1. Introduction

Driven by many emerging bandwidth-hungry applications, optical interconnects continue to scale up in capacity for years [1]. Among different schemes to realize the optical interconnects, the coherent detection system offers excellent performances with a high electrical spectral efficiency [2], but the narrow-linewidth lasers inevitably result in high cost. Achieving a high-performance and spectrally efficient optical interconnect system without requiring narrow-linewidth lasers remains a significant challenge in optical communications. To overcome this challenge, the study of a direct detection (DD) system that inherently excludes narrow-linewidth lasers has resurged and received considerable research interest in recent years.

The DD system requires only one low-cost laser in the transmitter, relative to the coherent scheme with two narrow-linewidth lasers. In addition, the DD system does not require a bulky temperature cooler for laser wavelength stabilization. However, the conventional DD system losses approximately half of the electrical spectral efficiency, as it can only detect the amplitude information of an optical signal.

Therefore, it is desirable for a DD system to achieve a similar spectral efficiency as the coherent system which can detect a complex-valued double-sideband (CV-DSB) signal. With a well-designed transceiver structure, the electrical bandwidth required by the CV-DSB DD system could be approximately half of the signal baud rate similar to that in a coherent system. This allows the use of low-bandwidth optoelectronics with reduced cost [3]. The CV-DSB DD system shows a doubled electrical spectral efficiency compared with the conventional intensity modulation DD (IMDD) system [4].

Compared with the carrier-free coherent system, the achievable performance of the CV-DSB DD system is limited by the carrier-induced power penalty. The CV-DSB DD scheme relies on the square-law photodetection of an optical signal consisting of a carrier and a broadband information-bearing signal. After the square-law photodetection, the carrier-signal beat term represents the desired electrical signal, which is distorted by the signal-to-signal beat interference (SSBI) term [5,6]. As the carrier takes a large proportion of the total optical signal power but bears no information, a low carrier-to-signal power ratio (CSPR) is highly desirable to maximize the achievable system performance determined by the power ratio of the information-bearing signal to the noise. Besides, a high CSPR may lead to a high optical power, which introduces severe nonlinearity effects in fiber. For a wavelength division multiplexing system, a high CSPR limits the number of channels since the output power of an Erbium-doped fiber amplifier (EDFA) is limited. However, under the same optical power, a lower CSPR leads to a larger information-bearing signal and thus a stronger SSBI, limiting the further reduction of CSPR. As a result, it is challenging to realize a low operating CSPR in a high-performance CV-DSB DD system.

For a CV-DSB DD system, besides the requirements of signal detection with a high tolerance to laser wavelength drift and a half-baud-rate receiver bandwidth, the recovery of the transmitted signal distorted by the SSBI is critical to the system design. Based on an iterative SSBI cancellation algorithm, a carrier-assisted differential detection (CADD) receiver achieves an optimum CSPR of ∼7 dB in simulation [3]. In [7], the CADD receiver based on quadrature phase shift keying (QPSK) modulation format was experimentally demonstrated. The extended versions of CADD were also reported. By using a modified version of the CADD, named generalized CADD (G-CADD), the optimized CSPR can be reduced to 3 dB in simulation, but at the expense of reduced tolerance to the laser wavelength drift down to several gigahertz due to an additional narrow-band optical filter [8]. In [9], a parallel dual delay-based CADD (PDD-CADD) was proposed to reduce the needed guard band. Moreover, a phase-retrieval DD scheme based on the Gerchberg-Saxton (GS) algorithm can recover the signal from the SSBI, thus eliminating the need for the optical carrier [10]. However, to fully detect the SSBI, the required receiver electrical bandwidth should be equivalent to the signal baud rate, which is an issue to tackle for the low-cost photonic integration. Recently, an asymmetric self-coherent detection (ASCD) scheme was proposed based on single chromatic dispersion (CD) element [11]. Besides, an ASCD receiver based on Mach-Zehnder interferometers was presented with an optimum CSPR of 8 dB [12].

To the best of our knowledge, the reported minimum operating CSPR is ∼7 dB in the experiment for a CV-DSB DD system with electrical bandwidth of approximately half of the baud rate [7]. Consequently, it is desired that the required CSPR can be significantly lowered. Also, the system should not require strict wavelength stabilization. Such a high-performance system, if implemented on large-scale photonic integrated circuits (PICs), will greatly benefit numerous cost-sensitive applications such as data center interconnects, metro networks, and mobile fronthaul systems.

Here, we present a deep-learning-enabled DD (DLEDD) scheme that can recover the full optical field of the transmitted signal at low CSPRs of ‒2 and 2 dB in simulation and experiment, respectively. In our scheme, the signal detection is based on a dispersion-diversity structure; the CD is exploited to extract the desired signal information, and a receiver diversity technique using more CD elements minimizes the effect of the CD-induced frequency-selective power fading. The dispersion-diversity receiver possesses a low electrical bandwidth of ∼61.0% signal baud rate and shows a high tolerance to the laser wavelength drift due to its all-pass frequency response characteristic. After the dispersion-diversity receiver, a deep convolutional neural network (CNN) based on the residual learning technique is applied to accurately recover the transmitted signal in the presence of a large SSBI under the low CSPR condition. We numerically compare the DLEDD scheme with the typical CADD using an iterative SSBI cancellation algorithm. The simulation results show that our proposal can reduce the optimum CSPR from 6 dB to ‒2 dB, leading to a 5.8-dB improvement in the signal-to-noise ratio (SNR). Compared with the ASCD in [11], the DLEDD scheme reduces the optimum CSPR from 7 dB to ‒2 dB by employing the dispersion-diversity structure and CNN. We also experimentally demonstrate the optical field reconstruction for a 28-GBaud 16-ary quadrature amplitude modulation (16-QAM) signal after 80-km single-mode fiber transmission based on DLEDD with a 2-dB optimum CSPR. The achievable CSPR can be further lowered by using more dispersive elements, while the cost would not increase significantly if large-scale photonic integration is implemented. The required receiver bandwidth of ∼61.0% baud rate can be further reduced with some degradation in the performance.

