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A novel method for measuring optical fiber’s nonlinear coefficient, based on phase mismatching four-wave mixing is proposed. Measurements for both high nonlinearity dispersion shifted fiber and low nonlinearity standard single mode fiber are demonstrated with simple setup. Chromatic dispersion is also measured with high precision simultaneously, and therefore its effect to the nonlinear coefficient measurement can be removed.
©2013 Optical Society of America
1. Introduction
For large capacity ultra-long haul transmission systems utilizing digital coherent technique, a major concern is its high sensitivity to nonlinear impairments caused by Kerr effects in optical fibers including self-phase modulation (SPM), cross-phase modulation (XPM) and four-wave-mixing (FWM). On the other hand, highly nonlinear fibers (HNLFs) have been widely applied to all-optical signal processing utilizing the efficient generation of nonlinearities [1]. Efficiency of the nonlinearity generation depends on the fiber nonlinear coefficient γ, which is defined as 2πn2/(λ·Aeff), where n2, λ, and Aeff are nonlinear refractive index, wavelength, and effective area, respectively. Therefore, accurate measurement of the γ is one of the important issues for designing transmission systems and HNLF-based devices.
A number of methods for measuring the nonlinear coefficient have been reported so far. Spectral broadening of short pulses due to SPM [2] or XPM [3] were employed to measure the nonlinear coefficient. However, critical assumption on the pulse shape and complex deconvolution calculation were required. Interferometric methods [4, 5] in which SPM- or XPM-induced phase shift were detected have relatively complicated setup, and can be unstable to environmental perturbations. CW-SPM method [6] has a simple setup, but its accuracy may be affected by the fiber chromatic dispersion. To neglect the effect of chromatic dispersion, the measured length would be limited to less than 500 m for standard single mode fiber (SSMF). The FWM idler power was also used to measure the γ for 12.5 km-long dispersion shifted fiber [7] or for 0.83 - 20 km-long reverse-dispersion fiber [8]. However, it might have been difficult to measure the γ for low nonlinearity fibers such as SSMF or shorter length fibers because of the low FWM efficiency.
In this paper, we propose a novel method to measure the γ by phase-mismatching four-wave-mixing (PM-FWM) using a simple setup. In this method, shifts of the frequencies satisfying the phase mismatching condition for FWM are used to measure the γ. The PM-FWM method can measure the chromatic dispersion with high precision simultaneously [9], and therefore its effect to the γ measurement can be removed. Using the proposed method, we demonstrate the γ measurements for highly nonlinear dispersion-shifted fibers (HNLDSFs) with the linear polarization state. In addition, the γ and chromatic dispersion of 1km-long SSMF having low nonlinearity and large chromatic dispersion are also measured with the same setup.
2. Principle of PM-FWM method
When pump and probe lights with their angular frequencies of ωpump and ωprobe, respectively, propagate together through a fiber, idler light is newly generated at a frequency of 2ωpump-ωprobe through the FWM process. For linearly co-polarized lights propagating in a fiber, the output power of the idler light, Pidler, can be described as below in the case of that the fiber loss is negligibly low [10],
where L, Ppump and Pprobe are the fiber length, launched pump and probe powers, respectively. Δβ is the phase mismatching parameter defined aswhere β2 represents the second-order dispersion at ωpump, and Δω = ωprobe – ωpump is a angular frequency difference between the pump and the probe.From Eqs. (1) and (2), we here propose two schemes for measuring γ based on the PM-FWM, depending on the β2 of the tested fiber. For a low β2 fiber, whose zero-dispersion wavelength is within or close to the measuring wavelengths, ωpump is scanned while keeping the Δω constant. On the other hand, for a high β2 fiber, ωpump and therefore β2 are kept constant while ωprobe is scanned to change Δω.
