Multimode fiber modal decomposition based on hybrid genetic global optimization algorithm

Lei Li; Jinyong Leng; Pu Zhou; Jinbao Chen

doi:10.1364/OE.25.019680

1. Introduction

Modal decomposition (MD) techniques are essential tools to reveal modal characteristics in high power fiber lasers. Recent years, MD has been widely applied for revealing optical field features and temporal dynamics, such as wavefront reconstruction [1], beam quality measurement [2], mode-resolved gain or bend loss analysis [3,4], mode instabilities temporal diagnose [5], and adaptive mode control [6], etc. Generally, modal decomposition methods can be divided into two classes. One way is to directly measure through experiments, such as spatially and spectrally resolved imaging [7], the use of ring-resonators [8], low-coherence interferometry [9], correlation filter method [10], and newly developed frequency domain cross-correlated imaging [11], etc. These techniques have shown outstanding merits, but require intense experiment conditions and temporal effort, or costly [5,6,13]. In addition, the other way is numerical MD based on measured optical intensity distribution, which stands out for its efficient realization and simple experimental effort [5,12,13].

Mapping two-dimensional intensity distribution onto one-dimensional superposition of fiber eigenmodes, optical field can be reconstructed with corresponding amplitudes and phases [12]. The main challenge for numerical MD is to avoid local minima problems from sensitivity to initial values (i.e. concerning the minimization methods, different solutions can be found with varying initial values), especially for multimode fiber (e.g. six guided modes or more). Although several reported numerical MD algorithms, such as Gerchberg-Saxton algorithm [13], simplex-search algorithm [5], and SPGD algorithm [6,14,15], suffer some degree of local minima problems caused by sensitivity to initial values in multimode fiber MD. On the contrary, genetic algorithm (GA), which is one kind of global optimization methods to resolve local minima problem, has not been applied for multimode fiber MD before. Based on evolutionary ideas of “natural selection” and “survival of the fittest”, GA firstly generates a large random solution population, and then iterates to next solution populations upon the rough direction of historical information [16,17]. Although it can effectively reduce such local minima problems, but takes much longer time to converge.

In this paper, we propose a hybrid genetic global optimization algorithm which combines the advantages of GA and SPGD algorithm to realize multimode fiber MD. Firstly, GA searches the rough global optimization position in the whole space of possible eigenmode combinations based on near- and far-field intensity distribution, which can avoid ambiguities caused by modal interference and local minima caused by sensitivity to initial values. Upon those initial values, the SPGD algorithm is afterwards used to find the exact optimization values based on the near-field intensity profiles with fast convergence speed. Numerical simulations validate the feasibility of complete modal decomposition. We believe that this reliable algorithm will play a major role in reconstructing multimode fiber optical field, evaluating beam quality, and revealing the mode dynamics process in multimode fiber (e.g. mode instability), etc.

2. Modal decomposition

An arbitrary normalized near-field U_NF(x, y) at the end of fiber can be projected to a combination of eigenmodes ψ (x, y):

U_{NF} (x, y) = \sum_{n = 1}^{N} ρ_{n} e^{i θ_{n}} ψ_{n} (x, y) 〈 ψ_{m} (x, y) | ψ_{n} (x, y) 〉 = δ_{m n}

\sum_{n = 1}^{N} ρ_{n}^{2} = 1 θ_{n} \in (- π, π)

Where, ρ_n and θ_n are the modal amplitude and relative phase differences between n^th eigenmode and fundamental mode, respectively. And ρ_n² means corresponding modal weight. In fact, the number of bound modes N is decided by fiber structure. Here, complete orthogonal linearly polarized (LP) modes are applied for describing eigenmodes based on weak-guidance approximation [18]. In addition, far-field U_FF(x₂,y₂) can be calculated from near-field distribution U_NF(x₁,y₁) by using Fourier Transform method [19]. In fact, the observation of the Fourier plane U_FF(x₂,y₂) represents for focal plane of near-field plane U_NF(x₁,y₁), and f is the corresponding focal length:

