Nonlinear Fourier transform receiver based on a time domain diffractive deep neural network

Junhe Zhou; Junhe Zhou; Junhe Zhou; Qingsong Hu; Qingsong Hu; Haoqian Pu

doi:10.1364/OE.473373

1. Introduction

Nonlinear Fourier transform (NFT) has been viewed as one of the promising ways to overcome the fiber nonlinearity constraint to further boost the system transmission capacity. The fundamental wave propagation model in the optical fibers, i.e., the nonlinear Schrodinger equation (NLSE), forms an integrable system and the beam converted by inverse NFT (INFT) can be used as a perfect information carrier, which will only alter its phase in the nonlinear spectral domain during transmission [1–11]. While the theoretical analysis shows that the system capacity can be significantly improved [4], NFT receiver design is non-trivial due to the high complexity to realize NFT numerically.

Machine learning technology has experienced rapid development and has been implemented in various areas, such as computing, image processing, video and audio signal recognition, signal coding and decoding, etc. [12]. Likewise, it has demonstrated the capability to achieve NFT decoding by symbol classification [5–11]. Like most of the machine learning implementations, they are done in the electronic domain, which is time consuming. All optical machine learning intends to process the data at the speed of light and aims to overcome the computational complexity. Diffractive deep neural network (D2NN) is one of the latest achievements in the field of all optical machine learning [13–20]. It implements cascaded free space propagation and amplitude/phase modulation to realize a tunable spatial beam transformer to enable a variety of all optical spatial signal processing tasks [13–20]. Since the D2NN is implemented in the spatial domain, it cannot be used directly for NFT decoding.

It is widely accepted that time and space are analogous and the concept in space can be extended to the one in time by simply implementing the time domain elements instead of the spatial domain ones [21]. Henceforth, we propose an all-optical time domain D2NN to classify the INFT symbols, by changing the spatial domain propagation into the time domain propagation in a dispersive element, and the spatial domain amplitude or phase modulation into the time domain amplitude or phase modulation.

It is already known that the cascaded dispersive propagation and phase modulation can be used to realize the time domain unitary transforms. It was suggested that all optical time lens could be formed by two dispersive elements and a quadratic phase modulator [22]. All optical fractional Fourier transform could be realized in this way as well [23]. In [24], an all-optical unitary transformer for quantum signal processing was proposed and discussed, which were realized by cascaded dispersive elements and phase modulators. While these pioneering researches are inspiring, the studies focus on the topic of all-optical unitary transform, and no signal recognition scenarios have been discussed. Furthermore, the existing researches usually achieve phase tuning by the global searching methods, such as the simulated annealing algorithm [24–25], while the more efficient gradient search methods have not been used due to the difficulty to find the close form expressions for the gradients. It is highly expected an efficient phase tuning algorithm can be derived.

In this work, we propose a novel time domain D2NN with cascaded dispersive elements and phase modulators for the automatic regression of INFT symbols. After the INFT symbol conversion by the D2NN, simple phase and amplitude measurement will determine the correct symbol while avoiding the time-consuming NFT. Since the time domain D2NN is fiber based, it is convenient to be integrated into the NFT transmission system. During the training stage, an all-optical back-propagation (BP) algorithm is proposed by inversely propagating the conjugated error signal in the D2NN. By measuring the inversely propagating conjugated error signal all optically, one may compute the gradient to adjust the phases layer by layer.

2. Time domain D2NN and all optical error back-propagation

The proposed time domain D2NN is composed of multiple layers, with each layer comprised by a neuron layer, which is realized by an optical phase modulator (OPM), followed by a dispersive element (Fig. (1a)). The dispersive element can be a chirped fiber Brag grating (CFBG) (Fig. (1b)), a piece of optical fiber (Fig. (1c)), or other dispersive elements, as is shown in Fig. 1.

Fig. 1. Basic structure of the D2NN. (a) Basic schematic for the time domain D2NN (b) the layer formed by a chirped fiber grating (CFBG) and the optical phase modulator (OPM) or (c) a piece of optical fiber and the optical phase modulator.

