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Abstract
We present a model that is capable of describing the polarization crosstalk in terms of nonlinear polarization rotation (NPR) of semiconductor optical amplifier (SOA) in wavelength conversion for polarization multiplexing (Pol-Mux) orthogonal frequency division multiplexing (OFDM) signal. Then a simple nonlinear polarization crosstalk canceled wavelength conversion (NPCC-WC) with polarization-diversity four-wave mixing (FWM) is proposed. The effectiveness is achieved successfully in the proposed wavelength conversion for the Pol-Mux OFDM signal by using simulation. In addition, we studied the influence of several system parameters on the performance, including the power of the signal, SOA’s injection current, frequency spacing, polarization angle of signal, laser linewidth, and modulation order. The results show that the proposed scheme has better performance due to crosstalk cancellation than the conventional scheme, such as wider wavelength tunability, lower polarization sensitivity, and wider laser linewidth tolerance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
5 June 2023: A typographical correction was made to the author affiliations.
1. Introduction
With the rapid development of new applications such as cloud computing, the internet of things, and artificial intelligence, the data traffic capacity of communication networks is increasing explosively. To satisfy this traffic growth and demand, various advanced modulation formats, such as polarization multiplexing (Pol-Mux) and orthogonal frequency division multiplexing (OFDM) techniques have received considerable attention. Polarization multiplexing technology doubles the capacity by transmitting two independent data information on two orthogonal polarization modes [1–3]. OFDM has high spectral efficiency, and dispersion tolerance and can flexibly allocate subcarrier power and bit rate, saving system resources [4–6].
All-optical signal processing functions in the optical network node, including optical cross-connection, optical add-drop multiplexing, regeneration, and all-optical wavelength conversion, would play a key role in enabling flexible routing between networks and improving the utilization of network capacity [7,8]. In particular, wavelength conversion is very important to solve the congestion caused by wavelength competition [9,10]. Several schemes, such as cross-absorption modulation (XAM) in electrical absorption modulators (EAM) [11], cross-gain modulation (XGM) [12], and cross-phase modulation (XPM) [13] in semiconductor optical amplifier (SOA), the four-wave mixing (FWM) in highly nonlinear fiber (HNLF) [14,15] and SOA [16–19], the sum frequency generation (SFG) and difference frequency generation (DFG) nonlinear effect in periodically-poled lithium-niobate (PPLN) [20], can be used to realize all-optical wavelength conversion (AOWC). Among these schemes, FWM is considered to be a promising scheme since it is independent of data rate and modulation format and can achieve multicast. Compared with HNLF [21,22], schemes based on FWM in SOA for Pol-Mux OFDM signals have the advantages of lower power consumption and easier integration [23,24]. However, conventional AOWC based on FWM in SOA for PM-OFDM signal has one issue, the Pol-Mux signals suffer from degradations arising from the interplay between the residual polarization dependency of the conversion processes and the nonlinear crosstalk among the Pol-Muxed channels [25]. It is a challenge to implement transparent AOWC of PM-OFDM signal, and the crosstalk needs to be eliminated before the polarization beam splitter (PBS) in the direct detection system.
In this paper, a theoretical model describing nonlinear polarization crosstalk in wavelength conversion based on FWM in SOA for Pol-Mux OFDM signals is deduced by using the Jones vector and 2 × 2 matrices. Then a simple nonlinear polarization crosstalk canceled (NPCC) scheme with polarization diversity FWM is proposed, where two FWM processes are implemented independently in SOA, so the residual polarization dependency induced polarization crosstalk is canceled out. In this way, it is not necessary to use the radio frequency (RF) power as the feedback signal in [26] to monitor the polarization state change of the optical signal and the crosstalk at the receiver, or use a complex receiver in [27,28].
This paper is organized as follows. In Section 2, we describe the co-polarized pumps scheme and NPR effect in SOA. Then we present the theoretical model for nonlinear polarization crosstalk that is based on the above principles and analyze the origin of the polarization crosstalk. An optimal scheme is proposed. In Section 3, we conduct simulations to verify the effectiveness of the proposed scheme. Finally, the conclusions are drawn in Section 4.
