PDM-SHD NFDM transmission with envelope-detection feedback polarization tracking and optical injection locking

Junda Chen; Zihe Hu; Zheng Yang; Yizhao Chen; Mingming Zhang; Yifan Zeng; Weihao Li; Chen Cheng; Ziwen Zhou; Xuefemg Wang; Deming Liu; Ming Tang; Ming Tang

doi:10.1364/OE.486833

1. Introduction

Coherent detection techniques with large capacity have been widely employed in optical fiber communications systems for decades, owing to the development of digital signal processing (DSP) algorithms [1]. One major foundation of the coherent DSP algorithms is that the optical link is considered as a linear channel while the impact of fiber nonlinearity is ignored. As a consequence, the fiber optical transmission capacity is limited by nonlinear distortions, especially the nonlinear Kerr effect, which is also called the nonlinear Shannon limit [2]. If we assume the optical fiber a nonlinear lossless channel, based on the inverse scattering transform (IST) and nonlinear Schrodinger equation (NLSE), the nonlinear Fourier division multiplexing (NFDM) has been widely investigated [3]. The information is modulated on the nonlinear spectrum by using nonlinear Fourier transform (NFT) [4], where the transmission in standard single mode fiber (SSMF) can be expressed with a single linear transfer function. In addition to the transmission form, the corresponding NLSE accounts for both the self-phase modulation (SPM) effect and the dispersion effect. As the Kerr effect is now considered to be a fundamental element of the fiber link, rather than the “nonlinear noise,” it is theoretically possible to completely eliminate the impact of such noise through the utilization of NFT algorithms [4–6]. The nonlinear spectrum $\lambda $ can be divided into a continuous spectrum which is located on the real axis of the complex plane, and a discrete spectrum which includes N discrete eigenvalues in the upper half of the complex plane and is also known as eigenvalue communication. Both spectra can be used in NFDM systems separately or jointly [5,7,8]. The derivation of the NFT algorithm solely considers the intra-channel nonlinear effect, specifically the SPM effect, while disregarding inter-channel nonlinear effects like cross-phase modulation (XPM). This limitation presents challenges in establishing WDM-NFDM transmission. Significant progress has been made by using advanced algorithms, such as deep learning [9]. However, to simplify the complexity, in this paper, we still focus on the single wavelength eigenvalue communications.

Due to the constraints of Gordon-Haus effect, the baud rate of eigenvalue communications is normally limited to a few GBaud [3,10], which is much less than the baud rate of linear spectrum communications (up to 100 GBaud) [11]. For the common coherent communication systems based on the intradyne detections (IDs), the transmitter laser's linewidth and frequency impairments would be converted to phase noise and carrier frequency offset (FO) at the receiver, and cause a cumulative phase change on received signal, which is proportional to the duration of each symbol (inversely proportional to the baud rate) [1]. Consequently, the impact of phase noise and FO can be hardly delt with by DSP in eigenvalue communications, especially under high-order modulation format conditions such as 64-ary amplitude phase shift keying (APSK) [12,13]. An alternative method is to use the ultra-narrow linewidth laser such as fiber laser (FL) as the light source, which is widely used in NFDM transmissions [7,14]. However, the use of FL comes with expensive cost. Therefore, we introduce the self-homodyne detection (SHD) into the eigenvalue communication link to overcome the impact of laser linewidth and FO by transmitting a pilot carrier originating from the transmitter laser to the receiver [15]. Furthermore, the pilot can be transmitted through the orthogonal polarization state of signal in standard single mode fiber (SSMF) [16], so that the polarization division multiplexed-SHD (PMD-SHD) eigenvalue communication system is established [17].

Considering the pilot carrier often provides the coherent receiver's local oscillator (LO) directly in SHDs [18], which should be at least 10 dB greater than the signal power [1] and requires a great carrier to signal power ratio (CSPR). The fiber nonlinearity such as stimulated Brillouin scattering (SBS) makes it difficult to transmit a high-power continuous-wave laser over long-haul transmission [19]. Therefore, the emission power of the pilot carrier should be restricted to realize a low CSPR PMD-SHD system, and an amplifier for the pilot should be utilized at the receiver which can be achieved by the optical injection locking (OIL) with an external light shined into a laser cavity [20–22]. As an optical frequency and phase synchronization technique, the frequency and phase of the OIL output light is identical to the input one, making OIL a high-gain amplifier with a robust filter which can amplify only the pilot carrier. By using the OIL, a phase noise variance of only 0.2 degrees can be achieved to support high order modulation coherent optical transmission such as 256-quadrature amplitude modulation (QAM) [23]. Up to −65 dBm low injection power has also been achieved [24], which is compatible to regenerate the LO of the low CSPR PMD-SHD system.