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We use the cascaded operators to characterize the propagation inside the time domain D2NN, which is

(1)$${{\mathbf U}_k} = {{\mathbf N}_k}{{\mathbf D}_k}{{\mathbf U}_{k - 1}},$$

where U_k stands for the time domain wave after propagation through the kth layer, D_k is the propagation operator for the dispersive element, N_k is an operator describing the impact of the phase modulation. If the system is discretized in time, U_k will be a vector, while D_k and N_k will be two matrices. Particularly, N_k will be a diagonal matrix with each of diagonal element as the controllable neuron n_k^l. Those neurons form a neuron vector n_k^T, which is in the form of exp(jφ_k), and the matrix and the vector are related by N_k= diag(n_k). The above-mentioned equation resembles the equation for the spatial domain D2NN [15]. The final output is related to the input by:

(2)$${{\mathbf U}_{N + 1}} = {{\mathbf D}_{N + 1}}\left( {\prod\limits_{k = N}^1 {{{\mathbf N}_k}{{\mathbf D}_k}} } \right){{\mathbf U}_0}.$$

where U₀ is the input signal and U_N+₁ is the output signal, which is transformed by the time domain D2NN.

It will be helpful to understand the D2NN concept by comparing it with the split step Fourier method (SSFM). Indeed, they are very similar. Both of them adopt the cascaded dispersive operators and phase operators. However, the phase operators are nonlinear for SSFM, i.e., the phase is dependent on the amplitude of the input wave in the nonlinear step. Meanwhile, the phase operators are linear in the D2NN case, i.e., the phase is fixed (albeit learned by the following BP algorithm) during the wave propagation. Therefore, there are significant similarity and difference between the two approaches.

One may define the loss function as

(3)$$L = {|{{{\mathbf U}_{N + 1}} - {\mathbf T}} |^2},$$

where T is the target wave. Here we have assumed the amplitude and phase are both included in the target wave; however, the loss function can be redefined if only the intensity is of concern as the spatial domain D2NN [19]. The adjustment of the neurons requires the gradient of the loss function to be evaluated:

(4)$$\begin{array}{l} \frac{{\partial L}}{{\partial {{\mathbf n}_k}^T}} = {({{{\mathbf U}_{N + 1}} - {\mathbf T}} )^H}\frac{{\partial {{\mathbf U}_{N + 1}}}}{{\partial {{\mathbf n}_k}^T}},\\ \frac{{\partial {{\mathbf U}_{N + 1}}}}{{\partial {{\mathbf n}_k}^T}} = {{\mathbf D}_{N + 1}}\left( {\prod\limits_{p = N}^{k + 1} {{{\mathbf N}_p}{{\mathbf D}_p}} } \right)\frac{{\partial {{\mathbf N}_k}}}{{\partial {{\mathbf n}_k}^T}}{{\mathbf D}_k}\left( {\prod\limits_{q = k - 1}^1 {{{\mathbf N}_q}{{\mathbf D}_q}} } \right){{\mathbf U}_0}\\ = {{\mathbf D}_{N + 1}}\left( {\prod\limits_{p = N}^{k + 1} {{{\mathbf N}_p}{{\mathbf D}_p}} } \right)diag\left( {{{\mathbf D}_k}\left( {\prod\limits_{q = k - 1}^1 {{{\mathbf N}_q}{{\mathbf D}_q}} } \right){{\mathbf U}_0}} \right). \end{array}$$

Due to the following symmetrical property of the matrices:

(5)$$\begin{array}{l} {{\mathbf D}_k}^T = {{\mathbf D}_k},\\ {{\mathbf N}_k}^T = {{\mathbf N}_k}, \end{array}$$

one has:

(6)$$\begin{array}{l} \frac{{\partial L}}{{\partial {{\mathbf n}_k}^T}} = {({{\mathbf q}_k^f \odot {\mathbf p}_k^b} )^T},\\ {\mathbf q}_k^f = \left( {\left( {{{\mathbf D}_k}\prod\limits_{q = k - 1}^1 {{{\mathbf N}_q}{{\mathbf D}_q}} } \right){{\mathbf U}_0}} \right),\\ {\mathbf p}_k^b = \left( {\left( {\prod\limits_{p = k + 1}^N {{{\mathbf D}_p}{{\mathbf N}_p}} } \right){{\mathbf D}_{N + 1}}({{{\mathbf U}_{N + 1}}^\ast{-} {{\mathbf T}^\ast }} )} \right), \end{array}$$

where ${\odot}$ denotes the element wise product. For the phase tunable neurons, n_k= exp(jφ_k), where φ_k stands for the phase vector. Hence, we have:

