Four-Wave Mixing Crosstalk Suppression Based on the Pairing Combinations of Differently Linear-Polarized Optical Signals

A new approach to suppressing the four-wave mixing (FWM) crosstalk by using the pairing combinations of differently linear-polarized optical signals was investigated. The simulation was conducted using a four-channel system, and the total data rate was 40 Gb/s. A comparative study on the suppression of FWM for existing and suggested techniques was conducted by varying the input power from 2 dBm to 14 dBm. The robustness of the proposed technique was examined with two types of optical fiber, namely, single-mode fiber (SMF) and dispersion-shifted fiber (DSF). The FWM power drastically reduced to less than −68 and −25 dBm at an input power of 14 dBm, when the polarization technique was conducted for SMF and DSF, respectively. With the conventional method, the FWM powers were, respectively, −56 and −20 dBm. The system performance greatly improved with the proposed polarization approach, where the bit error rates (BERs) at the first channel were 2.57 × 10−40 and 3.47 × 10−29 at received powers of −4.90 and −13.84 dBm for SMF and DSF, respectively.


Introduction
Four-wave mixing (FWM) is one of the phenomena that may lower the effectiveness of the transmitted signal in wavelength division multiplexing (WDM) systems under dense channel spacing and low chromatic dispersion. In a WDM system with equally spaced channels, the new frequencies generated by FWM will drop at the channel frequencies and will introduce crosstalk [1][2][3]. The FWM effect is a result of the change in the intensity dependence of the refractive index of optical fiber.
Few reports and methods have been proposed for solving the problems associated with FWM. The examples of such methods are the use of nonzero dispersion fibers, relatively low channel counts, and unequal channel spacing techniques [4][5][6]. However, dispersion causes the distortion of the transmitted signals and needs to be compensated to achieve a long-haul system. As the channel count increases, more channels have to be confined to the erbium-doped fiber amplifier gain band by reducing the channel spacing. This condition increases the FWM effects and has a negative effect on the FWM suppression methods. Increased channel separation would prevent the implementation of a dense WDM. Similarly, reducing the levels of FWM crosstalk by choosing unequal channel frequency spacing may not be a practical option because this technique also needs additional optical bandwidth.
By contrast, orthogonal polarization has recently been found to reduce the FWM crosstalk. The FWM time average power strongly depends on the relative polarization states of the mixing channels. The researcher has reduced the FWM by randomly adjusting the polarization state of the adjacent channels to be orthogonal to one another [7][8][9][10]. Nevertheless, adjusting the polarization state randomly will not surely reduce the FWM crosstalk in all optical channels. Furthermore, the bit error rate (BER) may be not improved in all users because the orthogonal polarization does not include all channel interactions.

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The Scientific World Journal In this work, we combined pairs of channels with different polarizations. The first channel was polarized by a linear polarization of 45 ∘ , while the second channel was polarized at 90 ∘ away by a linear polarization of (45 ∘ + 90 ∘ ). Both of the polarized channels were combined using a polarizer combiner. The proposed technique was investigated in both single-mode fiber (SMF) and dispersion-shifted fiber (DSF) with a 70 km fiber length and four channels. Through this approach, the FWM crosstalk significantly reduced and a good improvement was observed in system performance. The results confirm the robustness of the polarization technique against the FWM crosstalk and show that the FWM crosstalk has no dangerous influence on the system performance, even at a high value of input power.

