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.


Introduction
Few-mode fiber lasers are widely used in fiber laser communication [1][2][3][4][5][6], nonlinear optics research [7][8][9][10][11], high-power fiber laser processing [12][13][14][15][16], etc.As a core physical parameter, the mode composition of fiber lasers determines the brightness, pointing errors, etc., which is quite important for their practical application.Thus, it is important to give out an effective method that can precisely describe the mode information in few-mode fiber lasers, including the power share and relative phase of different modes, which is generally called mode decomposition (MD).
Fiber laser MD technology can be divided into two kinds of categories, i.e., direct measurement methods and numerical analysis methods.Direct measurement methods include imaging methods (such as spatially and spectrally resolved imaging [17][18][19], cross-correlated imaging methods [20,21]), correlation filter methods [22][23][24], and wavefront measurement methods [25][26][27].Generally, direct measurement methods require a broadband light source or reference beam, which may limit its application in some scenarios with dynamic changes in linewidth, e.g., in characterizing the spatial characteristics between linewidth and transversal mode instability (TMI) threshold [28].The second kind of MD methods i.e., numerical analysis MD methods are based on computer algorithms, in which only the near-field or far-field beam intensity distribution of the fiber output beams are required.Commonly, the MD algorithms contain the Gerchberg-Saxton (GS) algorithm (a phase-retrieval algorithm) [29], line search algorithm [30], stochastic parallel gradient descent (SPGD) algorithm [31][32][33][34][35], and neural networks [36][37][38][39][40][41][42].The first three algorithms (i.e., GS, line search and SPGD algorithms) require iterative calculation, resulting in time-consumption (e.g., the decomposition speed of MD technology based on the SPGD algorithm is ~10Hz), and the MD accuracy is also affected by the initial value selection of the algorithm.In recent years, the application of intelligent algorithms in fiber lasers has received widespread attention, and they have also shown great potential in fiber laser mode decomposition.In 2019, An et al., first reported an MD method based on convolutional neural networks, which achieved a decomposition speed of about 33Hz [36], In 2020 An et al., further reported a real-time MD method based on deep learning, which achieves a MD speed of 200Hz [37].Besides, in 2022, Jiang et al., reported another impressive result of intelligent mode decomposition technology, in which an unsupervised learning method is applied [38].This work demonstrates that, based on the unsupervised learning method, the speed of MD is significantly increased and the single decomposition time under different number of modes keeps nearly the same (such as in the case of 3-10 modes, the corresponding decomposition time is only 1.7 ms per frame).Moreover, in 2022, Zhang et al., introduce the mode transformer network, which can perform MD on 23 modes and has been trained offline using synthetic data [39].Intelligent algorithms are still an important development direction of MD technology, and how to balance MD speed, MD accuracy, and calculating sample data demand is an important issue.
Faster MD techniques are still urgently needed.For example, in the TMI effect the mode dynamics changes at frequencies up to 1-10 kHz, thus MD techniques with speeds above kHz are of great significance for the characterization of the TMI effect.In 2020, a new insight MD method based on matrix operation (MDMO) was proposed by Manuylovich et al. [43], which divided the complete nonlinear MD problem into a linear step (i.e., solve a system of linear equations) and a simple nonlinear step (i.e., solve a simple system of nonlinear equations).For 3-, 5-, and 8-mode fibers the speed of MDMO technique can reach up to 100,000 Hz, furthermore, the number of fiber modes that the MDMO method can accurately decompose is up to 49 [44].In 2022, Xu et al. used the results of the MDMO method as the initial value of the SPGD algorithm and further increased the number of patterns decomposition to 50, the image noise resistance of MD was also improved [45].In addition, our group analyzed the mechanism of MDMO technology's sensitivity to image noise based on matrix norm theory, derived the error upper bound formula of the MDMO technique, and proposed a strong antinoise method quite recently [46].It is important to note that the MDMO technique is a noniterative MD method, and its decomposition accuracy is extremely dependent on the measuring accuracy of the near-field or far-field beam intensity distribution.Such as the mismatch of optical imaging systems can lead to imaging errors and affect the accuracy of mode decomposition.Meanwhile, the deviation between the setting of the image coordinate origin and coordinate axis direction and eigenmodes coordinate origin and coordinate direction during MD will also lead to the decreases of the MD accuracy.In addition, mode distortion caused by fiber bending and aberrations will also lead to image distortion, resulting in a decrease in MD accuracy.Therefore, developing a comprehensive, quantitative error analysis study of the MDMO methods is of great significance for further improving their accuracy.
In this paper, the general error of the MDMO method was analyzed in detail.The section structure is arranged as follows.In Section 2, the design of a fast-mode decomposition system in high-power fiber laser systems was introduced, in which the MDMO method was employed.In Sections 3-6, the effects of installation errors of the imaging system, coordinate deviations, aberrations, and mode aberrations due to fiber bending on the MD accuracy are investigated, respectively.Finally, in Section 7, fitting formulas for the accuracy of the MDMO technique changes with different influencing factors are provided, where the position deviation of the optical imaging system, coordinate deviation of the image acquisition system, aberrations and mode distortion are taken into account.