The rest of this paper is organized as follows: Section 2 introduces the principle of signal recovery enabled by dispersion diversity and deep learning. Section 3 presents the simulation results. Section 4 shows the experimental setup and results. Section 5 compares the performance of the DLEDD scheme with other CV-DSB DD schemes. Finally, we draw the conclusions in Section 6.

2. Signal recovery enabled by dispersion diversity and deep learning

The basic principle to detect the CV-DSB signal is based on the dispersion-induced in-phase and quadrature (IQ) signal conversion, which is different from the dispersion-induced amplitude and phase conversion in [10]. With the propagation of an optical CV-DSB signal over a dispersive element, the CD can be modeled as an all-pass filter having a quadratic phase response, which has no impact on the carrier but affects the broadband information-bearing QAM signal. Therefore, the dispersive QAM signal is the convolution of the original QAM signal and the impulse response induced by dispersion. We represent the dispersive QAM signal s_d as follows:

 

Fig. 1. The CD-induced IQ signal conversion process, and the square-law detections are applied for both the original and dispersive QAM signals.

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Then, we show that the dispersion-induced IQ signal conversion can be used to detect the CV-DSB signal. Figure 1 illustrates the square-law detection processes for both original and dispersive QAM signals with the same optical carrier. Here, the carrier phase is assumed to be zero, which divides the optical field into an in-phase field and a quadrature field. After the square-law detection of the optical field using a photodetector (PD) followed by a direct current (DC) blocker, the photocurrent i₁ can be expressed as follows:

To detect the quadrature component of the QAM signal, a dispersive element is used in the other DD branch to introduce the IQ signal conversion. After the dispersive element, the in-phase field contains both the in-phase and quadrature components of the QAM signal, whose spectra show frequency-selective power fading effects induced by the in-phase and quadrature components of the dispersion-related impulse response, respectively. Then, the dispersive optical field is detected by a PD with a DC blocker. We describe the photocurrent i₂ as follows:

Next, we show our proposed method to reconstruct s. According to Eq. (2), the in-phase component of the QAM signal s_in-phase can be recovered as follows:

To recover s_quadrature from Eq. (3), the linear crosstalk 2As_in-phaseUh_in-phase should be eliminated. We consider two methods to eliminate 2As_in-phaseUh_in-phase: (i) we estimate 2As_in-phaseUh_in-phase based on i₁ and then remove it from i₂; (ii) we use another dispersive branch to eliminate 2As_in-phaseUh_in-phase. The first method can be expressed as follows:

The estimated s_quadrature is expressed as:

 

Fig. 2. (a) Frequency response of h_quadrature; (b) frequency response of the inverse h_quadrature.

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In the Eq. (6), the second term on the right-hand side represents the SSBI enhanced by the inverse response of h_quadrature. The SSBI enhancement significantly increases the recovery error for the quadrature signal s_quadrature. The field recovery error is as follows:

In the second method, we employ two dispersion elements with the same absolute dispersion value but different signs, for example, +185 and ‒185 ps/nm. In Fig. 3, we have

The estimated s_quadrature is expressed as:

 

Fig. 3. Receiver structure with two dispersion elements.

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It is difficult to compare the magnitudes of $\frac{1}{{2A}}[{{{|s |}^2} \otimes {h_{in - phase}} - {{|{s \otimes h} |}^2}} ]$ in Eq. (6) and $\frac{1}{{4A}}[{{{|{s \otimes {h^\ast }} |}^2} - {{|{s \otimes h} |}^2}} ]$ in Eq. (10). We plot the electrical spectra of the field recovery errors for magnitude comparison as shown in Fig. 4. The difference of the field recovery errors is slight for small dispersion values, while it is significant for large dispersion values. Thus, method 2 is better than method 1.

 

Fig. 4. Spectra of the field recovery errors without dispersion diversity: dispersive elements use (a) {+185-, and −185-}, (b) {+1008-, and −1008-} ps/nm.

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Due to the frequency-selective fading in the dispersive branch, the quadrature component of the signal close to the frequency nulls are blocked, severely degrading the system performance. To improve the transmission performance in the presence of fading, we propose the use of two or more dispersive elements to obtain the differently fading replicas of the quadrature component of the QAM signal at the receiver. Such a technique is referred to as dispersion diversity in this paper since it is similar to the antenna diversity that combines several wireless fading paths [13].

There are two problems to address in the design of the dispersion-diversity receiver: (i) how to select the dispersion values; (ii) how to combine the receiver outputs and then recover the original signal. Note that the dispersion diversity receiver is different from the parallel alternative projection receiver proposed in [10]. The dispersion diversity receiver is based on the dispersion-induced IQ signal conversion, but the parallel alternative projection receiver is based on the dispersion-induced amplitude and phase conversion. Therefore, the required electrical bandwidth of the two receivers is different. The criterion for choosing the dispersion values is also different.

