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Abstract
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.
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1. Introduction
Few-mode fibers (FMFs) and multimode fibers (MMFs) allow multiple different transverse modes of laser beam propagation in fibers, which is important for many scientific and industrial applications. For example, in optical fiber communication applications, compared with single-mode fiber (SMF), more modes of simultaneous propagation in FMFs and MMFs mean more communication channels, and more information can be carried to improve communication efficiency [1,2]. Moreover, the complex nonlinear effects in FMFs and MMFs are also very interesting for nonlinear dynamics (e.g., optical beam self-cleaning [3–6], spatiotemporal solitons [7], etc.). Furthermore, the mode area of FMFs and MMFs are larger than that of SMF which can suppress the nonlinear effect and improve the damage threshold in the high-power laser system, but this may result in an appearance of the transverse mode instability (TMI) effect [8–11]. Therefore, accuracy and fast mode information (i.e., mode weights and mode phases) measurement method, which is called mode decomposition (MD), is strongly required for investigating the TMI effect, multi-mode optical communication, multimode nonlinearity, etc.
To date, some MD methods for fiber laser studies with different techniques were carried out [12–27]. Some MD methods needed a broadband light source or a reference beam [12–15], such as the spatially and spectrally resolved imaging method [12,13] and correlation filter method [14,15]. However, for these methods, the experimental steps are complex, time-consuming, and the experimental device is not universal (i.e., measuring different systems requires replacing different elements). The second kind of MD method is the computer-algorithm-based MD method, using a computer algorithm to obtain mode information only from measured beam intensity, such as Gerchberg-Saxton [16], line-search [17] stochastic parallel gradient descent (SPGD) [18–22] and neural network MD methods [23–25]. In 2005, Shapira et al., successfully implemented MD using the Gerchberg-Saxton (GS) algorithm (a phase-retrieval algorithm) [16], but the computation time of the GS algorithm is very long because it requires heavy iterative calculation. The MD method based on line search [17] or SPGD algorithm [18–22], search and update the mode coefficients gradually, which had a faster decomposition speed (∼10 Hz) than that of the GS MD method. However, they were sensitive to the initial value and often falls to a local minimum. In 2019, the MD method based on neural networks, characterized by the need for high-performance computers and long network training time, has made a progress in decomposition speed (∼33 Hz) [23]. However, all of the MD methods mentioned above cannot satisfy the over kHz mode measurement of the fast-varying beam, e.g., the frequency of mode coupling in the TMI effect can reach 2 kHz, while the characteristic frequency of the nonlinear effect in multimode fiber is higher.
Quite recently, Manuylovich et al., proposed a novel ultrafast MD method based on matrix operation (MDMO method) which divided the complete nonlinear mode decomposition 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) [26,27]. In the noiseless case, the best results of the decomposition speed of the MDMO method can be up to 100,000 Hz for 3-, 5-, and 8-mode fibers [26], and the number of modes that MDMO can decompose is up to 49 [27], which is the fastest and the largest number of decomposition modes MD algorithm among the previously reported MD methods. Therefore, the MDMO method has great potential for scientific and industrial applications. However, we found that the accuracy of such MD algorithms based on beam intensity measurement will be severely limited by image noise.
In this paper, we proved that the accuracy of the MDMO method using either measured form near-field (NF) or far-field (FF) beam intensity will be greatly impaired by image noise. In addition, the conventional typical image filter for image denoising is almost impossible to improve the accuracy of the MDMO. Based on the matrix norm theory, the mechanism of image noise sensitivity for MOMD is investigated. Based on our analyses, we present an anti-noise MD method gives high accuracy even with strong image noise.
