## 2 Problem Description, Simulation Model and Methodology  
### Few-Mode Fiber Transmission Systems Description  
Figure 1 shows a diagram of the 3-mode FMF transmission system used as a model in this work. The transmitter consists of one laser used as light source that is split into three optical signals which are input to a set of three independent dual-polarization dual-parallel Mach-Zehnder modulators (DP-DPMZMs). Each DP-DPMZM generates a polarization-division multiplexing (PDM) M-QAM signal or stream to be carried on an orthogonal spatial channel. After modulation, a spatial multiplexer with three SMF inputs generates a linear combination of the three orthogonal modes in the FMF for each input stream. At the receiver, a demultiplexer couples three linear combinations of orthogonal modes present in the FMF into a set of three SMFs. We consider the multiplexer (MUX), the FMF and the demultiplexer (DEMUX) as a black box MIMO linear system with 3 single-mode inputs and 3 single-mode outputs, as remarked in Figure 1. Each SMF carries 2 polarizations that correspond to degenerate modes so the dimensions of the MIMO system are actually \(6\times 6\) over a complex space. The receiver consists of three polarization-diverse optical hybrids with four balanced photodiodes and digitizers each. The signal from a free-running laser used as LO is split into three equal parts and input to the optical hybrids and hence the phase noise in the three streams is equal and correlated. The receiver DSP first normalizes the signal and recovers the clock. Note that in this simulation it is not necessary to compensate for receiver imperfections like quadrature-skew or differential photodiode responsivities. After the clock recovery, the 6 complex signals are input to a complex \(6\times 6\) MIMO time-domain equalizer and CPR loop [25]. After CPR, the samples are hard-detected, used to compute the equalizer error-signal, and to update the taps with this error-signal. The equalizer  
loop initially uses a known training sequence to achieve convergence and then switches into blind decision-directed mode. The response of the black-box in Figure 1 can be modeled as a matrix \(\mathbf{M}(\omega)\), which includes the response of the MUX, the DEMUX and any linear impairments in the FMF transmission channel [17; 26]. In general, \(\mathbf{M}(\omega)\) is a time-varying matrix, due to external perturbations like fiber movement and temperature changes, but we will consider it as a time-invariant matrix through the rest of the paper for simplicity. In a 3-mode FMF system, \(\mathbf{M}(\omega)\) is a complex square matrix with rank 6. If there are no losses or gains in the system, i.e., the energy at the input is the same as the energy at the output, then \(\det\mathbf{M}(\omega)=\pm 1\) and \(\mathbf{M}(\omega)\) is unitary. Unitary matrices can be interpreted as rotational matrices with a possible reflection in a complex functional space. We can factorize the black-box response as,  
\[\mathbf{M}(\omega)=\mathbf{O}(\omega)\mathbf{F}(\omega)\mathbf{I}(\omega) \tag{1}\]  
where \(\mathbf{F}(\omega)\) is the response of the FMF, \(\mathbf{I}(\omega)\) is the response of the MUX and \(\mathbf{O}(\omega)\) is the response of the DEMUX. The singular value decomposition (SVD) of a unitary matrix is not unique. However, imposing the condition that \(\mathbf{U}(\omega)\) must be independent of frequency to the first order, \(d\mathbf{U}(\omega)/d\omega=0\), then the columns of \(\mathbf{U}(\omega)\) are the input principal modes [26], which is a generalization into higher dimensions of the principal states of polarization in an SMF [27].