When we choose the dispersion values, the frequency nulls should be properly allocated in spectra for different dispersive branches such that the signal components suppressed by one branch can be passed through other branches. In this way, the detrimental influence of the fading effect can be effectively mitigated. As an example of the dispersion diversity, Fig. 5 gives the fading shapes for the ‒1008- and ‒185-ps/nm dispersive branches, respectively. The corresponding quadrature impulse responses for ‒185- and ‒1008-ps/nm dispersive elements are h_quadrature and b_quadrature, respectively. As shown in Fig. 5, the signal components at 26 GHz are blocked by h_quadrature, but they can be passed through b_quadrature. In addition, the signal components close to the zero frequency are less suppressed by b_quadrature compared with h_quadrature. It can be noted that a zero-frequency null is inevitable regardless of the dispersion value. Thus, a small guard band centered at the zero frequency is needed for the information-bearing signal.

 

Fig. 5. Frequency responses of h_quadrature and b_quadrature, respectively. h_quadrature and b_quadrature correspond to ‒185- and ‒1008-ps/nm dispersions, respectively.

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Then, we take an example to show the combination of the receiver outputs and the recovery of the original signal. Figure 6 depicts a dispersion-diversity receiver with four dispersion elements.

 

Fig. 6. Dispersion-diversity receiver with four dispersion elements.

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Similarly, we have

If we combine i₃ ‒ i₂ and i₅ ‒ i₄ by orthogonal means, the estimation of s_quadrature can be derived as follows:

Frequency response curves of the (h_quadrature + jb_quadrature) and the inverse (h_quadrature + jb_quadrature) are shown in Figs. 7(a) and 7(b), respectively. The resulting field recovery error can be described as:

 

Fig. 7. (a) Frequency response of (h_quadrature + jb_quadrature); (b) frequency response of the inverse (h_quadrature + jb_quadrature); (c) spectra of the field recovery errors.

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However, the orthogonal combining method may not be an optimal combining method. It is challenging to find the optimum combining method by analytical means. In fact, it does not matter which combining method is optimal. Our goal is to minimize the field recovery error after combining the outputs of the dispersion diversity receiver and mitigating SSBI, thus maximizing the BER performance. Here, we employ a neural network to combine the receiver outputs and mitigate the SSBI, and use the power of the recovery error, i.e., the mean squared error (MSE), as the loss function of the neural network. After iteratively training the neural network to minimize the loss function, the trained network can be used to recover the optical full field with a minimized error power, thus maximizing the system performance.

In our scheme, a CNN is employed to combine the multi-channel receiver outputs, since the CNN shows a good performance in multi-channel data processing such as the three-channel RGB (red, green, and blue) image processing [14]. Firstly, we show how to combine the receiver outputs based on a CNN with a single convolutional layer in the absence of the activation function and the bias. The CNN structure is given in Fig. 8, where the receiver outputs are sent to the convolutional layer, leading to two network outputs.

 

Fig. 8. CNN structure with a single convolutional layer in the absence of the activation function and the bias.

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The two output channels of the CNN correspond to the estimated values of s_in-phase and s_quadrature, respectively.

It is observed that the CNN combines the receiver outputs by using a convolutional layer with multiple kernels. The resulting field recovery error is

The field recovery error would change by adjusting the kernels. As discussed previously, the field recovery error should be minimized to maximize the system performance. Thus, we optimize the kernels to minimize the field recovery error. We use the power of the field recovery error as the loss function of the CNN, and then optimize the kernels by minimizing the loss function.

Furthermore, a deep CNN with multiple convolutional layers and nonlinear activation functions can be used to achieve a powerful nonlinear modeling capability, which helps to mitigate the nonlinear SSBI. Our proposed dispersion-diversity receiver with deep-learning-enabled full-field direct detection is shown in Fig. 9, which can detect both the in-phase and quadrature optical fields (i.e., the full optical field) before the receiver. Different from the phase-retrieval GS algorithm which needs numerous iterations to refine the recovered phases [10], the iteratively trained neural network already has a high field recovery accuracy, and the practical implementation of the trained network does not require the iterative operation. Figure 9 provides the detailed CNN structure based on deep residual learning [15]. The residual learning adds shortcut connections to the conventional CNN structure, which can accelerate the training process of the network [15]. The deep residual network relies on the stacked residual blocks. Each residual block consists of two convolutional layers and a shortcut connection. In the deep CNN, the rectified linear units (ReLU) function is used as the activation function since it enables a fast-training process.

 

Fig. 9. Dispersion-diversity receiver enabled by the deep CNN. The fading shapes of two different dispersive branches are provided for the dispersion-diversity receiver. The deep CNN is based on the residual learning technique. The time-domain data channels in the deep CNN are schematically illustrated.

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At the input of the CNN, multiple channels of the receiver outputs are sent to a convolutional layer to extract the features of the data. Thus, the outputs of the convolutional layer are referred to as high-dimensional featured channels. Then, several cascaded residual blocks are applied. In each residual block, batch normalization (BN) is performed before the ReLU activation. After the last residual block, the high-dimensional featured channels are converted to two featured channels, which are combined with another shortcut connection to form the network outputs. The two output channels of the CNN correspond to the estimated in-phase and quadrature components of the original baseband information-bearing signal, respectively. After a training process to minimize the MSE between the estimated and original signals, the CNN can be used to recover the signal with high accuracy. Equivalently, the influence of strong SSBI is minimized, enabling a low-CSPR operation of the DD system. It is worth noting that the CNN with convolution operations is capable of compensating for the dispersion of the fiber link and equalizing the signal distortion from the nonideal optoelectronics. Thus, the dispersion values of the dispersion-diversity receiver can be optimized in the optical back-to-back (OBTB) case, and then applied for various transmission distances.

Figure 10 provides the spectrum of the field recovery error after the deep CNN. Note that the decrease of the field recovery error is the result of effectively combining the receiver outputs and mitigating the SSBI.

