Real-time determination of laser beam quality by modal decomposition

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.

Study on Laser Parameter Measurement System Based on Cone-Arranged Fibers and CCD Camera

This paper proposes a new laser parameter measuring method based on cone-arranged fibers to further improve the measurable spot size, allowable incident angle range, and spatial sampling resolution. This method takes a conical array composed of flexible fibers to sample and shrink the cross-section spot of the laser beam, facilitating low-distortion shooting with a charge-coupled diode (CCD) camera, and adopts homogenized processing and algorithm analysis to correct the spot. This method is experimentally proven to achieve high-accuracy measurements with a decimeter-level spot-receiving surface, millimeter-level resolution, and high tolerance in order to incite skew angle. Comparing the measured spot under normal incidence with the real one, the root mean square error (RMSE) of their power in the bucket (PIB) curves is less than 1%. When the incident angle change is between -8° and 8°, the RMSE is less than 2% and the measurement error of total power is less than 5% based on the premise that the fiber's numerical aperture (NA) is 0.22. The possibility of further optimizing the measurement method by changing the fiber parameters and array design is also reported.

Spatial tomography of light resolved in time, spectrum, and polarisation

Measuring polarisation, spectrum, temporal dynamics, and spatial complex amplitude of optical beams is essential to studying phenomena in laser dynamics, telecommunications and nonlinear optics. Current characterisation techniques apply in limited contexts. Non-interferometric methods struggle to distinguish spatial phase, while phase-sensitive approaches necessitate either an auxiliary reference source or a self-reference, neither of which is universally available. Deciphering complex wavefronts of multiple co-propagating incoherent fields remains particularly challenging. We harness principles of spatial state tomography to circumvent these limitations and measure a complete description of an unknown beam as a set of spectrally, temporally, and polarisation resolved spatial state density matrices. Each density matrix slice resolves the spatial complex amplitude of multiple mutually incoherent fields, which over several slices reveals the spectral or temporal evolution of these fields even when fields spectrally or temporally overlap. We demonstrate these features by characterising the spatiotemporal and spatiospectral output of a vertical-cavity surface-emitting laser.

The work harnesses principles of spatial state tomography to fully characterise an optical beam in space, time, spectrum, and polarisation. Analysis of the output of a vertical-cavity surface-emitting laser illustrates the technique’s capabilities.

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.

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.

Measuring the squared amplitudes of the Laguerre-Gaussian beams via a single intensity frame

We propose the use of an intensity technique to decompose superpositions consisting of two, three, or four basis Laguerre-Gaussian (LG) modes, and measure the orbital angular momentum (OAM) of such superpositions. The mode generation and decomposition are both accomplished only on a 2f optical imaging system. We demonstrate numerically and experimentally that the squared amplitudes of superpositions can be determined by recording a single frame of the intensity distribution. This is accomplished by measuring the intensity along certain circles and solving a linear set of equations relating the sampled intensities to squared amplitudes. The accuracy of better than 98% for composite beams consisting of two, and about 90% for composite beams consisting of more than two basis modes are achieved. Finally, we report the value of the measured OAM of the superpositions with excellent accuracy regarding theoretical values, for small and large integer and non-integer OAM.

Optical storage of Ince-Gaussian modes in warm atomic vapor

We report on the optical storage of Ince-Gaussian modes in a warm rubidium vapor cell based on electromagnetically induced transparency protocol, and we also qualitatively analyze how atomic diffusion affects the retrieved beams after storage. Ince-Gaussian modes possess very complex and abundant spatial structures and form a complete infinite-dimensional Hilbert space. Successfully storing such modes could open up possibilities for fundamental high-dimensional optical communication experiments.

Modal analysis of structured light with spatial light modulators: a practical tutorial

A quantitative analysis of optical fields is essential, particularly when the light is structured in some desired manner, or when there is perhaps an undesired structure that must be corrected for. A ubiquitous procedure in the optical community is that of optical mode projections-a modal analysis of light-for the unveiling of amplitude and phase information of a light field. When correctly performed, all the salient features of the field can be deduced with high fidelity, including its orbital angular momentum, vectorial properties, wavefront, and Poynting vector. Here, we present a practical tutorial on how to perform an efficient and effective optical modal decomposition, with emphasis on holographic approaches using spatial light modulators, highlighting the care required at each step of the process.