(7)$$\begin{array}{l} \frac{{\partial L}}{{\partial {{\mathbf \varphi }_k}^T}} = j\frac{{\partial L}}{{\partial {{\mathbf n}_k}^T}}{{\mathbf N}_k} + c.c.\\ = 2\textrm{Re} \left( {j\frac{{\partial L}}{{\partial {{\mathbf n}_k}^T}}{{N}_k}} \right)\\ = 2\textrm{Re} {({j{\mathbf p}_k^f \odot {p}_k^b} )^T},\\ {\mathbf p}_k^f = {{\mathbf N}_k}{\mathbf q}_k^f = \left( {\prod\limits_{q = k}^1 {{{\mathbf N}_q}{{\mathbf D}_q}} } \right){{\mathbf U}_0}. \end{array}$$

where c.c. denotes complex conjugate. It can be seen from Eq. (6-7) that the gradient of the loss function can be computed backwardly. More importantly, the vector p_k^f is actually the forward propagating optical field after the kth layer, which can be obtained directly through forward optical propagation. The vector p_k^b can be obtained through all-optical back-propagation of the conjugated error field at the same position. The conjugated error wave can be obtained through optical interference and all optical phase conjugation, which can be accomplished within two steps. Firstly, one needs to conduct all optical interference between the output signal of the DNN and the target waveform, so that the subtraction of the two waveforms can be obtained all optically. Secondly, one may obtain the conjugated error signal through parametric amplification, which is able to produce the phase conjugated signal with high fidelity on the idler wave. The analysis above demonstrates that an all-optical BP algorithm is realizable, and the detailed illustration is shown in Fig. 2.

Fig. 2. Forward and backward propagation of the vectors ${\mathbf p}_k^f$ and ${\mathbf p}_k^b$.

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After the gradient of each layer is obtained, one may update the phase in each layer:

(8)$${{\mathbf \varphi }_k}^{new} = {{\mathbf \varphi }_k}^{old} - \mu \frac{{\partial L}}{{\partial {{\mathbf \varphi }_k}^{old}}},$$

where µ is the learning rate. After rounds of iterations, the phases are tuned and optimized.

3. Results and discussions

In the simulation, the dispersive element has the chromatic dispersion of 160 ps²/nm, which equals the chromatic dispersion of a 10 km-long fiber [24]. Without loss of generality, a chirped fiber grating (CFBG) [24] is assumed as the dispersive element in our simulation. Within the time period of 1 ns, there exist 128 neurons, which suggest the phase modulator produces a tunable phase within the duration of about 8 ps. It is worth mentioning that each layer of the time domain DNN may have either different number of neurons or the same number of neurons. In the simulation, layers with the same number of neurons are assumed. Such phase modulation is possible as the 64 Gbaud coherent transceiver is commercially available, which can produce amplitude and phase modulated signals with 128 G samples per second. One phase modulator and one dispersive element form a D2NN layer.

NFT transmission is realized on a polarization multiplexed (Pol-Mux) systems with a discrete NFT spectrum, which has two eigen values of 0.3j and 0.6j [2]. The discrete eigen values are modulated with QPSK signals [2], with the radii as 0.14 and 5 on the two eigen values. A π/4 phase difference is inserted during the modulation for the two eigen values [2]. The baud rate of the transmission system is 1 G Baud/s, which suggests the time duration of each symbol is 1 ns [2]. Since we have two polarizations and two eigen values for QPSK modulation, one should have totally 256 different symbols. However, due to the property of NFT, many of them are alike in amplitude and phase and only differ from other symbols by a constant phase term on the both polarizations. Hence, one may find 16 “independent” symbols for the Pol-Mux system, which have the different amplitude or phase distributions from each other. Those independent symbols are shown in Fig. 3. It should be noted that only the power distribution is plotted in Fig. 3 and some of the pulses might look alike. However, they differ in phase distributions as shown in Fig. 4, e.g., symbol 1 and symbol 9 in Fig. 3 and Fig. 4. With the different input symbols, the target outputs are the rectangular pulses in different time slots. By evaluating the power within the time slot, one may conclude which symbol it is, while avoiding the time-consuming NFT. The D2NN phase optimization is realized by adjusting the phase with respect to the phase gradient, which is calculated from the forward propagation error vector p_k^f and the backward propagation error vector p_k^b. Both of the error vectors are obtained by all optical propagation. During the training process, 16 classes of symbols with the targets of one in the related time slots are used as the input and the output. 10000 iterations are conducted with the proposed BP algorithm to obtain the optimized phase distributions. The learning rate is 0.001.

