## 1 Introduction  
Coherent transmission systems using multi-mode fibers (MMFs), and in particular few-mode fibers (FMRs) as a subset thereof, have recently attracted much research attention due to the capacity enhancement enabled by transmitting multiple data streams on a set of orthogonal spatial modes [1; 2; 3]. The incorporation of digital signal processing (DSP) algorithms, including multiple-input multiple-output (MIMO) equalizers, enable the compensation in the digital domain of the differential mode delay (DMD) and the decoupling of the modes that were coupled both during propagation through the FMF channel and at the input/output multiplexing devices. Conversely, in incoherent optical communication systems, where DSP can be used with the squared modulus of the electrical field and not with the amplitude and phase thereof [4], the impairments caused by the MMF channel, e.g., the mode mixing/scrambling and the DMD, constitute a fundamental limitation to the system capacity and reach. Hence, incoherent MMF/FMF links that employ modulation formats such as ON-OFF keying (OOK) [5] or 4-way pulse amplitude modulation (PAM4) [6] are currently used predominantly for short reach applications. Without loss of generality, in this manuscript we focus on coherent FMF optical transmission systems as a particular case of MMF systems when the number of modes is small. In FMF systems, some authors claim that the DMD spread does not constitute a fundamental limitation because even large DMD spread values can be compensated with sufficiently large linear MIMO equalizers, although at a very high computational cost of the DSP circuitry [7]. However, in a practical implementation the phase noise in the transmitter and receiver lasers cannot be neglected and the interaction of the DMD with the phase noise of the local oscillator (LO) will cause equalization enhanced phase noise (EEPN), which may significantly degrade the system performance. The interaction of the DMD with the phase noise of the LO in FMF systems plays a similar role as the interaction of the chromatic dispersion (CD) with the phase noise of the LO in single-mode fiber (SMF) systems [8], and additionally in FMF transmission systems the CD also produces EEPN. EEPN has been studied in detail for dispersion-uncompensated SMF systems with digital compensation of the CD. [PERSON] and [PERSON] [9] studied the theoretical EEPN power penalty in SMF systems resulting from the interaction of the CD and the LO phase noise and found that the EEPN penalty increases for larger dispersion values and baud rates. [PERSON] [10] numerically demonstrated, using quadrature phase-shift keying (QPSK) as the modulation format, that the EEPN penalty increases linearly with the baud rate. [PERSON] et al. [11], both analytically and numerically, concluded that the effect of the EEPN induced phase noise is approximately twice as large as the EEPN induced amplitude noise, and also that the contribution of the transmitter phase noise is negligible compared to the LO for links with residual dispersion higher than 700 ps/nm. [PERSON] [12] presented an analytical model for differential N-phase shift keying (PSK) systems and concluded that the EEPN performance degradation increases significantly with the increment of the LO phase noise, the fiber dispersion and the order of the modulation format. [PERSON] et al. [13] conducted a frequency domain analysis and concluded that the EEPN penalty mainly originates from the low frequency noise components of the LO. The same authors [14] extended their analysis and concluded that the EEPN is similar to multi-path fading in wireless communication systems. [PERSON] et al. [15] demonstrated both numerically and experimentally that in systems with a Stokes-vector direct-detection (SV-DD) receiver the EEPN penalty increases with the LO laser linewidth, the accumulated dispersion and the order of the modulation format, and this penalty can be partially mitigated with a maximum-likelihood (ML) phase estimator.