2. Principle of the MDMO method and the effect of image noise on its accuracy
Reconstructing mode weights and phase only from the measured intensity distribution is one of the most attractive methods to characterize the complete spatial information of a fiber laser beam. As shown in Fig. 1, a Fourier lens or 4f system is usually used to measure the far-field or near-field intensity distribution of fiber laser in practical systems. The main idea of the MDMO method is to decompose the image intensity matrix captured by the camera into the product of the coefficient matrix and the linear polarization eigenmode pairwise products matrix determined by the fiber parameters. Specifically, for an image of beam intensity distribution with M × M pixels, measured from N-mode MMFs or FMFs, the relative intensity of the mth pixel I(m) can be expressed as [26]
From Eq. (4) it can be seen that the first step in a nonlinear MD problem is to solve a system of linear equations (i.e., Eq. (4)), and the second step is to solve a simple nonlinear system of equations (i.e., Eq. (3)). Equation (4) is easily solved by matrix operations, i.e., X = A-1I, and A-1 = (AHA)-1AH is the pseudoinverse matrix of A. Moreover, if the mode number is more than 10 in NF-MD, matrix A is not full of rank, and its pseudoinverse matrix can’t be obtained which requires recombining the linearly dependent column vectors in matrix A [26]. On the other hand, if the MD method is not based on NF beam intensity measurement but on FF beam intensity measurement, the eigenmodes need to be Fourier transformed as shown in Ref. [27].
Under noiseless conditions, the MDMO method is undoubtedly one of the fastest and most accurate MD methods at present. However, this situation will change when the image noise is considered. Due to the inherent physical limitations of various recording devices, images become prone to manifestation of some random noise during image acquisition [28], and according to the data we acquired in a real environment, the kind of image noise in MD is additive white Gaussian noise (AWGN), which primarily occurs in analog circuitry during image acquisition and transmission. Thus, a beam intensity distribution with image noise Inoisy can be expressed as [26]
Several simulation examples of the original MDMO method are shown in Fig. 2. Figures 2(a)–(d) are the true intensity distributions, generated by Eqs. (1), (7), and the mode weights and phases are randomly, Figs. 2(e)–(h) are the reconstructed intensity distributions by the MDMO method, and Figs. 2(i)–(l) are the discrepancy between Figs. 2(a)–(d) and Figs. 2(e)–(h), respectively. Compared with the case without noise (see the first two column figures), the decomposition accuracy is decreased in the noise case (as shown in the last two column figures, SNR = 20 dB), i.e., in the noise case, the discrepancy between true intensity and reconstructed intensity is greater than that of the noiseless case, which is also consistent with situation of the correlation coefficient C given Fig. 2(e)–(h).
Fig. 2. The results of the original MDMO method, N = 5. (a) - (d) are the true intensity distributions, (e) - (h) are the reconstructed intensity distributions by the original MOMD method, and (i) - (l) are the discrepancy between (a) – (d) and (e) – (h), respectively.
For quantitative analysis, the changes of the original MDMO accuracy (i.e., correlation coefficient C) versus the SNR with different mode number N (as shown in Fig. 3(a)) and different MD situations, NF-MD or FF-MD (see Fig. 3 (b)), are shown in Fig. 3. One can see that the MDMO accuracy decreases as SNR decreases (i.e., the increases of noise intensity). In addition, the MDMO accuracy decreases as the mode number N increases (see Fig. 3(a)), and the MDMO accuracy of FF-MD is higher than that of NF-MD (as shown in Fig. 3(b)). The reasons for the phenomenon will be explained in Section 3.
Fig. 3. The correlation coefficient C versus SNR, (a) for different mode number N; (b) for different MD positions.
Image filter is an important method for image denoising. Figure 4 shows that the changes of the MD accuracy versus SNR for adopting different conventional image filters (including average filter, Gaussian filter, 2-dimensional wiener filter, wavelet transform with hard threshold, and wavelet transform with soft threshold [28]). From Fig. 4 one can see that the use of the image filter can significantly improve the MD accuracy (i.e., the C value of the black dashed line is smaller than that of the curve using the image filter) only when the SNR is lower than about 6 dB, while when the SNR is higher than about 20 dB, the use of image filters does not improve the MD accuracy, but reduces it (i.e., the C value of the black dashed line is higher than that of the curve using the image filter). The reason is that the image filter inevitably destroys the real intensity distribution when reducing noisy intensity. When the noise is strong (i.e., the value of SNR is low), the effect of the filter on image denoising dominates and the MD accuracy is improved. When the noise is weak (i.e., the value of SNR is high), the effect of the filter on destroying the real intensity distribution dominate, which results in a decreased MD accuracy. It should be noted that even in the case that image filters can improve MD accuracy (SNR < 6 dB), their values of C are all less than 0.9, such a low accuracy is almost unacceptable in MD problems. In a word, the effect of conventional typical image filter denoising to improve the accuracy of the original MDMO method is almost ineffective.
