# What is Four-Wave Mixing (FWM) in Fiber Optic Communication Systems?

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### >> Nonlinear Effects in High Power, High Bit Rate Fiber Optic Communication Systems

When optical communication systems are operated at moderate power (a few milliwatts) and at bit rates up to about 2.5 Gb/s, they can be assumed as linear systems. However, at higher bit rates such as 10 Gb/s and above and/or at higher transmitted powers, it is important to consider the effect of nonlinearities. In case of WDM systems, nonlinear effects can become important even at moderate powers and bit rates.

There are two categories of nonlinear effects. The first category happens because of the interaction of light waves with phonons (molecular vibrations) in the silica medium of optical fiber. The two main effects in this category are *stimulated Brillouin scattering* (SBS) and *stimulated Raman scattering* (SRS).

The second category of nonlinear effects are caused by the dependence of refractive index on the intensity of the optical power (applied electric field). The most important nonlinear effects in this category are *self-phase modulation* (SPM) and *four-wave mixing* (FWM).

### >> Basic Principles of Four-Wave Mixing

**1. How the Fourth Wave is Generated**

In a WDM system with multiple channels, one important nonlinear effect is four-wave mixing. Four-wave mixing is an intermodulation phenomenon, whereby interactions between 3 wavelengths produce a 4th wavelength.

In a WDM system using the angular frequencies ω_{1}, … ω_{n}, the intensity dependence of the refractive index not only induces phase shifts within a channel but also gives rise to signals at new frequencies such as 2ω_{i}-ω_{j} and ω_{i} + ω_{j} – ω_{k}. This phenomenon is called four-wave mixing.

In contrast to Self-Phase Modulation (SPM) and Cross-Phase Modulation (CPM), which are significant mainly for high-bit-rate systems, the four-wave mixing effect is independent of the bit rate but is critically dependent on the channel spacing and fiber chromatic dispersion. Decreasing the channel spacing increases the four-wave mixing effect, and so does decreasing the chromatic dispersion. Thus the effects of Four-Wave Mixing must be considered even for moderate-bit-rate systems when the channels are closely spaced and/or dispersion-shifted fibers are used.

To understand the effects of four-wave mixing, consider a WDM signal that is the sum of n monochromatic plane waves. Thus the electric field of this signal can be written as

The nonlinear dielectric polarization P_{NL}(**r**,t) is given by

where χ^{(3)} is called the *third-order nonlinear susceptibility* and is assumed to be a constant (independent of t).

Using the above two equations, the nonlinear dielectric polarization is given by

Thus the nonlinear susceptibility of the fiber generates new fields (waves) at the frequencies ω_{i} ± ω_{j} ± ω_{k} (ω_{i}, ω_{j}, ω_{k}not necessarily distinct). This phenomenon is termed four-wave mixing.

The reason for this term is that three *waves* with the frequencies ω_{i}, ω_{j}, and ω_{k} combine to generate a *fourth* wave at a frequency ω_{i} ± ω_{j} ± ω_{k}. For equal frequency spacing, and certain choices of I,j, and k, the fourth wave contaminates ω_{i}. For example, for a frequency spacing Δω, taking ω_{1}, ω_{2}, and ω_{3} to be successive frequencies, that is, ω_{2} = ω_{1} + Δω and ω_{3} = ω_{1} + 2Δω, we have ω_{1}-ω_{2}+ω_{3} = ω_{2}, and 2ω_{2}-ω_{1}=ω_{3}.

In the above equation, the term (28) represents the effect of SPM and CPM. The terms (29), (31), and (32) can be neglected because of lack of phase matching. Under suitable circumstances, it is possible to approximately satisfy the phase-matching condition for the remaining terms, which are all of the form ω_{i} + ω_{j} – ω_{k}, I,j k (ω_{i}, ω_{j} not necessarily distinct).

For example, if the wavelengths in the WDM system are closely spaced, or are spaced near the dispersion zero of the fiber, then β is nearly constant over these frequencies and the phase-matching condition is nearly satisfied. When this is so, the power generated at these frequencies can be quite significant.

**2. Power Penalty Due to Four-Wave Mixing**

From the above discussion, we can see that the nonlinear polarization causes three signals at frequencies ω_{i}, ω_{j}, and ω_{k} to interact to produce signals at frequencies ω_{i} ± ω_{j} ± ω_{k}. Among these signals, the most troublesome one is the signal corresponding to

ω_{ijk} = ω_{i} + ω_{j} – ω_{k}, i k, j k

Depending on the individual frequencies, this beat signal may lie on or very close to one of the individual channels in frequency, resulting in significant crosstalk to that channel. In a multichannel system with W channels, this effect results in a large number (W(W-1)^{2}) of interfering signals corresponding to i ,j,k varying from 1 to W. In a system with three channels, for example, 12 interfering terms are produced, as shown in the following figure.

Interestingly, the effect of four-wave mixing depends on the phase relationship between the interacting signals. If all the interfering signals travel with the same group velocity, as would be the case if there were no chromatic dispersion, the effect is reinforced. On the other hand, with chromatic dispersion present, the different signals travel with different group velocities. Thus the different waves alternately overlap in and out of phase, and the net effect is to reduce the mixing efficiency. The velocity difference is greater when the channels are space farther apart (in systems with chromatic dispersion).

