore, because of perturbation of higher-order effects such as stimulated Raman scattering, self-steepening and higher-order dispersion, the spectrum structure is complicated and irregular. Especially, when the two ZDWs are designated closely, this phenomenon will be more obvious since higher-order dispersion effect is more susceptible in near-zero dispersion regions.
Figure 4 shows how the spectrums and temporal intensity vary for positive-chirped pulses at 15cm (0.1L D2) and negative-chirped pulses at 30cm (0.2L D2), respectively, where the two spectrums have saturated. Here we take the logarithm (base 10) of spectrum intensity for obvious observation of spectrum. From Fig. 4 (a) and (c), we can see that the spectrums of positive-chirped pulses are much more regular than those of zero or negative-chirped pulses,which is consistent with the Fig. 3. Especially, the spectrums are considerably flat in the visible region. So the spectrum can be tailored and the pulse can be compressed to be sub-femtosecond pulse efficiently by choosing positive chirps. As shown in Fig. 4 (b) and (d), for the negative-chirped pulses, the spectrums are difficult to be used in practice because they are complicated and irregular. The greater the absolute values of negative chirps, the more irregular the spectral shapes.
Moreover, another obvious phenomenon can also be found in Fig. 4. For negative-chirped pulses, there is still much energy left between the two ZDWs. However, for positive-chirped, the spectrum between two ZDWs depletes and the energy is almost thoroughly transferred to normal dispersion region provided that the chirp is large enough. The conversion efficiency, which is defined as the ratio of the energy in the normal dispersion region to the total input pulse energy, is proposed to evaluate the energy conversion. In Fig. 5, the conversion efficiencies for negative and positive chirps are plotted. Compared with the case of zero or negative chirps, the conversion efficiencies increase greatly in the case of minor positive chirps. For example, the efficiency is improved from 93.4% to 99.8% when the chirp increases from 0 to 8. As for the cases of larger positive chirps, the efficiencies are identical and can reach to 100% nearly. In contrast, the conversion efficiencies of negative-chirped pulses are less than those of positive-chirped pulses because some energy is still reserved in the anomalous region. For example, for chirped pulses of C=20 and C=-20, the conversion efficiencies are approximately 100% and 89%, respectively.
Numerical simulations under different pulse parameters such as the peak power P 0 , the center wavelength and the pulse width τp obtained similar results, with the spectrum width and conversion efficiencies depending on the simulation conditions. Figure 6 (a) and (b) illustrates the influence of P 0 on the spectrum width and on the conversion efficiency for different positive chirps, respectively. It can be found from Fig. 6 (a) that the spectrum width is sensitive to the fluctuation of peak power no matter how much the chirp value is. But the influence of positive chirps on spectrum width is not identical for different peak power. When P 0 is weak, the spectrum width is susceptive to the variation of the chirp. When P 0 is strong,the spectrum width is determined by peak power mainly and therefore the chirps have little influence on the spectrum width. So, in the case of small P 0 , we suggest increasing the value of positive chirp to expand the spectrum. From Fig. 6 (b), it can be observed the conversion efficiencies are sensitive to the fluctuation of peak power under the condition of small chirps.For larger chirps, the fluctuation of peak power has little influence on conversion efficiency which is kept constant approximately. Moreover, when P 0 is strong, the chirps have little influence on the conversion efficiency. When P 0 is weak, the conversion efficiency is susceptive to minor chirps. So, an appropriate positive chirp should be calculated to enhance the conversion efficiency. For example, when P 2 = 1kW, C=10 is sufficient to enhance the conversion efficiency.
Although the pulses are all chirped in temporal domain in the above simulations, which means the temporal shape can be kept, we also considered the pulses chirped in spectral domain obtained by adding quadratic phase to an unchirped input pulse. In this case the input temporal width is broadened and the peak power is reduced to keep the pulse energy conserved . So, compared with the pulses chirped in temporal domain, those chirped in spectral domain have narrower input spectrum width and lower peak power, which is disadvantageous in achieving broad SC spectrum and high conversion efficiency.
We have investigated the effects of frequency chirps on SCG in a PCF with two ZDWs closely spaced at 778 nm and 950 nm. Owing to the narrow span of the two ZDWs, SPM and phase-matched FWM are decisive in SCG among the nonlinear effects . The positive chirps facilitate SPM and FWM as discussed above. However, when the span of the two ZDWs is wide, typically above 1400nm, the nonlinear process of soliton self-frequency shift plays a crucial role in SCG . Therefore, the effect of frequency chirps on SCG will be another scenario in this case and needs to be investigated in detail. This will be discussed in future works.
The effect of initial frequency chirp on SCG in PCFs with two ZDWs is investigated numerically. Positive initial chirp results in regular SC spectrum while negative initial chirp gives birth to complicated and irregular spectrum. At the same time, by choosing appropriate positive chirp, nearly all the energy in the anomalous dispersion region between the two ZDWs can be transferred to the normal dispersion region thoroughly. While choosing the initial negative chirp, there is still much energy left between the two ZDWs. The conversion efficiency is enhanced greatly by employing the positive chirps. Moreover, the saturated length of the PCF corresponding to the maximal output spectrum width could be reduced by choosing initial positive chirp while the initial negative chirp would extend the length. Therefore, to obtain efficient SC spectrums, the positive chirps are preferable to the zero or negative chirps in PCFs with two ZDWs. This property has important potential applications in all optical wavelength conversion  and broadband parametric amplification .
We acknowledge financial support from Hi-Tech Research and Development Program of China (Grants No: 2006AA01Z246), National Natural Science Foundation (60602004), China Postdoctoral Science Foundation, Graduate Student innovation Foundation of BUPT and Program for New Century Excellent Talents in University (NCET-05-0112). We would also like to thank Peter Falk (Technical University of Denmark) for providing simulation data about dispersion parameters.
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