Polarization dependent gain in Raman fiber amplifiers with multiple pumps.

In this paper, the polarization dependent gain (PDG) in Raman fiber amplifiers (RFAs) with multiple pumps is studied thoroughly. A comprehensive model, which takes the random polarization mode dispersion, the nonlinear coupling between the pumps, and the degree of polarization (DOP) of the pumps into account, is proposed. The complex nonlinear and random coupling inside the optical fiber is described by a set of nonlinear stochastic differential equations (SDEs), which enable co-simulation of the polarized part and the depolarized part of the multiple pumps. It is revealed that the average PDG and the PDG standard deviation are linearly proportional to the DOP of the pumps, which agrees with the single mode case. More importantly, when the pump wavelength is far away from the signal amplification range (pump-signal wavelength difference larger than 100 nm), its DOP still affects the PDG of the signal. Such a phenomenon is caused by the fact that the pumps interact with each other and the pump DOP could transfer among the pumps, which could enhance the PDG of the RFA. The findings in the work will have important implications for the design of the optical transmission systems with the multi-pump RFAs.

Polarization evolution in Brillouin scattering with a partially polarized pump.

Polarization sensitivity has been a major issue in Brillouin scattering-based optical fiber sensing systems. Randomization of the polarization state of the pump is one of the ways to circumvent the problem. However, there could exist a residual degree of polarization (DOP) for the pump after polarization randomization, and hence, a model to characterize the polarization evolution in Brillouin scattering with a partially polarized pump is essential for the performance evaluation. In this work, a comprehensive theoretical model to characterize the beam variation with the partially polarized pump wave and Stokes wave is proposed, which is based on a set of stochastic differential equations (SDEs). The polarized part of the pump wave and the Stokes wave, as well as the total powers of the waves, are incorporated in the coupled SDE simultaneously, which enables the comprehensive simulation of the polarization evolution in the fiber. It is revealed in the study that the DOPs of the pump wave and the Stokes wave affect the gain stability and should be reduced simultaneously by polarization scrambling to ensure a fixed Brillouin gain without fluctuations.

Spatiotemporal diffractive deep neural networks.

A spatiotemporal diffractive deep neural network (STD2NN) is proposed for spatiotemporal signal processing. The STD2NN is formed by gratings, which convert the signal from the frequency domain to the spatial domain, and multiple layers consisting of spatial lenses and space light modulators (SLMs), which conduct spatiotemporal phase modulation. An all-optical backpropagation (BP) algorithm for SLM phase tuning is proposed, with the gradient of the loss function computed by the inner product of the forward propagating optical field and the backward propagating conjugated error field. As a proof of concept, a spatiotemporal word "OPTICA" is generated by the STD2NN. Afterwards, a spatiotemporal optical vortex (STOV) beam multiplexer based on the STD2NN is demonstrated, which converts the spatially separated Gaussian beams into the STOV wave-packets with different topological charges. Both cases illustrate the capability of the proposed STD2NN to generate and process the spatiotemporal signals.

Spherical Gauss-Laguerre beam propagation in 4D space-time.

In this paper, what we believe to be a novel class of beams, which are referred to as the spherical Gauss-Laguerre beams, are proposed. The beams propagate stably in the anomalous dispersive media, within which the second order derivative with respect to t could be combined with the two-dimensional (2D) Laplacian operator in the transverse direction and forms a three-dimensional (3D) Laplacian operator, which describes the beam propagation in the z direction within the four-dimensional (4D) x-y-z-t space-time. The wave equation is solved by the variable separation method and the analytical expression for the spherical Gauss-Laguerre beams is derived. The beams have a 3D Gaussian field distribution with a variable beam waist with respect to the propagation distance. Unlike any 2D spatial vortex beams, the 3D beams could possess either the spatial vortex or the spatiotemporal optical vortex (STOV) by choosing the vortex plane in the 3D x-y-t space-time. The derived spherical Gauss-Laguerre beam expression in the 4D space-time is verified by the numerical simulations with excellent agreement.

Small-angle measurement system enhanced by an optical phased array.

Small-angle measurement can be realized by embedding the laser beam in a reflective sector, within which multiple reflections enlarge the angle between the input and the output beams. However, the maximum detectable angle is limited by the detector aperture at the receiver side. In this work, we propose, to the best of our knowledge, a novel small-angle measurement system enhanced by an optical phased array (OPA), which is loaded on a spatial light modulator (SLM) to increase the maximum measurement range. The experimental results verify the effectiveness of the proposed system, and a wider measurement range with an unaffected measurement accuracy can be obtained. In the proof-of-concept demonstration, the measurement range of the system is enlarged by at least five times compared to the system without OPA, while maintaining the same measurement accuracy.