On the other hand, the random birefringence of SSMF results in a random rotation of the state of polarization (RSOP), making it difficult to separate the pilot carrier from the signal at the receiver of the PDM-SHD system. One promising solution is to adopt an adaptive polarization controller (APC) to track the time-varying RSOP at the receiver. There are some commercial [25] or home-made [26] products, whose tracking feedback is the optical power in a certain polarization state. In this way, it may track polarization by ensuring that the feedback maintains the optical power at its maximum or minimum value [27]. However, these products require high CSPR to provide a high polarization extinction ratio (PER), which is incompatible with our suggested low CSPR scheme. Therefore, a CSPR unrestricted APC is still a key challenge for long-haul PDM-SHD eigenvalue communication.

Our prior investigations have demonstrated the feasibility of constructing a low-CSPR-SHD communication system for the long-haul eigenvalue or conventional communication system to mitigate the impact of laser frequency impairment [17,28]. In this paper, a CSPR-unrestricted APC structure is proposed, based on which a PDM-SHD eigenvalue communication system is demonstrated with 16-APSK and 64-APSK constellations. The CSPR-unrestricted APC is realized by detecting the intensity fluctuation instead of direct optical power of a certain polarization state as the feedback with an envelope detector. Building upon the well-established single-polarization eigenvalue communication system, the enhancement of using PDM-SHD structure is investigated. The rest of this paper is organized as follows. Section 2 reviews the theory and basic property for the NFT. In section 3, the structure and performance of our proposed envelope-detection APC and OIL structure are introduced. It also showed that the distortion of signal-signal beat interference (SSBI) can be inhibited by the use of OIL. Section 4 describes the experimental structure and DSP flow. The experiment solely focuses on the NFDM modulation format for communication, which has demonstrated advantages over conventional communication approaches, both theoretically and experimentally [4–6]. In section 5, the impact of long-haul fiber link on the APC structure is investigated, and modifications are made. The influence of CSPR, FO, laser linewidth, transmission distance and launch power are also investigated. At last, the conclusions of this work are given in section 6.

2. Principle of the NFT theory

The evolution of a slow-varying signal in SSMF can be well modeled by the NLSE [4]:

(1)$$\frac{{\partial E(\tau ,\ell )}}{{\partial \ell }} ={-} i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}E(\tau ,\ell )}}{{\partial {\tau ^2}}} + i\gamma |E(\tau ,\ell ){|^2}E(\tau ,\ell ), $$

where $E(\tau ,\ell )$ denotes the scalar electric field, $\ell$ denotes the propagation distance, $\tau $ stands for the retarded time co-moving with the group velocity of the envelope, ${\beta _2}$ denotes the group velocity dispersion and $\gamma $ denotes the nonlinear parameter. As a linear partial differential equation (PDE), the NLSE can be solved by using the IST. In this way, the NLSE can be simplified by introducing the normalized variables

(2)$$q = \frac{E}{{\sqrt P }},\quad t = \frac{\tau }{{{T_0}}},\quad z ={-} \frac{\ell }{\mathrm{{\cal L}}}, $$

where $P = |{{\beta_2}} |/({\gamma T_0^2} )$, $\mathrm{{\cal L}} = 2T_0^2/|{{\beta_2}} |$ and ${T_0}$ is a free normalization parameter. The normalized NLSE is derived as

(3)$$i\frac{{\partial q(t,z)}}{{\partial z}} ={\pm} \frac{{{\partial ^2}q(t,z)}}{{\partial {t^2}}} + 2|q(t,z){|^2}q(t,z). $$

The normalized NLSE is a case of the Zakharov–Shabat spectral problem (ZSP), which can be analyzed with initial conditions [24]

(4)$$\frac{\partial }{{\partial t}}v = \left[ {\begin{array}{{cc}} { - j\lambda }&{q(t,z)}\\ { - {q^\ast }(t,z)}&{j\lambda } \end{array}} \right]v,\quad \mathop {\lim }\limits_{t \to - \infty } v(t,\lambda ) = \left[ {\begin{array}{{c}} 1\\ 0 \end{array}} \right]{e^{ - j\lambda t}}, $$

where $\lambda $ and v are eigenvalue and eigenvector respectively with $\textrm{v} = {[{{v_1},{v_2}} ]^T}$. If the signal power is distributed in the time interval of $[{{T_1},{T_2}} ]$, the scattering data can be defined as $a(\lambda ) = {v_1}({{T_2},\lambda } ){e^{j\lambda {T_2}}},b(\lambda ) = {v_2}({{T_2},\lambda } ){e^{ - j\lambda {T_2}}}$. Thus, the scattering data are two time-invariant parameters, which can be utilized to reconstruct the signal $q(t )$ uniquely. The nonlinear spectrum of a time domain signal $\textrm{q}(t )$ is defined as

