Linear Propagation Effects in Mode-Division Multiplexing Systems

TLDR: In this article, the authors present a concatenation rule for the accumulation of MD along a multisection link, including physical origins, models, and regimes of weak and strong coupling.

Abstract: In this paper, we review linear propagation effects in a multimode fiber (MMF) and their impact on performance and complexity in long-haul mode-division multiplexing (MDM) systems. We highlight the many similarities to wireless multi-input multioutput (MIMO) systems. Mode-dependent loss and gain (MDL), analogous to multipath fading, can reduce average channel capacity and cause outage in narrowband systems. Modal dispersion (MD), analogous to multipath delay spread, affects the complexity of MIMO equalization, but has no fundamental effect on performance. Optimal MIMO transmission uses a basis of the Schmidt modes, which may be obtained by a singular value decomposition of the MIMO channel. In the special case of a unitary channel (no MDL), an optimal basis is the set of principal modes, which are eigenvectors of a group delay operator, and are free of signal distortion to first order. We present a concatenation rule for the accumulation of MD along a multisection link. We review mode coupling in MMF, including physical origins, models, and regimes of weak and strong coupling. Strong mode coupling is a key to overcoming challenges in MDM systems. Strong coupling reduces the group delay spread from MD, minimizing the complexity of MIMO signal processing. Likewise, it reduces the variations of loss and gain from MDL, maximizing channel capacity. In the strong-coupling regime, the statistics of MD and MDL depend only on the number of modes and the variance of accumulated group delay or loss/gain, and can be derived from the eigenvalue distributions of certain Gaussian random matrices.

Figures

Fig. 1. Probability density of strongly coupled group delay, normalized by the STD of group delay, in MMFs with various numbers of modes. The semicircle distribution is shown as a dashed curve.

Fig. 4. Average channel capacity of MDM systems without CSI as a function of overall SNR ρt . Different numbers of modes D and different values of accumulated MDL ξ are represented by different line colors and styles, respectively.

Fig. 3. Probability density of strongly coupled MDL, normalized by the STD of overall MDL, for D = 6 modes for various values of the accumulated MDL ξ = √ Kσg . The exact distribution (27) is shown by solid curves. The approximation based on a Gaussian unitary ensemble is shown by dashed curves. Numerical simulations are shown as markers. The y-axis is in arbitrary linear units, with different curves shifted to improve visibility.

Fig. 2. Complementary cumulative distribution of group delay spread, normalized by the STD of group delay, in MMFs with different numbers of modes.

Frequently asked questions

Q1. What are the contributions in "Linear propagation effects in mode-division multiplexing systems" ?

In this paper, the authors review linear propagation effects in a multimode fiber ( MMF ) and their impact on performance and complexity in long-haul mode-division multiplexing ( MDM ) systems. The authors present a concatenation rule for the accumulation of MD along a multisection link. The authors review mode coupling in MMF, including physical origins, models, and regimes of weak and strong coupling. 

Q2. What is the effect of mode coupling on the MDL?

strong mode coupling minimizes the variation of gain or loss from MDL, minimizing any loss of capacity and minimizing the potential for outage [24]. 

Q3. What is the eigenmode of the input unitary matrix?

The input unitary matrix U(t)(ω) comprises D column vectors u1(ω),u2(ω), . . . ,uD (ω), each D × 1, which represent mutually orthogonal input eigenmodes of the system, and are eigenvectors of a phase-conjugate round-trip propagation operator M(t) ∗ (ω)M(t)(ω). 

Q4. What is the main effect of mode coupling on the system properties?

Only coupling between forwardpropagating modes is considered here, since it has a dominant effect on the system properties of interest, including MD and MDL. 

Q5. What is the eigenmode of a round-trip propagation operator?

In a laser resonator, the lasing mode is equivalent to the eigenmode of a round-trip propagation operator M(t)T (ω)M(t)(ω) that has the largest eigenvalue. 

Q6. What is the frequency dependence of g(t)()?

In MDM systems, the frequency dependence of g(t)(ω) may be exploited to provide frequency diversity [24] by using wideband single-carrier modulation or multicarrier modulation with error-correction coding across frequency. 

Q7. What is the polarization coupling state of the MMF?

After 5 to 15 meters, the two polarization modes of each spatial mode strongly couple with each other and the MMF enters the polarization coupling state [68], [69]. 

Q8. What is the correlation length in the multisection model?

In the context of the multisection model, the correlation length corresponds to the minimum section length such that successive unitary matrices U(k) and V(k−1) are mutually uncorrelated. 

Q9. How did the effectiveness of mode coupling be confirmed?

The effectiveness of mode coupling in reducing MDL and increasing channel capacity were confirmed in [92], [99]–[101] by simulation. 

Q10. What is the correlation length in the weak-coupling regime?

In the weak-coupling regime, the spread of eigenvalues describing quantities of interest, such as modal group delays or modal gains, scales linearly with the total system length [13], [73], [75], and each coupled eigenmode is a linear combination of a small number of uncoupled waveguide modes. 

Q11. How many modes does the Gaussian unitary ensemble approximation show?

The Gaussian unitary ensemble approximation can be used to show that the maximum MDL difference g(t)1 − g (t) D is 4 to 5 times σmdl , depending on the number of modes D, similar to Fig. 

Q12. What is the effect of the strong-coupling regime on the performance of MDM systems?

While MD impairs traditional MMF transmission systems using direct detection, it has no fundamental effect on performance in MDM systems using coherent detection and MIMO digital signal processing. 

Q13. What is the chi distribution for the delay spread?

For the values of D shown in Fig. 2 and forp of order 10−4 to 10−6 , uD (p) is of order 4 to 5, depending on the number of modes D.The delay spread was studied in [90], which generalized the concept of Stokes space to D > 2 and showed that in the strong-coupling regime, the distribution of the delay spread τ(t) 1 − τ (t) D is well-approximated by a chi distribution.