Fast modal analysis for Hermite-Gaussian beams via deep learning

The eigenmodes of Hermite-Gaussian (HG) beams emitting from solid-state lasers make up a complete and orthonormal basis, and they have gained increasing interest in recent years. Here, we demonstrate a deep learning-based mode decomposition (MD) scheme of HG beams for the first time, to the best of our knowledge. We utilize large amounts of simulated samples to train a convolutional neural network (CNN) and then use this trained CNN to perform MD. The results of simulated testing samples have shown that our scheme can achieve an averaged prediction error of 0.013 when six eigenmodes are involved. The scheme takes only about 23 ms to perform MD for one beam pattern, indicating promising real-time MD ability. When larger numbers of eigenmodes are involved, the method can also succeed with slightly larger prediction error. The robustness of the scheme is also investigated by adding noise to the input beam patterns, and the prediction error is smaller than 0.037 for heavily noisy patterns. This method offers a fast, economic, and robust way to acquire both the mode amplitude and phase information through a single-shot intensity image of HG beams, which will be beneficial to the beam shaping, beam quality evaluation, studies of resonator perturbations, and adaptive optics for resonators of solid-state lasers.

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.

Hermite-Gaussian mode detection via convolution neural networks

Hermite-Gaussian (HG) laser modes are a complete set of solutions to the free-space paraxial wave equation in Cartesian coordinates and represent a close approximation to physically realizable laser cavity modes. Additionally, HG modes can be mode-multiplexed to significantly increase the information capacity of optical communication systems due to their orthogonality. Because cavity tuning and optical communication applications benefit from a machine vision determination of HG modes, convolution neural networks were implemented to detect the lowest 21 unique HG modes with an accuracy greater than 99%. As the effectiveness of a CNN is dependent on the diversity of its training data, extensive simulated and experimental data sets were created for training, validation, and testing.

Modeling and Analysis of the Influence of an Edge Filter on the Combining Efficiency and Beam Quality of a 10-kW-Class Spectral Beam-Combining System

Filter-based spectral beam combining (FSBC) is a promising power-scaling concept for high-power, broad-linewidth fiber lasers, as it relaxes the requirements for linewidth control and also the sizes of the individual beams. As the combining element in the FSBC system, the steep-edge filter plays a major role in achievement of the combining efficiency and the beam quality. In this case, we combine the uncorrelated surface roughness model and the combining efficiency model, and we conduct a comprehensive analysis of the effects of surface roughness, thickness error, and incident angle on the filter’s optical properties and the combining efficiency, in order to determine the optimal configuration for the laser beam-combining system. The simulation results show a good agreement with the measured ones. Meanwhile, through the adoption of the angular spectrum theory, this paper has also conducted a preliminary analysis of the influence of the combining elements on the quality of the combined beam, and some theoretical instructions on the future design of the spectral beam-combining system are provided.

Interferometric spatial mode analyzer with a bucket detector

A spatial mode analyzer based on a Michelson interferometer with a bucket detector is experimentally implemented. The delay line in the interferometer is an optical implementation of the fractional Fourier transform (fFT) which enables the spatial mode analysis of a given input field in the Hermite-Gaussian (HG) mode basis. Modal weights for both 1D and 2D input fields are experimentally measured. Results for input fields comprising of multiple HG modes are also presented.

Suppressing the influence of charge-coupled device vertical blooming on the measurement of laser beam quality factor (M2) of a near-infrared laser

In this paper, a new method, which is based on reconstructing the original intensity distribution of a laser with images captured by a charge-coupled device (CCD) in two orthogonal directions, is proposed for suppressing the influence of CCD vertical blooming on the measurement of the laser beam quality factor (M²). A simplified theoretical model for the distribution of CCD blooming is also proposed. With the proposed method and model, the influence of CCD vertical blooming on the measurement of M² is simulated. The experimental results demonstrate that the new method can be an effective means to measure the M² of a near-infrared laser with a silicon CCD camera. The proposed method can be applied to a beam quality analyzer in order to suppress the influence of blooming on the measurement of M².