Characterization of PT-symmetric quantum interference based on the coupled mode theory.

In this paper, we propose a comprehensive quantum theoretical framework to formulate the quantum interference inside the parity-time (PT) symmetric waveguide system which is formed by two coupled optical waveguides with unequal losses. Based on the theory, the expression for the well-known Hong-Ou-Mandel (HOM) dip is derived, which is in an exact agreement with the published results. What's more, a novel one-photon quantum interference phenomenon is predicted according to the model, which suggests a quantum interference process similar to the HOM effect can be observed for the one-photon state, while the other photon is lost due to the waveguide attenuation. Such phenomenon cannot occur in a Hermitian system or in the system formed by the waveguides with equal losses.

Analytical formulation of quantum interference inside coupled waveguides with unequal losses.

In this paper, a theoretical framework is proposed to formulate the quantum interference inside the coupled waveguides with unequal losses. The quantum coupled mode equation is added with the Langevin noise terms to account for the impact of unequal losses, which can be solved analytically. A close form formula is derived for the correlation matrix of the Langevin noise terms, which provides full information for the density matrix of the propagation state. The theory is self-consistent and tested with a three-waveguide system, which is considered as anti-parity-time (PT) symmetric and simulated in the previous publications. An 89-waveguide system is analyzed afterwards to further demonstrate the applicability of the theory.

Analytical expressions for the crosstalk of super-modes in the tightly bounded multicore fibers.

In this paper, an analytical approach for the super-modes in the tightly bounded multicore fibers is proposed. The method considers deterministic and random inter-core coupling, and the analytical analysis is based on the ordinary differential equations (ODEs), which are derived from the stochastic differential equations (SDEs). It is theoretically found that the crosstalk level is directly proportional to the square of the ratio for the random inter-core coupling strength over the deterministic coupling strength, and is inversely proportional to the random coupling correlation length. The ODEs for the variance and the super-mode power correlations are also provided to further facilitate the analysis for the tightly bounded multicore fibers. Simple and explicit formulas for the super-mode crosstalk power and power covariance evaluation are provided in the weak super-mode crosstalk scenario.

Orbital angular momentum mode sorting based on a hybrid radial-angular hybrid lens.

Orbital angular momentum (OAM) modes have their phase distribution as exp (jlθ), which resembles the plane wave in the Cartesian coordinates. Like the traditional lens, which can focus the plane wave on the focal plane, the angular lens can focus the OAM beam in the angular domain, albeit with a relatively long tail due to the unsatisfied angular focal condition for the non-ring shape beams. In this paper, a hybrid lens in the angular domain and the radial domain is proposed. The radial lens with the specific radially distributed phase guarantees the angular focal condition is met for the beams with an arbitrary beam waist or radial field distribution, which significantly improves the performance for the OAM modes sorting by the angular lens. The discrimination of the different OAM modes can be achieved efficiently based on such a single optical component, i.e., the proposed hybrid radial-angular lens, with the OAM modes inter-mode crosstalk as 3.7% when the topological charge difference is 3.

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

A diffractive deep neural network (D2NN) is proposed to distinguish the inverse nonlinear Fourier transform (INFT) symbols. Different from other recently proposed D2NNs, the D2NN is fiber based, and it is in the time domain rather than the spatial domain. The D2NN is composed of multiple cascaded dispersive elements and phase modulators. An all-optical back-propagation algorithm is proposed to optimize the phase. The fiber-based time domain D2NN acts as a powerful tool for signal conversion and recognition, and it is used in a receiver to recognize the INFT symbols all optically. After the symbol conversion by the D2NN, simple phase and amplitude measurement will determine the correct symbol while avoiding the time-consuming NFT. The proposed device can not only be implemented in the NFT transmission system, but also in other areas which require all optical time domain signal transformation and recognition, like sensing, signal coding and decoding, beam distortion compensation and image recognition.

Modulating the Rise and Decay Dynamics of Upconversion Luminescence through Controlling Excitations.

Different from organic dye/quantum dot possessing one luminescent center, upconversion luminescence (UCL) is actually a statistic of temporal behaviors of countless individual activators. Our experimental results have shown that the rise and decay dynamics of UCL is directly associated with the relative contribution of sensitizer-to-activator energy transfer and energy migration among sensitizers, which can be physically modulated by simply tuning the excitation laser. Therefore, dynamic UCL with record-wide 20-fold lifetime, ≈70-fold red-to-green intensity ratio, and reversibly definable emission color is easily realized by just modulating the excitation laser. Moreover, this generally applicable strategy only requires a simplest-possible UCL system whereas prevalent material engineering such as complicated composition design, sophisticated core-shell construction, or tedious chemical synthesis, is no longer needed.