U_{F F} (x_{2}, y_{2}) = \frac{\exp (i k f) \exp (\frac{i k (x_{2}^{2} + y_{2}^{2})}{2 f})}{i λ f} \int_{- \propto}^{+ \propto} \int_{- \propto}^{+ \propto} U_{N F} (x_{1}, y_{1}) \exp (\frac{- i k (x_{2} x_{1} + y_{2} y_{1})}{f}) d x_{1} d y_{1}

Thus, the near-field and far-field intensity distribution can be described as:

I_{N F} = {| U_{N F} (x, y) |}^{2}

I_{F F} = {| U_{F F} (x, y) |}^{2}

Considering ambiguities caused by modal interference [13], merit function J₁ including near- and far-field correlation functions (ΔJ_NF and ΔJ_FF) is selected in the first step with GA method.

J_{1} = Δ J_{NF} + Δ J_{FF} \to \max

Upon the rough global optimization position, the SPGD algorithm costs less convergence time to find the exact superposition based on the near-field intensity distribution [14], thus merit function J₂ including near field correlation function (ΔJ_NF) is chosen in the second step:

J_{2} = Δ J_{NF} \to \max

The correlation function can be described as following:

Δ J = | \frac{\iint Δ I_{me} (x, y) Δ I_{re} (x, y) d x d y}{\sqrt{\iint Δ I_{m e}^{2} (x, y) d x d y \iint Δ I_{r e}^{2} (x, y) d x d y}} |

Where, the correlation function is to evaluate the difference between measured and reconstructed results, and

Δ I_{j} (x, y) = I_{j} (x, y) - \bar{I_{j} (x, y)}

(j = measured, reconstructed) and

\bar{I_{j} (x, y)}

is the corresponding mean value. The larger value of correlation function stands for smaller error between measured and reconstructed field [6,13].

3. Hybrid genetic global optimization algorithm

Hybrid genetic global optimization algorithm combines the advantages of GA and SPGD algorithm to realize multimode fiber numerical MD. Firstly, GA is used to search rough global optimization position based on near- and far-field intensity distribution. Then, SPGD algorithm is applied to find the exact optimization position based on near-field intensity distribution.

GA is a kind of population-based global optimized algorithm [16,17], and has been widely used in adaptive optics [20,21]. In the searching process, each individual is a potential solution for the whole optimization space. And an iteration scheme is outlined in Fig. 1.

Fig. 1 The iteration scheme of genetic algorithm.

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(1) To begin with, the variables of amplitudes { ρ_i | i = 1,2,…N} and relative phase {θ_i | i = 2,…N} are randomly generated and encoded to a L = 10-bit binary strings, which directly stores in a vector called chromosome. N is the number of eigenmodes decided by fiber structure. Thus, the initial generated population includes randomly yielding N₁ grating-like chromosomes, which is a group of MD solutions.
(2) Then, each “chromosome” is evaluated with merit function J₁, and corresponding fitness is calculated as following: $F_{i} = {\begin{array}{r} J_{i} \begin{matrix} / \sum_{i = 1}^{N} J_{i} & (J_{i} \geq F_{C}) \end{matrix} \\ 0 \begin{matrix} (J_{i} < F_{C}) \end{matrix} \end{array}$
Where, J_i means merit function of i^th chromosome, and F_c is the threshold to dissolve chromosomes, which means larger-than threshold chromosomes are saved, and the others die out. And it is the process of “nature selection” or “survival of the fittest”.
(3) Allotting more positions to larger “fitness” chromosome, a new population is generated, in which the “chromosome” is chosen with a probability P_i: $P_{i} = F_{i} / \sum_{i = 1}^{N_{C}} F_{i}$
Where, N_c means the number of non-zero-fitness chromosomes, and selection probability is proportional to its fitness.
(4) The selected chromosomes are crossover with a probability P₁, and the new individuals will possess some parts of both parent’s genetic fragment. Here, P₁ is random produced, if P₁<P_c, the process will carry on, and the crossover length is equal to L*P₁. Then, some positions in the chromosomes are mutated with a probability P₂ to ensure population variety. And P₂ is random produced, if P₂<P_m, one allele is replaced with the opposite one at the position of L*P₂. Thus, a newly population is generated.
(5) In the end, save the “fittest” chromosome to next population and repeat “selection, crossover and mutation” process until a rough global optimization “chromosome” comes up, which means reconstructed near- and far-field intensity distributions are mostly likely with measured near- and far-field intensity distributions. Local minima problems can be validly avoided from sensitivity to initial values due to global searching process.

