1. Introduction

Recently, dense wavelength division multiplexing (DWDM) has become widely adapted as a major transport method for next generation high capacity optical fiber communication systems. Because chromatic dispersion (CD) is the main factor in determining the bandwidth of a highspeed DWDM optical transmission system, it is considered as one of the most important property of an optical fiber or a photonic device. Accurate determination of both linear and nonlinear parameters for preinstalled fibers is essential for a practical DWDM system design. In order to replace old time division multiplexing (TDM) optical transmission systems with DWDM systems, a nondestructive method for determining chromatic dispersion, dispersionzero wavelength, and dispersion-zero wavelength mapping along an optical fiber is essential for the usage of already installed fiber links.

As the four-wave-mixing (FWM) efficiency curve of an optical fiber as a function of wavelength depends significantly on the chromatic dispersion, the dispersion-zero wavelength, and the nonlinear coefficient of an optical fiber, FWM experiment in an optical fiber has attracted much attention lately as a tool for measuring these parameters of an optical fiber [1–8]. There are several reports that this scanning FWM experiments can be used to measure zero-dispersion wavelength, chromatic dispersion slope, and non-linear refractive index both individually [4–6] and simultaneously [7, 8]. However, there are many limitations to using this scanning FWM measurement in an optical fiber as a tool for obtaining these parameters of an optical fiber. First of all, the generated FWM signal is normally very weak in a silica based optical fiber such that the optical powers of input lights must be very high, and a very sophisticated detection method must be used. Another major restriction of conventional degenerate FWM experiments for determining fiber parameters is that the dispersion-zero wavelength of a sample fiber under test must be within the tunable wavelength range of a tunable pump laser used in the FWM experiments. The scanning pump laser wavelength must pass over the dispersion-zero wavelength of the sample fiber in order to obtain a phase matched FWM efficiency curve for a test fiber. Because of these fundamental drawbacks of the conventional degenerate FWM measurement method for determining optical parameters of a fiber, it is impossible to measure the dispersion of a sample fiber whose dispersion zero wavelength is off from the tuning range of a tunable laser used as a pump light source in a FWM experiment. Therefore the conventional FWM measurement methods were only used to measure the properties of a DSF so far, and optical parameters for other fibers such as an NZDSF or a single mode fiber have not been measured with this FWM technique. In this paper, we propose a novel phase matching condition in FWM experiments which can be applied to the other fibers whose dispersion zero wavelengths are out of the tunable wavelength range of a pump light source in a FWM experiment. We have used another phase matching condition that can be satisfied in a degenerate FWM experiment when the wavelength of a pump laser passes over the wavelength of a probe laser. With our method, we have measured several FWM efficiency curve, and have shown that our method can be used to very accurately measure the dispersion-zero wavelength, the dispersion slop, and the nonlinear refractive index of a sample fiber, regardless of the fiber type.

2. The principle of FWM in an optical fiber

FWM is an interesting non-linear process that generates a new fourth optical wave (ω₄) in nonlinear optical materials when three other optical input waves (ω₁, ω₂, ω₃) are provided, where the frequency of the new optical wave is given as ω₄ = ω₁ + ω₂ - ω₃. FWM in an optical fiber can be explained with a classical expression derived by Hill et al. [9]. This well-known numerical formula was later modified by Shibata et al. [10] to include the dependence of phase-matching in a FWM efficiency curve. In the case of partially degenerate case (ω₁ = ω₂), the optical power of the generated wave at ω₄ (= 2ω₁ - ω₃) frequency due to FWM effect is given by [7, 11, 12]:

where P₁ and P₃ are the two input powers at λ₁ and λ₃ wavelengths respectively, L is the length of the optical fiber, α is the absorption coefficient. The effective fiber length L_eff is defined as $L_{eff}\equiv {\displaystyle {\int }_0^L\exp }$ (-αz)dz=(1-exp(-αL))/α, and the nonlinear coefficient γ is equal to 2πn₂/λA_eff. FWM efficiency η(Δβ) is a coefficient determining the FWM efficiency on phase mismatching term Δβ and is defined as η(Δβ) ≡ P ₄(L,Δβ)/P ₄(L,Δβ = 0). The phase-mismatching term Δβ is defined as the difference between the propagation constants of the four waves: Δβ ≡ β₄+β₃ - 2β₁. The FWM efficiency and the phase mismatching term are given as [10, 11]:

where λ₀ is the dispersion-zero wavelength, λ₁ is the pump wavelength, and λ₃ is the probe wavelength.