Introduction to the MOMD technique and practical MD system
The main idea of the MDMO method is to process the MD problem as a projection of the vector represented by the image pixel values, captured by a camera in near-field or far-field, onto the basis vector formed by the product of the linear polarization (LP) eigenmodes or its Fourier transform, namely, solve a system of linear equations.Taking NF-MD as an example, for an input intensity distribution image of a few-mode fiber laser beam with M × M pixels.The intensity of each pixel can be expressed as a linear superposition of the LP eigenmodes, i.e., where m = 1, 2, 3, …, M 2 , N is the mode number, ζ p = γ p exp(iθ p ) are the complex coefficients, γ p and θ p are the mode wights and phases of the pth eigenmode.Rewriting Eq. ( 1) as a matrix equation, i.e., , = AX I (2) where the vector X to be solved contains mode weights and mode phases information, the dimension of vector I is M 2 × 1, which is obtained by rearranging the intensity image matrix with M × M pixels, and the matrix A is composed of paired products of linear polarization (LP) eigenmodes φ p (p = 1, 2, 3, …, N), i.e., Vector X can be expressed as )) where superscript H is the transpose symbol.Noted that for far-filed mode decomposition (FF-MD), the matrix A is composed of paired products of Fourier transform of LP eigenmodes, and the form of vector X is also different from the near-field mode decomposition (NF-MD) [44].
Mode decomposition methods based on numerical analysis typically require a near-field or far-field beam intensity distribution acquisition optical system, such as a 4-f imaging system or a Fourier lens.In addition, in high-power fiber laser systems, the laser beam is usually output through a collimator, which can be equivalent to a focusing lens.In this paper, the focal length of the collimator is adopted as 120mm.However, if two lenses are used to form a 4-f system to amplify the output beam size of the fiber by tens of times (within the range of camera acquisition size), another lens with a focal length of several meters is required, this will result in the system becoming larger.Thus, double 4-f imaging system is feasible.The diagram of the near-field beam intensity distribution image acquisition optical system is shown in Fig. 1(a).The focal lengths of the collimator and Lenses 1-4 are 120mm, 400mm, 20mm and 200mm, respectively.The first 4-f imaging system composed of the collimator and lens 1, which can amplify the near-field intensity distribution image by 3.33 times, furthermore, lenses 2 and 3 form the second 4-f imaging system amplifies the intensity distribution image by 10 times, thus, these two 4-f imaging systems can amplify the near-field intensity distribution image by 33 times.Moreover, the far-field beam intensity distribution image acquisition optical system is shown in Fig. 1(b), the first 4-f system is the same as the near-filed imaging system, and a Fourier lens with a focal length of 100mm is used to convert near-field beam intensity distribution into the far-field distribution.It is feasible and simple to use the generalized scalar diffraction integral, i.e., Collins formula to calculate the envelope of the electric field propagation from the end of optical fiber E 1 (x 1 , y 1 ) through a complex optical system shown in Fig. 1 to the camera target E 2 (x 2 , y 2 ), as the entire complex optical system can be simply represented as a 2×2 ABCD matrix, i.e., AB CD