Furthermore, SPGD algorithm, a gradient-based optimized algorithm, and has been widely used in coherent beam combining [22], adaptive optics [23,24] and modal decomposition [6,14,15], etc., and deeply discussed in Ref [14]. Combining GA and SPGD algorithm, the hybrid genetic global optimization algorithm is outlined in Fig. 2. Firstly, initializing “grating-like” chromosomes with a pop size of N₁, and set the iterative number is N₂. Through selection, crossover and mutation processes, rough global optimization position are generated. Upon these selected initial values, SPGD algorithm is applied to find out the exact optimization position. When the results meet terminating condition (ΔJ_NF>0.999 and ΔJ_FF>0.999), the searching process will stop, otherwise it will come back to initialize a new population, and come across GA-SPGD algorithm again.

Fig. 2 The iteration scheme of GA-SPGD algorithm

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4. Simulations and results

Considering a multimode step-index fiber with core diameters of 20 μm and core NA number of 0.08, the normalized frequency V-number at 1064 nm is about 4.72. Thus the fiber can only support LP₀₁, LP₀₂, LP_11e, LP_11o, LP_21e, and LP_21o modes, where e and o stand for even and odd modes, respectively. Consequently, 11 parameters should be considered to describe fiber optical field, including six mode weights and five relative phase differences between LP₀₁ and LP₀₂, LP_11e, LP_11o, LP_21e, LP_21o. In fact, multimode fiber numerical MD is one kind of multi-local-minima problems, which need a suitable multi-dimensional optimization algorithm to realize global optimization. First of all, we will discuss the problems of ambiguity and initial values in numerical MD for multimode fiber. Then, numerical simulations are discussed to validate the feasibility and reliability of multimode fiber numerical MD with GA-SPGD algorithm.

4.1 The problems of ambiguity and initial values

Ambiguity comes from numerical modal decomposition based on near-field, which can be dived into two kinds. One is the uncertainty of all mode phase sign with the same modal weights, which means the reconstructed field is its conjugated field [13,14]. The other is the uncertainty of complete different mode weights mode and phase difference, from coherent superposition and associated modal interference [13]. Although similar near-field is reconstructed, the results are completely wrong. In the case of six guided modes or more, such kind of ambiguity takes places frequently. Table 1 shows typical ambiguity example for six guided mode.

Table 1. Given and convergence modal weights and phase differences

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Here, absolute modal weight error $| Δ ρ^{2} |$ means $| ρ_{c}^{2} - ρ^{2} |$ , and absolute phase difference error $| Δ θ |$ stands for $| θ_{c} - θ |$ .The mean absolute modal weight and phase difference error is 0.0478 and 0.4555, respectively. Figure 3(a) shows given near-field intensity distribution. Through numerical MD based on near-field, the reconstructed merit function convergent to 0.9974 as shown in Fig. 3(b), that is to say, reconstructed near-field (Fig. 3(c)) is quite similar with given one (Fig. 3(a)). Corresponding far-fields are calculated as shown in Fig. 3(d) (Given far-field) and Fig. 3(e) (Reconstructed far-field). Notably different far-fields with similar near-fields demonstrate that the modal decomposition falls into ambiguity. By adding another optical field intensity measurement (e.g. far-field intensity), the above two kind of ambiguity can be resolved [13].

Fig. 3 Ambiguity in the situation of numerical MD based on near-field intensity distribution due to modal interference: (a) shows initial near-field intensity distribution; (b) depicts the convergence merit function; (c) shows reconstructed near-field; (d) and (e) are the corresponding given and reconstructed far-field intensity, respectively.