As FWM efficiency in a silica based optical fiber is very low, the optical power of the generated wave in FWM process is practically measurable only near a phase matching condition: Δβ = 0. It is obvious from Eq. (2) that the FWM efficiency becomes the maximum of 1 when the phase-mismatching term becomes zero. As given in Eq. (3), there are two phase matching solutions for making the phase-matching term Δβ to be zero: λ₁ = λ₀ or λ₁ = λ₃. Most of the previous efforts have been devoted to the first phase matching solution of λ₁ = λ₀ [6, 7]; the phase-mismatching term Δβ goes to zero as the pump wavelength λ₁ becomes equal to the dispersion-zero wavelength λ₀ of a fiber. FWM efficiency curve η(Δβ) has been measured while the pump wavelength λ₁ is scanned across the dispersion-zero wavelength of a sample fiber λ₀ and the linear and nonlinear optical parameters of a sample fiber can be obtained by fitting the FWM efficiency curve. This is the reason why conventional FWM measurement methods were only used to measure the properties of a DSF whose dispersion-zero wavelength is inside the tuning range of a tunable laser source used in FWM experiments. The second phase matching condition λ₁ = λ₃ can be easily obtained when the pump wavelength λ₁ and the probe wavelength λ₃ becomes equal. Since the second phase matching solution is independent with the dispersion-zero wavelength of a sample fiber, it can be used for any fiber. If we rewrite the ω₄ = 2ω₁ - ω₃ with wavelengths it becomes

When the pump wavelength λ₁ becomes close to the probe wavelength λ₃ the generated wavelength λ4 also becomes close to the probe wavelength. Therefore it is very hard to exclusively measure the optical power of the generated wave out of the pump and the probe optical powers near this phase matching condition. This is the major reason why relatively little attention has been paid to the second phase matching condition in FWM experiments in an optical fiber. Here, we focus on the relation between FWM efficiency and phase-matching with the second phase matching solution. When λ₁ = λ₃ we also have λ4 = λ₁, which means that there exists only a single wavelength in an optical fiber. In this case, no new wavelengths of light are generated or observed. However, when the pump wavelength is very close but not equal to the probe wavelength, there exists a generated wavelength near the pump wavelength. If we use a very sharp optical bandpass filter we can measure the FWM coefficient η(Δβ) near the second phase matching condition. By scanning the pump wavelength over a wavelength range containing the probe wavelength, we were able to determine the FWM efficiency curve η(Δβ) needed to extract zero-dispersion wavelength, dispersion slope and nonlinear coefficient simultaneously.

3. Experiment and results

The schematic diagram of our experimental setup is shown in Fig. 1. Two commercially available tunable semiconductor lasers (TSLs) were used as input sources; TSL1 (Anritzu, MG9541A) and TSL2 (Santec, TSL-200), were used as the pump and probe light sources. The output powers of the two TSLs were combined with a 50/50 coupler and then pass through a fiber under the test. The output optical powers of the pump (λ₁) and the probe (λ₃) lights were 5 mW and 2 mW, respectively. Since no optical amplifier was used after the light source, precise control of optical powers for the input lights was possible, and it enhances the accuracy of our measurements. Two in-line polarization controllers (PCs) were used to adjust the polarization states of the two input lights carefully to maximize the power of generated FWM signal. The test fiber was put in a temperature-controlled chamber to keep a given temperature with an accuracy of 0.1 °C for 20 minutes, which is required for a complete FWM measurement. A tight temperature control is needed to suppress variation in the dispersion-zero wavelength of a sample fiber, which is extremely sensitive to its temperature changes. An optical spectrum analyzer (OSA, Agilent 86142B) was employed to monitor the power of the generated FWM signal.