2. Principle and theoretical model
2.1 Principle of wavelength conversion based on FWM with copolarized pumps
Figure 1 shows the schematic diagram of wavelength conversion based on co-polarized pumps FWM for Pol-Mux OFDM signal. Two pumps (pump1 and pump2) and signal lightwave generated from three continuous wave lasers (CW1, CW2, and CW3) can be expressed as: ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _i}({\omega _i},t) = {A_i}\exp j({k_i}z - {\omega _i}t + {\varphi _i})(i = 1,2,3)$, respectively. Here, ${A_i}$, ${k_i}$, ${\omega _i}$, and ${\varphi _i}$ are the amplitudes, wave vectors, angular frequencies, phases of the pumps, and the signal, respectively. Figure 1(a) represents the polarization state at the SOA input, and shows that the two pumps are co-polarized and aligned with the transverse electric (TE) or x mode of SOA’s waveguide. Signal lightwave is assumed, to have a polarization angle $\beta$ to PBS1’s principal axes, so that the input signal lightwave is separated into two orthogonal lightwaves. Electrical OFDM signals ${s_1}(t)$ and ${s_2}(t)$ are modulated on the two orthogonal lightwaves by two Mach-Zehnder modulators (MZMs) respectively. The OFDM signals in the time domain can be expressed as:
Fig. 1. Schematic diagram of wavelength conversion for Pol-Mux OFDM signal based on copolarized-pumps FWM. (a) input polarization state of pumps and signal lightwave; (b) spectrum before SOA; (c) spectrum after SOA. CW: continuous wave; PC: polarization controller; PBS: polarization beam splitter; MZM: Mach-Zehnder modulator; PBC: polarization beam combiner; SOA: semiconductor optical amplifier; OC: optical coupler; PD: photoelectric detector.
Pumps 1, 2, and the modulated signal lightwave are coupled by an optical coupler (OC) and then injected into SOA, shown by the spectrum in Fig. 1(b). According to the description of the FWM effect in SOA [16][19], pairs of input tones are imagined to beat to produce gain and phase gratings, which modulate or scatter the input fields to generate upper and lower sidebands. In this case, three beating gratings generated, including $|{{\omega_1} - {\omega_2}} |$, $|{{\omega_1} - {\omega_3}} |$ and $|{{\omega_2} - {\omega_3}} |$, and each grating will modulate the input fields to generate new fields. In this case, the beating grating $|{{\omega_1} - {\omega_2}} |$ will modulate signal ${\omega _3}$ and the beating grating $|{{\omega_2} - {\omega_3}} |$ will modulate pump1${\omega _1}$. Consequently, two converted signal at frequencies of ${\omega _1} - {\omega _2} + {\omega _3}$ and ${\omega _3} - {\omega _1} + {\omega _2}$ are generated on both sides of the signal respectively. The spectrum at the SOA output shown in Fig. 1(c) shows that the output optical frequency of interest in this scheme is ${\omega _c} = {\omega _1} - {\omega _2} + {\omega _3}$.
2.2 Nonlinear polarization rotation (NPR) effect in SOA
The incoming arbitrarily polarized electric field can be decomposed into TE or x (parallel to the waveguide direction) and transverse magnetic (TM) (perpendicular to the waveguide direction) or y modes, which correspond to the two principle axes of the SOA. We define two coordinate systems, assuming that $x,y,z$ is the reference coordinate system and $x^{\prime},y^{\prime},z^{\prime}$ is the SOA’s inherent coordinate system, the $x^{\prime}$ axis makes an angle $\theta$ with x axis, and $z^{\prime}$ axis is in the same direction as $z$ axis, as shown in Fig. 2. The electric field vector of an arbitrarily polarized light wave can be described as:
Due to the waveguide structure of SOA in a practical application being anisotropic, by solving the propagation equation, the polarization-dependent gain for TE and TM polarization directions can be obtained [29]:
where, ${\xi ^{TE}}$/${\xi ^{TM}}$ is the gain coefficient for the TE/TM mode, ${n_c}(z,t)$ denotes the number of electrons in the conduction band, ${n_x}(z,t)$ and ${n_y}(z,t)$ denote the number of holes involved in the x and y transitions, and ${N_0}$ is the total number of electronic states involved in the optical transition. The phase rotation for TE and TM modes due to the TE/TM asymmetry in the confinement factors and gain of SOA can be expressed as [29]:Here ${\alpha ^{TE}}$, ${\alpha ^{TM}}$ are the phase modulation coefficient, and $v_g^{TE}$, $v_g^{TM}$ are the corresponding group velocity taken at the central frequency of the wave, ${\Gamma ^{TE}}$, ${\Gamma ^{TM}}$ are the confinement factor. As these components ($E_{0x}^{\prime}$ and $E_{0y}^{\prime}$) pass through the SOA, polarization-dependent phase rotation introduces the corresponding component ($E_{1x}^{\prime}$ and $E_{1y}^{\prime}$) can be expressed as:
The x and y components (${E_{1x}}$ and ${E_{1y}}$) of the output state of polarization of the $x,y,z$ coordinate system can also be obtained from $E_{1x}^{\prime}$ and $E_{1y}^{\prime}$ by rotating the ($x^{\prime}$, $y^{\prime}$) coordinate system clockwise by an angle $\theta$ as:
Thus, the effect of nonlinear polarization rotation in SOA on a given state of polarization by multiplying the input Jones vector by three 2 × 2 matrices can be obtained:
The Jones matrix of the SOA is given by:
It can be seen from Eq. (16) that the Jones matrix can describe how the polarization of the light wave changes when it is passed through the SOA.