3. Mechanism of the MDMO method sensitivity to image noise
In Section 2, we prove that image noise will reduce the MDMO accuracy, while typical image filter denoising is not an ideal way to improve the MDMO accuracy. Therefore, it is very important to study the mechanism of the MDMO sensitivity to noise. In this Section, the norm theory of matrices [29] is used to solve this problem.
The total error of the MDMO method is from solving the linear matrix equation i.e., AX = I. Considering the measurement error of the practical system (i.e., ΔI and ΔA are the errors of I and A, respectively) linear matrix equation can be rewritten as
where $\tilde{\mathbf{X}} = \mathbf{X} + \Delta \mathbf{X}$, and ΔX indicates the error of the equation. The formula of ΔX is obtained, i.e.,Definition ||·|| is a vector L2-norm, and |||·||| is a matrix L2-norm. According to the triangular inequality of norm, Eq. (11) can be rewritten as
Using the inequality relation of [29], $||{{{({\mathbf{A} + {\Delta} \mathbf{A}} )}^{ - 1}}} ||\le {{|{||{{\mathbf{A}^{ - 1}}} ||} |} / {({1 - |{||{{\mathbf{A}^{ - 1}}{\Delta} \mathbf{A}} ||} |} )}}$, and $||\mathbf{I} ||= ||{\mathbf{AX}} ||\le ||{|\mathbf{A} |} ||||\mathbf{X} ||$, the formula of relative total upper-bound error of AX = I can be obtained, i.e.,
From Eq. (14) one can see that the relative upper bound error of the linear equations depends both on the measurement error of beam intensity ΔI (i.e., image noise) and the condition number κ(A) of the matrix A. Hence, the sensitivity of the MDMO method to noise depends on the condition number of the coefficient matrix of linear equation κ(A). The value of κ(A) versus the mode number N is shown in Table 1. It can be seen that as the mode number increases the value of κ(A) increases rapidly. Besides, the value of κ(A) in the FF-MD method is smaller than that in the NF-MD method. Predictably, according to Eq. (14), the MD accuracy of the original MDMO method will decrease as N increases, and the FF-MD accuracy of the original MDMO method will be higher than that of NF-MD, which is exactly the same as shown in Fig. 2. In addition, the value of ||ΔI||/||I|| can be used to qualitative describe the effect of image denoising. The changes of the relative error of beam intensity ||ΔI||/||I|| versus SNR with different image filter is shown in Fig. 5. It can be seen that in the low SNR case, the ||ΔI||/||I|| value can be compensated by the image filter. However, the opposite situation appears when SNR is high because the effect of the filter on destroying the real intensity distribution dominate, which is in agreement with the results obtained in Fig. 4.
Fig. 5. The changes of the relative error of beam intensity ||ΔI||/||I|| versus SNR with different image filter.
Table 1. The value of κ(A) versus the mode number N
In fact, the condition number of an orthogonal matrix is equal to 1, and the LP eigenmodes are exactly mutually orthogonal. However, in the MDMO method, matrix A is not directly composed of the eigenmodes but the pairwise product of the eigenmodes, which results in matrix A not being an orthonormal matrix. Moreover, when N = 10 in the NF-MD method the columns of matrix A become linearly dependent, so the value of κ(A) is very large (see Table 1). However, in the FF-MD method even if N = 10, the columns of matrix A are linearly independent because of a lack of symmetry in matrix A in FF-MD [27]. Therefore, the reason why the original MDMO method is sensitive to image noise is that the matrix A is not orthogonal due to the paired product of eigenmodes.
