Real-time mode decomposition for few-mode fiber based on numerical method

General error analysis of matrix-operation-mode decomposition technique in few-mode fiber laser

The mode decomposition based on matrix operation (MDMO) is one of the fastest mode decomposition (MD) techniques, which is important to the few-mode fiber laser characterization and its applications. In this paper, the general error of the MDMO technique was analyzed, where different influencing factors, such as position deviation of the optical imaging system, coordinate deviation of the image acquisition system, aberrations, and mode distortion were considered. It is found that the MDMO technique based on far-field intensity distribution is less affected by optical imaging system position deviation, coordinate deviation of the image acquisition system, and mode distortion than those based on direct near-field decomposition. But far-field decomposition is more affected by aberration than those based on near-field decomposition. In particular, the numerical results show that the deviation of the coordinate axis direction is an important factor limiting the accuracy of MD. In addition, replacing the ideal eigenmode basis with a distorted eigenmode basis can effectively suppress the decrease in mode decomposition accuracy caused by fiber bending. Moreover, based on detailed numerical analysis results, fitting formulas for estimating the accuracy of the MDMO technique with imperfections are also provided, which provides a comprehensive method for evaluating the accuracy of the MDMO technique in practical engineering operations.

All-fiber few-mode interference for complex azimuthal pattern generation

We report on an all-fiber setup capable of generating complex intensity patterns using interference of few guided modes. Comprised by a few-mode fiber (FMF) spliced to a multimodal interference (MMI) fiber device, the setup allows for obtaining different output patterns upon adjusting the phases and intensities of the modes propagating in the FMF. We analyze the output patterns obtained when exciting two family modes in the MMI device using different phase and intensity conditions for the FMF modal base. Using this simple experimental arrangement we are able to produce complex intensity patterns with radial and azimuthal symmetry. Moreover, our results suggest that this approach provides a means to generate beams with orbital angular momentum (OAM).

Mode attraction, rejection and control in nonlinear multimode optics

Novel fundamental notions helping in the interpretation of the complex dynamics of nonlinear systems are essential to our understanding and ability to exploit them. In this work we predict and demonstrate experimentally a fundamental property of Kerr-nonlinear media, which we name mode rejection and takes place when two intense counter-propagating beams interact in a multimode waveguide. In stark contrast to mode attraction phenomena, mode rejection leads to the selective suppression of a spatial mode in the forward beam, which is controlled via the counter-propagating backward beam. Starting from this observation we generalise the ideas of attraction and rejection in nonlinear multimode systems of arbitrary dimension, which paves the way towards a more general idea of all-optical mode control. These ideas represent universal tools to explore novel dynamics and applications in a variety of optical and non-optical nonlinear systems. Coherent beam combination in polarisation-maintaining multicore fibres is demonstrated as example.

On similarity metrics evaluating the performance of mode decomposition in few-mode optical fibers

Mode decomposition refers to a set of techniques aimed to recover modal content in multimode optical fibers. In this Letter, we examine the appropriateness of the similarity metrics commonly used in experiments on mode decomposition in few-mode fibers. We show that the conventional Pearson correlation coefficient is often misleading and should not be used as the sole criterion for justifying decomposition performance in the experiment. We consider several alternatives to the correlation and propose another metric that most accurately reflects the discrepancy between complex mode coefficients, given received and recovered beam speckles. In addition, we show that such a metric enables transfer learning of deep neural networks on experimental data and tangibly ameliorates their performance.