2.3 Theoretical model for polarization crosstalk
For a polarization multiplexing system, the input light wave can be decomposed into two orthogonal polarization states of light to simultaneously transmit two independent data information. From the preceding analysis, the birefringence effect in SOA introduces the phase rotation on the TE and TM modes. After FWM, the orthogonality of two polarized components can be broken, resulting in the nonlinear crosstalk among the two channels. Here, we present a model that is capable of describing the nonlinear polarization crosstalk in terms of NPR of SOA in wavelength conversion based on FWM for the Pol-Mux OFDM signal, as shown in Fig. 3.
Fig. 3. The model of nonlinear polarization crosstalk for wavelength conversion of Pol-Mux OFDM signal
The optical fields of SOA’s input (pumps1, 2, and signal lightwave) ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{1i\,}}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{2i\,}}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{3i\,}}$ can be represented in terms of a 2 × 1 matrix as follows:
The effect of SOA on a given state of polarization can be obtained by multiplying the input Jones vector by a 2 × 2 Jones matrix of SOA. Thus, the output optical fields of SOA can be expressed by the Jones matrix: ${\mathbf{E}_{1o}} = \mathbf{J}{\mathbf{E}_{1i}}$, ${\mathbf{E}_{2o}} = \mathbf{J}{\mathbf{E}_{2i}}$ and ${\mathbf{E}_{3o}} = \mathbf{J}{\mathbf{E}_{3i}}$. Thus, pump1 can be written as:
The electric field vector of pump1 can be described in terms of two orthogonal polarized components:
Likewise, pump2 and signal lightwave can be expressed as:
According to the description of the principle of FWM, the beating grating $|{{\omega_1} - {\omega_2}} |$, modulates signal ${\omega _3}$ and the beating grating $|{{\omega_2} - {\omega_3}} |$ modulates pump1${\omega _1}$, the converted signal at the frequency of ${\omega _c} = {\omega _1} - {\omega _2} + {\omega _3}$ generates [19]. The amplitudes of gratings $|{{\omega_1} - {\omega_2}} |$ and $|{{\omega_2} - {\omega_3}} |$ can be expressed as:
Here $r({\omega _1} - {\omega _2})$ and $r({\omega _3} - {\omega _2})$ are relative conversion-efficiency coefficient, which is inversely proportional to the frequency interval [19]. In our system, since the frequency spacing between pumps is much smaller than the frequency spacing between signal and pump 2, $r({\omega _1} - {\omega _2}) \gg r({\omega _3} - {\omega _2})$. The converted signal can be expressed as:
It can be shown from Eq. (26) that the x and y components of the converted lightwave separated by a PBS both carry ${s_1}(t)$ and ${s_2}(t)$. That is, the converted lightwave suffers from nonlinear crosstalk between the polarization-multiplexed channels due to polarization rotation, and will lead to what is termed intensity noise after direct detection. Therefore, the crosstalk-induced degradation should be eliminated before PBS in the Pol-Mux system.
2.4 Nonlinear polarization crosstalk canceled wavelength conversion
A scheme of nonlinear polarization crosstalk canceled wavelength conversion (NPCC-WC) for Pol-Mux OFDM signals is shown in Fig. 4. Figure 4 (a) shows that two pumps are polarized at an angle 45° to the PBS2’s principle axis by using PC1 and PC2, so that half of the pumps power launches to each SOA. The signal lightwave is polarized at an angle $\beta$ to the PBS1’s principle axis using PC3, and the signal lightwave is separated into two streams with orthogonal polarization states. The OFDM signals ${s_1}(t)$ and ${s_2}(t)$ are modulated on the orthogonal polarization states and recombined through PBC1. Pumps and the modulated signal lightwave are coupled by an OC and then launched into PBS2, which provides the polarization diversity, and sent to two SOAs. In this way, at the two output ports of PBS2, two pumps and signal lightwave are co-polarized, and single-polarization (x or y component) FWM performs in each SOA to give rise to the converted signals, which are shown in Fig. 4(c) and Fig. 4(e). The outputs coming from two SOAs are recombined by PBC2. According to the above analysis, in the proposed scheme, there is only ${\varphi ^{TE}}$ and ${s_1}(t)$ or ${\varphi ^{TM}}$ and ${s_2}(t)$ i input into each SOA. As a result, Eq. (26) transforms to:
Fig. 4. Scheme of the NPCC-WC for Pol-Mux OFDM signal. (a) Input state of polarization; Optical spectra (b) before and (c) after SOA1, (d) before and (e) after SOA2.