  
. The Collins formula can be written as [47] ( where A, B, C, D indicate the elements of the imaging system ABCD matrix, z is the total distance propagated along the axis, and (x 1 , y 1 ), (x 2 , y 2 ) are the coordinates of the initial and target surfaces, respectively.Thus, different optical system parameters can be reflected in the values of ABCD matrix elements.Note that for the conjugate imaging system, the value of matrix element B is equal to 0, using the relationship of AD-BC=1 and Eq. ( 5) can be rewritten as [47] ( ) ( ) ( )  (6) From Eqs. ( 5) and ( 6), it can be seen that the deviation of the ABCD matrix (i.e., the deviation of the positions of each lens and camera in the imaging optical system) will directly distort the far-field beam intensity distribution (i.e., I = E 2 E 2 * ) acquired by the camera.In addition, the deviation between the selection of the image coordinate system (x 2 , y 2 ) and the actual situation will also affect the accuracy of MD.Moreover, the mode distortion caused by fiber bending and phase distortion caused by aberrations in optical systems will also cause deformation of the acquisition beam intensity, leading to a decrease in MD accuracy.
In this paper, unless specified, the calculation parameters are taken as the wavelength λ = 1.064μm, the core diameter of step fiber is 30μm and numerical aperture NA = 0.6, modes number N = 3.Moreover, the image size used in the simulation in this article is 256 × 256 pixels.In addition, the correlation coefficient η is defined as the MD accuracy factor, i.e., ( ) ( ) where I m and I r represent the beam intensity distribution captured by a camera and the beam intensity distribution reconstructed from the MD results, respectively.The value of 0＜η≤1, and the larger the value of η, the higher the accuracy of MD.

The effect of installation error of imaging system on MD accuracy
The impact of the position deviation of various lenses and cameras in optical imaging systems on MD accuracy is still unveiled.The definition of ΔS i (i=1, 2, 3, 4, 5) represents the position deviation of the collimator, lenses 1 -4, and camera.In this section, the effect of ΔS i on the accuracy of NF-MD and FF-MD will be investigated in detail.

NF-MD
The MDMO accuracy i.e., correlation coefficients η versus the position deviation of different optical components ΔS i in NF-MD are shown in Fig. 2, and assuming that one of the optical components has a positional deviation, the other optical components are in the ideal position.In addition, negative numbers of ΔS i indicate that the deviation position of the optical element is on the left side of the ideal position, while positive numbers indicate that it is on the right side.As is visible in Fig. 2, the accuracy of the MDMO method changes rapidly with changes in S 1 and S 3 , i.e., the accuracy of MD is greatly affected by the deviation of lens 1 and lens 3 positions.Lens 1 and 3 are the first lenses of the 4-f systems 1 and 2, respectively.Without losing generality, we randomly set 500 sets of position deviation ΔS i values, each ranging from -2mm to 2mm, (Each set of deviations corresponds to different ABCD deviation values).For a certain set of deviations, 1000 beams with different mode component are randomly generated and propagated from the end of fiber to the camera.Then perform MD on each intensity image captured by the camera.By using Eq. ( 7) to calculate the correlation coefficient between the intensity image reconstructed by MD and the original image, each MD accuracy can be obtained.Take the average of 1000 MD accuracies under the same deviation parameter as the MD accuracy for each set of lens deviations.Noted that the accuracy of MD in these 500 randomly set lens deviations is also randomly distributed.In order to observe the linear correlation between the accuracy of MD and the deviation values of each element in the ABCD matrix, i.e., A-A 0 , B-B 0 , C-C 0 and D-D 0 (where A 0 , B 0 , C 0 , and D 0 are the element of ideal ABCD matrices), the accuracy of MD was arranged in a monotonically decreasing order from large to small (as shown in Fig. 3), and the corresponding distribution of the deviation values of the ABCD matrix elements is shown in Fig. 4. From Figs. 3 and 4, one can see that as the MD accuracy decreases (see Fig. 3) the deviation of B and D increases (see Figs.   Furthermore, the Pearson correlation coefficient is often used to describe the correlation between two variables, thus the potential influence of every installation and adjustment error on MD accuracy was accessed by applying the Pearson analysis.Noted that the ranging of Pearson correlation coefficient value from -1 to 1 (i.e., the absolute value of Pearson correlation coefficient from 0 to 1).The closer the absolute of Pearson correlation coefficient to 1, the higher the correlation between the two variables.Fig. 5 shows the heat map of the absolute value of the Pearson correlation coefficient between the MD accuracy and ΔS i , and the deviation of the ABCD matrix elements.From the first column of Fig. 5, one can see that the absolute value of Pearson correlation coefficient between η and |ΔS 1 | is 0.9569, between η and |B-B 0 | is 0.9594, between η and |D-D 0 | is 0.6544, i.e., the MD accuracy is extremely strong correlated with |ΔS 1 | and |B-B 0 |, and strongly correlated with |D-D 0 |.Namely, the correction of the position of collimating lens 1 is very important for achieving high-accuracy of NF-MD.In addition, for 4-f near-field imaging systems, the error of matrix element B in the ABCD matrix has the greatest impact on imaging quality.The relationship between B coefficient and ΔS 1 can be qualitatively explained.Namely, the equiphase plane curvature radius of a beam plays an important role in its propagation.Due to the size of the fiber in the micrometer range, we assume that the beam emitted from the fiber is a point light source (i.e.spherical wave).When the emitted position is in the front focal plane of the first lens, the laser beam will be collimated by the first lens.After that, the beam emitted from the first lens is approximately a plane wave.The curvature of the equiphase plane remains approximately unchanged, therefore it is insensitive to the positional deviation of lens 2. Then the beam emitted from the second lens becomes a converging spherical wave.At the rear focal plane of lens 2, the curvature radius of the beam's equiphase plane is infinite, corresponding to a B coefficient of 0 [47], and the optical system is an imaging optical system.However, if there is a deviation in the position of lens 1, and the beam emitted from lens 1 is no longer a collimated beam, it cannot be imaged on the rear focal plane of lens 2, and B is no longer 0. Thus, the mismatch of lens 1 is the reason why the 4f optical system is no longer an ideal imaging system, which results in the B coefficient no longer being 0. It is noted that the 4f system composed of lens 3 and lens 4 is similar to the first 4f system, hence it will not be repeated.