 

Fig. 10. Spectra of the field recovery errors.

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In addition, such a receiver possesses a high tolerance to the laser wavelength drift due to the all-pass characteristic of the dispersive element in the optical domain. For the low-cost photonic integration of the dispersion-diversity receiver, the dispersive element can be implemented in the form of on-chip Bragg grating, which offers a large dispersion value with a low insertion loss and small size [16]. According to [16], the insertion loss of the on-chip Bragg grating providing a 1-ps/nm chromatic dispersion value is 9.6×10⁻³ dB.

3. Numerical validation

A numerical simulation is performed to evaluate the performance of the proposed DLEDD scheme. In the DLEDD scheme, we generate a CV-DSB signal based on the dual-single-sideband (SSB) Nyquist 16-QAM format with a roll-off factor of 0.01 [17,18]. The CV-DSB signal has an aggregate baud rate of 28 GBaud and a guard band of ∼6 GHz, i.e., ‒3 GHz to 3 GHz. The sampling rate is set to 56 GSa/s. After the additive white Gaussian noise (AWGN) loading, the DLEDD scheme is used to detect the CV-DSB signal. We consider two dispersion-diversity receivers with the dispersion values of {‒1008, ‒185} ps/nm and {+1008, ‒1008, +185, ‒185} ps/nm, respectively. To avoid the spectral aliasing induced by the broadened spectrum after square-law detection, the optical field is resampled to 112 GSa/s before the square-law detection. After the square-law detection, the resulting electrical signal is resampled back to 56 GSa/s. Then, a rectangular-shaped low-pass filter removes the out-of-band SSBI components beyond 17.07 GHz, thus enabling a low receiver bandwidth of ∼61.0% (17.07 GHz / 28 GBaud) signal baud rate. The undesired signal components at the low-frequency guard band are also suppressed before the CNN. For the CNN based on residual learning, we use 3 residual blocks. Thus, there are 8 convolutional layers in the CNN according to Fig. 9. The corresponding featured channel numbers after the 8 convolutional layers are 32, 34, 34, 36, 36, 38, 38, and 2, respectively. The kernel sizes for the convolutional layers and the shortcut connections are 13 and 1, respectively. In the simulation, a binary stream generated by the Mersenne Twister algorithm is mapped into 1638400 16-QAM symbols [19]. We use 80% of the data symbols for training and the remaining 20% symbols for testing [20–22]. The training symbols are split into multiple groups to enable the mini-batch gradient descent algorithm in the training process, while the testing symbols do not need the group splitting operation. With a sampling rate of 2 samples per symbol, the sequence lengths used by the CNN are 512 and 655360 for the training and testing, respectively. During the training process, an Adam optimizer is employed to minimize the MSE loss function of the network [23]. The learning rate of the optimizer is 0.001 at the beginning and then decayed to 0.0001 after 20 epochs. With 30 training epochs, the optimized network is applied to the test set to recover the signal. Finally, the bit error ratio (BER) is used to characterize the performance of the DLEDD scheme. In the simulation, no field reconstruction failure was observed, and stable BER measurement was achieved.

As typical iterative SSBI cancellation schemes for benchmarking, the performances of the CADD and ASCD receivers are also investigated. For the same CV-DSB signal in the DLEDD system, the CADD receiver has an optimized delay of 48 ps and a sufficient iteration number of 4. The ASCD receiver uses a ‒185-ps/nm dispersive element.

In addition, we numerically study the performance of the phase-retrieval GS algorithm. Since the GS scheme does not require a signal guard band, we generate a single-carrier Nyquist 16-QAM signal rather than the dual-SSB Nyquist 16-QAM signal. The signal has a baud rate of 28 GBaud, and the Nyquist roll-off factor is 0.01. The sampling rate of the simulation is set to 56 GSa/s. To realize the carrier-less operation in the GS simulation, no optical carrier is added to the 16-QAM signal at the transmitter. After AWGN loading, a parallel-alternative-projection receiver in [10] is used to detect the carrier-less 16-QAM signal. The receiver has four dispersion elements with the dispersion values of +650, +1300, +1950, and +2600 ps/nm, respectively. These dispersion values are sufficient to introduce symbol mixing and intensity variations according to [10]. After the square-law detection, a 28-GHz rectangular-shaped low-pass filter is applied. Then, we run the GS algorithm iteratively between the four dispersion elements [10]. 500 iterations are applied for each dispersion element, leading to a total iteration number of 2000. Similar to [10], we use 20% pilot symbols. Both phase and amplitude of the pilot symbol are applied as the constraints. The phase reset is excluded [10].