(5)$$\begin{aligned} {Q_{c}}(\lambda ) &= b(\lambda )/a(\lambda ),\quad \lambda \in {\mathbb R}\\ {Q_{d}}({{\lambda_i}} )&= b({{\lambda_i}} )/{a^\prime }({{\lambda_i}} ),\quad {\lambda _i} \in {{\mathbb C}^ + },i = 1,\ldots ,N \end{aligned},$$

where ${Q_{c}}(\lambda )$ is the continuous spectrum, ${Q_{d}}({{\lambda_i}} )$ is the discrete spectrum while $a({{\lambda_i}} )= 0$, and $a^{\prime}({{\lambda_i}} )$ is the derivative of $a({{\lambda_i}} )$. Generally, the signals are modulated on ${Q_{c}}(\lambda )$ or ${Q_{d}}({{\lambda_i}} )$ while using continuous or discrete spectrum modulation. However, both $a(\lambda )$ and $b(\lambda )$ would be affected by noise during transmission. Alternatively, it has been demonstrated that the signal modulated directly on $b(\lambda )$ instead of $Q(\lambda )$ would reduce the influence from noise [29]. Hence, the evolution of scattering coefficients along transmission follows the rules below:

(6)$$a(\lambda ,z) = a(\lambda ,0),\quad b(\lambda ,z) = b(\lambda ,0)\exp ({ - 4j{\lambda^2}z} ). $$

Although SSMF is known as a nonlinear channel, Eq. (6) shows that in the nonlinear frequency domain, the evolutionary process of scattering data $b(\lambda )$ in SSMF can be described by a single linear transfer function as $H(z) = {e^{ - 4j{\lambda ^2}z}}$, similar to the effect of dispersion on the linear frequency domain. In other words, the nonlinear effect of the optical fiber is converted to a linear phase rotation defined as Eq. (6). There are two fundamental properties of the NFT signal on the relationship between the time domain signal and the scattering data $q(t) \leftrightarrow b(\lambda )$:

(7)$$\begin{array}{c} {e^{ - 2j\omega t}}q(t) \leftrightarrow b(\lambda - \omega )\\ {e^{j\varphi }}q(t) \leftrightarrow {e^{ - j\varphi }}b(\lambda ) \end{array}, $$

where $\omega $ and $\varphi $ are the frequency shift and phase shift respectively. It shows that FO and phase noise of a time domain signal leads to directly negative effect on the NFT domain. Thus, the impact of laser linewidth and FO should be eliminated.

The discrete spectrum NFDM transmission can be considered as some kind of soliton transmission employing adjustable solitons [10]. During the soliton transmitting over a long-distance link, the Gordon-Haus effect demonstrates that periodic optical power amplification and spontaneous emission noise would cause soliton jitter and a superposition on the neighboring pulses [30]. Generally, the linear superposition of two soliton pulses is not a solitonic solution. Consequently, adjacent pulses will interact with one another as they propagate depending on their relative phase as an additional timing jitter, which significantly degrades performance. Therefore, a guard period is required between each pulse to prevent interference and assure reliable detection, which would contrarily limit the baud rate of discrete NFDM transmissions. The baud rate constraint of NFDM transmission places excessive demand on the FO estimation and CPE algorithms. One can use the SHD structure to overcome the impact of laser linewidth and FO.

3. Polarization tracking and LO regeneration

3.1 APC structure for PDM-SHD

One main challenge in PDM-SHD transmissions is to separate the signal from pilot carrier at the receiver, who are combined due to the random RSOP of the optical fiber. The most typical component to separate the signal from the pilot carrier is the APC, which typically uses the optical power difference between the signal and pilot carrier as the controlling feature, and tracks the polarization state by controlling the optical power in a certain polarization state to its maximum or minimum value [16,27,31]. Consequently, such components require the optical power of pilot carrier at least 10 dB greater than the signal power, rendering them unsuitable for long-distance SHD transmissions because of the nonlinear effects especially SBS in the SSMF (causing the limited optical power of pilot carrier).