FF-MD
In FF-MD cases, similarly, assuming only one optical element has positional deviation, Fig. 6 shows the changes of the MD accuracy versus the position deviation of different optical components ΔS i , where i = 1, 2, 3, 4. As can be seen from

The effect of coordinate deviation on MD accuracy
In practice, the human-sets MD coordinates (see the red coordinate system in Fig. 8) usually deviate from the real coordinates (see the yellow coordinate system in Fig. 8), including the positional deviation of the coordinate origin and the direction deviation of the axes.The diagram of the coordinate deviation is shown in Fig. 8, and defined x shift /w and y shift /w as the coordinate origin deviation, where the normalized parameter w is the beam width on the CCD of a Gaussian beam with an initial beam width equal to the optical fiber core radius, and defined Δϕ as the deviation in coordinate axis direction.The correct position of the origin of coordinates and the direction of the axes are the ground trues of our simulation, while the deviation of the coordinate is artificially set for image pixel shift and image rotation.

Shift of coordinate origin
The contour lines of the correlation coefficient (i.e., MD accuracy) versus the shifts in the x-and y-directions of coordinate origin (i.e., x shift /w, y shift /w) are shown in Fig. 9, where the normalized parameter w is the beam width on the CCD of a Gaussian beam with an initial beam width equal to the optical fiber core radius.From Fig. 9, it is shown that as the shifts of the coordinate origin increase the MD accuracy decreases.In addition, in order to maintain MD accuracy of no less than 0.95, the shift value cannot exceed about 0.12w in the NF-MD case (see Fig. 9 (a)).The MD accuracy versus the shifts of coordinate origin in the FF-MD case is shown in Fig. 9 (b).Compared Fig. 9 (a) with Fig. 9 (b), one can see that FF-MD case has a stronger tolerance for the shift of the coordinate origin, e.g., the shift should not exceed 0.15w while ensuring the MD accuracy is not less than 0.95, namely, under the same dimensionless shift of the coordinate origin, the accuracy of FF-MD is higher than that of NF-MD.Additionally, the changes of MD accuracy versus the coordinate origin shift in FF-MD is not symmetrical, we speculate that is because the phases of far-field eigenmodes is more complex and no longer a simple left-right or up-down symmetry relationship, which results in the difference of MD accuracy when the coordinate origin is shifted in different directions.

