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Adaptive Modal Gain Equalization Techniques in Multi-Mode Erbium-Doped Fiber Amplifiers Reza Nasiri Mahalati, Daulet Askarov, and Joseph M

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Adaptive Modal Gain Equalization Techniques in Multi-Mode Erbium-Doped Fiber Amplifiers Reza Nasiri Mahalati, Daulet Askarov, and Joseph M JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 11, JUNE 1, 2014 2133 Adaptive Modal Gain Equalization Techniques in Multi-Mode Erbium-Doped Fiber Amplifiers Reza Nasiri Mahalati, Daulet Askarov, and Joseph M. Kahn, Fellow, IEEE Abstract—We demonstrate two adaptive methods to equalize reasons. First, adaptive methods may compensate for the afore- mode-dependent gain (MDG) in multi-mode erbium-doped fiber mentioned non-idealities in the doping profile or pumping and amplifiers (MM-EDFAs). The first method uses a spatial light mod- may track them if they change over time. Second, it is desirable ulator (SLM) in line with the amplifier pump beam to control the modal powers in the pump. The second method uses an SLM imme- to equalize not only the MDG in a single MM-EDFA, but also diately after the MM-EDFA to directly control the modal powers in the MDG accumulated in the link prior to the MM-EDFA, which the signal. We compare the performance of the two methods applied is not likely to be known apriori. Third, even if the MDG of to a MM-EDFA with a uniform erbium doping profile, supporting each amplifier is constant, the coupled MDG of a link can vary 12 signal modes in two polarizations. We show that root-mean- over time, since it depends on phase-dependent mode coupling squared MDGs lower than 0.5 and 1 dB can be achieved in systems having frequency diversity orders of 1 and 100, respectively, while along the link [5]. causing less than a 2.6-dB loss of mode-averaged gain. Among adaptive MDG equalization methods, the pump beam modal content may be adjusted by passing several variable- Index Terms—Mode-division multiplexing, multi-input multi- output (MIMO), multi-mode fiber (MMF), optical fiber amplifiers. power beams through phase masks and combining them using beam splitters [8]. The beam splitter losses make this method in- creasingly inefficient as the number of pump beams is increased, I. INTRODUCTION as might be required to accommodate an increasing number of S long-haul single-mode fiber systems approach signal modes. A information-theoretic limits [1], [2], spatial multiplexing We present two new methods for adaptive equalization of in multi-mode or multi-core fibers offers a possible route to MDG in MM-EDFAs. The first method is to control the modal higher throughput [3]. The properties of transmission fibers and content of the pump using a spatial light modulator (SLM) fiber amplifiers are crucial to the ultimate feasibility of spa- placed in the pump path. This method is more power-efficient tially multiplexed long-haul systems. In transmission fibers, low than the previous method of controlling the modal content of group delay spread minimizes receiver signal processing com- the pump [8] as it only requires one pump laser per pump polar- plexity [4], while large modal effective areas minimize nonlinear ization, independent of the number of signal modes. The second effects. In fiber amplifiers, low mode-dependent gain (MDG) method directly equalizes MDG in the signal by placing an SLM minimizes the loss of capacity and the potential for outage [5]. immediately after the MM-EDFA. Methods for equalizing MDG in multi-mode erbium-doped Here, we study MM-EDFAs for signals spatially multiplexed fiber amplifiers (MM-EDFAs) may be classified as fixed or adap- in 12 modes (six spatial modes in two polarizations). By numer- tive. Among fixed methods, the erbium doping profile in the ical solution of multi-mode rate equations, we compare MDGs MM-EDFA may be optimized to minimize MDG [6], [7]. Al- obtained before and after optimizing the SLM using the two pro- ternatively, a phase mask may be placed in the pump beam posed equalization schemes. We show that the first scheme is to optimize the pump modal content, or may be placed in the more power-efficient, adapts faster and requires simpler power output beam to provide mode-selective attenuation. Although monitoring method, but cannot perform per-wavelength MDG such fixed methods can greatly reduce MDG when implemented equalization. Therefore, if the first scheme is used, wavelength- perfectly, non-idealities in the erbium doping profile or in the dependent gain needs to be controlled either in the MM-EDFA pumping can lead to increased