Fast mode decomposition in few-mode fibers

Retrieval of the optical phase information from measurement of intensity is of a high interest because this would facilitate simple and cost-efficient techniques and devices. In scientific and industrial applications that exploit multi-mode fibers, a prior knowledge of spatial mode structure of the fiber, in principle, makes it possible to recover phases using measured intensity distribution. However, current mode decomposition algorithms based on the analysis of the intensity distribution at the output of a few-mode fiber, such as optimization methods or neural networks, still have high computational costs and high latency that is a serious impediment for applications, such as telecommunications. Speed of signal processing is one of the key challenges in this approach. We present a high-performance mode decomposition algorithm with a processing time of tens of microseconds. The proposed mathematical algorithm that does not use any machine learning techniques, is several orders of magnitude faster than the state-of-the-art deep-learning-based methods. We anticipate that our results can stimulate further research on algorithms beyond popular machine learning methods and they can lead to the development of low-cost phase retrieval receivers for various applications of few-mode fibers ranging from imaging to telecommunications.

(1) (1) The number of columns in the matrix T equals to the length of vector z and equals to Nz = Nm·(Nm+1)/2. The matrix can we written in the form: We investigated how the rank of the matrix and the condition number depend on the number of modes. We noticed that with the number of modes N = 10, there is a sharp jump in condition number to about the inverse accuracy of This means that with the number of modes N = 10, the matrix columns become linearly dependent. We calculated the angles between all pairs of columns and found that there are 2 types of anomalies: exactly collinear pairs of columns and approximately collinear (with an accuracy of 10 -3 -10 -2 ) pairs of columns. These second anomalous pairs of columns aroused the greatest interest, since one column from such a pair always contains the product of the fundamental mode LP01 and some other higher mode. This is especially important, because we directly calculate the phase difference between this higher mode and the fundamental mode using these pairwise products.
From a certain number of modes there are exactly coinciding columns t = ΨjΨk that correspond to pairwise products of sin-and cos-submodes, for example for LP11 и LP12 modes:

Supplementary Figure 2. Modes that form coincide columns in matrix T.
It is worth mentioning here that for each mode with L > 0 there are 2 orthogonal "submodes" sin LP11 and cos LP11, which we consider as separate modes. This leads to the appearance of coinciding columns in the matrix T.
In addition, there are very similar columns: Supplementary Figure 3. Modes that form close columns in matrix T.
It should be noted that for "almost coincident" (in the picture above) the discrepancies are due to the difference in the transverse wave numbers for the modes LP31, LP01 и LP61.
Such coinciding columns result in the matrix being ill-conditioned, and it becomes impossible to take the pseudoinverse matrix. Starting from the number of modes N = 10, the rank of the matrix T does not coincide with the number of columns, since for the first time a combination of the LP11 and LP12 modes appears, which give the same columns for sin-and cos-components (see Fig. SP2). However, so far this has little effect on the accuracy of the decomposition, since many other linearly independent columns remain, which include coefficients for these modes.
There is a way to get rid of the ill conditioning of the matrix T. The first idea is to leave only one of the two columns, add the corresponding components of the vector z and renumber it.
For example, for the number of modes N = 10, the columns that correspond to the modes sin LP11· cos LP12 and cos LP11· sin LP12 coincide (see Supplementary Figure 2a). But this trick doesn't lead neither to decreasing of condition number nor to increasing rank of the matrix T. It is another evidence for linear dependence of multiple columns in the matrix of pairwise products of eigenmodes which limits the maximum number of modes in a mode decomposition problem.

Supplementary Note 2: On error distribution
Non-symmetrical error distribution is observed for non-negative noise model. The thing is we can't use gaussian noise directly because when added to low-intensity pixels it can produce negative values of intensity.
That's why we use max[0, x] for each intensity distribution after adding gaussian noise. And it's something what is truly happening in an experiment. We found that the error distribution for decomposed weights is shifted: Supplementary Figure 6. Non-symmetrical error distribution for non-negative noise model Compared to "true" gaussian noise (which can't be observed since there are some pixels with negative intensity): Supplementary Figure 7. For symmetrical noise distribution the decomposition error is also symmetrical.
This induces additional errors in mode weight distribution. We believe that using general message passing algorithm can help to get rid of this because it allows to consider noise model that is different from standard additive Gaussian white noise.