Figure 11 depicts the simulated BER performances by varying the CSPR value. The optical SNR (OSNR) is defined as the optical signal power divided by the noise power in a 12.5-GHz bandwidth. In the OSNR definition, the optical signal includes both the carrier and the information-bearing signal. As shown in Fig. 11, an optimum CSPR of 6 dB is observed for the CADD receiver at the 18-dB OSNR. Further decreasing the CSPR leads to a higher BER, due to the limited performance of the iterative SSBI cancellation algorithm. At a lower CSPR, the power ratio of the SSBI to the carrier-signal beat term becomes larger, reducing the accuracy of the SSBI estimation in the iterative SSBI cancellation loop and thus degrading the performance of CADD. However, by using optical filters for CADD, the BER performance can be further improved [8]. The CADD receiver contains several power splitters, which would affect the system performance in the practical implementation. The SNR estimated from the recovered QAM constellation is 8.38 dB at the optimum 6-dB CSPR for the CADD receiver. By using the DLEDD schemes, the optimum CSPRs can be significantly reduced to 0 and ‒2 dB for the dispersion-diversity receivers with 2 and 4 dispersive elements, respectively. The corresponding SNRs are 12.44 and 14.21 dB, respectively. It can be seen that using more dispersive branches improves the BER performance. Considering the large-scale PICs, such as silicon photonics, adding two more dispersive branches results in a limited increase in the cost. For the DLEDD scheme, a further reduction in the BER by decreasing CSPR is limited due to the residual SSBI. As shown in Fig. 11, the optimized CSPR is higher with the increase of the OSNR. At a large OSNR value, the BER performance is limited by the SSBI instead of the noise. In this case, a high CSPR is needed to minimize the influence of SSBI, thus improving the BER performance. With the 25% forward error correction (FEC) threshold of 4 × 10^‒2 [24], an 18-dB OSNR is sufficient for the DLEDD scheme. In Fig. 11, the theoretical coherent BER curve for a 28-GBaud 16-QAM signal considering the carrier-induced power penalty is also provided as a benchmark [25]. The performance penalty for either the CADD or DLEDD scheme relative to the theoretical curve comes from the residual SSBI. As depicted in Fig. 11, the carrier-less GS method shows similar performance as the DLEDD with four dispersive elements. A further performance improvement for the GS scheme could be achieved by using some methods to accelerate the convergence of the GS algorithm [10]. In addition, the fiber transmission can help to lower the BER of the GS scheme [10]. Compared with the ASCD in [11], the DLEDD-II reduces the optimum CSPR from 7 dB to ‒2 dB by employing the dispersion-diversity structure and CNN.

 

Fig. 11. Simulated BER versus CSPR for different OSNRs. The ASCD scheme uses ‒185-ps/nm dispersive element. The DLEDD-I and DLEDD-II schemes employ {‒1008-, ‒185-} and {+1008-, ‒1008-, +185-, ‒185-} ps/nm dispersive elements, respectively. The GS method uses {+650-, +1300-, +1950-, and +2600-} ps/nm dispersive elements.

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Figure 12 shows the simulated BER versus CSPR for different schemes. For the same SSBI mitigation method using iteration algorithm, the orthogonal combining scheme can reduce the optimum CSPR from 8 dB to 4 dB compared with the ASCD. Thus, the dispersion diversity receiver contributes to the CSPR reduction. Compared with the ASCD scheme, ASCD-CNN can reduce the optimum CSPR from 8 dB to 6 dB. Compared with the CADD scheme, CADD-CNN can reduce the optimum CSPR from 6 dB to 5 dB. Thus, the CNN is better than the iteration algorithm regarding the mitigation of SSBI, and leads to a lower optimal CSPR. In conclusion, the CSPR reduction of the DLEDD scheme can be attributed to two aspects: (i) dispersion diversity technique and (ii) CNN-based SSBI mitigation algorithm. Compared with the ASCD-CNN (6-dB optimum CSPR) and the CADD-CNN (5-dB optimum CSPR), the DLEDD-II achieves the lowest optimum CSPR of ‒2 dB.

 

Fig. 12. Simulated BER versus CSPR for different schemes. Both the ASCD-CNN and the CADD-CNN schemes use CNN to mitigate the SSBI. Both the orthogonal combining and the DLEDD-II schemes employ {+1008-, ‒1008-, +185-, ‒185-} ps/nm dispersive elements. For all the schemes, the OSNRs are set to 21 dB.

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Figure 13 presents the simulated BER versus OSNR for different schemes. Compared with the coherent transmission system, DLEDD-II has a ∼4-dB OSNR penalty at the 25% FEC threshold, which is mainly induced by the carrier power and the residual SSBI.

 

Fig. 13. Simulated BER versus OSNR for different schemes. The ASCD scheme uses ‒185-ps/nm dispersive element. The DLEDD-I and DLEDD-II schemes employ {‒1008-, ‒185-} and {+1008-, ‒1008-, +185-, ‒185-} ps/nm dispersive elements, respectively. The GS method uses {+650-, +1300-, +1950-, and +2600-} ps/nm dispersive elements.

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Note that detecting the linear carrier-signal beat terms is sufficient to recover the optical field, thus the dispersion-diversity receiver in the proposed scheme does not require the detection of the out-of-band SSBI. Moreover, the receiver bandwidth can be further reduced by using a smaller guard band with certain performance degradation. However, the SSBI should be fully detected in all the previous dispersion-based receiver structures, such as the phase-retrieval GS scheme [10]. Therefore, the proposed method greatly reduces the receiver electrical bandwidth requirement compared with the phase-retrieval GS scheme, significantly lowering the cost of the system. Please refer to the Appendix for the detailed analysis of the required receiver bandwidth.

4. Experimental setup and results

To verify the feasibility of the proposed DLEDD scheme, we perform a proof-of-concept experiment, in which discrete components are used to realize a dispersion-diversity receiver with two dispersive branches. The experimental setup is shown in Fig. 14. An arbitrary waveform generator (AWG) (Keysight M8195A) at 64 GSa/s outputs a 28-GBaud dual-SSB 16-QAM signal with a 6-GHz guard band. After being amplified by the electrical amplifiers (EAs), the dual-SSB 16-QAM signal drives a 22-GHz IQ modulator (IQM) biased at the transmission null. A continuous-wave light from a tunable optical laser at 1550 nm is split into two paths. One path is modulated with the dual-SSB 16-QAM signal through the IQM, while the other provides the carrier. The carrier and the dual-SSB 16-QAM signal are polarization-aligned and then combined to form an optical CV-DSB signal, whose CSPR is changed by adjusting the variable optical attenuator (VOA) in the carrier path. After being boosted by an EDFA, the optical CV-DSB signal with a power of 0 dBm is launched into an 80-km single-mode fiber (SMF). Then, another EDFA is employed to amplify the optical signal, and a 1-nm optical band-pass filter (OBPF) is used to suppress the amplified spontaneous emission (ASE) noise before the dispersion-diversity receiver. Normally, for very short reach applications (< 2 km), the EDFAs may not be required. While for data center interconnects, metro networks, and mobile fronthaul systems, EDFAs are required to cover transmission distances of tens of kilometers or beyond. Heree, we use two dispersive elements, which are emulated by a waveshaper (WS) and a dispersion compensation module (DCM), respectively. The corresponding dispersion values are ‒185, and ‒1008 ps/nm, respectively. After three 43-GHz PDs, the electrical signals are sampled by a 36-GHz digital storage oscilloscope (DSO) (LeCroy 36Zi-A) at 40 GSa/s.

