## Abstract

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 OSA

## 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 2πn_{2}/(λ·A_{eff}), where n_{2}, λ, and A_{eff} 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 ω_{pump} and ω_{probe}, respectively, propagate together through a fiber, idler light is newly generated at a frequency of 2ω_{pump}-ω_{probe} through the FWM process. For linearly co-polarized lights propagating in a fiber, the output power of the idler light, P_{idler}, can be described as below in the case of that the fiber loss is negligibly low [10],

_{pump}and P

_{probe}are the fiber length, launched pump and probe powers, respectively. Δβ is the phase mismatching parameter defined as

_{2}represents the second-order dispersion at ω

_{pump}, and Δω = ω

_{probe}– ω

_{pump}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 β_{2} of the tested fiber. For a low β_{2} fiber, whose zero-dispersion wavelength is within or close to the measuring wavelengths, ω_{pump} is scanned while keeping the Δω constant. On the other hand, for a high β_{2} fiber, ω_{pump} and therefore β_{2} are kept constant while ω_{probe} is scanned to change Δω.

#### I. Low β_{2} fiber

For constant Δω in Eqs. (1) and (2), P_{idler} is a periodic function of ω_{pump} as shown in Fig. 1(a) because β_{2} monotonically increase or decrease with ω_{pump}. From Eq. (1), P_{idler} becomes maximum at ΔβL/2 = 0, and has minimal values at which Δβ satisfies

_{pump}to be zero, at a pump frequency of ω

_{Z}

^{(N)}providing the minimal P

_{idler}, β

_{2z}

^{(N)}can be expressed as [9]In the case where P

_{pump}> 0, the pump frequency indicating the minimal P

_{idler}will shift to ω

_{P}

^{(N)}as shown by solid curve in Fig. 1(a). Here, β

_{2}at ω

_{P}

^{(N)}, β

_{2P}

^{(N)}can be approximated using the third order dispersion at ω

_{Z}

^{(N)}, β

_{3Z}

^{(N)}asP

_{P}

^{(N)}can be described as

_{P}

^{(N)}becomes a linear function of P

_{pump}, and the nonlinear coefficient γ can be calculated as

_{P}

^{(N)}can be determined from the slope of the ω

_{P}

^{(N)}versus P

_{pump}graph. Here, β

_{3Z}

^{(N)}in Eq. (7) can be also obtained using Eq. (3). Although the condition of P

_{pump}= 0 in Eq. (3) would be impossible for actual measurement, ω

_{Z}

^{(N)}can be estimated by calculating the y-intercept on the ω

_{p}

^{(N)}versus P

_{pump}graph in Eq. (6). β

_{2}spectrum can be also determined with β

_{2z}

^{(N)}for several N from Eq. (4), and β

_{3Z}

^{(N)}can be then calculated by a biquadratic approximation of the β

_{2}spectrum. The definitions of β are summarized in Table 1.

#### 2.1 High β_{2} fiber

When we measure high β_{2} 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, β_{3Z}^{(N)} in Eq. (7) cannot be obtained. Here, keeping ω_{pump} constant, only Δω and P_{pump} become the variable in Eq. (2). In this case, P_{idler} 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 P_{idler} to be minimum with N and N-1 are given by

_{P}

^{(N)}is the N-th frequency difference providing the minimal P

_{idler}for P

_{pump}>0. From two adjacent frequency differences of Δω

_{P}

^{(N)}and Δω

_{P}

^{(N-1)}in Eqs. (8) and (9), β

_{2}at ω

_{pump}can be determined independently of N as

_{z}

^{(N)}providing the minimal P

_{idler}for the limit of P

_{pump}to be zero is given by

_{P}

^{(N)}]

^{2}also becomes a linear function of P

_{pump}, and γ can be evaluated by the slope of the [Δω

_{P}

^{(N)}]

^{2}versus P

_{pump}graph, as

_{2}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 β_{2}. Respective frequencies of the pump and probe lights were scanned together with a constant difference of Δω = 7.8 × 10^{12} rad/s. Examples of the generated idler power P_{idler} against the pump frequency are shown in Fig. 3(a). The periodical minima are observed, and we confirmed that the ω_{P}^{(N)} were shifted with increase of the pump power from 1.6 to 26mW, as theoretically predicted in Fig. 1(a). Measured ω_{P}^{(N)} as a function of P_{pump} 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), ω_{P}^{(N)} are linearly decreasing with the increasing of P_{pump}.