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Another common problem for numerical MD with line-searching methods is initial values, i.e. different results can be found with sensitivity to initial values due to algorithm stops in a fixed value or flat regions far away from global optimization position, not only for corr NF method, but also for corr NF + FF method [13]. The global optimization can be found through repeating often enough with random initial values. It is somewhat inefficient for distinguishing local minima due to kick one possible solution in one-time searching process. On the contrary, GA-SPGD algorithm is quite a robust algorithm to cope with local minima. Firstly, GA, a kind of global optimization searching algorithms, searches the rough global optimization position in the whole space of possible eigenmode combinations based on near- and far-field intensity. With fast convergence speed, SPGD algorithm is afterwards used to find the exact superposition based on the near-field intensity profiles. If it falls into local minima, the GA-SPGD algorithm will terminate quickly, and come back to initialize a new population, and repeat GA-SPGD algorithm. Here, the termination condition is set as (ΔJ_NF>0.999 and ΔJ_FF>0.999).

4.2 Modal decomposition with GA-SPGD algorithm

Firstly, based on near- and far-field intensity distribution, GA searches the rough global optimization position in the whole solution space. The selective threshold F_c of 1.45 is chosen to ensure that reconstructed near- and far-field intensity distribution for selected population are mostly similar with given or measured near- and far-field field. Set the initial population number N₀ and evolution number are 250 and 50, respectively. To ensure the population variety, the crossover probability P_c and mutation probability P_m are 0.75 and 0.05, respectively. Given mode weights and phase differences are described in Table 1, a rough global optimization position (i.e. reconstructed near- and far-field intensity distributions are mostly similar with given near- and far-field) is calculated as shown in Table 2. The mean absolute modal weight and phase difference error is 0.04 and 0.2894, respectively. Here, the merit function is the sum of reconstructed near-field and far-field intensity distribution correlation function, as shown in Fig. 4. The initial merit function is 1.676 and end merit function is 1.9012, which demonstrated that “fittest” chromosome is easily selected with GA searching method.

Table 2. Modal weights and phase differences of the rough global optimization position

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Fig. 4 Merit function as a function of iteration numbers (GA searches the rough global optimization position based on near- and far-field intensity distribution).

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Upon the rough global optimization position, SPGD algorithm is afterwards used to find the exact modal weights and phase differences based on near-field intensity distribution, as shown in Table 3. The mean absolute modal weight and phase difference error is 0.00082 and 0.0038, respectively.

Table 3. Modal weights and phase differences revealing from GA-SPGD algorithm

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Figures 5(a)-5(e) depict given near-field intensity distribution, convergence merit function, reconstructed near-field intensity distribution, given and reconstructed far-field intensity distribution, respectively. The near-field and far-field merit functions are as high as 0.99999 and 0.99999, respectively. The lowest fraction of mode weights is decided by merit function, i.e., if the merit function of reconstructed optical field is as high as 0.99999, the lowest fraction of mode power is close to 0.1%. The larger merit functions, the smaller lowest fraction of mode power. Limited by experimental effort, the lowest fraction of mode power can be improved with multi-measurement of near- and far-field intensity distribution to reduce experimental noise and multi-numerical MD to reduce algorithm error. The total searching time for GA-SPGD algorithm is about 150 s on a computer with Inter Core i5-4460 and 4 GB RAM with 256x256 pixels image, and the mainly time-cost is GA process which costs about 140 s.

Fig. 5 Global optimization of numerical MD with GA-SPGD algorithm: (a) shows initial near-field intensity distribution; (b) depicts the convergence merit function; (c) shows reconstructed near-field; (d) and (e) are the corresponding given and reconstructed far-field intensity, respectively.

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In addition, we have continued numerical MD for 100 times under the same condition as above without termination conditions. The modal weights differ from more than 5% from the given modal weights in Table 1 are defined as outliers. Figure 6(a) shows the outliner numbers for each mode in continuing 100-times numerical MD with GA-SPGD algorithm, in which initial population number is 250, the maximum number of outliner (LP_21o) is 4 in 100-times, in which inappropriate rough global optimization positions leads to SPGD algorithm convergence to a local minima.