Fig. 3. The 16 independent pulses for the Pol-Muxed NFT systems with two discrete eigen values. The QPSK signals are modulated on the discrete nonlinear spectrum.

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Fig. 4. The phase distributions of the 16 independent pulses in Fig. 3.

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Two transmission systems are considered. The first system has a relatively short span length of 40 km, while the second system has a longer span length of 80 km, which resemble the systems in [2], with the fiber span lengths as 41.5 km and 83 km. Standard single mode fibers are assumed for the spans, with the chromatic dispersion as 16 ps²/nm/km and the nonlinear coefficient as 1.2 W^-1/km. Optical amplifiers are inserted after the spans to compensate for the fiber attenuation, with the noise figures (NFs) to be adjustable to tune the SNR at the output. The short fiber span ensures the transmission system to approximate the integrable system while the long fiber span propagation may deviate from the propagation in the perfect integrable system. The INFT symbol propagation in the related optical links is simulated by the coupled Manakov equation.

The short span transmission has a relatively lower power level and the signals have propagated for 5 spans with the total length of 200 km. Five optical amplifiers with the NFs of 5 dB are inserted after the spans. The demultiplexing results for the 16 independent signals are shown in Fig. 5. The number of layers for the D2NN is 14, with each layer containing 128 neurons per nanosecond. Since the targets for the time domain D2NN conversion are the rectangular pulses, the converted signals in Fig. 5 are the stair shape pulses. In this way, the signal is converted to an intensity modulated signal and can be easily recognized. As discussed earlier, one may tell the correct symbol by evaluating the powers within the time slots. For example, the first sub-figure in Fig. 5 demonstrates the converted pulse for the first independent symbol in Fig. 3, which has the highest power concentration on the first time slot both for the x and y polarizations. Comparing with the powers within other time slots, one may conclude that this is the first symbol in Fig. 3 after the conversion by the time domain D2NN.

Fig. 5. The 16 independent pulses demultiplexing results with the fiber-based time domain D2NN. The targets are the rectangular pulses during the D2NN training. One may collect the powers within the time slots to determine which INFT symbol it should be.

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While the 16 independent symbols can be recognized easily by the power measurement, one may distinguish the signals with the same amplitude and phase distributions for the x and y polarizations (except for a constant phase term) through the relative phase measurement. The signals are coherently measured to obtain the phases. Since they are modulated with QPSK signals, 4 different phases exist for each polarization and there exist 16 different phase combinations regarding x-y polarizations for one symbol class. According to the phase, the received INFT symbols are finally categorized as 256 different types. Carrier frequency offset as well as phase offset estimation and compensation are performed before decision [8]. As such, the INFT symbols are recognized all optically without the complex NFT. It is also worth noting that special attention should be paid in case there is phase noise on the laser source and the related estimation and compensation algorithms should be used [8].

Totally 256000 symbols are transmitted in the 5-span system with the overall length of 200 km for symbol error rates (SERs) evaluation. The SER of the demultiplexed signals in Fig. 5 with respect to the different SNRs are shown in Fig. 6. The missing point suggests that the related SER is zero for the 256000 symbols. It is worth noting that since accurate SER estimation requires at least 100 errors to be counted, the slight fluctuations in the figures could be numerical artifacts. The SNR can be adjusted by changing the NFs of the optical amplifiers. It can be seen from the figure that the SER increases as the SNR degrades. The SER is also related to the number of neuron layers for the D2NN. The minimum required layer number to achieve successful demultiplexing is 7. When the layer number reaches 11, the performance becomes stable and can be adopted for the practical receivers. As a comparison, the NFT results are presented, which is achieved by the layer peeling method in combination with the Newton method to obtain the discrete spectrum values. The NFT is conducted with 128 samples per symbol and the related SER is much lower than the results by the time domain D2NN (If 64 samples per symbol are used in NFT as the case in [2], the SER performance will be much worse than the 11-layser D2NN). However, considering the great computational efforts saved during the decoding process, the proposed all optical approach can be viewed as a promising substitution for NFT. Furthermore, the NFT results will deteriorate when the imperfection accumulates during the propagation while the D2NN does not suffer from such penalty, which will be shown in the long-span system propagation analysis below.

Fig. 6. The SER of the de-multiplexed signals with respect to different SNRs. The fiber span length is 40 km. The number of spans is 5.