In this paper, we propose a novel CSPR-unrestricted APC structure. Theoretically, the intensity of pilot carrier is stable, while the intensity of the signal fluctuates. Consequently, the direct-current (DC) and fluctuation properties can be utilized to separate the signal and carrier pilot. We can employ the fluctuation of the intensity envelope of a certain polarized light as the feedback of APC, instead of the power of the specific polarization state as usual. We deploy a high-speed DC-blocked photodetector (PD) to recognize the pilot carrier or signal by detecting the optical power fluctuation of a given polarization state. The proposed CSPR-unrestricted APC structure is duplicated in Fig. 1(a). The PDM light is first injected into a microcontroller-controlled Lithium niobate-polarization controller (LN-PC) to control the polarization state of the injected light. Then, a set of orthogonal polarized light is divided by using a polarization beam splitter (PBS) with integrated polarization rotator, so that the polarization state of the two output ports are X-axis aligned. After a polarization-maintained beam splitter (PM-BS) and a PD, an envelope detector (ADL5511) is used to generate the feedback signal. The envelope detector can convert the power fluctuation of the polarized light in to a feedback voltage, which is recognizable by the drive circuit. By controlling the LN-PC to minimize the output of the envelope detector, the carrier and signal are divided. Meanwhile, the circuit device is connected to an external trigger, enabling the APC to operate in a loop structure.

Fig. 1. (a). Proposed APC structure based on optical intensity fluctuation feedback. (b) OIL structure.

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To indicate the performance of APC, the PER is presented as:

(8)$$PER = 10 \times {\log _{10}}({{\raise0.7ex\hbox{${{P_X}}$} \!\mathord{/ {\vphantom {{{P_X}} {{P_{XY}}}}}}\!\lower0.7ex\hbox{${{P_{XY}}}$}}} ), $$

where ${\textrm{P}_{XY}}$ is the crosstalk power from X-polarization to Y-polarization and ${\textrm{P}_X}$ is the optical power of the X-polarization. Furthermore, we introduce CSPR to express the optical power ratio of the carrier and the signal as $\textrm{CSPR} = 10{\log _{10}}\left( {{{\left\langle {{P_{PC}}} \right\rangle } / {\left\langle {{P_{Sig}}} \right\rangle }}} \right)$, where ${\textrm{P}_{Sig}}$ is the signal optical power, ${P_{PC}}$ is carrier optical power and $\left\langle {{\kern 1pt} \cdot {\kern 1pt} } \right\rangle $ is the ensemble average. The blue curve in Fig. 2 depicts the normalized optical spectrum of the light injected into the APC at -7 dB CSPR (signal power greater than carrier power), which consists the spectrum of the pilot carrier and the signal. The red curve in Fig. 2 depicts the normalized spectrum of the light at the carrier output of the APC, which consists the spectrum of the pilot carrier and the residual signal power after APC. The pilot carrier is located in the center of the spectrum, while the rest is the signal. Each data point represents the optical power integral within a 100 MHz range. The spectra are normalized to their respective maximum power, which represents the optical power of the pilot carrier approximatively. Considering that the signal inhabits on the X-polarization of the light while the pilot carrier inhabits on the Y-polarization, the signal power represented by the blue line in Fig. 2 may be regarded as ${\textrm{P}_X}$, and the residual signal power in the carrier port, represented by the red line, can be regarded as ${\textrm{P}_{XY}}$. Consequently, the logarithmic optical power difference for the signal portion between the blue and the red line can be considered as the PER, which is approximately 18.2 dB in this experiment.

Fig. 2. Blue line shows the optical spectrum of PDM-SHD light injected into the APC with both signal and pilot carrier. Red line shows the optical spectrum of light measured at the carrier output of APC with the pilot carrier and the residual signal. Yellow shows the optical spectrum of light measured at the output port of the OIL laser.

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In contrast to conventional APC approaches, our suggested method uses signal and pilot carrier fluctuations as feedback, whereby the SHD connection is not restricted to the CSPR. After converted by the envelope detector, the device circuit just has to manage the minimum feedback voltage. By using current algorithms, our proposed APC structure might possibly provide high speed [32], high reliability, and rest-free [33] polarization control. Our suggested CSPR unconstrained APC structure permits a simple method for the use of PDM-SHD in long-distance fiber communications.

3.2 Optical injection locked laser based LO regenerator

For optical coherent detection, the power of LO is generally 10 dB higher than the signal [1]. Hence, the pilot carrier needs to be amplified due to the low CSPR. However, the widely used Erbium-doped fiber amplifier (EDFA) with broad gain spectral width acts on both the pilot carrier and amplifier spontaneous emission noise (ASE) at the receiver, severely reducing the received signal-noise ratio (SNR). As a result, a device with narrow-bandwidth filtering effect and frequency selective amplification is essential for the LO regeneration.

OIL is an optical frequency and phase synchronization technique based on photon-photon interaction with an external light shined into a laser cavity [20], which is widely used in SHD systems [22]. A common OIL setup is illustrated in Fig. 1(b), which contains a master laser and a slave laser. The light of the master laser is injected into the slave laser via a circulator as the seeding light, so that the phase of the slave laser will be forced to synchronize with the master laser. Within the proper range of the frequency deviation, the slave laser will track the frequency change of the master laser. The OIL laser can be considered as a low-cost, high-gain amplifier with an ideal narrow-bandwidth filter [21]. In this experiment, the slave laser is a customized distributed feedback (DFB) semiconductor laser with the output optical power of 13 dBm, and the injection light as the master is the carrier after APC. The output optical spectrum is also illustrated in Fig. 2. It can be demonstrated that the signal-to-carrier crosstalk ratio is decreased by 14.6 dB using OIL, resulting in a sharp difference of 32.8 dB considering APC and OIL.