 

Fig. 1. Experimental set-up for simultaneous measurement of linear and nonlinear optical properties of an optical fiber using FWM experiments.

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The 3 dB spectral bandwidth of the OSA used to exclusively measure the power of the FWM signal at λ4 wavelength was 0.06 nm. First, the wave length of the probe light λ₃ was fixed for a FWM measurement. Then, the TSL1 was controlled by a computer with a General Purpose Interface Bus (GPIB) interface to scan its wavelength λ₁ for about 10 nm wavelength range across the probe wavelength λ₃. For each scanning wavelength of the pump laser, the OSA was synchronously controlled by the computer to detect only the optical power of the generated wave at λ4 wavelength. Due to proper number of averaging in measurements, wavelength adjustment in TLS1, OSA setting and readout time for each scanning wavelength, it took about 20 minutes to obtain a FWM efficiency curve for about a 10 nm wavelength range with this fully computerized measurement setup. The dispersion-zero wavelength and dispersion slope of the sample fiber was obtained by a nonlinear least-square curve fit to a measured FWM efficiency curve with a formula given in Eq. (2). The Levenberg-Marquardt (LM) algorithm, a widely-used technique for nonlinear fitting processes, was used as a nonlinear least squares fitting method.

 

Fig. 2. Generated optical power of FWM signal for a DSF.

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To demonstrate the feasibility of our proposed method we have measured the FWM efficiency curve for a 5 km long DSF. Figure 2 shows the optical power of generated FWM light at λ₄ = λ₁λ₃/(2λ₃-λ₁) wavelength obtained by our novel phase-matching method. The pump wavelength was scanned from 1550 nm to 1557 nm, and the probe wavelength was set to 1554 nm. As is shown at the center of Fig. 2 we have observed a power surge in the detected FWM signal around the probe wavelength. This surge is an error caused by the optical powers of the pump and the probe lights in the detected FWM signal. It is because the three wavelengths (λ₁, λ₃, λ₄) involved in our FWM experiment become almost the same when the pump wavelength λ₁ approaches the probe wavelength λ₃. Although this power surge near the probe wavelength is inevitable in our measurement, it does not degrade the accuracy of the measurement system. The pump light was placed very close to the probe wavelength as much as the OSA can clearly discern the input lights and newly generated FWM light. As shown in Eq. (1), the FWM efficiency is almost equal to 1 with a fully matched phase matching condition. In order to obtain good fitting parameters for the FWM efficiency curve, the surge portion of in Fig. 2 was removed. Figure 3 shows the FWM efficiency curve in linear scale without the power surge portion of the data shown in Fig. 2. By fitting this FWM efficiency curve η with a formula shown in Eqs. (1), (2), and (3), we have λ₀ = 1547.1 nm, dD_C/dλ = 0.066 ps/nm²-km. Note that the dispersion-zero wavelength of the sample fiber is off from the scanning wavelength range of the pump laser. The solid line is the fitted curve with these parameters and shows a good agreement with the experimental data in small white dots in Fig. 3.

 

Fig. 3. FWM power curve obtained by novel phase-matching method.

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Fig. 4. FWM power curve obtained from NZDSF

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We have also measured the dispersion-zero wavelength and the dispersion slop of an NZDSF, which was manufactured by Samsung Electronics. Figure 4 shows the FWM efficiency curve for this fiber. The probe wavelength was set to 1551 nm in this time, and the pump wavelength was scanned from 1549 nm to 1553 nm with a 0.01 nm step size. After performing a least-square curve fitting process, we successfully have obtained λ₀ = 1504.1 nm and dD_C/dλ = 0.069 ps/nm²-km. As far as we know, this is the first demonstration of measuring λ₀ and dD_C/dλ for an NZDSF by using a FWM measurement technique. We compared the measured FWM data with a numerically fitted data in Fig. 4. Solid line shows the numerically fitted result and the empty circles are the measured data. Figure 4(a) is the enlarged view of small oscillations in Fig. 4 near 1550 nm wavelength. It should be noted that the two curves match very well with each other. We have also measured the dispersion-zero wavelength and the dispersion slope using a conventional modulation phase shifting method, and the results are λ₀ = 1504.7 nm and dD_C/dλ = 0.068 ps/nm²-km.