 

Fig. 14. Experimental setup. AWG: arbitrary waveform generator; EA: electrical amplifier; IQM: IQ modulator; PC: polarization controller; EDFA: erbium-doped fiber amplifier; SMF: single-mode fiber; OBPF: optical band-pass filter; WS: waveshaper; DCM: dispersion compensation module; DSO: digital storage oscilloscope.

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In the experiment, a laser source with a narrow linewidth of 15 kHz is employed due to the length mismatch between the carrier branch and the information-bearing signal branch [26]. Such a length mismatch can be avoided in the PIC with a proper design. Then, the carrier and the information-bearing signal possess the same phase noise, which is canceled in the carrier-signal beating process at the receiver, thus relaxing the linewidth requirement on the laser source and reducing the cost.

In the transmitter-side DSP, we generate 409600 16-QAM symbols, which is limited by the equipment memory. Since training the CNN requires a large number of data samples, 80%, and 20% symbols are used for training and testing, respectively. In the practical implementation, the training symbols are only used in the beginning to optimize the CNN parameters. Then, the trained CNN can be employed to recover the field and no more training symbols are needed. We add 2048 16-QAM symbols for synchronization. After dual-SSB signaling, resampling, and clipping, the signal is sent to the AWG. At the receiver, the signals are resampled to 56 GSa/s and then digitally filtered to block the signal components beyond 17.07 GHz. In addition, the signal components within the guard band are removed. After synchronization, the three digital signals are processed by a deep CNN with the same network structure and training process in the simulation. The sequence lengths used by the CNN are 512 and 163840 for the training and testing, respectively. The CNN operates in a window-sliding manner, and pilot symbols are not needed. Also because in-phase and quadrature components are simultaneously recovered at the CNN output, there does not exist a phase ambiguity problem. Finally, the recovered dual-SSB signal is decomposed, followed by the BER calculation. In the experiment, the BER is calculated by error counting over ∼1.3 million bits, which are sufficient for testing BERs around 10^‒3 [27,28].

Figure 15 shows the BER performances for both OBTB and fiber transmission cases. Compared with the simulated OSNR sensitivity of ∼18 dB, a higher OSNR of 21 dB is required in the experiment to achieve a BER lower than the FEC threshold. This can be mainly attributed to the low-pass filtering effect of the cascaded optoelectronics, quantization noise, and other electrical noises in the experiment. At the 21-dB OSNR, the optimum CSPR for the DLEDD scheme is ∼2 dB, achieving error-free operation after the FEC. To the best of our knowledge, this is a record-low optimum CSPR among the experimentally demonstrated DD receivers requiring a bandwidth much lower than the signal baud rate. A lower operating CSPR and a better BER performance can be expected by using a dispersion-diversity receiver with more dispersive elements like the one in the simulation. The constellation diagrams of the recovered 16-QAM signals are also provided in Fig. 15.

 

Fig. 15. BER versus CSPR for different OSNRs in the experiment.

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5. Discussion

Compared with the coherent transmission system, CV-DSB DD system saves a costly narrow-linewidth local oscillator, thus lowering photonic circuit cost. Table 1 compares the DD schemes that are capable of recovering CV-DSB signals based on simulation results. For a fair comparison, all the results in Table 1 are based on the 16-QAM format. As evidenced by Table 1, DLEDD, CADD, and ASCD enable receiver bandwidths much lower than the signal baud rates and provide the simultaneous advantage of strong tolerance to laser wavelength drift. Thus, the cost of the system can be reduced by using low-bandwidth optoelectronics, and the power consumption can be minimized without requiring a temperature controller for wavelength stabilization. Furthermore, compared with the CADD receiver, the DLEDD scheme possesses a much-reduced optimum CSPR, resulting in a 5.8-dB higher SNR and thus a significant performance improvement. In addition, the DLEDD method can eliminate the need for iterative loop, fiber dispersion compensation, and signal equalization.

Table 1. Comparison of simulation results for CV-DSB DD systems

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As the implementation details in these algorithms are different, we do not precisely know the numbers of multiplications per symbol for all the algorithms. The approximate orders of the required real-valued multiplications per symbol can be derived as follows. Most multiplications in the CADD, GS, and ASCD schemes come from frequency-domain operations such as dispersion and filtering. With a 4096-point fast Fourier transform/inverse fast Fourier transform length, one frequency-domain operation requires ∼10² real-valued multiplications per symbol. There are several frequency-domain operations in each iteration for the CADD, GS, and ASCD schemes. Considering the ∼4 iterations for both the CADD and the ASCD, the required real-valued multiplications per symbol could be ∼10³, which include the CD compensation, equalization, and decision operations in each iteration. For the GS, because ∼10³ iterations are needed, the required real-valued multiplications per symbol for the GS could be ∼10⁵. As for the multi-layer structure of the deep CNN, the 8 convolutional layers contain ∼7800 kernels, and the kernel size is 13. Thus, the DLEDD scheme requires ∼10⁵ real-valued multiplications per symbol. The order of the multiplication number for the DLEDD could be similar to that of the GS algorithm. To simplify the network structure and decrease the computation complexity, a pruned algorithm can be employed.