On the other hand, considering a defective APC in the PDM-SHD systems without OIL, while APC leaves the remaining RSOP angle of $\theta $, the received electric signal of the integrated coherent receiver (ICR) ${{I}_{Detected}}$ and ${Q_{Detected}}$ can be given as [26]:

(9)$$\begin{array}{c} {{I}_{Detected}} = 2{k^2}R\left[ { - \sqrt \varepsilon \cos (\beta )A_{LO}^2 + \sqrt \varepsilon \cos (\beta ){{|{{A_S}(t)} |}^2}} \right] + I\\ {Q_{Detected}} = 2{k^2}R\left[ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sqrt \varepsilon \cos (\beta )A_{LO}^2 - \sqrt \varepsilon \cos (\beta ){{|{{A_S}(t)} |}^2}} \right] + Q \end{array}, $$

where ${I}$ and ${Q}$ represent the intended electric signal for final processing while the others are the distortions, with $k = {1 / {\cos \theta }}$, $\textrm{R}$ is the response ratio for balanced PD, $\varepsilon = {({\tan \theta } )^2}$ is the crosstalk parameter, ${\textrm{A}_S}$ and ${A_{LO}}$ are the amplitude of signal and LO, respectively. The first terms are carrier-carrier beat interference, which are DC components and can be easily filtered out. The second terms are the SSBI, which act as a nonlinear intensity interference. As the main interference in PDM-SHD system, SSBI is directly affected by the signal-to-carrier crosstalk ratio, which may be reduced by using OIL. Given that OIL is the carrier amplifier in the carrier (LO) path, we assume the gain of LO is $G_{\textrm{LO}}^{LO}$, the gain of signal in the carrier path is $G_{Sig}^{LO}$ with $\Delta \textrm{SCR = }10{\log _{10}}({{{G_{Sig}^{LO}} / {\textrm{G}_{LO}^{LO}}}} )$, where $\Delta \textrm{SCR}$ shows the decrease of signal-to-carrier crosstalk ratio introduced by the OIL which is -14.6 dB as shown in Fig. 2. On the other hand, the light in signal path is not amplified. Therefore, Eq. (9) becomes:

(10)$$\begin{array}{c} {{I}_{Detected}} = 2{k^2}R\left[ { - \textrm{G}_{LO}^{LO} \cdot \sqrt \varepsilon \cos (\beta )A_{LO}^2\textrm{ + G}_{Sig}^{LO} \cdot \sqrt \varepsilon \cos (\beta ){{|{{A_S}(t)} |}^2}} \right] + \textrm{G}_{LO}^{LO} \cdot I\\ {Q_{Detected}} = 2{k^2}R\left[ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{G}_{LO}^{LO} \cdot \sqrt \varepsilon \cos (\beta )A_{LO}^2 - \textrm{G}_{Sig}^{LO} \cdot \sqrt \varepsilon \cos (\beta ){{|{{A_S}(t)} |}^2}} \right] + \textrm{G}_{LO}^{LO} \cdot Q \end{array}. $$

Equation (10) shows that the gain on the SSBI term is $G_{Sig}^{LO}$, which is much smaller than the gain of signal term which is $G_{\textrm{LO}}^{LO}$. Thus, the impact of SSBI is greatly inhibited by the ratio of $\Delta \textrm{SCR}$ which is -14.6 dB in this experiment.

In general, there are several benefits by using OIL in PDM-SHD systems:

• To increase the LO power. The pilot carrier can be significantly amplified at the receiver, so that the carrier is no longer suffered from the nonlinear effects such as SBS.
• To be a strong narrow bandwidth amplifier and mitigate the effects of SSBI.
• The slave laser in OIL is a DFB laser without isolator, which is much cheaper than normally used EDFA in SHDs or external cavity laser (ECL) in IDs.

It should be noted that these advantages are not only applicable for eigenvalue PDM-SHD systems, but suitable for all PDM-SHD systems.