Determination of the laser beam quality factor (M2) by stitching quadriwave lateral shearing interferograms with different exposures

A complete complex amplitude reconstruction method for the determination of the laser beam quality factor M² based on the multiple exposure of a quadriwave lateral shearing interferometer (QWLSI) is presented. The theoretical analysis and simulation of the influence of the information in the small signal area on the calculation of the M² factor is provided. The experimental results demonstrate that the new method can be an accurate means to measure the M² factor. The proposed method can avoid the influence of phase inaccuracy in the small signal area of the interferogram, during the measurement of the M² factor.

Real-time complex amplitude reconstruction method for beam quality M2 factor measurement

We present a real-time complex amplitude reconstruction method for determining the beam propagation ratio M² of laser beams based on the transport of intensity equation (TIE). In this work, a synchronous acquisition system consisting of two identical CCDs is established. Once two beam intensity images at different cross-section positions along the optical axis are captured simultaneously by the system, the complex amplitude of the laser beam can be rapidly reconstructed using TIE algorithm. Then the beam intensity distribution at any section position along its propagation direction can be obtained by using angular spectrum (AS) theory. The beam quality M² factor is therefore calculated utilizing the second-order moments and hyperbola fitting methods, which conform to the ISO standard. The suitability of this method is verified by the numerical analysis and experiments with the He-Ne and high-power fiber laser sources, respectively. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.

Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor

A real-time complex amplitude reconstruction method for determining the dynamic beam quality M² factor based on a Mach-Zehnder self-referencing interferometer wavefront sensor is developed. By using the proposed complex amplitude reconstruction method, full characterization of the laser beam, including amplitude (intensity profile) and phase information, can be reconstructed from a single interference pattern with the Fourier fringe pattern analysis method in a one-shot measurement. With the reconstructed complex amplitude, the beam fields at any position z along its propagation direction can be obtained by first utilizing the diffraction integral theory. Then the beam quality M² factor of the dynamic beam is calculated according to the specified method of the Standard ISO11146. The feasibility of the proposed method is demonstrated with the theoretical analysis and experiment, including the static and dynamic beam process. The experimental method is simple, fast, and operates without movable parts and is allowed in order to investigate the laser beam in inaccessible conditions using existing methods.

Measurement of M²-Curve for Asymmetric Beams by Self-Referencing Interferometer Wavefront Sensor

For asymmetric laser beams, the values of beam quality factor Mx2 and My2 are inconsistent if one selects a different coordinate system or measures beam quality with different experimental conditionals, even when analyzing the same beam. To overcome this non-uniqueness, a new beam quality characterization method named as M²-curve is developed. The M²-curve not only contains the beam quality factor Mx2 and My2 in the x-direction and y-direction, respectively; but also introduces a curve of Mxα2 versus rotation angle α of coordinate axis. Moreover, we also present a real-time measurement method to demonstrate beam propagation factor M²-curve with a modified self-referencing Mach-Zehnder interferometer based-wavefront sensor (henceforth SRI-WFS). The feasibility of the proposed method is demonstrated with the theoretical analysis and experiment in multimode beams. The experimental results showed that the proposed measurement method is simple, fast, and a single-shot measurement procedure without movable parts.

Smartphone laser beam spatial profiler

A simple, low-cost, portable, smartphone-based laser beam profiler for characterizing laser beam profiles is reported. The beam profiler utilizes a phosphor silica glass plate to convert UV light into visible (green) light that can be directly imaged onto an existing smartphone CMOS chip and analyzed using a customized app. 3D printing enables the ready fabrication of the instrument package. The beam's diameter, shape, divergence, beam quality factor, and output power are measured for two UV lasers: a CW 244 nm frequency-doubled Ar ion laser and a pulsed 193 nm ArF exciplex laser. The availability of specialized phosphor converters can extend the instrument from the UV to the near infrared and beyond, and the smartphone platform extends the Internet of Things to map laser beam profiles simultaneously in different locations.

Scattered light imaging method (SLIM) for characterization of arbitrary laser beam intensity profiles

A laser beam characterization method is reported, which is applicable to arbitrary and ideal laser beam intensity profiles. This method, called the scattered light imaging method (SLIM), is based on scattered light imaging of a laser beam and provides a complete visualization of it in the region of interest. The method was applied to characterize an arbitrary pedestal-shaped beam and compared with a conventional method (camera scanning). The results we presented show that, for arbitrary beams, it seems much more meaningful to know the intensity profile evolution than to determine an M2 value. Therefore the SLIM is a powerful tool for a new and more complete type of laser beam characterization.