Equation (27) suggests that the converted Pol-Mux OFDM signals can be split by PBS3 without polarization crosstalk. This can be explained that in the proposed scheme, polarization independency of the conversion processes maintains the ideal orthogonal condition between the polarization-multiplexed channels, which results in nonlinear crosstalk to be canceled.
3. Simulation and results
3.1 Simulation setup
To verify the theoretical analysis, the simulation platforms of the conventional scheme and the proposed scheme are built respectively by Optisystem software. Three CW lasers have been used to generate two parallel pumps and signal lightwaves. The frequency and power of pumps and signal lightwave are 193.24THz, 193.2THz, 193.05THz, 10 mW, 10 mW and 0.2 mW, respectively. OFDM signal is generated offline by MATLAB program. Figure 5(a) shows the digital signal processing (DSP) for OFDM modulation at OFDM transmitter. A serial pseudo-random binary sequence (PRBS) with a length of 55296 bits is transformed into parallel data steam by serial to parallel conversion. A parallel 216-bit sequence is mapped into 54 16-QAM complex-valued symbols, which are modulated to the 1st to 54th subcarriers (SCs). The direct-current (DC) subcarrier, the Nyquist SC and other 19 high-frequency SCs are set as 0. The 54 negative-frequency SCs are the complex conjugates of the 54 positive-frequency SCs, which makes the input vector to 128-point IFFT satisfy Hermitian symmetry for the real-valued OFDM signal generation. A 16-point cyclic prefix (CP) is applied to avoid inter-symbol interference (ISI) and a training sequence (TS) is inserted for timing synchronization and channel estimation. The OFDM frame consists of a TS and 256 OFDM symbols. The effective sampling rate is 20 GSa/s, so the bandwidth of OFDM signal is [(54 + 1)/128] × 20 GHz ≈8.59 GHz, and the net signal bit rate is (256 × 54 × 4)/[(1 + 256)(128 + 16)] × 20 ≈30 Gbit/s. The injection currents of two SOAs are both 0.32A. The differential gain of SOA is 2.78 × 10−20 m2, the carrier density at transparent is 1.4 × 1024 m3 and the initial carrier density is 3 × 1024 m3. The optical spectra before and after FWM are depicted in Fig. 6. Converted signal at 193.01THz is selected by an optical band-pass filter (OBPF), and then separated by PBS3. The separated optical signals are directly detected by PDs before sending into OFDM receivers. The offline DSP procedure at OFDM receiver mainly includes TS-based symbol synchronization, serial to parallel conversion, the removal of CP, fast Fourier transformation (FFT), channel equalization and 16-QAM de-mapping et al.
3.2 Results and discussion
The system performances of the conventional scheme and proposed scheme are tested under different conditions. Figure 7 shows the BER performance versus signal power for conventional and proposed schemes. In the single polarization conversion, we tested the BER and constellation diagrams of received signal on x/y polarization state when there is no data on y/x polarization state. As we can see from the Fig. 7(a), in conventional scheme, the single polarization conversion performs well, while Pol-Mux OFDM signal conversion suffers from some apparent distortion that leads to BER increase. However, in the proposed scheme as shown in Fig. 7(b), the performance difference between single polarization conversion and PM-OFDM signal conversion is very small, which implies that the nonlinear polarization crosstalk is canceled. It can also be seen from the figure that the influence of the signal optical power on the system: as the signal power increases from 0.2 mw to 0.5 mW, the third-order nonlinear effect also enhances, which results in the increase of power of the converted signal and decrease in BER. Continue to increase the signal power, due to the gain saturation effect of SOA, which result in a decrease of conversion efficiency and increase of BER. On the other hand, FWM occur among subcarriers of optical OFDM signals, with the continuous increase of signal optical power, the increase of in band noise induced by enhanced FWM effect among subcarriers leads to the increase of BER.
Figure 8 shows the BER performance versus SOA’s injection currents for conventional and proposed schemes. Because as the current increases, the SOA’s gain increases, and power of the converted signal increases, resulting in the decrease of BER with the increasing current. But as the SOA is saturated, the SOA’s gain and the power of converted signals will not continue to increase. The optimal SOA injection current is nearly 0.32 A. Compared with Fig. 8(a)-(b), it can be seen that the BER in the polarization diversity scheme is lower than that in the conventional scheme. Moreover, in conventional scheme, single polarization conversion performs better than Pol-Mux OFDM signal conversion, because the latter is affected by polarization crosstalk, while in proposed scheme, the BER performance is improved due to the cancellation of polarization crosstalk.