4. Transmission scheme

Figure 3(a) shows the experimental construction of the PDM-SHD eigenvalue communication system. At the transmitter, a DFB laser (∼10 MHz linewidth), an ECL (∼100 kHz linewidth), and a FL (∼100 Hz linewidth) are respectively used to compare system performance under varied laser linewidth conditions. Then, 95% of the optical power is given to an I/Q modulator (20 GHz bandwidth), which is driven by an arbitrary waveform generator (AWG, 93 GSa/s) and 5% is used as the pilot carrier. A polarization-maintaining optical delay line (ODL) is utilized in the carrier path to compensate for the optical path length difference between the signal and the carrier, while a variable optical attenuator (VOA) is employed to regulate the CSPR and a polarization rotator is used to rotate the X-polarization aligned pilot carrier light to the Y-polarization. The signal and carrier are then combined by a polarization beam combiner (PBC) on the two orthogonal polarizations of an SSMF. The long-haul transmission link is demonstrated with a loop structure. The loop consists of four 75 km SSMF spans and four EDFAs with a noise figure of 4.2 dB while an additional EDFA and a VOA are used to adjust the optical power in the loop. Two 100 GHz optical bandpass filters are used to filter out the out-of-band ASE noise. The loop is driven by two program-controlled acousto-optic modulators (AOMs) with zero frequency shifting. AOM1 has an insertion loss of 5.5 dB, while AOM2 has an insertion loss of 3 dB. The APC and OIL structures are then used to divide the carrier from the transmitted signal, which can also amplify the carrier to 13 dBm serving as the LO in the receiver, as shown in section 3. Since the APC is synchronized with the AOMs, it can stabilize the polarization state for a certain loop output. On the signal link, a single-mode ODL is utilized to composite the length difference between the signal and the LO. The optical path length difference between the signal and the carrier can be less than 10 ps by using time-domain calibration method. Lastly, the signal is detected by a 40 GHz ICR and sampled by a digital storage oscilloscope with a sampling rate of 80 GSa/s.

Fig. 3. (a) Experimental structure of the PDM-SHD eigenvalue communication system. (b) Experimental DSP processes. (c) Received constellation after 450 km transmission using 16-APSK and 64-APSK modulation

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The DSP flows are depicted in Fig. 3(b). As a result of the nonlinear noise distribution observed in eigenvalue transmission, it has been demonstrated that utilizing APSK constellations can achieve superior performance compared to QAM constellations [7]. After signal generation at the transmitter, a geometry mapper is employed to map the generated signal into 16-APSK and 64-APSK constellations, respectively. The APSK constellation can be geometrically shaped based on the channel response by transmitting a regularly distributed APSK constellation through the link [34]. During the inverse NFT procedure, the APSK constellations are transformed into a series of optical pulses using the normalized time unit with a single eigenvalue $\lambda \textrm{ = }0.25\textrm{j}$, and the denormalization procedure can convert the signal to time domain, with a baud rate of 1.5 GBaud resulting in 6 Gbit/s while using 16-APSK constellation and 9 Gbit/s while using 64-APSK constellation. At the receiver, after frame synchronization and normalization, the signal is then transformed into the nonlinear spectrum by the direct NFT procedure. An ID link with FO compensation and carrier phase estimate (CPE) is also employed as a performance reference, which are not required in the SHD link. The received constellation after 450 km transmission is shown in Fig. 3(c). To facilitate separate identification of each constellation point, each point is depicted using a distinct color. The FO estimation and CPE are based on the fast Fourier transform (FFT) and blind phase search (BPS) algorithms [1], respectively. Cycle slip (CS) is a prevalent concern in CPE procedures. The first symbol of each frame (51 symbols) is used as a pilot to alleviate the impact of CS. Finally, in the Rx DSP, we compute the bit error rate (BER) using the support vector machines (SVM) demodulation approach [35].

5. Results and discussions

Based on the power fluctuation characteristic on signal and carrier, the APC structure is firstly introduced in the back-to-back (B2B) condition, as depicted in section 3.1. Theoretically, our suggested APC approach is suitable to a wide CSPR range. However, the effect of long-haul transmission on the RSOP recovery performance must be investigated. During the long-haul transmission, the light on orthogonal polarization state induces a nonlinear phase shift on each other, which is known as the cross-polarization modulation (XPolM) [19]. In eigenvalue communications, the signal is pulse shaped, which shares a same pulse shape with a small temporal offset with the repetition frequency of 1.5 GHz. While the pilot carrier is transmitted via the orthogonal polarization state of optical fiber as the signal, a nonlinear phase shift would be initially introduced by the XPolM effect from the signal with the same repetition rate as the signal. As a nonlinear interaction effect, the intensity of XPolM is related to the optical power of the light on its orthogonal polarization state. Consequently, the impact of XPolM on pilot carrier can be diminished by employing a high CSPR. After that, as a minor perturbation, the phase shift may cause the modulation instability (MI) on the carrier and deteriorates the continuous feature of pilot carrier. Given that the intensity of MI is proportional to the optical power of the carrier, the impact of MI on the pilot carrier can be mitigated by employing low CSPR. Consequently, the influence of CSPR on communication performance should be investigated.

