Frequency-domain adaptive MIMO filter with fractional oversampling using stochastic gradient descent for long-haul transmission over coupled 4-core fibers: erratum.

This erratum corrects errors in Fig. 9(b) and Fig. 14 of our published paper [Opt. Express31, 13104 (2023)10.1364/OE.486032]. Other results, descriptions, and conclusions are not affected by this correction.

Frequency-domain adaptive MIMO filter with fractional oversampling using stochastic gradient descent for long-haul transmission over coupled 4-core fibers.

We propose a fractionally spaced frequency-domain adaptive multi-input multi-output (MIMO) filter architecture in which the sampling rate of input signals is below 2× oversampling with a non-integer oversampling factor for mode demultiplexing in long-haul transmission over coupled multi-core fibers. The frequency-domain sampling rate conversion to the symbol rate, i.e., 1× sampling, is placed after the fractionally spaced frequency-domain MIMO filter. The filter coefficients are adaptively controlled by stochastic gradient descent and gradient calculation with back propagation through the sampling rate conversion from the output signals on the basis of deep unfolding. We evaluated the proposed filter through a long-haul transmission experiment of 16-channel wavelength-division multiplexed and 4-core space-division multiplexed 32-Gbaud polarization-division-multiplexed quadrature phase shift keying signals over coupled 4-core fibers. The fractional oversampling frequency-domain adaptive 8×8 filter with 9/8× oversampling provided little performance penalty after 6240-km transmission compared to the conventional 2× oversampling frequency-domain adaptive 8×8 filter. The computational complexity in terms of the required number of complex-valued multiplications was reduced by 40.7%.

Compensation and monitoring of transmitter and receiver impairments in 10,000-km single-mode fiber transmission by adaptive multi-layer filters with augmented inputs.

We propose an adaptive multi-layer (ML) filter architecture to compensate for linear impairments that occur in transmitter (Tx) and receiver (Rx) components in ultra-long-haul optical fiber transmission systems, in which large chromatic dispersion (CD) accumulates in the received signal. The architecture consists of strictly linear (SL) and widely linear (WL) filter layers, and the coefficients of the ML filters are adaptively controlled by gradient calculation with back propagation and stochastic gradient descent. Static CD compensation is performed on the received signal and its complex conjugate before the adaptive ML filters. These augmented signals are then the inputs of the first 2×1 SL filter layer of the ML filters, for compensation of in-phase (I) and quadrature (Q) impairments on the Rx side. Tx IQ impairments and polarization effects as well as Rx IQ impairments are adaptively compensated in the ML filters. By sweeping CD compensation filters before the ML filters, this architecture mitigates the computational complexity for back propagation of the ML filters especially for ultra-long-haul transmission, while mutual non-commutativity between the WL filter for IQ impairment compensation and the CD compensation filter is appropriately solved. We evaluated the proposed adaptive ML filter architecture with augmented inputs through both simulation and wavelength-division multiplexed transmission experiments of 32-Gbaud polarization-division-multiplexed 64-quadrature amplitude modulation-based probabilistic constellation shaped signals over 10,000 km of single-mode fiber (SMF). The results demonstrated that the proposed adaptive ML filter architecture effectively compensates for Tx and Rx IQ skews in ultra-long-haul SMF transmission, and that impairments can be monitored individually from the converged filter coefficients of the corresponding layers.

Adaptive multi-layer filters incorporated with Volterra filters for impairment compensation including transmitter and receiver nonlinearity.

We propose a receiver-side signal processing to compensate for nonlinearity that occurs in transmitter (Tx) and receiver (Rx) components of coherent optical fiber transmission systems. Nonlinear effects in transmission systems are not mutually commutative with any linear effects in general. Considering the order in which all the relevant impairments occur, we adopt a multi-layer (ML) filter architecture. The ML filters consist of strictly-linear and widely-linear filter layers to compensate for relevant linear impairments that occur in a transmission system and two Volterra filter layers to compensate for Rx and Tx nonlinearity. The coefficients of the ML filters including Volterra filter layers are adaptively controlled by using a gradient calculation with back propagation, which is similar to that used in the learning of neural networks, from the last layer and stochastic gradient descent to minimize a loss function that is composed of the last layer outputs. We evaluated the compensation performance of Tx and Rx nonlinearity using the proposed adaptive ML filters including Volterra filter layers both in simulations and experiments of the transmission of a 23 Gbaud polarization-division-multiplexed 64-quadrature amplitude modulation signal over a 100-km single-mode-fiber span. The results demonstrated that the Volterra filter layers in the ML filter architecture could compensate for the nonlinearity that occurs in Tx and Rx simultaneously and effectively even when other impairments such as chromatic dispersion coexist.

Transmitter and receiver impairment monitoring using adaptive multi-layer linear and widely linear filter coefficients controlled by stochastic gradient descent.