Analysis of an image noise sensitivity mechanism for matrix-operation-mode-decomposition and a strong anti-noise method

Mode decomposition (MD) based on the matrix operation (MDMO) is one of the fastest mode decomposition methods in fiber laser which has great potential for optical communications, nonlinear optics and spatial characterization applications. However, we found that the image noise sensitivity is the main limit to the accuracy of the original MDMO method, but improving the decomposition accuracy by using conventional image filtering methods is almost ineffective. By using the norm theory of matrices, the analysis result shows that both the image noise and the coefficient matrix condition number determine the total upper-bound error of the original MDMO method. Besides, the greater the condition number, the more sensitive of MDMO method is to noise. In addition, it is found that the local error of each mode information solution in the original MDMO method is different, which depends on the L2-norm of each row vector of the inverse coefficient matrix. Moreover, a more noise-insensitive MD method is achieved by screening out the information corresponding to large L2-norm. In particular, selecting the higher accuracy among the original MDMO method and such noise-insensitive method as the result in a single MD process, a strong anti-noise MD method was proposed in this paper, which displays high MD accuracy in strong noise for both near-filed and far-filed MD cases.

Theoretical and experimental studies on intermodal nonlinear effects of a high-power near-single-mode CW Yb-doped fiber laser

An intermodal-nonlinearity-induced time-frequency evolution model of high-power near-single-mode continuous-wave fiber lasers (NSM-CWHPFLs) was proposed to simulate the evolution of spectral characteristics and beam quality under the combined action of intermodal and intramodal nonlinear effects. The influence of fiber laser parameters on intermodal nonlinearities was analyzed, and a suppression method involving fiber coiling and seed mode characteristic optimization was proposed. Verification experiments were conducted with 20/400, 25/400, and 30/600 fiber-based NSM-CWHPFLs. The results demonstrate the accuracy of the theoretical model, clarify the physical mechanisms of nonlinear spectral sidebands, and demonstrate the comprehensive optimization of intermodal-nonlinearity-induced spectral distortion and mode degradation.

Seeing the strong suppression of higher order modes in single trench fiber using the S2 technique

The single trench fiber (STF) is a promising fiber design for mode area scaling and higher order mode (HOM) suppression. In this Letter, we experimentally demonstrate the strong HOM-suppression in a homemade STF using the spatially and spectrally resolved imaging (S²) technique. This STF has a 20-µm core and its performance is compared to a conventional step-index fiber with almost the same parameter. Results show that the bending loss of the HOM in STF is 8-times larger than conventional fiber at a bend radius of 7 cm. In addition, when severe coupling mismatch is introduced at the input end of the fiber, the STF can keep the fundamental-mode output while the conventional fiber cannot. To the best of our knowledge, this is the first time to experimentally analyze the HOM content in an STF and compare its performance with that of a conventional fiber. Our results indicate the great potential of the STF for filtering the HOM in fiber laser applications.

Complete modal decomposition of a few-mode fiber based on ptychography technology

An exact modal decomposition method plays an important role in revealing the modal characteristics of a few-mode fiber, and it is widely used in various applications ranging from imaging to telecommunications. Here, ptychography technology is successfully used to achieve modal decomposition of a few-mode fiber. In our method, the complex amplitude information of the test fiber can be recovered by ptychography, and then the amplitude weight of each eigenmode and the relative phase between different eigenmodes can be easily calculated by modal orthogonal projection operations. In addition, we also propose a simple and effective method to realize coordinate alignment. Numerical simulations and optical experiments validate the reliability and feasibility of the approach.

M2 factor estimation in few-mode fibers based on a shallow neural network

A high-accuracy, high-speed, and low-cost M² factor estimation method for few-mode fibers based on a shallow neural network is presented in this work. Benefiting from the dimensionality reduction technique, which transforms the two-dimension near-field image into a one-dimension vector, a neural network with only two hidden layers can estimate the M² factor directly. In the simulation, the mean estimation error is smaller than 3% even when the mode number increases to 10. The estimation time of 10000 simulation test samples is around 0.16s, which indicates a high potential for real-time applications. The experiment results of 50 samples from the 3-mode fiber have a mean estimation error of 0.86%. The strategies involved in this method can be easily extended to other applications related to laser characterization.