Table 1. Zero dispersion wavelength, dispersion slop and nonlinear refractive index measurement.

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We have also tested several fibers whose dispersion-zero wavelength is far from the tuning range of 1510~1640 nm for the TLS (Anritzu, MG9541A) used in our experiment, and the results are shown in table 1. The non-linear refractive indices of a DSF and various NZDSFs were also measured [6, 11]. Fibers #1 to #6 are commercially available NZDSFs and Fiber #7 is a DSF. Fibers #1 through #4 were provided by Samsung Electronics, fiber #5 is from Alcatel, and fiber #6 and #7 are made by Sumitomo Electric. Loss and length of each fiber were measured with an OTDR and are used for a fitting parameter in Eq. (2). We have obtained the dispersion-zero wavelength (DZW), the dispersion slope (D-Slope), and the nonlinear optical parameter (n₂/A_eff) of each sample fiber by using a least square curve fitting method. The optical powers for the pump (P₁) and the probe (P₃) were measured simultaneously with the generated FWM signal power by the optical spectrum analyzer during measurements. This was possible because no optical amplifier is used in our experiment, and only a few mW of optical powers were used both for the pump and probe lights. It shows that a fiber with its DZW as low as 1388 nm can be measured with our new phase matching method. As FWM signal data can be obtained so accurately with our fully computer controlled measurement setup, we have obtained the repeatability or the standard deviation of repeated measurements in DZW less than 1 nm, and that of D-slope less than 1%. Those experimental results are in good agreement with those obtained using a conventional phase shifting measurement method.

4. Discussion

It was believed that the phase-matching condition of a FWM process in an optical fiber can be satisfied only near the zero-dispersion wavelength of the fiber [5, 13]. However, we have shown that this is not the only phase-matching condition of a FWM process in an optical fiber. By making the pump wavelength and the probe wavelength very close in a partially degenerate FWM experiment we have demonstrated that we can obtain a good phase matched FWM signal data well off from the dispersion-zero wavelength of a fiber. An accurate separation for a generated FWM signal from the pump or the probe input lights in wavelength was possible because of recently available optical components such as a programmable tunable laser and a computer controlled optical spectrum analyzer. By using our method, we were able to extend the usage of FWM technique for retrieving dispersion-zero wavelength, dispersion slope, and nonlinear optical coefficient for an optical fiber with arbitrary dispersion characteristics.

FWM efficiency can be influenced by the intensity fluctuation of input optical powers, the dispersion fluctuation, and the polarization mode dispersion of a sample fiber [14–15]. When the input optical powers are high, intensity-dependent phase matching condition owing to the nonlinear refractive index change of a sample fiber needs to be considered in a FWM process. The nonlinear refractive index change is due to self-phase modulation (SPM) and cross-phase modulation (XPM) effects caused by the pump and the probe waves in a sample fiber. The nonlinear correction in the phase matching term in our experiment can be expressed as

