Fast mode decomposition in few-mode fibers

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.

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.

Observation of Light Thermalization to Negative-Temperature Rayleigh-Jeans Equilibrium States in Multimode Optical Fibers

Although the temperature of a thermodynamic system is usually believed to be a positive quantity, under particular conditions, negative-temperature equilibrium states are also possible. Negative-temperature equilibriums have been observed with spin systems, cold atoms in optical lattices, and two-dimensional quantum superfluids. Here we report the observation of Rayleigh-Jeans thermalization of light waves to negative-temperature equilibrium states. The optical wave relaxes to the equilibrium state through its propagation in a multimode optical fiber-i.e., in a conservative Hamiltonian system. The bounded energy spectrum of the optical fiber enables negative-temperature equilibriums with high energy levels (high-order fiber modes) more populated than low energy levels (low-order modes). Our experiments show that negative-temperature speckle beams are featured, in average, by a nonmonotonic radial intensity profile. The experimental results are in quantitative agreement with the Rayleigh-Jeans theory without free parameters. Bringing negative temperatures to the field of optics opens the door to the investigation of fundamental issues of negative-temperature states in a flexible experimental environment.

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.

Contribution to the Improvement of the Correlation Filter Method for Modal Analysis with a Spatial Light Modulator

Modal decomposition of light is essential to study its propagation properties in waveguides and photonic devices. Modal analysis can be carried out by implementing a computer-generated hologram acting as a match filter in a spatial light modulator. In this work, a series of aspects to be taken into account in order to get the most out of this method are presented, aiming to provide useful operational procedures. First of all, a method for filter size adjustment based on the standard fiber LP-mode symmetry is presented. The influence of the mode normalization in the complex amplitude encoding-inherent noise is then investigated. Finally, a robust method to measure the phase difference between modes is proposed. These procedures are tested by wavefront reconstruction in a conventional few-mode fiber.

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.

High-performance mode decomposition using physics- and data-driven deep learning

A novel physics- and data-driven deep-learning (PDDL) method is proposed to execute complete mode decomposition (MD) for few-mode fibers (FMFs). The PDDL scheme underlies using the embedded beam propagation model of FMF to guide the neural network (NN) to learn the essential physical features and eliminate unexpected features that conflict with the physical laws. It can greatly enhance the NN's robustness, adaptability, and generalization ability in MD. In the case of obtaining the real modal weights (ρ2) and relative phases (θ), the PDDL method is investigated both in theory and experiment. Numerical results show that the PDDL scheme eliminates the generalization defect of traditional DL-based MD and the error fluctuation is alleviated. Compared with the DL-based MD, in the 8-mode case, the errors of ρ2 and θ can be reduced by 12 times and 100 times for beam patterns that differ greatly from the training dataset. Moreover, the PDDL maintains high accuracy even in the 8-mode MD case with a practical maximum noise factor of 0.12. In terms of adaptation, with a large variation of the core radius and NA of the FMF, the error keeps lower than 0.43% and 2.08% for ρ2 and θ, respectively without regenerating new dataset and retraining NN. The experimental configuration is set up and verifies the accuracy of the PDDL-based MD. Results show that the correlation factor of the real and reconstructed beam patterns is higher than 98%. The proposed MD-scheme shows much potential in the application of practical modal coupling characterization and laser beam quality analysis.

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.

Intensity-only-measurement mode decomposition in few-mode fibers

Recovery of optical phases using direct intensity detection methods is an ill-posed problem and some prior information is required to regularize it. In the case of multi-mode fibers, the known structure of eigenmodes is used to recover optical field and find mode decomposition by measuring intensity distribution. Here we demonstrate numerically and experimentally a mode decomposition technique that outperforms the fastest previously published method in terms of the number of modes while showing the same decomposition speed. This technique improves signal-to-noise ratio by 10 dB for a 3-mode fiber and by 7.5 dB for a 5-mode fiber.

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.

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.

3D time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers

Characterization of the complex spatiotemporal dynamics of optical beam propagation in nonlinear multimode fibers requires the development of advanced measurement methods, capable of capturing the real-time evolution of beam images. We present a new space-time mapping technique, permitting the direct detection, with picosecond temporal resolution, of the intensity from repetitive laser pulses over a grid of spatial samples from a magnified image of the output beam. By using this time-resolved mapping, we provide, to the best of our knowledge, the first unambiguous experimental observation of instantaneous intrapulse nonlinear coupling processes among the modes of a graded index fiber.