As mentioned above, the β_{3} should be known to obtain the nonlinear coefficient γ. β_{2Z}^{(N)} can be calculated from Eq. (4), and ω_{Z}^{(N)} can be determined from the y-intercept of ω_{P}^{(N)}-P_{pump} graphs in Figs. 3(b) and 3(c), as described with Eq. (6). We drew a β_{2} spectrum by determining the β_{2Z}^{(N)} at several ω_{Z}^{(N)} related to respective N in Eq. (4), and β_{3} was calculated as + 0.034 ps^{3}/km by the biquadratic approximation of the β_{2} spectra. Then, we obtained the slopes of ω_{P}^{(N)}-P_{pump} 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 n_{2} of 3.6 × 10^{−20} m^{2}/W for A_{eff} of 8.5*μ*m^{2}.

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 ω_{P}^{(N)} for N = −1 at wavelength of around 1558 nm is shown in Fig. 3(d), where Δω was fixed to 15.3 × 10^{12} 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 β_{2} (about-22 ps^{2}/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 × 10^{15} rad/s (wavelength of 1550 nm). Figure 4(a) shows the periodic change of the generated idler power P_{idler} against the squared frequency difference (Δω)^{2} for P_{pump} of 208 mW. β_{2} at ω_{pump} was determined from Eq. (10) to be −21.4 ps^{2}/km. Measured Δω_{P}^{(N)} against P_{pump} is also shown in Fig. 4(b) for N = −1. We successfully determined the nonlinear coefficient γ and n_{2} as 1.3 W/km and 2.7 × 10^{−20} m^{2}/W, respectively. In addition, the chromatic dispersion was also measured as 16.5 ps/nm/km at the wavelength of 1550 nm.

Finally, we compared the nonlinear coefficient γ and n_{2} 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 n_{2} 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/nm^{2}/km in disersion slope.

## 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.

## References and links

**1. **M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-based highly nonlinear fibers and their application,” IEEE J. Sel. Top. Quant. **15**(1), 103–113 (2009). [CrossRef]

**2. **R. H. Stolen and L. Chinlon, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A **17**(4), 1448–1453 (1978). [CrossRef]

**3. **M. Monerie and Y. Durteste, “Direct interferometric measurement of nonlinear refractive index of optical fiber by cross-phase modulation,” Electron. Lett. **23**(18), 961–963 (1987). [CrossRef]

**4. **F. Wittl, “Interferometric determination of the nonlinear refractive index n2 of optical fibers,” Proc. Symposium on Optical Fiber Measurements’96, 71–74 (1996).

**5. **C. Vinegoni, M. Wegmuller, and N. Gisin, “Measurement of the nonlinear coefficient of standard SMF, DSF, and DCF fiber using a self-aligned interferometer and a Faraday Mirror,” IEEE Photon. Technol. Lett. **13**(12), 1337–1339 (2001). [CrossRef]

**6. **A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n_{2} in various types of telecommunication fiber at 155 µm,” Opt. Lett. **21**(24), 1966–1968 (1996). [CrossRef] [PubMed]

**7. **L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by Four-Wave Mixing,” IEEE Photon. Technol. Lett. **5**(9), 1062–1065 (1993). [CrossRef]

**8. **O. Aso, M. Tadakuma, and S. Namiki, “Four-Wave Mixing in Optical Fibers and Its Applications,” Furukawa Review, No. **19**, 63–68 (2000).

**9. **M. Hirano and T. Sasaki, “Straightforward chromatic dispersion measurement based on phase mismatching FWM,” Proc. ECOC’2009, Vienna, Austria, Paper 4.1.6.

**10. **G. P. Agrawal, *Nonlinear Fiber Optics*, 4th Edition, Academic Press, (2007).

**11. **T. Dennis and P. A. Williams, “Achieving high absolute accuracy for group-delay measurements using the modulation phase-shift technique,” J. Lightwave Technol. **23**(11), 3748–3754 (2005). [CrossRef]