Table 2 compares the state-of-the-art experimental results of various CV-DSB DD schemes. As shown in Table 2, only DLEDD and PDD-CADD provide experimental results in the 16-QAM format. The optimum CSPR of the DLEDD scheme is the lowest out of the DD schemes that require electrical bandwidths of approximately half of the baud rates, except for the GS scheme which does not need the carrier but the required electrical bandwidth is the baud rate. For the large-scale PICs, the cost of an integrated circuit is mainly dominated by the electrical bandwidth of the circuit but not the number of components [3]. These observations show that our proposal offers a promising solution towards integrated DD receivers with the attractive features of high performance, low cost, and minimized power consumption.

Table 2. Comparison of experimental results for CV-DSB DD systems

View Table | View all tables in this article

6. Conclusion

In conclusion, we have proposed a novel deep-learning-enabled DD scheme to recover the full optical signal field with a low operating CSPR and high system performance. In the DLEDD scheme, an optical CV-DSB signal is detected with a dispersion-diversity receiver and is then recovered by a deep CNN in the presence of a strong SSBI. Compared with the conventional iterative SSBI cancellation scheme, the DLEDD receiver shows a significant reduction of 8 dB in the optimum CSPR, leading to a significantly improved system performance with a ∼5.8-dB higher SNR in the simulation. Based on the dispersion-diversity receiver with two discrete dispersive elements, a low optimum CSPR of 2 dB is experimentally demonstrated. The achievable CSPR of the DLEDD scheme can be further reduced by using more dispersive elements. The increased cost of the passive dispersive components in a receiver could be minor in mass production considering the benefits brought by photonic integration. The required receiver bandwidth of ∼61.0% baud rate can be further reduced with a certain degradation in the performance.

Therefore, the proposed DLEDD scheme shows a unique combination of performance enhancement from the low-CSPR operation, minimized electrical bandwidth requirement, and high tolerance to laser wavelength drift. The structure of the DLEDD is also well-suited for photonic integration by using low-cost PICs, such as silicon photonics. Implementing the DLEDD scheme with the PICs opens up many application possibilities in cost-sensitive scenarios, such as data center interconnects, metro networks, and mobile fronthaul systems.

Appendix

The DLEDD scheme recovers the full optical field from the linear carrier-signal beat term, which has an electrical bandwidth of ((Baud rate + Guard band) / 2). Thus, the receiver bandwidth required by the DLEDD scheme is ((Baud rate + Guard band) / 2). A receiver with a cut-off bandwidth lower than ((Baud rate + Guard band) / 2) would suppress the linear carrier-signal beat term, thus significantly degrading the system performance. On the other hand, since the SSBI has a wider bandwidth relative to the linear carrier-signal beat term, filtering the out-of-band SSBI components beyond ((Baud rate + Guard band) / 2) does not affect the system performance significantly.

We consider a low-bandwidth receiver in Fig. 16, where a rectangular-shaped low-pass filter is used to emulate the bandwidth limitation of the receiver. Based on the mathematical derivations in the previous section, the receiver outputs in Fig. 16 can be expressed as:

(20)$${i^{\prime}_1} =({2A{s_{in - phase}} + {{|s |}^2}} )\otimes l = 2A{s_{in - phase}} + {|s |^2} \otimes l,$$

(21)$$\begin{aligned} {{i^{\prime}}_2} &= ({2A{s_{in - phase}} \otimes {h_{in - phase}} - 2A{s_{quadrature}} \otimes {h_{quadrature}} + {{|{s \otimes h} |}^2}} )\otimes l\\ &= 2A{s_{in - phase}} \otimes {h_{in - phase}} - 2A{s_{quadrature}} \otimes {h_{quadrature}} + {|{s \otimes h} |^2} \otimes l, \end{aligned}$$

(22)$$\begin{aligned} {{i^{\prime}}_3} &= ({2A{s_{in - phase}} \otimes {h_{in - phase}} + 2A{s_{quadrature}} \otimes {h_{quadrature}} + {{|{s \otimes {h^\ast }} |}^2}} )\otimes l\\ &= 2A{s_{in - phase}} \otimes {h_{in - phase}} + 2A{s_{quadrature}} \otimes {h_{quadrature}} + {|{s \otimes {h^\ast }} |^2} \otimes l, \end{aligned}$$

(23)$$\begin{aligned} {{i^{\prime}}_4} &= ({2A{s_{in - phase}} \otimes {b_{in - phase}} - 2A{s_{quadrature}} \otimes {b_{quadrature}} + {{|{s \otimes b} |}^2}} )\otimes l\\ &= 2A{s_{in - phase}} \otimes {b_{in - phase}} - 2A{s_{quadrature}} \otimes {b_{quadrature}} + {|{s \otimes b} |^2} \otimes l, \end{aligned}$$

(24)$$\begin{aligned} {{i^{\prime}}_5} &= ({2A{s_{in - phase}} \otimes {b_{in - phase}} + 2A{s_{quadrature}} \otimes {b_{quadrature}} + {{|{s \otimes {b^\ast }} |}^2}} )\otimes l\\ &= 2A{s_{in - phase}} \otimes {b_{in - phase}} + 2A{s_{quadrature}} \otimes {b_{quadrature}} + {|{s \otimes {b^\ast }} |^2} \otimes l. \end{aligned}$$

 

Fig. 16. Dispersion-diversity receiver with bandwidth limitation.