I. Low β2 fiber
For constant Δω in Eqs. (1) and (2), Pidler is a periodic function of ωpump as shown in Fig. 1(a) because β2 monotonically increase or decrease with ωpump. From Eq. (1), Pidler becomes maximum at ΔβL/2 = 0, and has minimal values at which Δβ satisfies
For the limit of Ppump to be zero, at a pump frequency of ωZ(N) providing the minimal Pidler, β2z(N) can be expressed as [9]In the case where Ppump > 0, the pump frequency indicating the minimal Pidler will shift to ωP(N) as shown by solid curve in Fig. 1(a). Here, β2 at ωP(N), β2P(N) can be approximated using the third order dispersion at ωZ(N), β3Z(N) asPSubstituting Eqs. (2) and (5) into Eq. (3), ωP(N) can be described asAs a corollary, ωP(N) becomes a linear function of Ppump, and the nonlinear coefficient γ can be calculated asTherefore, γ at ωP(N) can be determined from the slope of the ωP(N) versus Ppump graph. Here, β3Z(N) in Eq. (7) can be also obtained using Eq. (3). Although the condition of Ppump = 0 in Eq. (3) would be impossible for actual measurement, ωZ(N) can be estimated by calculating the y-intercept on the ωp(N) versus Ppump graph in Eq. (6). β2 spectrum can be also determined with β2z(N) for several N from Eq. (4), and β3Z(N) can be then calculated by a biquadratic approximation of the β2 spectrum. The definitions of β are summarized in Table 1.
Fig. 1 Idler output power as a function of (a) pump angular frequency with a constant frequency difference, and (b) squared angular frequency difference with a constant pump frequency.
Table 1. Definitions of β
2.1 High β2 fiber
When we measure high β2 fibers such as SSMF at wavelengths around 1550 nm, it is difficult to exactly know N in Eq. (4) because the measured frequencies are far away from the frequency at which ΔβL/2 = 0 and N becomes a large value. Therefore, β3Z(N) in Eq. (7) cannot be obtained. Here, keeping ωpump constant, only Δω and Ppump become the variable in Eq. (2). In this case, Pidler also be a periodic function of squared frequency difference (Δω)2 as shown in Fig. 1(b). From Eqs. (2) and (3), the conditions for Pidler to be minimum with N and N-1 are given by
where ΔωP(N) is the N-th frequency difference providing the minimal Pidler for Ppump>0. From two adjacent frequency differences of ΔωP(N) and ΔωP(N-1) in Eqs. (8) and (9), β2 at ωpump can be determined independently of N asSimilar to Eq. (4), a frequency difference Δωz(N) providing the minimal Pidler for the limit of Ppump to be zero is given byFrom Eqs. (8) and (11), the following equation is obtainedThis eauation shows that, [ΔωP(N)]2 also becomes a linear function of Ppump, and γ can be evaluated by the slope of the [ΔωP(N)]2 versus Ppump graph, aswhere β2 can be obtained from Eq. (10).3. Experimental setup and results
We conducted experiments to demonstrate the proposed PM-FWM method. The measurement setup is schematically shown in Fig. 2. CW tunable lasers were employed for pump and probe lights at wavelengths of around 1550 nm. The pump light was amplified by an EDFA to a sufficient power level for the measurement, and the generated ASE from high poser EDFA was removed with a following 4nm-bandwidth optical bandpass filter (OBPF). The polarization states of pump and probe lights were adjusted to be linear and parallel with each other by polarization controllers (PC), and we confirmed the co-polarized states of the pump and probe lights by a power meter (PM) to maximize the optical power after a polarization-maintained (PM)-3dB-optical coupler (OC) that had a function of eliminating one linear polarization. The pump and probe lights were then launched together into a tested fiber via the PM-3dB-OC. In this measurement setup, the generated idler output power was measured with an optical spectral analyzer (OSA). In order to evalute the nonlinear coefficient γ accurately, absolute power of respective lightwaves are required. We used an OSA to measure the power, which was carefully calibrated with an optical power meter. The input and output polarization states of the probe and pump lights were monitored with a polarization analyzer (PA) at the fiber output, and we confirmed that the states of polarization were very stable even after propagation through 1km-long HNLDSF and SSMF.
First, we have measured a 1km-long HNLDSF that has low β2. Respective frequencies of the pump and probe lights were scanned together with a constant difference of Δω = 7.8 × 1012 rad/s. Examples of the generated idler power Pidler against the pump frequency are shown in Fig. 3(a). The periodical minima are observed, and we confirmed that the ωP(N) were shifted with increase of the pump power from 1.6 to 26mW, as theoretically predicted in Fig. 1(a). Measured ωP(N) as a function of Ppump are shown in Figs. 3(b) and 3(c), satisfying the condition of Eq. (3) with (b) N = 1 and (c) N = 2, which correspond to the wavelength of about 1535 nm and 1539 nm, respectively. In accordance with Eq. (6), ωP(N) are linearly decreasing with the increasing of Ppump.