Analytical study on the evolutionary behavior of transfer matrix element moments in strongly coupled multimode systems.

In this paper, the evolutionary behavior of the transfer matrix element moments in strongly coupled multimode systems is studied analytically. The randomly coupled multimode system is modeled by a set of coupled stochastic differential equations (SDEs), which can be used to find the coupled ordinary differential equations (ODEs) for the arbitrary order moments of the matrix elements. Since the ODEs are with the constant coefficients, it is possible to obtain the analytical solutions. The asymptotic behaviors of the solutions are investigated by comparing with the existing results derived from the property of the Haar matrix, and a perfect agreement is observed. The evolutionary behaviors of the transfer matrix element moments computed by the analytical formulas have excellent match with the Monte Carlo simulation results. The analytical method can be highly beneficiary for the multimode system design and analysis.

Simple phase retrieval method based on two intensity measurements on a single plane.

In this work, a simple phase retrieval method is proposed by observing two intensity patterns on a single plane, which are generated with and without a lens. Rigorous theoretical derivations show that the two fields constitute the Fourier transform pairs, and a modified Gerchberg-Saxton algorithm is proposed to recover the phase patterns from the Fourier pairs. The proposed method does not require the intensity patterns to be measured on two different planes along the propagation distance, and this is quite beneficial in a system with a phase tuning element like a spatial light modulator, which can form a virtual lens by creating a parabola-like phase distribution. Experiments are conducted to demonstrate the effectiveness of the proposed phase retrieval method.

Higher order statistics of the Mueller matrix in a fiber with an arbitrary length impacted by PMD.

The higher order (such as the 2nd order and the 4th order) moments of the Mueller matrix elements are important to estimate the polarization mode dispersion (PMD) induced power fluctuations for the forward propagation and the backward scattered signals (e.g. fluctuation of the Raman gain and the Brillouin gain). Current knowledge about the Mueller matrix is limited to the 2nd order moments of its elements in a sufficiently long fiber. In this work, the higher order moments of the Mueller matrix elements of a fiber with arbitrary length is studied analytically. The stochastic differential equations (SDEs) for the moments of the Mueller matrix elements are derived and converted to the related ordinary differential equations (ODEs). Since the ODEs are with the constant coefficients, it is possible to obtain the analytical solutions. The predicted 2nd order moments in a sufficiently long fiber agree well with the existing results. The results of the 4th order moments of the Mueller matrix elements in an arbitrarily long fiber are validated by the numerical simulations with excellent agreement.

Blind back-propagation method for fiber nonlinearity compensation with low computational complexity and high performance.

In this paper, a blind back-propagation (BP) method for fiber nonlinearity compensation with low computational complexity and high performance is proposed. The BP method compensates the fiber chromatic dispersion step by step. Between two linear steps, the proposed method compensates the fiber nonlinearity with the nonlinear tap coefficients optimized by the nonlinear least square method (NLSM). Unlike the traditional BP method, the proposed method takes into account the SPM, the intra-channel XPM and the intra-channel FWM effects while it is purely blind and requires no prior information of the transmission link except the total accumulated chromatic dispersion, e.g., the BP step in the proposed algorithm can be set as an arbitrary value which has no connection to the physical span length. The computational complexity of the proposed method is much lower (less than 50%) than the conventional BP method with one step per span, because of the reduction of the total number of steps. Meanwhile, the method improves the nonlinearity compensation performance in comparison to the standard BP method with one step per span at the optimal input power while maintaining the same computational complexity.

Blind time domain nonlinear compensator embedded in the constant modulus algorithm.

In this paper, a blind nonlinear compensator is proposed for the compensation of fiber nonlinearity. The nonlinear compensator is embedded in the constant modulus algorithm (CMA) which consists of two stages. In the first stage, the linear filter coefficients are calculated while the nonlinear CMA is performed in the second stage. After the nonlinear coefficients reach the stable state, the two-stage CMA switches to the conventional CMA followed by a fixed time domain nonlinear equalizer because the fiber nonlinearity within the transmission link does not vary as the polarization rotation does. This greatly reduces the digital signal processing (DSP) complexity and saves the computational efforts. In comparison with the existing nonlinear compensation techniques such as the back-propagation (BP) method and the Volterra series based nonlinear equalizer (VSNE), the proposed method does not need fast Fourier transform (FFT) or iterative procedures, which significantly reduces the computational complexity. Since the method is performed blindly, no prior information about the transmission link is required, which greatly facilitates the implementation of the nonlinear compensation technique. In addition to these, the coefficients of the nonlinear compensator are optimized adaptively, and it outperforms the existing methods. Finally, the method can work in conjunction with the existing nonlinear compensators, such as the BP method, and significant performance improvement is found over the original nonlinear compensator.