Programmable high-order mode control method based on acoustically induced fiber grating

We demonstrate a programmable high-order mode control method that can be implemented in high-power fiber lasers. 2 W average-power mode-locked pulses are obtained based on a mode-locked fiber laser working in dissipative soliton resonance regime. The fundamental mode (LP₀₁) is fully or partially converted to the high-order modes (LP_11a/b) via an acoustically-induced fiber grating. The mode-superposition fields are recorded using an optical 4f system, and mode components are subsequently analyzed by a mode decomposition algorithm. Our experiments suggest that the mode patterns are stable and dynamically switchable. The method is expected to possess good application value in optical tweezers, fiber communication, laser material processing and other research fields.

Modal decomposition of an incoherent combined laser beam based on the combination of residual networks and a stochastic parallel gradient descent algorithm

With the increase of the superimposed eigenmodes number, the traditional numerical modal decomposition (MD) technique will inevitably suffer from ambiguity and local minima problems and thus is typically unsuitable for conducting modal decomposition of an incoherent combined laser beam. In this paper, we propose a novel, to the best of our knowledge, MD algorithm, named ResNet-SPGD, which combines the advantages of residual networks (ResNet) and stochastic parallel gradient descent (SPGD) algorithm. Via setting the modal mode coefficients obtained from the CNN model as the initial value of the SPGD algorithm, such algorithm shows an attractive solution to mitigate the problem of modal ambiguity. The proposed algorithm is preliminarily applied to the modal decomposition of an incoherent combined laser beam, and the feasibility is demonstrated via numerical simulations. Complete MD is performed with high accuracy, and the only cost is the sacrifice of some real-time capacity.

2D least-squares mode decomposition for mode division multiplexing

We investigate a fast and accurate technique for mode decomposition in multimode optical fibers. Initial decomposition task of near-field beam patterns is reformulated in terms of a system of linear equations, requires neither machine learning nor iterative routines. We apply the method to step and graded-index fibers and compare the decomposition performance. We determine corresponding application boundaries, propose an efficient algorithm for phase retrieval and carry out a specific preselective procedure that increases the number of decomposable modes and makes it possible to handle up to fifteen modes in presence of realistic noise levels.

Sub-sampled modal decomposition in few-mode fibers

Retrieving modal contents from a multimode beam profile can provide the most detailed information of a beam. Numerical modal decomposition is a method of retrieving modal contents, and it has gained significant attention owing to its simplicity. It only requires a measured beam profile and an algorithm. Therefore, a complicated setup is not necessary. In this study, we conceived that the modal decomposition can be notably improved by data-efficiently sub-sampling the beam image instead of using full pixels of a beam profiler. By investigating the window size, the number of pixels, and algorithm for sub-sampling, the calculation time for the algorithm was faster by approximately 100 times than the case of full pixel modal decomposition. Experiments with 3-mode and 6-mode beams, which originally span 201×201 and 251×251 pixels, respectively, confirmed the remarkable improvement of calculation speed while maintaining the error function at a level of ∼10^-3. This first demonstration of sub-sampling for modal decomposition is based on the modified stochastic parallel gradient descent algorithm. However, it can be applied to other numerical or artificial intelligence algorithms and can enhance real-time analysis or active control of beam characteristics.

Seeing the beam cleanup effect in a high-power graded-index-fiber Raman amplifier based on mode decomposition

Due to the beam cleanup effect, brightness enhancement (BE) can be achieved in a Raman fiber amplifier (RFA) based on multimode (MM) graded-index (GRIN) fiber. In this Letter, a novel, to the best of our knowledge, diagnostic tool of mode decomposition (MD) based on a stochastic parallel gradient descent algorithm is demonstrated to observe the beam cleanup effect in a GRIN-fiber-based RFA for the first time, to our knowledge. During output power boosting up to 405 W at 1130 nm, the output beam quality factor M² improves from 3.45 to 2.88, with a BE factor of 10.5. The MD results based on the near-field beam profiles from RFA indicate that the modal weight of the fundamental mode increases from 74.5% to 87%, confirming that the fundamental mode dominates with higher Raman gain. Moreover, the beam quality is found to be limited by the existence of a higher-order (Laguerre-Gaussian) LG₁₀ mode, which is insensitive to the beam cleanup effect. The correlation coefficient reaches over 0.98 for all MD results. Thus, the accuracy of the MD method is high enough to provide further valuable insight into the physics of spatiotemporal beam dynamics in MM GRIN fiber.

