Statistical description of transport in multimode fibers with mode-dependent loss

We analyze coherent wave transport in a new physical setting associated with multimode wave systems where reflection is completely suppressed and mode-dependent losses together with mode mixing are dictating the wave propagation. An additional physical constraint is the fact that in realistic circumstances the access to the scattering (or transmission) matrix is incomplete. We have addressed all these challenges by providing a statistical description of wave transport which fuses together a free probability theory approach with a filtered random matrix ensemble. Our theoretical predictions have been tested successfully against experimental data of light transport in multimode fibers.


Introduction
Random matrix theory (RMT) has been successfully applied over the years in a variety of physics areas ranging from nuclear and atomic physics to mesoscopic physics of disordered and chaotic systems [1][2][3][4][5][6]. Its applicability relies on the assumption that in complex systems the underlying wave interference impose universal statistical rules which govern their transport characteristics. Along these lines of thinking, random matrix models allowed us to uncover some of the most fundamental properties of disordered/chaotic systems, including the structure and statistical properties of their eigenstates [7,8] and eigenvalues [9,10], the conductance [11][12][13], the resonance widths and delay times [14], etc. It turned out that many of the universal features of transport are directly connected with the various symmetries (time-reversal, chiral, etc) that a specific complex system satisfies [15]. In all these studies, nevertheless, it was always assumed that the scattering process does not involve any additional constrains and has both a backward (reflection) and a forward (transmission) component.
Recently, the interest in wave transport has extended to new physical settings with practical relevance, namely, a class of complex multimode systems where reflection processes are absent [16,17]. Obviously, the zero reflectivity condition, imposes new constraints to the wave scattering process, thus constituting the previous RMT predictions void. These type of transport problems have emerged naturally in the framework of multimode (or coupled multi-core) fiber optics. In these systems, fiber imperfections (core ellipticity and eccentricity) and external perturbations (index fluctuations and fiber bending) cause coupling and interference between propagating signals in different spatial modes and orthogonal polarizations. At the same time, the effect of mode-dependent loss (MDL) (or gain due to optical amplifiers) in wave propagation is another important feature whose ramifications are not yet completely understood [16][17][18][19]. In the framework of multimode fibers (MMFs), for example, it leads to fundamental limitations in their performance since extremely high MDL can reduce the number of propagating modes and thus the information capacity of MMFs. It is, therefore, imperative to develop statistical theories that take into consideration the modal and polarization mixing and MDL and provide a quantitative description of light transport in realistic MMFs (and other multimode systems that demonstrate similar challenges). Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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Here, we develop a statistical theory of light transport in MMFs, where both MDL and modal and polarization mixing are considered. Our theory is accounting for the fact that in experiments, the degree of modal and polarization control is limited. To this end, we have combined free probability theory and the filtered random matrix (FRM) ensemble and took into consideration the finite length of the MMFs. Unlike the telecommunication fibers, which are typically very long, finite length MMFs are common in medical applications (endoscopy), sensing, local-area networks and data-center interconnects etc. As an example, we have implemented our theoretical formalism in order to derive two important statistical measures: (a) the distribution of transmission eigenvalues for polarization maintenance (and/or conversion); and (b) the absorbance distribution of a monochromatic light propagating in a MMF with MDL and strong mode and polarization coupling. Our theoretical results have been validated via direct comparison with experimental measurements using MMFs.

Fiber model
We consider a MMF supporting N propagating linearly polarized (LP) modes, with each LP mode being twofold degenerate corresponding to the horizontal (H) and vertical (V) polarizations. We model the fiber of interest as consisting of a concatenation of K independent and statistically identical segments [16], with the linear propagation through the MMF described by aŃ where the elements of the N×N block matrices ( ) t K HH ( ( ) t K VH ) are the transmission amplitudes into the H (V) polarization when the incident light is H-polarized. Each segment is modeled via a matrix v k Λ, where v k is á N N 2 2 unitary matrix describing the polarization and mode mixing in the segment, and L = L L ( ) diag , H V is a diagonal matrix describing the free propagation and attenuation in the absence of such mixing. We consider MMFs in the strongly mixed regime where every mode is coupled to every other mode in one segment, with v k being random unitary matrices drawn from the circular unitary ensemble (CUE) [3]. Hence, one segment corresponds to one transport mean free path l t of light scattering in fiber mode basis. The number of segments gives the ratio of the fiber length (L) to the transport mean free path K=L/l t . We assume that the two polarizations have the same propagation constants and loss, so that d The higher-order LP modes take longer paths and impinge on the core-cladding interface at steeper angles, so they typically experience more attenuation than the lower-order modes; we model such MDL as  b = ( ) ( ) m n s N 2 n with n=1, L, N, characterized by the coefficient s (s>0 for loss). The real parts of β n describe the mode-dependent propagation phase shifts and are not important in the context of this paper as they can be absorbed into v k .
In actual experimental circumstances, the preparation and measurement of a waveform in all modes is technically challenging. In this respect, one needs to analyze portions of the total transmission matrix = where P in and P out are projections to the controlled incoming and outgoing modal subspace. Specifically, given an incident wavefront yñ | which belongs to the P in -subspace, the measured transmittance in the P out -subspace (summed over the spatial/polarization modes) after propagating through the MMF is . It is therefore obvious that the eigenvalues of the matrix ( ) out in out in dictates the transport properties of such MMFs. For example, the extremal eigenvalues (and corresponding eigenvectors) are associated with the maximal and minimal transmittances achieved in such set-ups and can be used in order to design waveform schemes with extreme transport characteristics. Along these lines of reasoning, of particular interest is the eigenvalue statistics  t ( ) HH HH . In this case the preparation (associated with P in ) and measurement (associated with P out ) subspaces correspond to the set of modes with horizontal (H) polarization. The maximum eigenvalue τ (and the associated eigenvector) indicate the optimal polarization retention that can be achieved when light propagates in the system.
Another interesting statistics is  t ( ) HH HH VH VH . In this case P in corresponds to the subspace of horizontally polarized modes while P out is the identity matrix i.e. the whole modal space including both polarizations. The eigenvalues of T H provide information about the total transmissivity summed over the two polarization states at the output, given a H-polarized incident light. The complementary matrix A H ≡1−T H provides information about the amount of absorption during propagation inside the MMF.