Fig. 3 Experimental results for HNLDSFs. (a): Examples of generated idler power Pidler against pump frequency with Δω = 7.8 × 1012 rad/s for 1km-long HNLDSF. (b)-(d): Pump angle frequency providing the minimal idler power against pump power with (b) N = 1 and (c) N = 2 for 1.0km-long HNLDSF, and (d) N = 1 for 0.14km-long HNLDSF.
As mentioned above, the β3 should be known to obtain the nonlinear coefficient γ. β2Z(N) can be calculated from Eq. (4), and ωZ(N) can be determined from the y-intercept of ωP(N)-Ppump graphs in Figs. 3(b) and 3(c), as described with Eq. (6). We drew a β2 spectrum by determining the β2Z(N) at several ωZ(N) related to respective N in Eq. (4), and β3 was calculated as + 0.034 ps3/km by the biquadratic approximation of the β2 spectra. Then, we obtained the slopes of ωP(N)-Ppump graph in Figs. 3(b) and 3(c) by means of least squares approximation. Finally, γ at wavelenths of 1535 nm and 1539 nm were calculated as 17.2 and 17.3/W/km, respectively. These correspond to a nonlinear index n2 of 3.6 × 10−20 m2/W for Aeff of 8.5μm2.
In order to know the effect of the length, we also evaluated a 0.14 km-long HNLDSF acquired from the same spool. The measured ωP(N) for N = −1 at wavelength of around 1558 nm is shown in Fig. 3(d), where Δω was fixed to 15.3 × 1012 rad/s. From Fig. 3(d), the measured γ was 19 /W/km, which is closely accorded with the one in the 1km-long HNLDSF.
Next, in order to verify the PM-FWM method can be applied to a low nonlinearity (less than one-tenth of that in HNLDSF) and high β2 (about-22 ps2/km or chromatic dispersion of about + 17 ps/nm/km) fiber, γ of a 1km-long SSMF was measured using Eqs. (10) and (13). The probe frequency was scanned while pump frequency is kept to be 1.2 × 1015 rad/s (wavelength of 1550 nm). Figure 4(a) shows the periodic change of the generated idler power Pidler against the squared frequency difference (Δω)2 for Ppump of 208 mW. β2 at ωpump was determined from Eq. (10) to be −21.4 ps2/km. Measured ΔωP(N) against Ppump is also shown in Fig. 4(b) for N = −1. We successfully determined the nonlinear coefficient γ and n2 as 1.3 W/km and 2.7 × 10−20 m2/W, respectively. In addition, the chromatic dispersion was also measured as 16.5 ps/nm/km at the wavelength of 1550 nm.
Fig. 4 Experimental results for 1km-long SSMF. (a) The generated idler power Pidler against squared frequency difference. (b) Squared frequency difference [ΔωP(N)]2 providing the minimal idler power against pump power Ppump with N = −1.
Finally, we compared the nonlinear coefficient γ and n2 measured with the PM-FWM method to the ones with CW-SPM method [6], in which the factor of 8/9 was considered in the evaluation for the random input polarization states. It is noted that γ of SSMF was measued with a 0.40 km-long fiber to neglect the effect of the chromatic dispersion on the CW-SPM method. The measurement results are summarized in Table 2, and we found that the PM-FWM method gives results well consistent with those by the CW-SPM method. The n2 measured with PM-FWM method are 5%-larger than that with CW-SPM method for HNLDSFs and 2%-smaller for SSMF. The chromatic dispersion and dipersion slope measured with PM-FWM method are also shown in Table 2, which agree well with ones with commercially available modulation phase shft (MPS) method [11]. The errors are less than 0.02 and 0.1 ps/nm/km in dispersion for HNLDSF and SSMF respectively, and less than 0.002 ps/nm2/km in disersion slope.
Table 2. Measured results of n2
4. Conclusion
We have proposed a novel fiber nonlinear coefficient γ measuring method employing PM-FWM. With simple setup, we confirmed that nonlinear coefficient γ can be determined for 0.1- to 1-km long HNLDSF and 1km-long low-nonlinearity SSMF in the straightforward way. The measured results agree well with the ones with CW-SPM method. Chromatic dispersion and dispersion slope can be simultaneously measured with high precision, and therefore its effect to the nonlinear coefficient γ measurement can be removed.
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