Deviation of the coordinate axis direction
In the previous text, we used the correlation coefficient between CCD-captured images I CCD and reconstructed beam intensity distribution images I re to describe MD accuracy, i.e., η(I re -I CCD ).It should be noted that this is consistent with the conclusion obtained by using the correlation coefficient between ideal beam intensity distribution images I true and reconstructed images I re to describe MD accuracy, i.e., η(I re -I true ).However, when discussing the influence of coordinate direction deviation Δϕ on MD accuracy, the conclusions drawn from these two methods of describing MD accuracy will be inconsistent.).The reason is that the LP eigenmodes are orthogonal and complete, and even if the image is rotated, its distribution can still be described by the linear combination of LP eigenmodes (i.e., when MD accuracy is described in terms of η(I re -I CCD ), MD accuracy can remain high as the coordinate angle deviation becomes larger).On the other hand, the LP eigenmodes consist of rotationally symmetric modes (e.g., LP01 mode) and rotationally asymmetric modes (e.g., LP11e and LP11o modes), which leads to a large error in decomposition of rotationally asymmetric modes when there is a deflection in the direction of the coordinate axes, which is the reason for the low accuracy when using η(I re -I true ) to describe MD accuracy.Furthermore, whether the MD error caused by the deviation in the direction of the coordinate axis comes from the error in mode weight or phase error is still unveiled.In this paragraph, NF-MD will be used to study this problem.The error of mode weight and phase are defined as where γ and θ are the mode weights and relative phases reconstructed by MD, respectively, and γ 0 and θ 0 are the true mode weights and relative phases, and ||•|| is the L-2 norm of a vector.Fig. 11 shows the mode weights error (see the black curve) and mode phases error (see the blue curve) of NF-MD.Fig. 11 indicates that the mode phase error caused by the deviation of the coordinate axis direction is greater than the mode weight error (i.e., the blue curve is above the red curve).In addition, in MD, the relative mode weights between the fundamental and higherorder modes are usually more concerned.The red curve in Fig. 11 shows the changes of the fundamental mode (FM) weight error ε(γ LP01 ) versus Δϕ, it can be seen that the mode weight of the FM is least affected by Δϕ.Namely, the measurement error of relative weights between FM and HOMs is smaller, while the reason for the larger measurement error of all weight coefficients (see the black curve) is that the higher-order degenerate modes are symmetry when the direction of the coordinate axis deviates, there is coupling between the weight coefficients of the measured degenerate higher-order modes, such as LP11e and LP11o modes.Although the FM is rotationally symmetric, incorrect estimation of the components of HOM in the complex beam intensity distribution formed by linear superposition with HOMs can also lead to prediction errors in the FM components.It also can be inferred from Eq. ( 3) that the coefficient matrix A is composed of paired products of both FM and HOMs, the FM and HOMs are coupled.Thus, the modal weight of the symmetric FM changes with the coordinate axis direction (as shown in the red curves of Fig. 11) is reasonable.Moreover, when Δϕ = π(2n-1)/2 (n = ±1, ±2), the intensity distribution exchange between LP11e mode and LP11o mode, and the measurement results of mode weight and mode phase have the maximum error (see the black and blue curves in Fig. 11).
We also calculated the effect of coordinate axis direction deviation on the measurement error of beam quality M 2  factor, which was obtained by using the reconstructed beam field simulation to calculate beam propagation.The results show that the maximum of percentage measurement M 2 factor error does not exceed 7%.In summary, the deviation of the coordinate axis direction has a relatively small impact on the measurement of the M 2  factor, but it has a significant impact on the measurement of mode (in especially the high-order mode) coefficient and phase.Thus, calibration of the coordinate axis direction is particularly important in the MDMO technique.