The normalized optical spectrums of the received signal and pilot carrier in simulations with a CSPR of 0 dB and -7 dB after 2100 km transmission are shown in Fig. 4(a) using VPIphotomics design suit. The simulation is illustrated by employing the identical setup as the experiment but without ASE noise, while the impact of fiber loss, dispersion, fiber nonlinearity, polarization rotation and PMD are considered. The received pilot carrier is found to be disturbed by nonlinear noise, which is mostly concentrated on various sidebands with frequency offsets of multiples of 1.5 GHz. When the spectrum is normalized by the central frequency, the normalized power of the first sidebands of the pilot carrier with -7 dB CSPR is 7.4 dB less than that of the 0 dB CSPR. The normalized optical power of the sideband is almost proportional to the optical power of the pilot carrier, indicating that these sidebands are mainly caused by MI on the carrier rather than XPolM from the signal and the influence can be diminished by using low CSPR. Although the fact that these sidebands can be filtered out by the OIL, which have negligible effect on the detection procedure, it can be detected as intensity fluctuations on the carrier in time domain by the high-speed PD in APC, resulting in the failure of the adaptive polarization tracking. As a result, sidebands on the pilot carrier must be filtered out by using a low-speed PD or a low-pass filter following the PD in APC. The intensity fluctuation on the carrier and signal using -7 dB and 0 dB CSPR are illustrated in Fig. 4(b) and 4(c) respectively, when a PD with a bandwidth of 1 GHz is used. It is demonstrated that the signal and the carrier can be simply separated by using the suggested APC structure under -7 dB CSPR condition. But even after using 1 GHz PD, the continuous-wave operation state of the pilot carrier is severely disturbed under 0 dB CSPR condition, which may impair the RSOP recovery accuracy. Clearly, the CSPR parameter should be optimized.

Fig. 4. (a) Normalized optical spectrum of received signal and pilot carrier with 0 dB or -7 dB CSPR in simulations with the signal as contrast. (b) Intensity fluctuation on pilot carrier and signal with -7 dB CSPR. (c) Intensity fluctuation on pilot carrier and signal with 0 dB CSPR

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The experimental impact of CSPR on transmission performance is shown in Fig. 5, as assessed by the Q-factor defined with $Q_{dB}^2 = 20{\log _{10}}\left[ {\sqrt 2 {{\textrm{erfc}}^{ - 1}}({\textrm{BER}} )} \right]$. Due to the fact that the 16-APSK and 64-APSK constellations have different noise tolerances, the experimental distances for the 16-APSK and 64-APSK constellation links are 1500 km and 900 km, employing 5 and 3 loops respectively, to ensure that the optimal Q-factor is close to the soft-detection forward error correction limit (SD-FEC). The signal launch power is set to approximately 3.6 dBm. It can be demonstrated that similar performance is obtained no matter FL, ECL, or DFB lasers are used. The Q-factor is insensitive to CSPR over a wide range of values from -14 to 0 dB. The performance worsens with increasing or decreasing CSPR. Because the intensity of MI introduced distortion is transmission distance related, the Q-factor of 16-APSK constellations with 1500 km transmission degrades rapidly under high CSPR circumstances. Meanwhile, due to the fact that the transmission distance of 64-APSK is only 900 km, its Q-factor parameter has limited degradation. Under low CSPR conditions, regardless of modulation format or transmission distance, the Q-factor performance tends to be unstable. The signal power remains constant while the carrier power is reduced as CSPR decreasing. With the PER of APC stays constant, under the same crosstalk power from signal path to the carrier path, the signal-to-carrier crosstalk ratio increases as the carrier power decreases, resulting in an increase in SSBI. Consequently, the extremely low CSPR would worsen the SSBI distortion hence is not recommended.

Fig. 5. Experimental impact of CSPR on the Q-factor

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The FO is a linear phase impairment originating from the laser frequency mismatch between the transmitter and LO, which have similar impact on the constellation in the linear frequency domain and NFT domain. In the ID systems, there are several FO estimation algorithms such as Viterbi-Viterbi (V-V) or FFT based algorithms [1]. The theoretical estimation range for such algorithms are $[{ - {R / {2N}},{R / {2N}}} ]$, where R is the baud rate and $\textrm{N}$ is the constellation point number per circle. The theoretical estimate range for this experiment is $[{{ - 93}{.8MHz,93}{.8MHz}} ]$, which is a tight range for the lasers. In SHD systems, the LO is regenerated by using OIL, which can be regarded as the slave laser output of the transmitter laser, ensuring that the received signal is unaffected by the laser frequency mismatch. However, the FO is in-turn restricted by the OIL locking range related to the injection power. As shown in Fig. 6, when the injected optical power and frequency offset fall within the shaded region, the output frequency of the slave laser aligns with the injected frequency. For instance, if the injection optical power is -20 dBm, the FO tolerance range for SHD is $[{ - 1.85GHz,1.66GHz} ]$, which is more than 17 times larger than the ID one.

