Space division multiplexing in standard multi-mode optical fibers based on speckle pattern classification

In optical communications the transmission bandwidth of single mode optical fibers is almost fully exploited. To further increase the capacity of a telecommunication link, multiplexing techniques can be applied across 5 physical dimensions: amplitude, quadrature, polarization, frequency and space, with all but the latter being nearly exhausted. We experimentally demonstrate the feasibility of an original space division multiplexing technique based on the classification of speckle patterns measured at the fiber’s output. By coupling multiple optical signals into a standard multimode optical fiber, speckle patterns arise at the fiber’s end facet. This is due to quasi-random interference between the excited modes of propagation. We show how these patterns depend on the parameters of the optical signal beams and the fiber length. Classification of the speckle patterns allows the detection of the independent signals: we can detect the state (i.e. on or off  ) of different beams that are multiplexed in the fiber. Our results show that the proposed space division multiplexing on standard multimode fibers is robust to mode-mixing and polarization scrambling effects.

Let us collect the indices of the n active lasers in set J such that For example, if {s 1 s 2 s 3 s 4 } = {0101} then J has two elements, namely J 1 = 2 and J 2 = 4. The expression for the speckle pattern P then becomes We know that the covariance operator is a bilinear, i.e.
Cov (R i we then find the sample cross-correlation coefficient between reference pattern R i and speckle pattern P Since we made sure the individual signal beams generate uncorrelated patterns, we have Cov(R J j , R J k ) = σ 2 δ J j J k and through the construction of J we have that δ J j J k = δ jk . When comparing one of the reference patterns R i with a pattern P generated by n active signal beams, one of which generated the reference pattern R i , we have i ∈ J. In this case, we find Alternatively, when comparing one of the reference patterns R i with a pattern P generated by n active signal beams, none of which generated the reference pattern R i , we have i / ∈ J. In this case, we find µ ρ = Corr(R i , P) = 0. As we combine up to 4 signals (n ≤ 4), this motivates the division of our measurement results into 5 groups: we expect to find 48 correlation coefficients close to 1, 144 coefficients close to 1/ √ 2, 144 coefficients close to 1/ √ 3, 48 coefficients close to 1/ √ 4 and 384 coefficients close to 0.
We can go a step further and construct the non normal probability density functions (PDF) for these correlations per group. We can do this using the Fisher transformation F, requiring only the expected value of the correlation coefficient and the sample size, which in this case corresponds to the number of speckle spots m in each pattern. The Fisher transform z is a useful tool in determining the distribution of the sample correlation coefficient ρ between normally distributed bivariate sample pairs with expected correlation µ ρ . The Fisher variable z is obtained as z = F(ρ) = atanh(ρ) and by good approximation it follows a normal distribution with mean and variance We cannot use these results directly as our sample pairs represent speckle intensities which are not normally distributed. Taking into account the correct distribution of n (> 1) overlapping speckle intensities (assumed equally strong and having m speckle spots), we calculated that the Fisher variable z now has mean and variance We will assume that these characteristics are still sufficient to accurately describe the distribution of z. Since F −1 (z) is monotonic we can obtain the non-normal PDF of ρ from the normal PDF of z using Since the inverse Fisher transfom is given by F −1 (z) = tanh(z) and since dF −1 (z)/dz = 1 − tanh 2 (z) = 1 − (F −1 (z)) 2 this expression can be further simplified. We arrive at the PDF of the sample correlation coefficient ρ as a function of the number of active lasers n and the number of speckle spots m.
These PDFs are compared with the measurement groups where partial or zero cross-correlations are expected. For the group of results with expected unity correlation, the PDF should in theory be a δ -function at ρ = 1. Only measurement noise and temporal drift of the speckle patterns is expected to induce some variance to these measurements.

Speckle correlation statistics: inputs coherently related
When all the signal beams are coherently related, the speckle patterns generated by the individual lasers will interfere coherently at the fiber's end facet. As such, the expected correlation between these combined patterns and the reference patterns scales differently with the number of active lasers n. In the case where the signal beams are not coherently related, we found µ incoh ρ (n) = 1/ √ n. We will show that in the coherent case we have µ coh ρ (n) = 1/n (assuming all patterns have equal mean intensities).
In this coherent regime we cannot simply work with recorded intensities. Instead we must keep track of the amplitude A and phase θ of the electric field E, or equivalently the real and imaginary part, denoted X and Y respectively, with The joint probability density function of the field components X and Y corresponding to a fully-developed speckle pattern is known to be where σ 2 is related to the mean intensity of the speckle pattern, as can be seen from the corresponding probability density function of the speckle intensity I p I (I) = 1 2σ 2 exp − with mean intensity I = 2σ 2 = I 0 . In the incoherent case, we said that speckle spot intensities were considered to be sets of samples X i of the (then unspecified) speckle intensity distribution. Now, in the coherent case, we will consider the field components at these speckle spots X i and Y i to be a set of bivariate samples from p X,Y (X,Y ). Note that the field components are uncorrelated since p X,Y (X,Y ) can be written as the product of the marginal distributions p X (X) and p Y (Y ). X and Y are both normally distributed with mean X = Y = 0 and variance Var(X) = Var(Y ) = σ 2 . Due to the coherent superposition, the intensity pattern P generated by n coherent beams can now be written in terms of the composing field components X j and Y j ( j = 1, ..., N). We now consider the i th correlation classifier which has to detect wether the i th beam is on. This means we are comparing the reference pattern R i based on X i and Y i with a pattern P based on the square of the sum Z X of all X j and the square of the sum Z Y of all Y j for j ∈ J with (the same definition as in the previous section) Note that Z X = Z Y = 0 and Var(Z X ) = Var(Z Y ) = Nσ 2 . We are establishing the expected cross-correlation coefficient between patterns R i and P Applying the bilinear covariance operator to R i and P we find Let us start by examining the first term of Cov(R i , P) We have If i ∈ J we split Z X in two parts to calculate X 2 i Z 2 Since X i and X j are independent random variables for i = j, it follows that X i is independent from W i .

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Note that W i = 0 and Var(W i ) = (n − 1)σ 2 , resulting in and further leading to Analagously, we find the second term of Cov(R i , P) to be Cov(Y 2 i , Z 2 Y ) = 2σ 4 . Since X i is independent from Y j for all j, and is thus independent from Z Y , we find the third term of Cov(R i , P) to be Analagously we have Cov(Y i , Z X ) = 0, resulting in Cov(R i , P) = 4σ 2 . Since we know R i and P to be fully developed speckle patterns, their variances equal their mean intensity squared Thus, when comparing one of the reference patterns R i with a pattern P generated by n coherent beams, one of which is the i th beam, we find Corr(R i , P) = 4σ 4 (4σ 4 )(4n 2 σ 4 ) = 1 n Similarly, when i / ∈ J we find Corr(R i , P) = 0

Speckle pattern cross-correlation confidence bound
Here we construct a confidence interval to decide with 95% probability wether a measured cross-correlation coefficient ρ belongs to a zero-mean distribution µ ρ = 0. For our purposes, a one-sided confidence interval is most suited. Since correlation coefficients are in general not normally distributed we use the Fisher transform z defined as z = atanh(ρ).
When testing the hypothesis µ ρ = 0, the corresponding Fisher variable z has a zero-mean normal distribution with variance σ 2 z . Thus we can easily construct the required confidence bound on z as P(z ≤ 1.645σ z ) = 95%.
For our purposes σ z is calculated in function of the number of speckle spots, as described in previous sections.