4. Strong anti-noise MD method
In Section 3 we prove that the condition number of the coefficient matrix composed of the paired product of the eigenmodes determines the total error of the MDMO method. However, the total error only represents the error of solution X in Eq. (4), whether the larger total error leads to larger local errors (i.e., the errors of specific mode weights and phases information) is still unveiled. In this Section, the local errors of the MDMO method are studied and a strong anti-noise MD method is proposed. In addition, unless specified, NF-MD of the MDMO method with N = 5 is adopted as an example.
The total error comes from the process of solving $\mathbf{A\tilde{X}} = \mathbf{I} + \Delta \mathbf{I}$, while the local error comes from solving the mode information from an inaccuracy $\tilde{\mathbf{X}}$, i.e., solving mode information with Eq. (3).
For quantification, the local error of ζth elements of X is defined as
Fig. 6. With average of 5000 calculations, (a) the values of different local error Erlocal(ζ), (b) the values of ||A-1(ζ)||. For 5-mode fiber and NF-MD, SNR = 20 dB.
The explanations for the difference of element noise sensitivity in X are shown as follows. The ζth element in X, i.e., X(ζ), can be expressed as
As can be seen from Eq. (16), the sensitivity of X(ζ) value to noise depends on the sensitivity of each row vector A-1(ζ) of matrix A-1 to noise. Obviously, the noise sensitivity of the A-1(ζ) increases as its magnitude (i.e., ||A-1(ζ)||) increases. Figure 6 (b) shows the values of ||A-1(ζ)||. It is found from Fig. 6 (b) that, the values of L2-norm of A-1(2), A-1(3), A-1(8), A-1(9), and A-1(10), are much larger than others. Predictably, the values of X(2), X(3), X(8), X(9), and X(10) are more affected by image noise, which is in agreement with the results obtained in Fig. 5(a).
To sum up, even in a strong image noise case, there are still 10 elements, i.e., X(1), X(4) - X(7), and X(11) - X(15), that can be solved quite accurately (as shown in Fig. 6(a)), which are enough to solve 9 coefficients of mode information. Namely, it is possible to improve the anti-noise ability of the original MDMO method by using the characteristic of information redundancy, named as MDMO-CIR method. In particular, if the MDMO-CIR method is adopted, a different nonlinear equation needs to be solved for the second step of the MDMO method, e.g., when N = 5 for NF-MD, the nonlinear equation of MDMO-CIR is
Fig. 7. The MD results, (a)-(c): by using the original MDMO method; (d)-(f): by using the MDMO-CIR method; (a) (c): true intensity with noise, (b) (e) their reconstructed beam intensity profiles, (c) (f) discrepancy between true intensity and reconstructed intensity. For 5-mode fiber in NF-MD case.
The decomposition accuracy versus SNR with different MD methods is shown in Fig. 8, and the data is the average value obtained from 5000 calculations. Compared the black curve (i.e., results of original MDMO method) with the red curve (i.e., results of MDMO-CIR method) from Fig. 8, one can see that the decomposition accuracy of the original MDMO method changes more rapidly with SNR than that of MDMO-CIR method. Namely, the MDMO-CIR method is more insensitive to image noise than that of the original MDMO method. Furthermore, in strong noise case (e.g., SNR is lower than 36 dB, in Fig. 8), the MD accuracy of the MDMO-CIR method is significantly higher than that of the original MDMO method, while if SNR is higher than 36 dB the opposite situation appears. The reason is that it may fall to a local minimum when numerically solving Eq. (17) in the MDMO-CIR method, which results in a decrease of MD accuracy. However, the original MDMO method is to solve a sample nonlinear equation and has an analytical solution. Hence, in the case of high SNR, the accuracy of the original MDMO method is still higher than that of the MDMO-CIR method.
Fig. 8. The changes of the correlation coefficient C versus SNR with different MD methods. For 5-mode fiber in NF-MD.
