Higher-order four-wave mixing and its effect in WDM systems

doi:10.1364/OE.7.000166

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Higher-order four-wave mixing and its effect in WDM systems

Shuxian Song »View Author Affiliations

Optics Express, Vol. 7, Issue 4, pp. 166-171 (2000)
http://dx.doi.org/10.1364/OE.7.000166

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▸ Article Outline
– Abstract
1. Introduction
2. A simple higher-order FWM model and experimental results
3. Measuring system performance degradation due to higher-order FWM
4. Conclusions
– References and links

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Abstract

Higher-order four-wave mixing effects are evaluated in WDM systems. Calculated and measured results show that higher-order FWM crosstalk, though small compared to the first-order FWM crosstalk, could be significant in unequal channel-spacing WDM systems where the first-order FWM is not a problem.

[Optical Society of America ]

1. Introduction

To suppress FWM-induced crosstalk in wavelength-division multiplexed (WDM) systems in dispersion-shifted fiber (DSF), the unequal channel-spacing scheme was proposed and worked quite well for most cases since it avoids generating FWM products to fall on to any channels [1

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “WDM system with unequally spaced channels,” J. Lightwave Technology , 13, 889–897, (1995). [CrossRef]

]. However, the newly produced FWM products can mix with channel signals or themselves to produce higher-order FWM products which can overlap with channels and result in crosstalk. In this paper, we study these FWM products and show that these products can become a serious problem when channel power become large or channel spacing becomes narrow.

Fig. 1. Higher-order FWM products produced from two channels

2. A simple higher-order FWM model and experimental results

Higher-order FWM is a quite complex process in multiple channel systems. To simplify analysis, we use a two-channel model. The two channel signals serve as two pumps to produce FWM products, as shown in Fig. 1, where P₁ and P₂ represent the channel power for the two channels. These two channels produce two first-order FWM products with power P_F11 and P_F11. The rest of signals are higher-order FWM products produced by the first-order FWM products and the channel signals. We use P_F21 and P_F22 to represent the second order FWM power, P_F31 and P_F32 the third order FWM power, and so on. FWM products to the left side of the channels are labeled with 1 and those to the right side are labeled with 2. For the FWM products that fall on to the two channels, a subscript c is added. These notations are illustrated in Fig. 1.

Fig. 1 shows that each FWM product depends on the channel signal power and other FWM products. The FWM power in each frequency results from multiple mixing products of channel signals and mixing products, in which complex relations are involved. Here, we make some reasonable approximations by including only the dominant products to the FWM power at each frequency. We call these products as primary FWM products. The power at each frequency is the sum of the power from all primary FWM products and is written as following,

First order:

P_{F 11} = η_{11} P_{1}^{2} P_{2}

(1a)

P_{F 12} = η_{12} P_{2}^{2} P_{1}

(1b)

Second order:

P_{F 21 c} = η_{21 c - 1} P_{2}^{2} P_{F 12} + η_{21 c - 2} P_{1} P_{2} P_{F 11}

= η_{21 c - 1} η_{12} P_{1} P_{2}^{4} + η_{21 c - 2} η_{11} P_{1}^{3} P_{2}^{2}

(2a)

P_{F 22 c} = η_{22 c - 1} P_{1}^{2} P_{F 11} + η_{22 c - 2} P_{1} P_{2} P_{F 12}

= η_{22 c - 1} η_{11} P_{1}^{4} P_{2} + η_{22 c - 2} η_{12} P_{1}^{2} P_{2}^{3}

(2b)

P_{F 21} = η_{21 - 1} P_{1}^{2} P_{F 12} + η_{21 - 2} P_{1} P_{2} P_{F 11}

= η_{21 - 1} η_{12} P_{1}^{3} P_{2}^{2} + η_{21 - 2} η_{11} P_{1}^{3} P_{2}^{2}

(2c)

P_{F 22} = η_{22 - 1} P_{2}^{2} P_{F 11} + η_{22 - 2} P_{1} P_{2} P_{F 12}

= η_{22 - 1} η_{11} P_{1}^{2} P_{2}^{3} + η_{22 - 2} η_{12} P_{1}^{2} P_{2}^{3}

(2d)

Third order:

P_{F 31} = η_{31 - 1} P_{1}^{2} P_{F 22} + η_{31 - 2} P_{F 11}^{2} P_{2} + η_{31 - 3} P_{1} P_{F 11} P_{F 12} + η_{31 - 4} P_{1} P_{2} P_{F 21}