Modal decomposition of fiber modes based on direct far-field measurements at two different distances with a multi-variable optimization algorithm

We present a novel method for modal decomposition of a composite beam guided by a large-mode-area fiber by means of direct far-field pattern measurements with a multi-variable optimization algorithm. For reconstructing far-field patterns, we use finite-number bases of Hermite Gaussian modes that can be converted from all the guided modes in the given fiber and exploit a stochastic parallel gradient descent (SPGD)-based multi-variable optimization algorithm equipped with the D4σ technique in order for completing the modal decomposition with compensating the centroid mismatch between the measured and reconstructed beams. We measure the beam intensity profiles at two different distances, which justifies the uniqueness of the solution obtained by the SPGD algorithm. We verify the feasibility and effectiveness of the proposed method both numerically and experimentally. We have found that the fractional error tolerance in terms of the beam intensity overlap could be maintained below 1 × 10^-7 and 3.5 × 10^-3 in the numerical and experimental demonstrations, respectively. As the modal decomposition is made uniquely and reliably, such a level of the error tolerance could be maintained even for a beam intensity profile measured at a farther distance.

Vortex soliton oscillation in a mode-locked laser based on broadband long-period fiber grating

Optical vortex beams (OVBs) have attracted much attention in diverse applications and spatiotemporal mode locking for optical soliton formation. In this Letter, a compact mode-locked (ML) vortex fiber laser is demonstrated based on a broadband long-period fiber grating near the dispersion turnaround point, where the group velocities between the core modes of ${{\rm LP}_{01}}$ and ${{\rm LP}_{11}}$ in a two-mode fiber are matched. The OVB pulses with first-order orbital angular momentum are oscillated through broadband mode conversion inside the cavity. The time-stretch dispersive Fourier transform method is also employed for the observation of vortex soliton buildup dynamics. The study of vortex soliton oscillation motivates the development towards controlling vortex modes in the ML fiber lasers.

Fast fiber mode decomposition with a lensless fiber-point-diffraction interferometer

Recently, the growing interest in few-mode fibers in telecommunications and high-power lasers has stimulated the demand for fiber mode decomposition (MD). Here we present a fast fiber MD method with a lensless fiber-point-diffraction interferometer. The complex amplitude at the fiber end is achieved by the polarization phase-shifting technique and the lensless imaging technique. Then, the eigenmode coefficients are determined by the mode orthogonal operations of the complex amplitude. In the experiment, the SMF-28e fiber containing 10 linear polarized modes at the wavelength of 632.8 nm is studied for MD. The decomposition of the 50 * 50 pixels interferograms takes only 0.0168 s. The similarity of the intensity patterns of the testing light is larger than 97% before and after the MD. This new, to the best of our knowledge, method can achieve fast and accurate 10-mode MD without using any imaging systems.

Fast mode decomposition in few-mode fibers

Retrieval of the optical phase information from measurement of intensity is of a high interest because this would facilitate simple and cost-efficient techniques and devices. In scientific and industrial applications that exploit multi-mode fibers, a prior knowledge of spatial mode structure of the fiber, in principle, makes it possible to recover phases using measured intensity distribution. However, current mode decomposition algorithms based on the analysis of the intensity distribution at the output of a few-mode fiber, such as optimization methods or neural networks, still have high computational costs and high latency that is a serious impediment for applications, such as telecommunications. Speed of signal processing is one of the key challenges in this approach. We present a high-performance mode decomposition algorithm with a processing time of tens of microseconds. The proposed mathematical algorithm that does not use any machine learning techniques, is several orders of magnitude faster than the state-of-the-art deep-learning-based methods. We anticipate that our results can stimulate further research on algorithms beyond popular machine learning methods and they can lead to the development of low-cost phase retrieval receivers for various applications of few-mode fibers ranging from imaging to telecommunications.