Fig. 8. BER as a function of injection currents of SOA in (a) conventional scheme and (b) proposed scheme.
Figure 9 shows the BER performance versus frequency spacing between pumps for conventional scheme and proposed scheme. Since the conversion efficiency function $r({\omega _1} - {\omega _2})$ is inversely proportional to the frequency spacing ${\omega _1} - {\omega _2}$, the power of the converted signals degrade with the increase of frequency spacing, and the BER becomes worse with the increase of the frequency spacing. Besides, compared Fig. 9(a) with Fig. 9(b), we can see that the BER of proposed scheme is lower than that of conventional scheme, because polarization crosstalk is cancelled in proposed scheme. And when the BER achieves the forward error correction (FEC) limitation of 3.8 × 10−3, the frequency spacing ${\omega _1} - {\omega _2}$ varies from 0.03 THz to 0.045THz in conventional scheme, while the frequency spacing varies from 0.03 THz to 0.06THz in proposed scheme. It implies that the proposed scheme has wider wavelength tunability than that of the conventional scheme.
Figure 10 shows the BER curves versus polarization angle of signal in the conventional and proposed schemes. We can see that the variation trend of the BER of x-polarized signal is opposite to that of y-polarization state. The BER is below the FEC limitation of 3.8 × 10−3 when polarization angle varies from 40° to 60° in conventional scheme and from 25° to 65° in proposed scheme, and reaches minimum at 45° because the converted signals power reach the maximum. The BER performance of proposed scheme is better than that of the conventional scheme. This fact is due to that the polarization crosstalk induced impairment can be effectively reduced, and thus the proposed scheme is less sensitive to polarization.
Fig. 10. BER as a function of polarization angle of signal in (a) conventional scheme and (b) proposed scheme.
Figure 11 shows the BER curves versus laser linewidth in conventional scheme and proposed scheme. It can be seen that compared to the conventional scheme, such proposed scheme can effectively reduce polarization crosstalk both in case of 16-QAM and 64-QAM modulation. Moreover, the BER performance and constellation is deteriorates with the increase of laser linewidth, as the laser phase noise characterized by laser linewidth cause a phase rotation of all subcarriers, which results in serious inter-carrier interference (ICI) [30]. In addition, higher-order modulation with small distance between constellation points is more sensitive to laser phase noise. In convention scheme, 16-QAM format requires a laser linewidth of 9 MHZ at BER of 3.8 × 10−3 Hard-Decision FEC (HD-FEC) limitation, 64-QAM format may be require a narrow linewidth laser (nearly few kHz), but commercial lasers cannot meet the requirement. In proposed scheme, 16-QAM format requires a laser linewidth of 16 MHZ at BER of 3.8 × 10−3, and 64-QAM format a laser linewidth of 5 MHZ requires at BER of the Soft-Decision FEC (SD-FEC) limitation of 2.7 × 10−2. Consequently, the tolerable linewidth for 16-QAM and 64-QAM format are increased by nearly 7 MHz and 5 MHz respectively by using the proposed scheme when compared to conventional scheme.
4. Conclusion
In this paper, a theoretical model that is capable of describing nonlinear polarization crosstalk in AOWC for Pol-Mux OFDM signals is presented, and we found that polarization crosstalk induced by NPR of SOA could cause severe impairment. Then a simple NPCC-WC with polarization diversity configuration is proposed for Pol-Mux OFDM signals. Simulations have demonstrated that compared to the conventional scheme, the proposed scheme provides performance improvements under various conditions, such as the power of signal, injection currents, frequency spacing, polarization angle of signal, and laser linewidth. Moreover, due to crosstalk cancellation, the proposed scheme has wider wavelength tunability (0.03THz), lower polarization sensitivity (25°∼65°), and wider linewidth tolerance for 16-QAM (16 MHz) and 64-QAM (5 MHz) compared to the conventional scheme. All results suggest that the performance of AOWC for the Pol-Mux OFDM signal is improved by applying the proposed model system. The proposed system also can be applied to AOWC for polarization multiplexing signals with different modulation formats, which may be a promising candidate to realize future transparent all-optical networks.
Funding
National Natural Science Foundation of China (61701180); National Key Research and Development Program of China (2018YFB1801503); Natural Science Foundation of Hunan Province (2020JJ4433, 2021JJ40360).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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