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As shown in Fig. 6, the 11-layer D2NN seems to be a good tradeoff between the performance and the system complexity, and it is adopted in the analysis for the second transmission system with the fiber span of 80 km. In order to evaluate the impact of the span number on the SER, the SNR is fixed as 10 dB and 15 dB. In Fig. 7, the curves suggest the SER by the time domain D2NN does not simply increase as the span number increases if the impact of SNR is fixed. This is because the performance of D2NN to demultiplex the signals depends on the orthogonality between the symbols, while the nonlinear transformation like NFT does not maintain the orthogonality between the symbols and causes degradation. If the overlap integral between the symbols is small, the D2NN will be more likely to provide a promising demultiplexing result. The overlap integral may change during the propagation in the NFT system, and therefore, we observe the irregularity for the SER curves with respect to the span number. For comparison, the NFT results are also shown in Fig. 7. The SER by the NFT, however, increases as the number of span increases, which are consistent with the NFT system experimental measurement in [2]. This is because the imperfect fiber channel contains attenuation and the fiber transmission system becomes less close to the integrable system as the span number increases. Therefore, the waveforms are less close to the perfect INFT symbols after the long-distance propagation, while the proposed D2NN based receiver shows better performance in this case.

Fig. 7. The SER of the de-multiplexed signals with respect to fiber spans under the SNRs of 10 dB and 15 dB. The fiber span length is 80 km. A 11-layer D2NN is used for INFT symbol recognition.

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Finally, the impact of phase neuron accuracy is studied. The SNR is assumed as 15 dB. In Fig. 8, the phase resolutions for the phase modulators are assumed to be 1 degree and 5 degrees respectively. The SER curves show that negligible difference can be observed between the results by the D2NN with 1-degree phase resolution and the D2NN with no phase error. Even the SER curve by the D2NN with 5-degree phase resolution is close to the other two curves. Since the testing symbols and training symbols are different (with different noise realizations), it is possible that the training optimal might not be the testing optimal, and Fig. 8 shows slight performance gains by the 5-degree precision D2NN at some specific span numbers. Fortunately, the performance fluctuations are very small. The results in Fig. 8 suggest that the proposed D2NN is quite robust with respect to the modulator phase errors.

Fig. 8. The SER of the de-multiplexed signals with respect to fiber spans under the SNRs of 15 dB. The fiber span length is 80 km. Impact of the phase neuron resolution is shown in the figure.

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It is worth noting that the performance is also related to the number of independent symbol classes to be recognized. In this case, the method will have a better performance using a single eigenvalue, because the independent symbol classes will be much fewer. If 3 eigenvalues are used, more independent symbol classes exist and the performance will be worse.

Hence, the proposed time domain D2NN has demonstrated its capability for INFT symbol recognition. The complex NFT is no longer required at the receiver side and conventional phase/amplitude measurement is sufficient for the INFT symbol classification after the signal conversion process by the D2NN. In addition to that, the proposed D2NN is quite robust with respect to the transmission system imperfection and phase resolution limit.

Finally, we would like to comment on the possibility to apply the method on the INFT signals with a continuous spectrum. The proposed time domain D2NN is acting as a unitary transformer which can be used for symbol recognition. For the continuous spectrum, the performance will be degraded because NFT is not a unitary transform and the continuous spectrum signal can not be fully converted by the time domain D2NN. However, the performance for the continuous spectrum as well as the discrete spectrum recovery can be improved by adding the nonlinear activation function (through the Kerr effect) to make the transform non-unitary. The related works are under way.

4. Conclusion

In summary, we have proposed a time domain D2NN, which is realized by cascaded dispersion elements and phase modulators. An optical BP algorithm is proposed which has the gradient of phase to be calculated by the inverse propagation of the conjugated error signal. The newly proposed D2NN has been tested as an all optical INFT symbol recognizer, which achieves promising performance for a 1 GBaud Pol-Muxed NFT system. The proposed scheme can not only be implemented in the NFT receiver, but also for other areas which requires all optical time domain signal transformation and recognition, like sensing, signal coding and decoding, beam distortion compensation and image recognition.

Funding

Science and Technology Commission of Shanghai Municipality (2021SHZDZX0100); Fundamental Research Funds for the Central Universities.

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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Nonlinear Fourier transform receiver based on a time domain diffractive deep neural network

Abstract

1. Introduction

2. Time domain D2NN and all optical error back-propagation

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (8)

Optics Express