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Abstract
We investigate a fast and accurate technique for mode decomposition in multimode optical fibers. Initial decomposition task of near-field beam patterns is reformulated in terms of a system of linear equations, requires neither machine learning nor iterative routines. We apply the method to step and graded-index fibers and compare the decomposition performance. We determine corresponding application boundaries, propose an efficient algorithm for phase retrieval and carry out a specific preselective procedure that increases the number of decomposable modes and makes it possible to handle up to fifteen modes in presence of realistic noise levels.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years mode division multiplexing (MDM) has drawn attention due to its potential for overcoming the capacity crunch of single-mode fibers [1,2]. From that perspective, mode decomposition (MD) represents the very first step in decoding output signals sent via multimode fibers (MMF) and has to be done both fast and accurately [3]. Various MD approaches that have been developed over the past decade can be divided into two categories. The first one relies solely on the experiment and involves additional equipment, such as spatial light modulators [4–7], cross-correlated imaging [8], spatially and spectrally resolved imaging [9], ring-resonators [10], low-coherence interferometry [11], digital holography [12] or correlation filter [13]. These methods require high experimental effort and complex data post-processing.
The second category is based on implementation of numerical algorithms, e.g. optimization procedures [14–18], line-search [19], Gerchberg–Saxton [19], stochastic parallel gradient descent [20]. These approaches are known to suffer either from initial value sensitivity and local minima problems or to be restricted with respect to the number of modes and computation time due to the necessity of beam reconstruction or introducing additional minimization routines.
Additionally, deep neural networks have demonstrated sound performance, especially regarding execution time [21–25]. However, existing solutions provide phases up to their signs, which makes it necessary to reconstruct beam patterns that correspond to all possible sign configurations. This approach requires a reference beam and becomes inefficient for higher number of modes. Analytical techniques for mode decomposition can compete with neural networks in both execution time and number of modes they can successfully deal with [3]. Nevertheless, such methods are scarce and their implementation is severely restricted by noise and beam misalignment. These algorithms operate in terms of large matrices which become ill-conditioned for higher number of modes and cannot successfully carry out decomposition. Finally, all works focus only on step-index fibers (SIFs) and do not take graded-index fibers (GRIN) into consideration.
In this paper, we reformulate the analytical approach to mode decomposition in terms of the least-squares based method. In this formulation of the task, decomposition is reduced to a small system of linear equations and can be accomplished fast and accurately. We provide first to our knowledge examples of decomposition applied to GRIN fibers and compare them to those of SIFs. We demonstrate that the method, to a certain extent, is agnostic with respect to the number of modes and can account for beam rotation. We propose a novel phase recovery algorithm and show that a beforehand mode selective procedure can increase the number of decomposable modes. We find a set that doubles this number compared to state-of-the-art results and demonstrates noise robustness that is $20$ dB higher than the set of consecutive modes.
2. Algorithm
Given a MMF of the length $L$ with an arbitrary refractive index profile function, the near-field beam pattern at the receiver reads as
Introducing $\tilde {N}\equiv N(N+1)/2$ and two $\tilde {N}$-vectors
As we mentioned above, in addition to the amplitude weights, Eq. (6) provides the phases $\phi _i$, $i=2\dots N$, via components
There are $2^{N-1}$ possible sign patterns and half of them are equivalent due to the sign ambiguity of the electric field, i.e. we exclude those with $\phi _2<0$. Then we build up a $(N-1)\times 2^{N-2}$ boolean matrix that covers every possible sign configuration (the matrix might seem large but it does not require a lot of memory though. e.g. for $N=15$ modes it takes only $0.11$ Mb). With help of $X_{N+i - 1}$ we determine raw phases and for each phase $i=2\dots N$ we pick two rows $i$ and $j>i$ from the sign matrix, assign the corresponding signs to the phases and find out which sign rows fulfil $\cos (\phi _i - \phi _j)=X_{2N+i-2}$. Each iteration bisects the configuration space and ideally only $(N-2)$ comparisons are required. If the algorithm does not find the appropriate sign pattern, one can change the reference cross-terms from $\varphi _1\varphi _i$ and repeat the procedure. However, once we get errors in the modal weights, no accurate decomposition is possible, as absolute values of the phases are obtained from Eq. (12).
3. Implementation of the algorithm
We numerically model the experimental setup given in Fig. 1. In this setup a multimode signal with arbitrary amplitudes and phases propagates through a MMF. The resulting near-field beam pattern is then captured by a camera. We simulate the images of the camera based on the known transverse eigenmodes of the fiber. The data is processed by the proposed algorithm for obtaining the modal amplitudes and phases of the output multimode signal. In this Section, we compare decomposition performance of the algorithm applied to a SIF fiber with NA $=0.2$ and a parabolic GRIN fiber
Fig. 1. Scheme of the modeled experimental setup. MMF, multimode fiber; L, lens; PBS polarization beam splitter.