To quantify the power penalty due to four-wave mixing, we can start from the following equation

This equation assumes a link of length *L* without any loss and chromatic dispersion. Here P_{i}, P_{j}, and P_{k} denote the powers of the mixing waves and P_{ijk} the power of the resulting new wave, is the nonlinear refractive index (3.0x 10^{-8} μm^{2}/W), and d_{ijk} is the so-called degeneracy factor.

In a real system, both loss and chromatic dispersion are present. To take the loss into account, we replace *L* with the effective length *L _{e}*, which is given by the following equation for a system of length

*L*with amplifiers spaced

*l*km apart.

The presence of chromatic dispersion reduces the efficiency of the mixing. We can model this by assuming a parameter η_{ijk}, which represents the efficiency of mixing of the three waves at frequencies ω_{i}, ω_{j}, and ωk. Taking these two into account, we can modify the preceding equation to

For on-off keying (OOK) signals, this represents the worst-case power at frequency ω_{ijk}, assuming a 1 bit has been transmitted simultaneously on frequencies ω_{i}, ω_{j}, and ω_{k}.

The efficiency η_{ijk} goes down as the phase mismatch Δβ between the interfering signals increases. We can obtain the efficiency as

Here, *Δβ* is the difference in propagation constants between the different waves, and *D* is the chromatic dispersion. Note that the efficiency has a component that varies periodically with the length as the interfering waves go in and out of phase. In this example, we will assume the maximum value for this component. The phase mismatch can be calculated as

Δβ = β_{i} + β_{j} – β_{k} – β_{ijk}

where β_{r} represents the propagation constant at wavelength λ_{r}.

Four-wave mixing manifests itself as intrachannel crosstalk. The total crosstalk power for a given channel ω_{c} is given as

Assume the amplifier gains are chosen to match the link loss so that the output power per channel is the same as the input power. The crosstalk penalty can therefore be calculated from the following equation.

Assume that the channels are equally spaced and transmitted with equal power, and the maximum allowable penalty due to Four-Wave Mixing (FWM) is 1 dB. Then if the transmitted power in each channel is P, the maximum FWM power in any channel must be < εP, where ε can be calculated to be 0.034 for a 1 dB penalty using the above equation. Since the generated FWM power increases with link length, this sets a limit on the transmit power per channel as a function of the link length. This limit is plotted in the following figure for both standard single mode fiber (SMF) and dispersion-shifted fiber (DSF) for three cases

(1) 8 channels spaced 100 GHz apart

(2) 32 channels spaced 100 GHz part

(3) 32 channels spaced 50 GHz apart

For standard single mode fiber (SMF) the chromatic dispersion parameter is taken to be D = 17 ps/nm-km, and for DSF the chromatic dispersion zero is assumed to lie in the middle of the transmitted band of channels. The slope of the chromatic dispersion curve, dD/dλ, is taken to be 0.055 ps/nm-km^{2}.

We can get several conclusions from the above power penalty figure.

1). The limit is significantly worse in the case of dispersion-shifted fiber than it is for standard single mode fiber. This is because the four-wave mixing efficiencies are much higher in dispersion-shifted fiber due to the low value of the chromatic dispersion.

2). The power limit gets worse with an increasing number of channels, as can be seen by comparing the limits for 8-channel and 32 channel systems for the same 100 GHz spacing. This effect is due to the much larger number of four-wave mixing terms that are generated when the number of channels is increases. In the case of dispersion-shifted fiber, this difference due to the number of four-wave mixing terms is imperceptible since, even though there are many more terms for the 32 channel case, the same 8 channels around the dispersion zero as in the 8 channel case contribute almost all the four-wave mixing power. The four-wave mixing power contribution from the other channels is small because there is much more chromatic dispersion at these wavelengths.

3) The power limit decreases significantly if the channel spacing is reduce, as can be seen by comparing the curves for the two 32-channel systems with channel spacing of 100 GHz and 50 GHz. This decrease in the allowable transmit power arises because the four-wave mixing efficiency increases with a decrease in the channel spacing since the phase mismatch *Δβ* is reduced. (For SMF, though the efficiencies at both 50 GHz and 100 GHz are small, the efficiency is much higher at 50 GHz than at 100 GHz.)

**3. Solutions for Four-Wave Mixing**

Four-wave mixing is a severe problem in WDM systems using dispersion-shifted fiber but does not usually pose major problem in systems using standard fiber. In face, it motivated the development of None-Zero Dispersion-Shifted Fiber (NZ-DSF). In general, the following actions alleviate the penalty due to four-wave mixing.

**1) Unequal channel spacing.** The positions of the channels can be chosen carefully so that the beat terms do not overlap with the data channels inside the receiver bandwidth. This may be possible for a small number of channels in some cases but needs careful computation of the exact channel positions.

**2) Increases channel spacing.** This increases the group velocity mismatch between channels. This has the drawback of increasing the overall system bandwidth, requiring the optical amplifiers to be flat over a wider bandwidth, and increases the penalty due to Stimulated Raman Scattering (SRS).

**3) Using higher wavelengths beyond 1560nm with DSF.** Even with DSF, a significant amount of chromatic dispersion is present in this range, which reduces the effect of four-wave mixing. The newly developed L-band amplifiers can be used for long-distance transmission over DSF.

**4)** As with other nonlinearities, reducing transmitter power and the amplifier spacing will decrease the penalty

**5) If the wavelengths can be demultiplexed and multiplexed in the middle of the transmission path**, we can introduce difference delays for each wavelength. This randomizes the phase relationship between the different wavelengths. Effectively, the FWM powers introduced before and after this point are summed instead of the electric fields being added in phase, resulting in a smaller FWM penalty.