Fig. 6 Outliner numbers for each mode in continuing 100-times numerical MD with GA-SPGD algorithm: (a) initial population number = 250; (b) initial population number = 300.

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As the initial population number increased, the rough global optimizations will be more exact, so the possibility of outliner will notably decrease. Figure 6(b) shows the outliner numbers for initial population number increased to 300. The computation time for 300 initial population number and 50-times evolutions is about 180s, which is the proportion with product of initial population number and evolution number. Compared Figs. 6(a) and 6(b), it can be concluded that initial population number of GA is important for rough global position quality, and the larger number leads to less possibilities of local minima. As total number of modes increased, the computation time will become longer and results accuracy scales will become lower. We believe either technique has its upper limit. Present technique is more suitable for analyzing multimode fiber mode instabilities (3 ~10 modes). A generalization of this technique to multimode fibers i.e. 100 microns core, 0.22 NA supporting a few hundreds of modes is difficulty to realize.

To further testify the robustness of GA-SPGD algorithm, another 3 arbitrary complete different mode combinations (as shown in Table 4) with each 30-times numerical decomposition and related statistical evaluation are discussed. In addition, corresponding near- and far-field intensity distributions are shown in Fig. 7. For speeding up the total searching time of GA-SPGD algorithm, the initial population number N₀ and evolution number changed to 300 and 10, respectively. And other algorithm parameters are the same as above. Here, the termination condition is set as (ΔJ_NF>0.999 and ΔJ_FF>0.999) to assure the accuracy of complete modal decomposition.

Table 4. Complete different mode combinations

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Fig. 7 Given near- and far-field intensity distribution for complete different mode combinations of Group 1, 2 and 3.

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Continuing numerical MD over 30 times for each combination, corresponding near- and far-field merit functions are shown in Fig. 8. We can see that all the merit functions are over 0.999, which demonstrate the accuracy of reconstructed optical field. The mean near-field merit functions are 0.99998, 0.99993 and 0.99997 for Group 1, 2 and 3, respectively. And the mean far-field merit functions are 0.99989, 0.99957 and 0.99956 for Group 1, 2 and 3, respectively. Moreover, corresponding average modal weight and phase differences are shown in Table 5. Here, absolute modal weight error $| Δ \bar{ρ^{2}} |$ means $| \bar{ρ^{2}} - ρ^{2} |$ , and absolute phase difference error $| Δ \bar{θ} |$ stands for $| \bar{θ} - θ |$ . The mean absolute modal weight errors are 0.00047, 0.0026 and 0.0005 for Group 1, 2 and 3 respectively. And the mean absolute phase difference errors are 0.0074, 0.0024 and 0.0073 for Group 1, 2 and 3 respectively. Since the modal weight of LP₀₂ mode is less than 0.001, LP₀₂ mode has litter contribution for the whole optical field. Thus, related absolute phase difference is as large as 1.2272.

Fig. 8 The results of near- and far-field merit functions in continuing 30-times-operation with GA-SPGD algorithm for Group 1, 2 and 3.

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Table 5. Average modal weights and phase differences

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Figure 9 shows corresponding searching time for each operation of Group 1, 2 and 3. The mean searching times are 103 s, 252 s and 635 s. The cost time depends on the number and density of local minima. In limited terminated operation times, the global optimization can be found out with GA-SPGD algorithm.

Fig. 9 Corresponding searching time for each operation of Group 1, 2 and 3.

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5. Conclusion

In this paper, a hybrid genetic global optimization algorithm is firstly proposed in multimode fiber modal decomposition, which combines the advantages of GA and SPGD algorithm. Numerical simulations validate the feasibility and reliability to resolve ambiguity and local minima problems caused by sensitivity to initial values. In the future, the algorithm will be further speeded up by using parallel programming and GPU programming. Furthermore, the influences of misalignments between the optical axes, noise, and the worse signal-to-noise relations of measured near- and far-field intensity will be taken into consideration in practice.

Funding

National Natural Science Foundation of China (NSFC) (61605246).

Acknowledgments

We thank Dr. Liangjin Huang for providing useful advice in modal decomposition algorithm.