Linear equations method for modal decomposition using intensity information

The linear equations method is proposed to calculate the complete modal content of the partially coherent laser beam using only the intensity information. This method could give not only the incoherent expansion coefficients of the modal decomposition but also the cross-correlation expansion coefficients using the intensity profiles in several planes of finite distance along the propagation direction. A simulation is also presented to verify the validity of this theory. In our algorithm, the minimum and maximum mode orders should be known a priori, so we provide an estimation method for the two parameters.

Holographic spatial coherence analysis of a laser

We show that the second-order coherent-mode representation of a stationary quasi-monochromatic scalar light beam can be experimentally characterized by dual-mode holographic interference using an arbitrary basis. Analysis of the laser beam emitted from a stable spherical mirror cavity, using a mismatched Hermite-Gaussian basis, recovered the profiles and powers of a set of cavity modes with the expected spot size, including a hybrid of frequency degenerate modes. Observed near- and far-field irradiance transverse profiles and associated M2 parameter measures confirmed the results.

Measurement of the M² beam propagation factor using a focus-tunable liquid lens

We demonstrate motion-free beam quality M² measurements of stigmatic, simple astigmatic, and general astigmatic (twisted) beams using only a focus-tunable liquid lens and a CCD camera. We extend the variable-focus technique to the characterization of general astigmatic beams by measuring the 10 second-order moments of the power density distribution for the twisted beam produced by passage through multimode optical fiber. Our method measures the same M² values as the traditional variable-distance method for a wide range of laser beam sources, including nearly TEM(00) (M²≈1) and general astigmatic multimode beams (M²≈8). The method is simple and compact, with no moving parts or complex apparatus and measurement precision comparable to the standard variable-distance method.

Modal decomposition without a priori scale information

The modal decomposition of an arbitrary optical field may be done without regard to the spatial scale of the chosen basis functions, but this generally leads to a large number of modes in the expansion. While this may be considered as mathematically correct, it is not efficient and not physically representative of the underlying field. Here we demonstrate a modal decomposition approach that requires no a priori knowledge of the spatial scale of the modes, but nevertheless leads to an optimised modal expansion. We illustrate the power of the method by successfully decomposing beams from a diode-pumped solid state laser resonator into an optimised Laguerre-Gaussian mode set. Our experimental results, which are in agreement with theory, illustrate the versatility of the approach.

Beam-quality measurements using a spatial light modulator

We present a fast and easy technique for measuring the beam propagation ratio, M(2), of laser beams using a spatial light modulator. Our technique is based on digitally simulating the free-space propagation of light, thus eliminating the need for the traditional scan in the propagation direction. We illustrate two approaches to achieving this, neither of which requires any information of the laser beam under investigation nor necessitates any moving optical components. The comparison with theoretical predictions reveals excellent agreement and proves the accuracy of the technique.

Mode analysis with a spatial light modulator as a correlation filter

A procedure for the real-time analysis of laser modes using a phase-only spatial light modulator is outlined. The procedure involves encoding into digital holograms by complex amplitude modulation a set of orthonormal basis functions into which the initial field is decomposed. This approach allows any function to be encoded and refreshed in real time (60 Hz). We implement a decomposition of guided modes propagating in optical fibers and show that we can successfully reconstruct the observed field with very high fidelity.

Azimuthal decomposition with digital holograms

We demonstrate a simple approach, using digital holograms, to perform a complete azimuthal decomposition of an optical field. Importantly, we use a set of basis functions that are not scale dependent so that unlike other methods, no knowledge of the initial field is required for the decomposition. We illustrate the power of the method by decomposing two examples: superpositions of Bessel beams and Hermite-Gaussian beams (off-axis vortex). From the measured decomposition we show reconstruction of the amplitude, phase and orbital angular momentum density of the field with a high degree of accuracy.

Fast M2 measurement for fiber beams based on modal analysis

We report on a fast and experimentally easy technique for measuring the beam propagation ratio M(2) of light guided by optical fibers. A holographic filter enables us to determine amplitudes and phases of the excited fiber eigenmodes. The coherent superposition of modes allows the reconstruction of the optical field. With this information at hand, we are able to simulate the free-space propagation of the beam and to perform a virtual caustic measurement. Associated beam propagation ratios M(2) accurately agree with ISO-standard measurements.