System Description and Theoretical Background
Figures 1(a)-1(b) describe the proposed and conventional system configuration of the transmitter and receiver. At the transmitter part, the array of continuous wave lasers (L 1 -L 4 ) is used to generate the carrier signal. The frequency of the first user is set to 193 THz, and the spacing between each user is 100 GHz. Each user is modulated with a 10 Gb/s data rate. Therefore, the total data rate of the system was 40 Gb/s. The array laser sources are connected to an external modulator. The external modulator comprised a Pseudo-Random Bit Sequence (PRBS), which is connected to a pulse generator to modulate the optical signals using an NRZ modulation format. It is then connected to the Mach-Zehnder modulator (MZM), which acts as an intensity modulator. In the proposed system simulation, each two channels are linearly polarized 90 ∘ apart and then combined together. As shown in Figure 1(a), the first channel is polarized using a linear polarization of ( ), while the second channel is polarized using a linear polarization of ( + 90 ∘ ). Each of the two channels is combined using a polarizer combiner that combines the two input signals to one output port. The polarization angle has been selected at = 45 ∘ . Then, the four signals are collected using a polarizer combiner with a 0 ∘ polarization angle. The combined signals pass through optical fiber with a 70 km length. In the conventional system Figure 1(b), the state of polarization of each transmitted channel is 0 ∘ . Two types of optical fiber were used such as SMF and DSF and the standard parameters of each one were in Table 1. At the receiver, the signal is demultiplexed. The signal is detected by a PIN photodiode for direct detection. It is then passed through the low-pass Bessel filter. Finally, the signal is then connected directly to the system performance analyzer, which is used to generate the graph.
The nonlinear light amplitude ENL describes the FWM light; FFWM = + − . The total nonlinear amplitude is [10]: where | (0)| ( = , , ) are the amplitudes at = 0. Relative polarization states can be represented by normalized Jones vectors | ⟩ , which are assumed to be maintained throughout the fiber. The orthogonal polarization effect on FWM efficiency can be classified into the following cases.
(3) In the case of | ⟩ = | ⟩ ⊥ | ⟩, both ⟨ | ⟩ = 0, ⟨ | ⟩ = 1; in this case the value of 2 111 = 1/4, so the square of the nonlinear amplitude now becomes  In a WDM system, the power transferred to new frequencies due to FWM after light propagation within a distance in the fiber can be estimated using equation [11]: where , , and are the input power values at central frequencies , , and , respectively. is the degeneracy factor that is equal to 3 for two-tone and 6 for three-tone systems, 111 is third-order susceptibility that is equal to 6 × 10 −15 (m 3 /w.s), eff is the effective area, is the speed of light, is the laser wavelength, is the fiber loss coefficient, is the total fiber length, is the refractive index of the fiber, and eff is the nonlinear effective length that can be calculated using the following equation: The efficiency ( ) of four-wave mixing is given by [3] where Δ represents the phase mismatch and may be expressed in terms of signal frequency differences: where is the fiber chromatic dispersion and / is a derivative dispersion coefficient of the optical fiber. The right term of (6) and (7) where (Δ , Δ ) is the channel spacing. Under the effect of polarization, FWM efficiency becomes The Scientific World Journal where FWM(polarized) is FWM efficiency attained by polarization technique. 1111 is a factor that represents polarization dependency of the FWM process and changes from 0 to 1 according to SOP between channels, as shown in (1) to (3). is the total number of channel and is the FWM efficiency in the conventional system.
Using (8), FWM efficiency ( ) can be rewritten as follows: 6 The Scientific World Journal By substituting (10) into (9), we can derive (11) as the following: With the polarization effect, FWM power in (8) can be modified as follows: By substituting (11) into (12), the general FWM power is as follows: In the Gaussian approximation, [8,9], error probability is written as To calculate system performance under the effect of FWM, shot, and thermal noises, we used the following equations: where is the factor, ( ) IM is the effective FWM crosstalk in intensity modulation-direct modulation transmission, is the FWM power generated at frequency fs from a frequency combination satisfying + − = , where = , is the FWM power at identical and , where = ̸ = , is the received power at the receiver, is the electron charge (1.6×10 −19 ), th is thermal noise, and sh is the shot noise.
To calculate the received power and to achieve a given BER = 10 −9 , = 6, without FWM, and ( ) IM = 0, (15) is modified as follows: The effect of shot and thermal noises can be neglected because these noises have smaller values than those of FWM noise. can be obtained using (15) as follows: BER is calculated using the following equation:

Analysis Results and Discussions
The proposed polarization technique was compared with the conventional method (without using the polarization) and examined with SMF and DSF. The comparison was conducted at an input power range of 2 dBm to 14 dBm as follows.

Effect of Proposed Technique on FWM Behavior and BER Using DSF.
For further investigation, the proposed polarization technique was tested with DSF, using the standard parameters in Table 1. In the DSF, FWM can strongly influence the transmission performance, because most of the FWM interaction occurs near zero dispersion wavelengths.
At a high input power, the FWM crosstalk increased dramatically and superimposed with the transmitted channels. Figure 5 shows that for the conventional system at a 2 dBm input power, the FWM power was −52 dBm. At a high input power of 14 dBm, the FWM power significantly increased with the number of FWM interfacing with channels, and