The effect of the aberrations on MD accuracy
In this section, the effect of the aberration on MD accuracy is investigated.The influence of aberration is usually considered as phase distortion of the initial field E in , i.e., ( ) where Z n is the Zernike polynomial [48], and C zernike is the Zernike coefficient that indicates the strength of phase distortion.
The simulation mode of the influence of aberration on MD is shown in Fig. 12.For simplify assuming the beam size has been magnified 30 times and two lenses with focal length of 100mm were used for near-field imaging and a Fourier lens with a focal length of 300mm was used for far-field imaging.Furthermore, consider all aberrations in the optical system as additional distorted phases after the last lens in the imaging system.and (c) γ 1 2 = 0.9.Note that Z 1 indicates piston phase distortion, which does not cause distortion in the light intensity distribution and has no impact on the MD accuracy.Therefore, in this paper, only the influence of Z 2 -Z 15 aberration on MD is calculated.In comparison with the NF-MD case (see Fig. 13), the accuracy of FF-MD is more affected by aberrations (see Fig. 14), e.g., when the accuracy of MD is 0.9, the value of the Zernike polynomial coefficient C zernike corresponding to FF-MD is much smaller than that of NF-MD.The physical reason is that aberration mainly causes phase distortion of the light beam, and the impact of phase on the beam profile mainly manifests in the far field after a certain propagation distance.As seen in Fig. 13, for NF-MD the Z 11 aberration has the greatest impact on the accuracy of MD, while for different fundamental mode weights, the effect of the aberrations on MD accuracy has some different, e.g., for Z 4 aberration, as the fundamental mode weights increases the MD accuracy increases.For FF-MD (see Fig. 14) Z 7 and Z 8 aberration has the greatest impact on the accuracy of MD.It is particularly important to clarify the upper tolerance limit of MD for different aberrations.Fig. 15 shows the maximum aberration coefficients C zernike-max (η~0.95) when the MD accuracy is not less than 0.95.As can be seen in Fig. 15, generally, MD has a higher tolerance for aberrations when the ratio of fundamental modes is higher (i.e., i.e., the values of blue lines are mostly greater than those of red and black lines).This is because the fundamental mode is rotationally symmetric and is less affected by phase distortion.More practically, the surface irregularity (peak to valley, PV) value of the lens can be used to characterize the aberration caused by the lens.Fig. 16 shows the MD accuracy for different types of aberrations with PV = λ/4 (i.e., a typical value, and lenses can be controlled within this range, generally).As is visible in Fig. 16 that when PV = λ/4 the impact of aberration on MD accuracy is acceptable, namely, MD accuracy is all higher than 0.95.

The effect of mode distortion induced by fiber bending on MD accuracy
In high-power fiber laser systems, usually, it is necessary to place the fibers on a water-cooled plate with a certain winding radius R, while also suppressing higher-order modes.However,  shows that the correlation coefficient between distorted LP modes with ideal LP modes is decreases as the bending radii decreases, and the FM is most affected by the bending distortion.We used the ideal LP eigenmodes distribution as the basis to perform MD on these beam intensity images, and the results as shown in Fig. 18.It implies that the MD accuracy increases as the bend radius R increases, i.e., the MD accuracy decreases as the mode field distortion becomes severe.In compared with the NF-MD case (see Fig. 18(a)), the influence of fiber bend on MD accuracy is weaker (see Fig. 18(b)).The physical reason is that the mode field distortion caused by fiber bending will improve as the beam propagates in free space to the far field.In some applications, such as using fiber bending to suppress beam quality degradation caused by TMI effect, the bending radius R is usually less than 10cm [49].While in order to avoid serious mode field distortion, the fiber bending diameter should not be less than about several centimeters (~8cm), thus the accuracy of FF-MD is higher than 0.98 (see Fig. 18(b)), However, the NF-MD accuracy will be lower than 0.95 (see Fig. 18(a)).To alleviate the decreases of the MD accuracy caused by the mode distortion in NF-MD, we adopted the distorted LP eigenmodes as the basis to decompose the distorted intensity distribution images.A typical comparison of MD results for distorted beam intensity distribution images using two different bases (i.e., the ideal LP eigenmodes and distorted LP eigenmodes) is shown in Fig. 19.One can see that the classic MDMO method has a lower decomposition effect on distorted beam images, i.e., the discrepancy between the input image and the reconstructed image is significant.While by using the distorted LP eigenmodes as the basis for MD is quite accurate.Furthermore, Fig. 20 shows the changes of MD accuracy versus the NF-MD accuracy by using the distorted LP eigenmodes distribution (i.e., LP(R)) as the basis to perform MD.Compared to the MD results in Fig. 18(a), using the distorted LP mode as the basis, the MD accuracy is almost unaffected by fiber bending.Thus, the decrease in MD accuracy caused by fiber bending can be improved by transforming the basis of MD.Thus, we propose an effective method to suppress the influence of mode distortion on MD accuracy.

Fitting formulas of MD accuracy changes with different factors
In Sections 3-6, we discussed in detail the factors that affect the accuracy of fast mode decomposition.In this Section, the fitting formula of MDMO technique accuracy changes with these factors are presented.
We have shown that the variation of MD accuracy with positional deviation of the optical system is extremely strongly correlated with the deviation value of the matrix element B (i.e., ΔB) in the ABCD matrix of the imaging system.In NF-MD and FF-MD, the random position deviations were set to ± 100 μm and ± 5 mm, respectively, and the changes of the MD accuracy versus ΔB as shown in Fig. 21 (see the blue dots in Figs.21(a Based on Eq. ( 11) the fitting curve is also given in Fig. 21 (see red curves in Fig. 21).In addition, the R 2 factor is used to examine the fitting model, and its value is closer to 1 indicating a better fit.The values of R 2 factor of equations (11a) and (11b) are 0.991and 0.8505, respectively.Thus, our fitting Eq. ( 11) is reliable and simple.In this Section, all the calculated data are shown as dots in Figs.