Fig. 2, illustrates the original and recovered beam speckles for one of $10000$ generated configurations, as well as their discrepancy for $N=3,~6,~10,~15$ modes from $S_1$. The complex coefficients are the same for both fibers. The only case with $C=0.8933$ and noticeable imperfections is that of GRIN with $15$ modes, whereas the other numerical experiments reveal flawless decomposition with $C=0.9999$. Corresponding averaged errors $\epsilon _v$ can be found in Table 1. $N=10$ represents the threshold of the ill-posed problem, which is highlighted by a substantial growth of $\epsilon _X$. However, the overall decomposition performance is decent due to the redundancy in cross-terms we mentioned in the previous Section. If the difference between $\text {rank}(P)$ and $\tilde {N}$ is moderate and no components of $X$ referring to the modal weights are affected, we can always extract consistent phases without sacrificing the accuracy.
Fig. 2. Examples of beam patterns for SIF and GRIN for the Set $S_1$ from Eq. (14): original (left), recovered (middle) beam speckles and their discrepancy (right). The complex coefficients $A_i$ are the same for both cases.
Table 1. Averaged decomposition errors for both sets.
Fig. 3 demonstrates comparison of SIF and GRIN in the case of the Set $2$. This scenario enables decomposition of higher numbers of modes, as in the case of SIF (GRIN) the rank incompleteness occurs for $N>15$ ($N>12$) and this number can be doubled if both spatial orientations ($a$ and $b$) are processed independently.
Fig. 3. Examples of beam patterns for SIF and GRIN for the Set $S_2$ from Eq. (15): original (left), recovered (middle) beam speckles and their discrepancy (right). The complex coefficients $A_i$ are the same for both cases.
It is worth noticing that on the regular workstation the algorithm operates within hundreds of microseconds depending on the number of modes, which correlates with time performance of the related analytical scheme [3] and outperforms state-of-the-art deep neural networks solutions [21].
4. Accounting for rotation
According to the proposed decomposition algorithm, the matrix $P$, constructed for a given number of modes $N=N_1$, can be applied to systems with $N_2<N_1$. In other words, if we consider a set of $N_2$ modes and decompose resulting beams with $N_1(N_1+1)/2\times N_1(N_1+1)/2$ matrices, we obtain zero modal weights and phases for modes indexed between $N_2$ and $N_1$. It makes it possible to account for random beam rotations for speckles including only modes of the same spatial orientation, i.e. those from $S_2$. For $N_2$ modes from $S_2$ we take $N_1$ from $S_1$ so that the latter comprises both $N_2$ modes and their dual orientations labeled as $(a)$. Solving Eq. (4) with the corresponding decomposition matrix yields
where $|A_{i,a}^{(n)}|$ refers to the modal weight of the mode $i$ with the orientation $a$ from the set of $n$ modes. Naturally, this approach is valid unless Eq. (4) yields wrong modal weights.Here we provide a simple example that demonstrates applicability of the algorithm to rotated speckles. We consider a SIF and its first three modes from $S_2$: $LP_{01}$, $LP_{11b}$, $LP_{21b}$. We randomly rotate them and build up beam patterns (see Fig. 4(a)). Obviously, the decomposition algorithm with unrotated fields cannot handle this task. Then we arrange the decomposition matrix for five modes $LP_{01}$, $LP_{11a}$, $LP_{11b}$, $LP_{21a}$, $LP_{21b}$ and reapply the algorithm. Solving Eq. (4) yields projections of the modal weights of each mode onto those of both orientations, which with help of Eq. (18) become the modal weights of the unrotated beam. Figure 4(b) demonstrates the initial beam, the reconstructed one, and their discrepancy in this case. Accuracy of the algorithm turns out to be equivalent to that for $N=5$.
Fig. 4. The rotated, the recovered beams for $N=3$ and their discrepancy obtained a) by means of the straightforward implementation of the algorithm and b) by the orientation-extended matrix.
5. Selective approach
As we mentioned above, the rank problem is the stumbling stone for the method. Having fixed that, one could expand the application boundaries of the algorithm. However, as it was correctly pinpointed in [3], starting with the mode $LP_{12}$ either identical field configurations or very similar ones are encountered. That makes it impossible for any method to cope with decomposition.
On the other hand, it is known that there are ways to obtain a specific modal content in optical fibers. For instance, using mode multiplexing techniques, such as photonic lanterns [27] or multi-plane light conversion [28,29]. These methods imply conversion of SIF modes into those of a MMF. In this light, we show that such a speckle ambiguity can be eliminated by pre-selecting a suitable modal content and demonstrate that the rank problem can be rectified by means of an appropriate mode filtering procedure. In this Section, we imply $\lambda =850$ nm, as it provides more guided modes compared to $\lambda =1550$ nm given $R=25~\mu$m.