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	Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
Given values	$ρ^{2}$	0.3	0.25	0.0417	0.0833	0.25	0.075
Given values	$θ$	0	0	0.3927	2.0944	−0.7854	0.7854
Convergence values	$ρ_{c}^{2}$	0.3179	0.2736	0.1437	0.0705	0.1813	0.0129
	$\| Δ ρ^{2} \|$	0.0179	0.0236	0.102	0.0128	0.0687	0.0621
	$θ_{c}$	0	−0.511	0.7184	1.8953	−0.8945	−0.3474
	$\| Δ θ \|$	0	0.511	0.3257	0.1991	0.1091	1.1328

Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
$ρ^{2}$	0.2875	0.2643	0.1238	0.0235	0.2741	0.0268
$\| Δ ρ^{2} \|$	0.0125	0.0143	0.0821	0.0598	0.0241	0.0482
$θ$	0	−0.0031	1.0226	1.5570	−0.648	0.9244
$\| Δ θ \|$	0	0.0031	0.6299	0.5374	0.1374	0.1390

Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
$ρ^{2}$	0.3017	0.2498	0.0424	0.0834	0.2494	0.0733
$\| Δ ρ^{2} \|$	0.0017	0.0002	0.0007	0.0001	0.0006	0.0017
$θ$	0	−0.0079	0.3968	2.0932	−0.7851	0.7802
$\| Δ θ \|$	0	0.0079	0.0041	0.0012	0.0003	0.0052

	Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
Group 1	$ρ^{2}$	0.0656	0.1795	0.5944	0.0675	0.0655	0.0275
Group 1	$θ$	0	0.0934	−2.0835	2.4017	2.5976	−1.8834
Group 2	$ρ^{2}$	0.3281	0.0001	0.3917	0.2163	0.0412	0.0226
Group 2	$θ$	0	0.6776	−0.6107	−0.84	2.1945	−1.2577
Group 3	$ρ^{2}$	0.2419	0.299	0.0059	0.304	0.1457	0.0035
Group 3	$θ$	0	−1.3917	0.2946	2.8746	2.921	−2.1513

	Parameters	LP₀₁	LP₀₂	LP_11e	LP_11o	LP_21e	LP_21o
Group 1	$\bar{ρ^{2}}$	0.0662	0.1787	0.5952	0.0672	0.0653	0.0274
	$\| Δ \bar{ρ^{2}} \|$	0.0006	0.0008	0.0008	0.0003	0.0002	0.0001
	$\bar{θ}$	0	0.0869	-2.0892	2.3991	2.5968	-1.912
	$\| Δ \bar{θ} \|$	0	0.0065	0.0057	0.0026	0.0008	0.0286
Group 2	$\bar{ρ^{2}}$	0.3286	0.001	0.3848	0.2155	0.0411	0.0289
	$\| Δ \bar{ρ^{2}} \|$	0.0005	0.0009	0.0069	0.0008	0.0001	0.0063
	$\bar{θ}$	0	-0.5496	-0.6076	-0.8404	2.0646	-1.1908
	$\| Δ \bar{θ} \|$	0	1.2272	0.0031	0.0004	0.1299	0.0669
Group 3	$\bar{ρ^{2}}$	0.2423	0.2976	0.0058	0.3051	0.1457	0.0035
	$\| Δ \bar{ρ^{2}} \|$	0.0004	0.0014	0.0001	0.0011	0	0
	$\bar{θ}$	0	-1.3891	0.2834	2.8765	2.9216	-2.1786
	$\| Δ \bar{θ} \|$	0	0.0026	0.0112	0.0019	0.0006	0.0273
* $\bar{ρ^{2}}$ means the average modal weights, and $\bar{θ}$ means the average relative phase difference.

Multimode fiber modal decomposition based on hybrid genetic global optimization algorithm

Abstract

1. Introduction

2. Modal decomposition

3. Hybrid genetic global optimization algorithm

4. Simulations and results

4.1 The problems of ambiguity and initial values

4.2 Modal decomposition with GA-SPGD algorithm

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (9)

Tables (5)

Equations (10)

Optics Express