= η_{31 - 1} (η_{22 - 1} η_{11} + η_{22 - 2} η_{12}) P_{1}^{4} P_{2}^{3} + η_{31 - 2} η_{11} P_{1}^{4} P_{2}^{3}

+ η_{31 - 3} η_{11} η_{12} P_{1}^{4} P_{2}^{3} + η_{31 - 4} (η_{21 - 1} η_{12} + η_{21 - 2} η_{11}) P_{1}^{4} P_{2}^{3}

(3a)

P_{F 32} = η_{32 - 1} P_{2}^{2} P_{F 21} + η_{32 - 2} P_{F 12}^{2} P_{1} + η_{32 - 3} P_{2} P_{F 11} P_{F 12} + η_{32 - 4} P_{1} P_{2} P_{F 22}

= η_{32 - 1} (η_{21 - 1} η_{12} + η_{21 - 2} η_{11}) P_{1}^{3} P_{2}^{4} + η_{32 - 2} η_{12} P_{1}^{3} P_{2}^{4}

+ η_{32 - 3} η_{11} η_{12} P_{1}^{3} P_{2}^{4} + η_{32 - 4} (η_{22 - 1} η_{11} + η_{22 - 2} η_{12}) P_{1}^{3} P_{2}^{4}

(3b)

where η_mn is the FWM efficiency that can be calculated from the formulas in [2

N. Shibata, R. P. Braun, and R. G. Warrts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode fiber,” IEEE J. of Quantum Electronics , QE-23, 1205–1211, (1987). [CrossRef]

S. Song, C. Allen, K. Demarest, and R. Hui, “Intensity-dependent effects on FWM in optic fibers,” J. of Lightwave Technology , 17, 2285–2290, (1999). [CrossRef]

]. If the polarization states of the two channel signals are not aligned, a polarization transfer function has to be added in the calculation [4

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. of Quantum Electronics , 28, 883–895, (1992). [CrossRef]

S. Song, C. Allen, K. Demarest, and R. Hui, “A novel nonlinear method for measuring polarization mode dispersion,” J. of Lightwave Technology , 17,12, 2530–2533, (1999).

]. Here, m (m=1, 2, 3…) represents the order of the mixing product and n (n=1, 2, 1c, 2c) represents left or right FWM products (or Stokes and Anti-Stokes waves). The index k in η _mn-k is the k-th primary FWM product that contributes to the power P _Fmn.

Assuming equal channel power for the two channels, i.e., P₁=P₂=P₀, then, we can give a general formula to represent the FWM power including both the first-order FWM and higher-order FWM as following,

P_{Fmn} = {\bar{η}}_{mn} P_{0}^{2 m + 1},

(4)

where η̄_mn is the effective FWM efficiency as defined in (1)–(3). One critical point this equation shows is that higher-order FWM power grows much faster than the lower-order FWM power when channel power increases.

Experiments on a two-channel system were performed. Fig.2 gives the experiment setup. In the experiment, each channel was modulated by a non-return-to-zero (NRZ) signal at 2.5 Gb/s. The channel wavelength separation is 0.8 nm. Dispersion-shifted fiber (DSF) of 25 km is used with zero-dispersion wavelength at 1551 nm. Two attenuators were used to adjust the input power to the fiber without changing the total system loss. The FWM crosstalk is the power ratio of FWM power to the signal power.

Fig. 3 gives results on the calculated and measured first-order and second-order FWM crosstalk as functions of input channel power (channel 1) to fiber. The calculated results from (1) agree well with measured data. When FWM power is small, measured FWM power is limited by ASE and measured slopes tend to become smaller than predicated. When FWM power is too large, the FWM induced power depletion on channel signals become large and make the slopes steep. From Fig. 2(b), we know that the second-order FWM crosstalk is quite large for large channel power.

Since the second-order FWM products, 21c and 22c, are always overlapped with the two channel signals, it is impossible to measure their power. Here we give calculated results on these two products in Fig. 3. The crosstalk they induced on the two channels is comparable with the other second-order FWM products given in Fig. 2.