where, P₁ is the optical power of the pump signal and P₃ is the optical power of the probe signal. We have tested the effect of this nonlinear phase matching term on our FWM experiment by doing some numerical simulation for various pump and probe signal powers. We have used λ₀ = 1412 [nm], λ₁ = 1554 [nm], L = 2 [km], α = 0.19 [dB/km], $\frac{n_2}{A_{\mathrm{eff}}}=7\times {10}^{-10}\left[\frac{1}{W}\right],\frac{d{D}_c}{\mathrm{d\lambda }}=0.07\left[\frac{\mathrm{ps}}{\mathrm{km}}-\mathrm{nm}\right]$, while λ₃ is scanned from 1551 to 1556 nm. When the pump, signal and the generated wavelengths are almost the same (λ ≈ λ₁ ≈ λ₂ ≈ λ₃) we can simplify Eq. (5) such as ${\mathrm{\Delta \beta }}_{\mathrm{NL}}\approx \frac{2\pi }{\lambda }\bullet \left(\frac{n_2}{A_{\mathrm{eff}}}\right)\bullet \left[2P_1-P_3\right]$._Figure 5 shows FWM signals near our proposed phase matching condition for various pump and probe powers with ΔP = 2P₁ - P₃. Graph (a) is when P₁ = 5 mW and P₃ = 2 mW such that ΔP = 8 mW. This graph is almost identical to the case without the nonlinear phase matching term. Graphe (b) is when P₁ = 2 mW and P₃ = 402 mW such that ΔP = -400 mW. Its peak FWM efficiency and the width of the phase matching curve is decreased. Graphe (c) is when P₁ = 2 mW and P₃ = 602 mW. Its peak FWM efficiency and the width of the curve is even further decreased. Graphe (d) is when P₁ = 101 mW and P₃ = 2 mW, which gives ΔP = 200 mW, Graphe (e) is when P₁ = 251 mW and P₃ = 2 mW such that ΔP = 500 mW. Through out this simulation we have concluded that that the nonlinear phase matching term becomes noticeable in our experiment only when ΔP becomes larger than about 50 mW. As the optical powers for the pump the probe signals were only 5 mW and 2 mW, respectively the intensity dependent nonlinear phase matching effect is negligible in our experiments.

 

Fig. 5. Calculated degenerate FWM efficiency curves for various pump and probe powers.

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Magnus Karlsson [14] observed that a random variation of the dispersion-zero wavelength in optical fibers degrades the performance of the FWM efficiency curve, and small deviations of the dispersion zero point along the propagation length of the fiber is unavoidable because of imperfections in fiber fabrication process. We believe that small discrepancies between experimental data and curve fitting in Fig. 3 and Fig. 4 are mostly due to this random variation of dispersion in sample fibers. There are also time-varying environmental effects on FWM experiments. We have reduced the contribution of temperature-dependent refractive index changes of a test fiber by putting the fiber samples in a temperature controlled chamber.

Polarization mode dispersion (PMD) is another possible source of error in our FWM experiments causing extra phase mismatching in a FWM process and possibly changing the shape of the FWM efficiency curve [16]. In previously reported FWM experiments [2–6], a probe signal (λ₃) needs to be fixed outside of the wavelength tuning range of a pump wavelength (λ₁) while the DZW (λ₀) of a sample fiber needs to be within the tuning range of the pump wavelength. When the probe wavelength is far from the DZW of a sample fiber, full width half maximum (FWHM) of a resonance peak in the FWM efficiency curve becomes narrower, and this improves the accuracy of DZW in measurements. However, as the separation between the pump and probe wavelengths increases, the PMD effect in a FWM experiment becomes larger, and it enhances discrepancy between an experimentally obtained FWM efficiency curve and the formula given in Eq. (2). As the relative wavelength difference between the probe and pump signals was within 2 nm, and the PMD of a normal fiber is less than 0.02 ps/km^1/2, the polarization effect was negligible in our experiments. We believe that the slight discrepancies between the formula and measured results in Fig. 4(a) are partly because of polarization mode dispersion effect. Further investigations on the influence of PMD effect in a FWM experiment are in progress and will be addressed for practical application soon.

5. Conclusion

A novel phase matching method for a FWM experiment for obtaining linear and nonlinear optical parameters of an optical fiber is proposed and demonstrated experimentally. Measured FWM curves are in good agreement with a simple classical formula. By fitting the FWM efficiency curve with a given formula with a least square curve fitting algorithm, we have demonstrated that zero-dispersion wavelength, dispersion slop, and non-linear coefficient of an optical fiber can be retrieved very accurately. Our measured values are in good agreement with those obtained using a conventional phase shifting method. Our proposed method can be used regardless of the dispersion-zero wavelength of a sample fiber it overcoming the major limitations of the conventional FWM measurement method. As this method is very simple and can easily apply to any fiber, we believe that this method can be used for a practical optical fiber test and measurement instrument.