Characterizing the modes at the output of a multimode fiber is time consuming due to computational cost. Here the authors present an algorithm for few-mode-fiber mode decomposition with a fast processing time and using only intensity measurements.

Modal decomposition for the fiber beams with arbitrary degree of coherence based on the Wigner distribution function

Modal decomposition (MD) plays an increasingly important role in characterizing fiber beams. Several promising MD techniques have been proposed in literature, all of which are based on a common hypothesis that the modal field is coherently superposed by transverse modes. However, the partially coherent conditions have to be expected in general. In order to take account of this ordinary case, a novel MD scheme employing the Wigner distribution function (WDF) is introduced, which allows the decomposition of fiber beams without any restrictions regarding their degree of coherence. The four-dimensional (4D) WDF (two spatial and two spatial frequency dimensions) of the 2D fiber beam is reconstructed using the coded aperture technique. Based on the measured WDF and orthogonal property of transverse modes, the modal coefficients as well as the mutual modal degree of coherence will be determined unambiguously. The validity and reliability of the proposed approach are illustrated with numerical examples.

Deep learning enabled superfast and accurate M2 evaluation for fiber beams

We introduce deep learning technique to predict the beam propagation factor M² of the laser beams emitting from few-mode fiber for the first time, to the best of our knowledge. The deep convolutional neural network (CNN) is trained with paired data of simulated near-field beam patterns and their calculated M² value, aiming at learning a fast and accurate mapping from the former to the latter. The trained deep CNN can then be utilized to evaluate M² of the fiber beams from single beam patterns. The results of simulated testing samples have shown that our scheme can achieve an averaged prediction error smaller than 2% even when up to 10 eigenmodes are involved in the fiber. The error becomes slightly larger when heavy noises are added into the input beam patterns but still smaller than 2.5%, which further proves the accuracy and robustness of our method. Furthermore, the M² estimation takes only about 5 ms for a prepared beam pattern with one forward pass, which can be adopted for real-time M² determination with only one supporting Charge-Coupled Device (CCD). The experimental results further prove the feasibility of our scheme. Moreover, the method we proposed can be confidently extended to other kinds of beams provided that adequate training samples are accessible. Deep learning paves the way to superfast and accurate M² evaluation with very low experimental efforts.

Adaptive phase correction of dynamic multimode beam based on modal decomposition

We propose and demonstrate a method for the adaptive phase correction of dynamic multimode fiber beams. The phase of incident beam is reconstructed in real-time based on the complete modal information, which obtained by using the modal decomposition of correlation filter method. For the proof of principle, both of the modal decomposition and the phase correction are implemented using the same computer-generated hologram, which was encoded into a phase-only spatial light modulator. We demonstrate the phase correction of dynamic multimode beam at a rate of 5 Hz and achieve a 1.73-fold improvement on the average power-in-the-bucket. The experimental results indicate the feasibility of the real-time phase correction for the large mode area fiber laser by adaptive optics.

Modal decomposition for few mode fibers using the fractional Fourier system

Modal decomposition (MD) has become an indispensable diagnostic tool for optical fibers. A novel MD method using the fractional Fourier system is developed in this paper. Firstly, the existing phase retrieval (PR) algorithm based on the the fractional Fourier transform (FrFT) power spectra is extended to account for the effect of optical vortex. The extended capability can then be employed to reconstruct the phase of modal field to high fidelity in a non-iterative and non-interferometric manner. Combining the reconstructed phase with the measured near-field (NF) intensity, the modal field could be obtained and based upon which the complete MD (involving modal weights and phases) is performed. The validity and reliability of the method are demonstrated through several numerical examples including the noisy signals with different signal-to-noise ratio (SNR) levels.