Fig. 2. Experiment setup for measuring FWM power DFB—DFB laser, MZM—Mach-Zehnder Modulator, MUX—Multiplexer, EDFA—Erbium-doped fiber-amplifier, DSF—Dispersion-shifted fiber, Att-1, 2---Tunable attenuators, DEMUX—Demultiplexer, OSA—Optical spectrum analyzer,

Fig. 2. Measured and calculated the first-order and second-order FWM power

Fig. 3. Calculated FWM power for the second-order FWM products overlapped with the two channels

3. Measuring system performance degradation due to higher-order FWM

The system set-up described in last section was used to measure the Q values. Fig. 4 shows Q as a function of channel power with different OSNR. When channel power is low (<5 dBm), no significant Q degradation occurs. But when channel power increases from 6 dBm to about 10 dBm, Q values begin to drop fast. For a normal situation with a 20-dB OSNR and a system penalty limit of 0.5 dB on Q, we found that the corresponding channel power is roughly 6 dBm.

Fig. 4. Measured Q as a function of input power to fiber

Fig. 5 show measured optical signal-to-noise-ratio (OSNR) that changed with the input channel power to fiber. The initial OSNR is the OSNR at low channel power. When channel power is increased, the OSNR decreased due to FWM induced signal power depletion. Therefore, the Q drop shown in Fig. 4 was caused by two effects, the higher-order FWM crosstalk and signal power depletion. To separate these two effects, we plot measured Q values vs. the system OSNR for different channel power in Fig. 6. From the figure, we observed that increasing channel power reduces both Q-factor and the OSNR, which results in a different slope from the pure OSNR induced Q drop. Comparing the two slopes, we find that the Q penalty induced by the higher-order FWM crosstalk is larger than penalty induced by the power depletion due to the first-order FWM. For example, when OSNR is close to 20 dB and channel power was increased from 1.42 dBm to 9.42 dBm, the Q penalty from the higher-order FWM crosstalk (ΔQ₁) is about 2.0 and the Q penalty from the power depletion (ΔQ₂) is about 1.0, as shown in Fig. 6.

Fig. 5. Measured OSNR as a function of input power to fiber

Fig. 6. Measured Q as a function of system OSNR

4. Conclusions

In conclusion, higher-order FWM induced crosstalk was studied by using a two-channel system in dispersion-shifted fiber (DSF). Calculated and measured results showed that higher-order FWM crosstalk, though small compared to the first-order FWM crosstalk, could be significant in unequal channel-spacing WDM systems where the first-order FWM is not a problem.

Acknowledgments:

The author likes to thank Vipul Bhatnagar, Michael Frankle and Steve Marlow for their help in the experiments.

References

1.	F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, “WDM system with unequally spaced channels,” J. Lightwave Technology , 13, 889–897, (1995). [CrossRef]
2.	N. Shibata, R. P. Braun, and R. G. Warrts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode fiber,” IEEE J. of Quantum Electronics , QE-23, 1205–1211, (1987). [CrossRef]
3.	S. Song, C. Allen, K. Demarest, and R. Hui, “Intensity-dependent effects on FWM in optic fibers,” J. of Lightwave Technology , 17, 2285–2290, (1999). [CrossRef]
4.	K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. of Quantum Electronics , 28, 883–895, (1992). [CrossRef]
5.	S. Song, C. Allen, K. Demarest, and R. Hui, “A novel nonlinear method for measuring polarization mode dispersion,” J. of Lightwave Technology , 17,12, 2530–2533, (1999).

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Research Papers

History
Original Manuscript: June 5, 2000
Published: August 14, 2000

Citation
Shuxian Song, "Higher-order four-wave mixing and its effect in WDM systems," Opt. Express 7, 166-171 (2000)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-4-166

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References

F. Forghieri, R. W. Tkach, A. R. Chraplyvy, "WDM system with unequally spaced channels," J. Lightwave Technology, 13, 889-897, (1995). [CrossRef]
N. Shibata, R. P. Braun, and R. G. Warrts, Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode fiber," IEEE J. of Quantum Electronics, QE-23, 1205-1211, (1987). [CrossRef]
S. Song, C. Allen, K. Demarest, R. Hui, "Intensity-dependent effects on FWM in optic fibers," J. of Lightwave Technology, 17, 2285-2290, (1999). [CrossRef]
Inoue, K., "Polarization effect on four-wave mixing efficiency in a single-mode fiber," IEEE J. of Quantum Electronics, 28, 883-895, (1992). [CrossRef]
S. Song, C. Allen, K. Demarest, R. Hui, "A novel nonlinear method for measuring polarization mode dispersion," J. of Lightwave Technology, 17,12, 2530-2533, (1999).

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