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In each receiver output, the second-order SSBI has a doubled electrical bandwidth relative to the linear carrier-signal beat term. Here, l means the time-domain impulse response of the low-pass filter, whose bandwidth is larger than the bandwidth of the linear carrier-signal beat term but smaller than the SSBI bandwidth. Thus, the rectangular-shaped low-pass filter l has no impact on the linear carrier-signal beat terms such as 2As_in-phase, 2As_quadratureUh_quadrature and 2As_quadratureUb_quadrature, but suppresses the out-of-band components of the SSBI terms such as |s|², |sUh|², and |sUb|². Then, we take the orthogonal combining method as an example to show the field recovery process.

(25)$${\hat{s}_{{in - phase, low - bandwidth receiver}}} =\frac{{{{i^{\prime}}_1}}}{{2A}} = {s_{in - phase}} + \frac{1}{{2A}}{|s |^2} \otimes l.$$

(26)$$\begin{aligned} {{\hat{s}}_{{quadrature, low - bandwidth receiver}}} & =\frac{1}{{4A}}[{({{i^{\prime}}_3} - {{i^{\prime}}_2}) + j({{i^{\prime}}_5} - {{i^{\prime}}_4})} ]\otimes {({h_{q{uadrature}}} + j{b_{q{uadrature}}})^{inv}}\\ & = {s_{q{uadrature}}} + \frac{1}{{4A}}[{({{{|{s \otimes {h^\ast }} |}^2} - {{|{s \otimes h} |}^2}} )+ j({{{|{s \otimes {b^\ast }} |}^2} - {{|{s \otimes b} |}^2}} )} ]\\ &\otimes {({h_{q{uadrature}}} + j{b_{q{uadrature}}})^{inv}} \otimes l. \end{aligned}$$

It can be observed that s_in-phase and s_quadrature can be successfully recovered from the linear carrier-signal beat terms, while the SSBI terms are not required in the field recovery process. Thus, filtering the out-of-band SSBI does not affect the performance significantly. Similar to the orthogonal combining method, the DLEDD scheme based on the CNN combining method can also recover the field from the linear carrier-signal beat terms even when the out-of-band SSBI is suppressed by the receiver.

We perform a simulation to investigate the BER performance of the DLEDD scheme by varying the receiver bandwidth, as shown in Fig. 17(a). In the simulation, we use a 28-GBaud Nyquist dual-SSB signal with a roll-off factor of 0.01 and a guard band of ∼6 GHz, i.e., ‒3 GHz to 3 GHz. Thus, the electrical bandwidths for the linear carrier-signal beat term and the SSBI after the direct detection are 17.07 and 34.14 GHz, respectively. After the direct detection and direct current (DC) blocking, a rectangular-shaped low-pass filter is applied to emulate the bandwidth limitation of the receiver. Then, the DLEDD scheme is used to recover the signal field. In addition, the required receiver bandwidth for the carrier-less Gerchberg-Saxton (GS) scheme is studied based on the simulation setup in Fig. 17(b). Most parameters of the simulation are the same as those in Section 3. A higher sampling rate of 84 GSa/s is used since the receiver bandwidth is larger.

 

Fig. 17. Simulation setups for (a) the DLEDD, and (b) GS schemes, respectively.

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The simulation results for the DLEDD and GS schemes are given in Figs. 14(a) and 14(b), respectively. It can be seen that filtering the out-of-band SSBI does not affect the performance of the DLEDD scheme significantly, while the GS method shows a severe performance degradation when the SSBI is suppressed. In addition, it is observed that detecting the out-of-band SSBI can slightly improve the BER performance of the DLEDD scheme in Fig. 18(a). This can be attributed to the powerful nonlinear modeling capability of the deep CNN, where the nonlinear out-of-band SSBI could be used to improve the accuracy of the field recovery. But suppressing the out-of-band SSBI does not lead to significant performance degradation.

 

Fig. 18. Simulated BER versus receiver bandwidth. The DLEDD-II scheme employs {+1008-, ‒1008-, +185-, ‒185-} ps/nm dispersive elements. The GS method uses {+650-, +1300-, +1950-, and +2600-} ps/nm dispersive elements.

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It is worth noting that the 6-GHz guard band for the 28-GBaud signal in the DLEDD scheme can be further reduced with a certain performance penalty. We perform a simulation to investigate the BER performance of the DLEDD scheme by varying the bandwidth of the guard band. In the simulation, we use a 28-GBaud dual-SSB 16-QAM signal. The CSPR and OSNR are set to ‒2 and 21 dB, respectively. The receiver bandwidth is emulated by a rectangular-shaped low-pass filter with a bandwidth of ((Baud rate + Guard band) / 2). Decreasing the guard band can reduce the receiver bandwidth requirement, while the BER performance would be degraded accordingly. As shown in Fig. 19, the BER increases from 1.43 × 10^‒3 to 3.50 × 10^‒3 when varying the guard band from 6 to 3 GHz. This can be explained by the stronger SSBI enhancement induced by the zero-frequency null at a smaller guard band. The zero-frequency null is inevitable in each dispersive branch regardless of the dispersion value.

 

Fig. 19. Simulated BER versus guard band. The DLEDD-II scheme employs {+1008-, ‒1008-, +185-, ‒185-} ps/nm dispersive elements.

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Funding

National Key Research and Development Program of China (2019YFB1803602); National Natural Science Foundation of China (61835008, 61860206001).

Disclosures

The authors declare no conflict 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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