Learning to decompose the modes in few-mode fibers with deep convolutional neural network

We introduce a deep-learning technique to perform complete mode decomposition for few-mode optical fibers for the first time. Our goal is to learn a fast and accurate mapping from near-field beam patterns to the complete mode coefficients, including both modal amplitudes and phases. We train the convolutional neural network with simulated beam patterns and evaluate the network on both the simulated beam data and the real beam data. In simulated beam data testing, the correlation between the reconstructed and the ideal beam patterns can achieve 0.9993 and 0.995 for 3-mode case and 5-mode case, respectively. While in the real 3-mode beam data testing, the average correlation is 0.9912 and the mode decomposition can be potentially performed at 33 Hz frequency on a graphic processing unit, indicating real-time processing ability. The quantitative evaluations demonstrate the superiority of our deep learning-based approach.

Multimode fiber modal decomposition based on hybrid genetic global optimization algorithm

Numerical modal decomposition (MD) is an effective approach to reveal modal characteristics in high power fiber lasers. The main challenge is to find a suitable multi-dimensional optimization algorithm to reveal exact superposition of eigenmodes, especially for multimode fiber. A novel hybrid genetic global optimization algorithm, named GA-SPGD, which combines the advantages of genetic algorithm (GA) and stochastic parallel gradient descent (SPGD) algorithm, is firstly proposed to reduce local minima possibilities caused by sensitivity to initial values. Firstly, GA is applied to search the rough global optimization position based on near- and far-field intensity distribution with high accuracy. Upon those initial values, SPGD algorithm is afterwards used to find the exact optimization values based on near-field intensity distribution with fast convergence speed. Numerical simulations validate the feasibility and reliability.

Fast modal decomposition for optical fibers using digital holography

Eigenmode decomposition of the light field at the output end of optical fibers can provide fundamental insights into the nature of electromagnetic-wave propagation through the fibers. Here we present a fast and complete modal decomposition technique for step-index optical fibers. The proposed technique employs digital holography to measure the light field at the output end of the multimode optical fiber, and utilizes the modal orthonormal property of the basis modes to calculate the modal coefficients of each mode. Optical experiments were carried out to demonstrate the proposed decomposition technique, showing that this approach is fast, accurate and cost-effective.

Ultrafast, all-optical control of modal phases in a few-mode fiber for all-optical switching

The phase differences between the transverse modes of an optical fiber can be altered all-optically by intermodal cross-phase modulation. In this Letter, we experimentally demonstrate this effect with ultrashort laser pulses. An ultrashort probe pulse, guided in both modes of a two-mode fiber, is co-propagating and temporally overlapping with an ultrashort control pulse, guided in the fundamental mode only and centered at a separate wavelength. The use of ultrashort pulses allows for a notable phase shift at a 33-fold reduced control pulse energy and a 173-fold reduced fiber length, compared to previous experiments. A total phase shift of 0.285π between the two probe modes was achieved at a 9 nJ control pulse energy in a 19 cm long two-mode graded-index fiber. Additionally, the capability of this scheme to switch ultrashort pulses in an all-optical manner was investigated. A modulation depth of 50% was achieved, limited by temporal nonlinear effects.

Adaptive mode control of a few-mode fiber by real-time mode decomposition

A novel approach to adaptively control the beam profile in a few-mode fiber is experimentally demonstrated. We stress the fiber through an electric-controlled polarization controller, whose driven voltage depends on the current and target modal content difference obtained with the real-time mode decomposition. We have achieved selective excitations of LP01 and LP11 modes, as well as significant improvement of the beam quality factor, which may play crucial roles for high-power fiber lasers, fiber based telecommunication systems and other fundamental researches and applications.

Focus issue introduction: Advanced solid-state lasers (ASSL) 2014

The editors introduce the focus issue on "Advanced Solid-State Lasers (ASSL) 2014," which is based on the topics presented at a congress of the same name held in Shanghai, China, from October 27 to November 1, 2014. This focus issue, jointly prepared by Optics Express and Optical Materials Express, includes 28 contributed papers (21 for Optics Express and 7 for Optical Materials Express) selected from the voluntary submissions by attendees who presented at the congress and have extended their work into complete research articles. We hope this focus issue offers a useful snapshot of the variety of topical discussions held at the congress and will contribute to the further expansion of the associated research areas.