Fig. 6. OIL locking range as a function of the injection power and frequency offset

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Figure 7(a) and (b) show the Q-factor performance for 16-APSK and 64-APSK constellations when the signal launch power is 3.6 dBm. In the ID link, the block length of CPE algorithm is set to 31, which is the ideal block length under 100 kHz linewidth (ECL) and 900 km transmission utilizing 64-APSK constellation, due to the fact that most of long-haul transmission employs ECL as the light source. Given that FL have a negligible linewidth and FO, the use of CPE and FO estimation algorithms will introduce additional noise because of the CS. A black curve is added in the figures to be used as the ultimate performance without the impact of laser frequency impairment and such algorithms, which can be considered as the benchmark performance of the transmission link. The Q-factor performances of SHD transmissions employing FL, ECL, and DFB are almost identical when both constellations are used, indicating that our proposed PDM-SHD eigenvalue communication structure is insensitive to the laser linewidth. Simultaneously, the SHD curves exhibit performance approaching that of the benchmark curve, thereby indicating that the SHD structure is associated with a negligible reduction in capability. In particular, for the 64-APSK constellation transmissions, the additional noise introduced by SHD leads to performance degradation in less than 750 km when the ID-FL link has better performance than the SHD-FL link. Due to the interaction of system noise and phase noise on the BPS algorithm, the Q-factor performance of ID-ECL links deteriorated faster than the SHD links. For SHD curves, the use of SHD eliminates the need for the BPS algorithm in the experiment thus the bit errors in SHD systems are primarily introduced by system noise rather than laser phase noise. When employing different lasers, any variations in the phase noise of the lasers will not significantly affect the Q-factor results. Consequently, the Q-factor performance remains consistent across different lasers in SHD-based systems. By using 64-APSK constellation, the Q-factor of ECL-SHD link is 3.37 dB higher than the ECL-ID link after 900 km transmission. On the other hand, the interrelation could also lead to serious CS, which would cause a Q-factor drop on each ID curve. With greater Euclidean distance on the intensity dimension, longer transmission distance may be accomplished employing the 16-APSK constellation, while the phase noise tolerance of the 16-APSK constellation remains the same, which makes the impact of phase noise more decisive and results in better Q-factor performance of SHD links than ID links under all conditions. It shows that under the impact of CS, the Q-factor difference between the ECL-ID curve and ECL-SHD curve can be up to 8.73 dB after 1500 km transmission using the 16-APSK constellation, at the SD-FEC limit. Figure 6(c) and (d) shows influence of signal launch power on Q-factor using 16-APSK constellation after 1500 km transmission and 64-APSK constellation after 900 km transmission respectively. It shows that using SHD did not change the optimal launch power for both constellations. Due to the fact that the CPE algorithm is not required in PDM-SHD links, their Q-factor performance is less affected by the inappropriate launch power than the ID ones.

Fig. 7. Q-factor performance under the scenario of 3.6 dBm signal launch power verse transmission distance for (a) 16-APSK and (b) 64-APSK constellations. Influence of signal launch power on Q-factor using (c) 16-APSK after 1500 km transmission and (d) 64-APSK after 900 km transmission

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6. Conclusions

This work demonstrated PDM-SHD eigenvalue communication system utilizing APC and OIL. A CSPR-unrestricted APC approach is proposed by employing the intensity fluctuation of the light in a certain politization state as feedback rather than the optical power, which is also applicable to linear spectrum coherent PDM-SHD systems. The impact of XPolM and MI is investigated and inhibited by using a low-speed PD. The results indicated that the FO tolerance range is around 3.5 GHz, which is 17 times greater than the ID tolerance range. Identical performance is obtained by using DFB, ECL, and FL after 1500 km with a 16-APSK constellation or after 900 km with a 64-APSK constellation, which demonstrates that our proposed PDM-SHD eigenvalue communication structure is insensitive to the laser linewidth. Furthermore, the suggested PDM-SHD configuration could also be employed for transmissions utilizing WDM technique.

Funding

National Natural Science Foundation of China (61931010, 62225110).

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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PDM-SHD NFDM transmission with envelope-detection feedback polarization tracking and optical injection locking

Abstract

1. Introduction

2. Principle of the NFT theory

3. Polarization tracking and LO regeneration

3.1 APC structure for PDM-SHD

3.2 Optical injection locked laser based LO regenerator

4. Transmission scheme

5. Results and discussions

6. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (10)

Optics Express