) As is visible in Fig. 22(a) and Eq. ( 12), the slope of η versus |x shift | of NF-MD is bigger than that of FF-MD, which indicates that the FF-MD method is less affected by coordinate deviation than the NF-MD method.In addition, the trend of MD accuracy with Δϕ in NF-MD and FF-MD is the same, and the fitting model is shown in Eq. ( 13) and Fig. 22(b).The main limited aberrations in high-power fiber laser systems are defocus (Z 4 ), astigmatism (±45°) (Z 5 ), astigmatism (0°/90°) (Z 6 ), coma-y (Z 7 ), coma-x (Z 8 ), vertical trefoil (Z 9 ), oblique trefoil (Z 10 ), and spherical aberrations (Z 11 ) [50].Considering the Z 4 -Z 11 aberration, the relationship between MD accuracy and Zernike coefficient C zernike can be represented by the following formula, i.e., where the polynomial coefficients and fitting accuracy R 2 factors of different aberration fitting formulas are shown in Table 1.Noted that the applicable range of fitting formula ( 14) is η~ 0.9-1.Besides, the larger the higher-order coefficient in the polynomial in fitting formula (14), indicates the greater the effect of this aberration on the MD accuracy.From Table 1, one can see that the three-order coefficients of the formula (14) a 3 are both equal to 0, i.e., the FF-MD is more sensitive to aberrations than that of NF-MD.In addition, spherical has a greater impact on the NF-MD accuracy, while both coma and spherical have a greater impact on the FF-MD accuracy.The influence of fiber bending on MD accuracy can be described by the fitting formula (15), and the calculated data and fitting curve are shown in Fig. 23.The error scaling law of the effect, such as installation error of imaging system, coordinate deviation, aberrations and mode distortion induced by fiber bending on MD accuracy are given by fitting formulas ( 11) - (15), which provided an effective method for evaluating the MD systems accuracy.

Concluding
In this paper, the error of the MDMO method was analyzed in detail.The effect of installation error of the imaging system, coordinate deviation, aberrations, and mode distortion induced by fiber bending on MD accuracy were investigated.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.While the MD accuracy of FF-MD technique is more affected by aberration than that of NF-MD technique.Specifically, by using the Pearson analysis method, we found that the impact of installation error of the imaging system on MD accuracy is extremely strongly correlated to the deviation of matrix element B in the ABCD matrix of the imaging system.In particular, the numerical results also show that the deviation of the coordinate axis direction is an important factor limiting the accuracy of MD.Furthermore, 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 by us.
In summary, we present a comprehensive method for evaluating the MD accuracy of MDMO technique in practical systems, which has important significance for promoting the MDMO technique from theory to practical applications.Disclosures.The authors declare no conflicts of interest.

Fig. 1 .
Fig. 1.The diagrams of (a) near-field and (b) far-field beam intensity distribution image acquisition optical system.
4 (b)   and (d)), but there is no significant correlation between the deviation of C and A and MD accuracy (see Figs.4 (a) and (c)).

Fig. 5 .
Fig. 5. Heat map of the Pearson correlation coefficient absolute value between NF-MD accuracy, position deviation and deviation of the ABCD matrix elements.
Fig 6 in FF-MD cases, the position deviation of a single optical element in an optical system has a small impact on the accuracy of MD.Adopted the same analysis method as near-field MD, randomly setting 500 sets of position deviation ΔS i (i= 1,2,3,4) values, each ranging from -1mm to 1mm.The Pearson analysis results of FF-MD are shown in Fig. 7.It is visible in Fig. 7 that the MD accuracy is extremely strongly correlated with |B-B 0 |, and strongly correlated with |ΔS 1 |, |ΔS 4 |, |A-A 0 |, |D-D 0 |.Thus, the correction of collimating lens 1 and camera position both has a significant impact on FF-MD accuracy.In addition, in far-field imaging systems, the error of matrix element B in the ABCD matrix plays a primary role in imaging quality, while the errors of matrix elements A and D play a secondary role.