Since decomposition potential depends mostly on the transverse field distribution of the modes, we could find such a set of modes that would guarantee solvability of Eq. (6). The most illustrative way of explaining the motivation behind the simplest decomposable set for a GRIN fiber can be found in Fig. 5. The columns represent groups formed with respect to the mode group number $g=2p +|m|+1$, where $p$ and $m$ refer to the radial and the angular numbers respectively. The rows, on the other hand, contain indices of the modes with the same $p$. One can show that modes of the same orientation considered row-wise, represent a recipe for a successful decomposition by means of the algorithm, i.e. we obtain a full rank matrix for an arbitrary number of guided modes taken from these sets. Additionally, including all $LP_{0i}$ preserves the rank and does not affect the decomposition performance. It makes it possible to decompose $27$ modes of one spatial orientation ($a$ or $b$).
Fig. 5. Tabulation of guided modes for the GRIN fiber at $\lambda =850$ nm. Each cell contains two integers $i,~j$ that refer to $LP_{ij}$ mode. Columns represent mode group number $g$, whereas rows contain modes with the same radial number.
Fig. 6 demonstrates decomposition examples for the first four rows from Fig. 5. In all cases we get correlation $C=1.0$, while the errors do not exceed $\epsilon _v\sim 10^{-10}$. Having a full rank matrix provides another advantage, namely the fact that the phase recovery algorithm can be simply accomplished via Eq. (12) with the minimal number of sign comparisons, i.e. one can efficiently retrieve the phases in $2N-3$ steps.
Fig. 6. Selective approach: the original (left), the recovered (middle) beam speckles and their discrepancy (right) for modes selected from different rows in Fig. 5.
These decomposable sets are not unique and the above examples are merely the most intuitive ones. For example, one can build up another decomposable set of $30$ modes by considering $LP_{0i}$ with $i=1\dots 4$, both orientations of $LP_{j1}$ with $j=1.\dots 7$ and any orientation of $LP_{12}$, $LP_{13}$, $LP_{22}$, $LP_{23}$, $LP_{32}$ and $LP_{42}$. In this case, the matrix $P$ has full rank as well and the decomposition performance does not differ from that of the previous examples.
The mode appearance in fibers can be altered through changes in the shape of the refractive index profile [26]. One could find such a profile function that would leave only necessary decomposable modes. However, in such profiles the transverse fields and the resulting beam speckles might differ dramatically from those of SIF or GRIN.
6. Noise
In practice, performance of all decomposition algorithms is strongly affected by noise. This fact severely restricts their practical implementation regarding the number of modes [3,30]. Therefore, we test our method on noise robustness by adding Gaussian noise to beams of the SIF at $\lambda =1550$ nm. We vary the noise level so that the corresponding signal-to-noise ratio (SNR) changes between $10$ and $80$ dB.
We benchmark the set $S_1$ from Eq. (14) against a decomposable set that consists of modes from the first row in Fig. 5. In what follows, the latter is referred to as the set $S_3$. Table 2 provides values of the correlation coefficient for $N=3,6,10,15$ modes from $S_1$ and $S_3$ at different noise levels. Each cell represents averaging over $10000$ launches. Modern receivers, like CCD cameras, are able to capture beam images with SNR up to $60$ dB, which does not enable potential implementation of the proposed algorithm for decomposition of even the first $10$ consecutive modes. The set $S_2$ turns out to be more sensitive to noise compared to $S_1$ and is not listed in Table 2. On can see that $S_3$ shifts the decomposition limit to $20$ dB and, thus, makes potential decomposition of $15$ modes possible. The other proposed decomposable configurations demonstrate identical noise robustness, as the set $S_3$. Figure 7 displays the decomposition performance of $S_1$ and $S_3$ for $N=10$.
Table 2. Averaged correlation coefficients for two sets of modes of the SIF in presence of noise. Dashes refer to $C<0.8$.
7. Summary
In this paper, we investigated a $2D$ least-squares based method for mode decomposition in MMFs. We applied the algorithm to SIF and the parabolic GRIN fiber profiles, and compared their decomposition performance. We showed that the algorithm can account for random rotations of beams comprising degenerate modes of the same spatial orientation. We proposed a recipe for constructing a set of an arbitrary number of guided modes that is always decomposable in the framework of the least-squares. Finally, we investigated the influence of noise on the decomposition performance and proved our selective sets of modes to be several orders of magnitude more noise-resistant than the consecutive one.
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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