We propose a monitoring method for individual impairments in a transmitter (Tx) and receiver (Rx) by using filter coefficients of multi-layer strictly linear (SL) and widely linear (WL) filters to compensate for relevant impairments where the filter coefficients are adaptively controlled by stochastic gradient descent with back propagation from the last layer outputs. Considering the order of impairments occurring in a Tx or Rx of coherent optical transmission systems and their non-commutativity, we derive a model relating in-phase (I) and quadrature (Q) skew, IQ gain imbalance, and IQ phase deviation in a Tx or Rx to the WL filter responses in our multi-layer filter architecture. We evaluated the proposed method through simulations using polarization-division multiplexed (PDM)-quadrature phase shift keying and a transmission experiment of 32-Gbaud PDM 64-quadrature amplitude modulation over a 100-km single-mode fiber span. The results indicate that both Tx and Rx impairments could be individually monitored by using the filter coefficients of adaptively controlled multi-layer SL and WL filters precisely and simultaneously, decoupled by chromatic dispersion and frequency offset, even when multiple impairments existed.

Adaptive equalization of transmitter and receiver IQ skew by multi-layer linear and widely linear filters with deep unfolding.

We propose a multi-layer cascaded filter architecture consisting of differently sized strictly linear (SL) and widely linear (WL) filters to compensate for the relevant linear impairments in optical fiber communications including in-phase/quadrature (IQ) skew in both transmitter and receiver by using deep unfolding. To control the filter coefficients adaptively, we adopt a gradient calculation with back propagation from machine learning with neural networks to minimize the magnitude of deviation of the filter outputs of the last layer from the desired state in a stochastic gradient descent (SGD) manner. We derive a filter coefficient update algorithm for multi-layer SL and WL multi-input multi-output finite-impulse response filters. The results of a transmission experiment on 32-Gbaud polarization-division multiplexed 64-quadrature amplitude modulation over a 100-km single-mode fiber span showed that the proposed multi-layer SL and WL filters with SGD control could compensate for IQ skew in both transmitter and receiver under the accumulation of chromatic dispersion, polarization rotation, and frequency offset of a local oscillator laser source.

Wide range rate adaptation of QAM-based probabilistic constellation shaping using a fixed FEC with blind adaptive equalization.

We investigate the rate adaptability of quadrature amplitude modulation (QAM)-based probabilistic constellation shaping (PCS) using a fixed forward error correction (FEC) scheme over a wide range of information rates (IRs). Blind adaptive equalization that does not sacrifice any of the IRs was adopted. We show that the conventional decision directed least mean square (DDLMS) algorithm can cause a problem of mis-convergence when it is applied to the PCS of a low IR. To avoid the mis-convergence of DDLMS, we propose a DDLMS-based algorithm that simultaneously minimizes the error between the average symbol power of filter outputs and that of a transmitted PCS signal. Using this technique, we conducted a wavelength-division multiplexed transmission experiment with 32-Gbaud 16/64QAM-based PCS and a fixed FEC of a low-density parity-check code for DVBS-2, where the IR of PCS was optimized at each transmission distance. We confirmed that the data rate of PCS with a fixed FEC and DSP configuration could be improved up to 1.9 times compared with that of QAM-only rate adaptation and that 64QAM-based PCS could provide a wider range of transmission distance and IR while almost covering that of the 16QAM-based one.

Performance of mode diversity reception of a polarization-division-multiplexed signal for free-space optical communication under atmospheric turbulence.

We investigated the performance of mode diversity reception of a polarization-division-multiplexed (PDM) signal with few-mode-fiber (FMF) coupling for high-speed free-space optical communications under atmospheric turbulence. Optical propagation through eigenmodes of a FMF yields coupling between different linearly polarized (LP) modes in orthogonal polarizations, which causes power imbalance and loss of the orthogonality of multiplexed signals within each individual LP mode. Due to this phenomenon, the architecture of mode diversity combining affects the receiver performance. We numerically simulated the power fluctuation coupled to each LP mode after atmospheric propagation and FMF propagation in the condition of an optical downlink from a low-Earth-orbital satellite to the ground. We found that full receiver-side multiple-input multiple-output (Rx-MIMO) architecture in three-mode diversity reception improved the performance by 5 dB compared with selection combining (SC) of signals decoded individually in LP modes, and that it mitigated the required transmitted power by 6 dB compared with reception with single mode fiber (SMF) coupling. We also experimentally confirmed in three-mode diversity reception of a 128 Gb/s PDM-quadrature phase-shift keying with a diffuser plate as a turbulence emulator, that full Rx-MIMO with adaptive filters could work under severe fading and that it outperformed SC.

Storage and retrieval of a squeezed vacuum.

Storage and retrieval of a squeezed vacuum was successfully demonstrated using electromagnetically induced transparency. The squeezed vacuum pulse having a temporal width of 930 ns was incident on the laser cooled 87Rb atoms with an intense control light in a coherent state. When the squeezed vacuum pulse was slowed and spatially compressed in the cold atoms, the control light was switched off. After 3 mus of storage, the control light was switched on again, and the squeezed vacuum was retrieved, as was confirmed using the time-domain homodyne method.

Ultraslow propagation of squeezed vacuum pulses with electromagnetically induced transparency.

We have succeeded in observing ultraslow propagation of squeezed vacuum pulses with electromagnetically induced transparency. Squeezed vacuum pulses (probe lights) were incident on a laser-cooled 87Rb gas together with an intense coherent light (control light). A homodyne method sensitive to the vacuum state was employed for detecting the probe pulse passing through the gas. A delay of 3.1 micros was observed for the probe pulse having a temporal width of 10 micros.