Null space method for tap value initialization in constant modulus algorithm for mode division multiplexing systems.

In this paper, we propose a constant modulus algorithm (CMA) for mode division multiplexing (MDM) systems with improved convergence performance. In order to adapt to sparse channels with large differential mode group delay (DMGD) in MDM systems, the CMA adopts a variable step size similar to the improved proportionate normalized least-mean-square (IPNLMS) algorithm. In additional to that, when a singularity problem is encountered or the tap values fail to converge, it reinitializes the tap coefficients according to the tap vectors of the successfully de-multiplexed data tributaries. The proposed initialization approach is based on the fact that the channel matrix is unitary in the frequency domain in the absence of mode dependent loss (MDL), which means the channel coefficient vectors for each data tributary should be orthogonal to each other. By limiting the initial values of the taps within the null space of the complex conjugate vectors of the successfully de-multiplexed channels, singularity can be effectively avoided and the convergence of the taps is guaranteed. When the number of modes is two, the proposed algorithm becomes the constrained CMA, which has been commonly implemented in polarization division multiplexing (PDM) systems. Although the algorithm has been derived under zero MDL assumption, it is found that the proposed CMA can be quite resilient to MDL. No singularity/tap convergence failure problem occurs when the MDL is below 4 dB at both the input and the output ports.

Capacity analysis for free space coherent optical MIMO transmission systems: with and without adaptive optics.

This paper analyzes the capacity of the free space coherent optical MIMO transmission system. Two scenarios are considered (i.e., with and without the adaptive optics compensation). It is generally accepted that the adaptive optics compensation can significantly improve the system performance, which is rigorously true when the MIMO algorithm is not implemented. However, it might not be the case in the coherent MIMO systems. When the turbulence strength is weak or moderate, this work demonstrates that the phase-only wave-front corrector will increase the mean eigen value of the coherent system capacity matrix HHH and make the eigen value distribution more even, i.e. it will decrease the maximal eigen value while increasing the average eigen value. Hence, the capacity of the system with adaptive optics increases when the channel information is not available, because the sub-channels are placed with equal powers. When the channel information is perfectly available and the water filling algorithm is used to optimize the power allocation, the system with adaptive optics could have a deteriorated performance as the capacity is more related to the large eigen values especially in the low signal to noise ratio (SNR) regime. When the turbulence strength is strong, it is found that adaptive optics will decrease both the mean and maximal eigen values for the capacity matrix HHH, and therefore the system capacity degrades, whether the channel information is available or not.

Tunable arbitrary unitary transformer based on multiple sections of multicore fibers with phase control.

In this paper, we propose a novel tunable unitary transformer, which can achieve arbitrary discrete unitary transforms. The unitary transformer is composed of multiple sections of multi-core fibers with closely aligned coupled cores. Phase shifters are inserted before and after the sections to control the phases of the waves in the cores. A simple algorithm is proposed to find the optimal phase setup for the phase shifters to realize the desired unitary transforms. The proposed device is fiber based and is particularly suitable for the mode division multiplexing systems. A tunable mode MUX/DEMUX for a three-mode fiber is designed based on the proposed structure.

Optimal transmission modes under atmosphere turbulence with transmitter/receiver aperture size constraint.

In this paper, an optimal mode set is proposed to maximize the receiving power for free space transmission under atmosphere turbulence with transmitter/receiver aperture size constraint. The optimal beam profiles are evaluated through eigenmode analysis of the Fredholm integral equation, which is mathematically equivalent to the eigen vector analysis of an infinitely large matrix. The matrix is formed by orthonormal basis expansion, and its element is the overlap integral of the orthonormal basis functions and the Fredholm kernel. If circular aperture is implemented, then it is rigorously proven in this work that the eigenmodes possess certain topological charges (i.e., they are the OAM modes). These OAM modes have specific radial beam profiles, which have been optimized to minimize the power loss and the inter-mode crosstalk. While the traditional OAM beams, such as the Laguerre-Gauss (LG) beams, suffer significant energy loss and inter-radial-mode crosstalk, the optimized beam profiles will remarkably reduce the penalties.