Transmittance eigenvalue distribution of concatenated fiber segments
Our theoretical investigation capitalizes on the multiplicative structure of the transmission matrix. Specifically, we use free probability theory [20][21][22] which predicts the spectral properties of a product of random matrices from the spectral properties of its constituents. Based upon the probability distribution  t = = ( ) associated with the eigenvalues of ( ) ( ) † ( ) t t 1 1 for a single segment, we can construct a recursion relation from the model definition in equation (1). Statistically every segment is equivalent, so we can write where the equality is in the statistical sense. Using the free probability theory [21,22], when  ¥ N we get from equation where S Q denotes the S transform for an arbitrary Hermitian matrix Q. The S transform is ultimately related to the Green's function In the case of one section K=1, we have the eigenvalue density  Typically we resort to numerical method to obtain the eigenvalue density by combing equation (5) and the implicit formula for the Green's function equation (9). However, for the first few moments, explicit results can be easily obtained. For example, using equations (3) and (4)  We further calculate the probability distribution  t ( ) ( ) K using equation (5). In figure 1 we show the theoretical results together with the outcome of simulations. We find that for finite number of concatenated segments K and finite MDL ¹ s 0, the distribution  t ( ) ( ) K deviates from the standard semicircle expected from standard RMT considerations, see figures 1(a) and (b). The explicit knowledge of the first two moments allow us to analyze the scenario of many concatenated fiber segments  ¥ K with a loss-per-segment s→0, such that the mean  . We find, using the Bhatia-Davis inequality, that in this case the variance of the probability distribution  t ( )

Transmittance distribution of filtered channels
To evaluate the transmission eigenvalue distribution for any portion of the transmission matrix, we use G ( K ) to derive the Green's function ( ) G P P . For this derivation one needs to consider that the input and output fraction of total mode space is half. Combining equations (9), (11), we derive an implicit equation    predictions and the numerical simulations is perfect. The above analysis also captures the effect of incomplete modal control, e.g. when only parts of the N spatial modes are modulated or measured.
Using equation (11) and the fact that  = -¥ ( ) ( ) G z z 1 (see discussion at the end of previous section), we reduces to a bimodal distribution with confined support  t Î ( ) 0, . In this case the information about the number of concatenation segments K and the loss-per-segment s is 'hidden' in the upper bound of the transmittance support . In the limiting case of zero losses s=0, it is easy to show, that the eigenvalue distribution  t ( ) HH reduces to a bimodal distribution It is tempting, at this point, to establish an analogy between the s=0 and ¹ s 0 cases (  ¥ K ). In both cases there are essentially only two groups of propagating channels-open channels associated with τ-values close to 1 or , and closed channels with τ-values in the neighborhood of zero. One then can understand the results for  ¥  K s , 0in the following way: when the MMF is long enough (large K ) such that complete mode and polarization mixing happens many times across the fiber, the mixing equalizes the mode dependence and turns MDL into mode-independent loss with the transmittance of the open channels being renormalized to . These analytic predictions are nicely confirmed by numerical simulations of the concatenated MMF model, as shown in figure 2(c).
We note that because of the strong mode and polarization mixing, this analysis applies equally to  t ( ) ( ) K VH or other quarters of the transmission matrix. We stress that the calculation strategy that we have used here is not bounded by the specific choice of MDL (constant increase) and can be easily generalized to any type of MDL distribution. Moreover, the same scheme can be utilized for the case of mode-dependent gain.
Using the same approach as above, we can also evaluate the eigenvalue distribution  t ( )

Conclusion
We have developed a theoretical formalism that utilizes a free probability theory together with a FRM approach in order to derive theoretical expressions for the probability distribution of transmittances and absorbances in multimode scattering set-ups where reflection mechanisms are absent (paraxial approximation) and the information about the transmission matrix is incomplete. The motivation for this study is drawn by the recent interest to understand light transport in MMFs with MDL and strong mode and polarization mixing. The resulting probability distributions are different from any known results found for lossy disordered or chaotic systems [26][27][28][29] indicating that the paraxial constraint, and/or the presence of MDLs can dramatically affect light transport. The validity of our predictions have been tested both with simulations and via direct comparison with experimental data. We stress that our scheme can take into account any type of MDL (or gain) distribution. It will be interesting to extend this study to the case of weak mode mixing where the mode mixing matrices v k do not belong to the CUE.