Fig. 9 .
Fig. 9. Contour lines of correlation coefficient versus the shift of coordinate origin, where w is the normalized parameter, (a) NF-MD, (b) FF-MD.
The MD accuracy described by η(I re -I CCD ), and by η(I re -I true ) versus the deviation angle of coordinate axis direction in NF-MD and FF-MD are shown in Figs. 10 (a) and 10 (b), respectively.As is visible in Fig. 10, both NF-MD and FF-MD, the MD accuracy of η(I re -I true ) changes rapidly with the changes of the direction of the coordinate axis Δϕ (see the red lines of Figs.10(a) and 10(b)).While η(I re -I CCD ) remains almost unchanged as Δϕ changes (see the blue lines of Fig. 10(a) and 10(b)

Fig. 10 .
Fig. 10.The changes of MD accuracy versus rotation angle of coordinate axis direction, in (a) NF-MD case and (b) FF-MD case.

Fig. 11 .
Fig. 11.The changes of the decomposition errors of mode coefficient (black line), phase (blue line) and fundamental mode coefficient (red line) versus rotation angle of coordinate axis direction, in NF-MD case.

Fig. 12 .
Fig. 12. Diagram of the simulation model for the effect of aberration on MD.The changes of the MD accuracy versus the Zernike coefficient C zernike with different aberrations of NF-MD and FF-MD are shown in Figs. 13 and 14, respectively, where each data is the average value obtained from 1000 MD calculations.In addition, different subfigures in Figs. 13 and 14 represent different proportions of fundamental mode weights, i.e., (a) γ 1 2 = 0.1, (b) γ 1 2 = 0.5,and (c) γ 1 2 = 0.9.Note that Z 1 indicates piston phase distortion, which does not cause distortion in the light intensity distribution and has no impact on the MD accuracy.Therefore, in this paper, only the influence of Z 2 -Z 15 aberration on MD is calculated.In comparison with the NF-MD case (see Fig.13), the accuracy of FF-MD is more affected by aberrations (see Fig.14), e.g., when the accuracy of MD is 0.9, the value of the Zernike polynomial coefficient C zernike corresponding to FF-MD is much smaller than that of NF-MD.The physical reason is that aberration mainly causes phase distortion of the light beam, and the impact of phase on the beam profile mainly manifests in the far field after a certain propagation distance.As seen in Fig.13, for NF-MD the Z 11 aberration has the greatest impact on the accuracy of MD, while for different fundamental mode weights, the effect of the aberrations on MD accuracy has some different, e.g., for Z 4 aberration, as the fundamental mode weights increases the MD accuracy increases.For FF-MD (see Fig.14) Z 7 and Z 8 aberration has the greatest impact on the accuracy of MD.
fiber bending will cause distortion in the distribution of the light field.By using the multi physics field simulation software, COMSOL Multiphysics, the beam mode field distribution of LP eigenmodes under different bending radii were calculated.For example, the intensity distribution of LP eigenmodes without bending and with bending radius R = 8 cm are shown in Fig.17 (a).One can see that the distorted eigenmodes are no longer symmetrical.Additionally, the changes of correlation coefficient between distorted LP modes with ideal LP modes versus bending radii are shown in Fig.17(b).Fig. 17(b)

Fig. 17
Fig. 17(a) the intensity distribution of ideal and distorted LP eigenmodes, (b) the correlation coefficient between distorted LP modes with ideal LP modes versus bending radii

Fig. 18 .
Fig. 17(a) the intensity distribution of ideal and distorted LP eigenmodes, (b) the correlation coefficient between distorted LP modes with ideal LP modes versus bending radii

Fig. 19 .
Fig. 19.A typical MD results for distorted beam intensity distribution images by using ideal eigenmodes and distorted LP eigenmodes as the basis for decomposition, R=8cm.

Fig. 20 .
Fig. 20.The changes of MD accuracy versus bend radius R based on distorted LP eigenmode (i.e., LP(R)) for NF-MD.
21 -23 and the fitting formulas (11) -(15) and fitting curves in Figs.21 -23 are both obtained based on these dots.The scaling laws for the effects of coordinate origin position deviation and coordinate axis direction deviation on MD accuracy are shown in Fig.22(a) and (b), respectively.Besides, the fitting formulas of η respect to |x shift | and Δϕ are Eqs.(12) and (13), respectively, i.e.,