Generally, a continuous change of physical effects in fiber lasers, such as the TMI effect, the noise intensity of the image acquired by the camera also changes dynamically. Therefore, an anti-noise MD method, in a large SNR range, is strongly required. In this paper, we propose a new insight MD method, i.e., the optimal-MDMO method, which selects the one with higher accuracy among the original MDMO and MDMO-CIR methods as the result of MD in a single MD process. The changes of the correlation coefficient C versus SNR with the optimal-MDMO methods are shown as the blue curve in Fig. 8. It can be seen that the optimal-MDMO method can significantly improve the anti-noise ability of MD in both low SNR and high SNR (i.e., the C-value of blue curve is always higher than that of the red and black curves). For example, in this case, in comparison with the original MDMO method the SNR needed for optimal MDMO with accuracy C = 0.95 decreases from 37.2 dB to 3.67 dB.
We have proved that the optimal MDMO method is an anti-noise MD method in NF-MD case. Similarly, in FF-MD case, the optimal MDMO method can also effectively improve the accuracy of MD, as shown in Fig. 9, the decomposition accuracy versus SNR with different MD method. For example, in comparison with original MDMO method the SNR needed for optimal MDMO with accuracy C = 0.95 decreases from 16.5 dB to 10.5 dB.
Fig. 9. The changes of the correlation coefficient C versus SNR with different MD methods. For 5-mode fiber and FF-MD.
Taken N = 5 for NF-MD as an example, the decomposition times of MDMO-CIR method and optimal MDMO method compared to the original MDMO method as shown in Table 2, where T0, T1, T2 are the decomposition time of the original MDMO method MDMO-CIR method and optimal MDMO method, respectively. All test have been performed on PC with CPU AMD Ryzen 5 5600 H and the data is the average value obtained from 5000 calculations. One can see that the decomposition speed of MDMO-CIR and optimal MDMO are about 10 times slower than the original MDMO method. As a comparison, the mode decomposition speed based on deep learning is about 1000 times slower than the original MDMO method [26].
Table 2. The decomposition time of MDMO-CIR method and optimal MDMO method compared to the original MDMO method, T0, T1, and T2 are the decomposition times of original MDMO, MDMO-CIR, and optimal MDMO methods, respectively. N = 5 for NF-MD, and image size is 128 × 128 pixels.
5. Conclusions
The fast and accurate MDMO method is needed for applications related to MMFs or FMFs. In this paper, we proved that the accuracy of original MDMO method is very sensitive to image noise, and the effect of conventional typical image filter denoising to improve the accuracy of MDMO method is very limited. Based on the matrix norm theory, the error upper bound formula of linear equations solution is derived. It is found that the total error of the original MDMO method increases as the coefficient matrix condition number of linear equations increases. In addition, the condition number increases as mode number increases and the condition number of FF-MD is smaller than that of NF-MD.
In particular, we found that the local error of each mode information solution in the original MDMO method is positively related to the L2-norm of each row vector of the inverse coefficient matrix. Besides, screening out the information more sensitive to noise, the remnant information is enough to obtain all mode weights and phases. A more noise-insensitive MD method (MDMO-CIR method) is proposed, which had much higher MD accuracy than the original MDMO method in NF-MD with a strong image noise case. However, as a result of full in a local minimum, the accuracy of the MDMO-CIR method is lower than that of the original MDMO method in a weak image noise situation. To obtain a more robust anti-noise MD method in a large SNR range, a new insight MD method, i.e., the optimal-MDMO method is presented, which selects the one with higher accuracy among the original MDMO and MDMO-CIR methods as the result of MD in a single MD process. It is shown that the optimal MDMO method can significantly improve the anti-noise ability in strong noise cases for both NF-MD and FF-MD. As a typical case, for 5-mode fiber MD, the SNR needed for optimal MDMO with accuracy C = 0.95 decreases from 37.2 dB to 3.67 dB for NF-MD, and 16.5 dB to 10.5 dB for FF-MD.
Funding
National Natural Science Foundation of China (61705264, 62075242).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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