MDG [7]. using gain-flattening filters [9], or elsewhere in the network us- While fixed methods are appealing for their simplicity, adap- ing appropriately designed multi-mode dynamic gain equalizers. tive MDG equalization methods may be necessary for several The second scheme can perform per-wavelength MDG equal- ization if it is used inside liquid crystal-on-silicon wavelength selective switches (LCOS WSSs) [10]. We study the impact of frequency-dependent MDG accumu- Manuscript received November 3, 2013; revised December 31, 2013, February 2, 2014, and March 18, 2014; accepted March 19, 2014. Date of lated in the link before the MM-EDFA on these two MDG equal- publication April 7, 2014; date of current version May 21, 2014. This work was ization schemes. By analyzing links with low and high frequency supported by the National Science Foundation under Grant ECCS-1101905 and diversity orders [11], we show that the schemes remain highly Corning, Inc. The authors are with the E. L. Ginzton Laboratory, Department of Elec- effective even in the presence of highly frequency-dependent trical Engineering, Stanford University, Stanford, CA 94305 USA (e-mail: accumulated MDG. [email protected]; [email protected]; [email protected]). The remainder of this paper is as follows. In Section II, Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. we describe the mathematical model for a multi-span mode- Digital Object Identifier 10.1109/JLT.2014.2314746 division multiplexed system and quantify MDG in such systems. 0733-8724 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 2134 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 11, JUNE 1, 2014 noted, however, that neither of our MDG equalization schemes depends on strong coupling to work. In fact, our methods work in systems where there is a combination of weak and strong coupling or no coupling at all. The MDG reduction obtained in Fig. 1. Long-haul system with K spans, each including an amplifier and an such systems might be different from what we report here for a (optional) equalizer for mode-dependent gain. strongly coupled system. Assuming each fiber span is a concatenation of N short sec- (k) In Section III, we introduce our two methods for MDG equal- tions, Mfiber(ω) is given by ization and describe the adaptive optimization algorithms used N for each method. In Section IV, we give simulation results for (k) (k,l) (k,l)H Mfiber(ω)= VfiberΛfiber(ω)Ufiber , (2) the two techniques. We provide discussion and conclusions in l=1 Sections V and VI, respectively. We provide a detailed mathe- H (k,l) (k,l) matical model of the MM-EDFA in the Appendix. where denotes Hermitian conjugate. The Vfiber and Ufiber are frequency-independent Ds × Ds random unitary matrices II. QUANTIFYING MDG IN A MULTI-SPAN SYSTEM representing mode coupling in the lth section of the kth MMF span. Λfiber(ω) is a Ds × Ds diagonal matrix that represents In order to describe MDG and modal dispersion in a long- the uncoupled modal dispersion in one fiber section, which is haul mode-division multiplexed system, we model the link as assumed to be the same in all sections. It is given by the concatenation of numerous spans. Fig. 1 shows the block −jωτ1 −jωτD diagram of a system with K spans. Each span of multi-mode Λfiber(ω)=diag e ,...,e s , (3) fiber (MMF) is followed by a MM-EDFA to compensate for the mode-averaged loss of the fiber. After each MM-EDFA there where τ1 ,...,τD s describe the uncoupled modal group delays is an optional MDG equalizer that minimizes the root-mean- in each MMF section. (k) × squared (RMS) MDG at the amplifier output. MDG from each The Λamp is a frequency-independent Ds Ds matrix rep- amplifier depends weakly on frequency, but propagation makes resenting MDG in the kth MM-EDFA. As it is shown is the MDG frequency-dependent [11] and hence, MDG is best equal- Appendix, mode coupling is negligible in the MM-EDFA and (k) ized in or right after the MM-EDFA. hence Λamp is a diagonal non-unitary matrix given by Assuming a MMF supporting D orthogonal propagating sig- s 1 ( k ) 1 ( k ) (k) 2 g1 2 gD nal modes (including polarization and spatial degrees of free- Λamp = diag e ,...,e s , (4) dom) and neglecting nonlinearity and noise, the total propaga- tion operator of the multi-span system is a D × D matrix that where g(k),...,g(k) describe frequency-independent uncou- s s 1 D s multiplies complex baseband modal envelopes at frequency ω, pled modal power gains in the kth MM-EDFA. (k) and is given by Meq is a frequency-independent Ds × Ds matrix repre- K senting MDG and mode coupling in the kth equalizer. It is a M(ω)= M(k)Λ(k) Mk (ω), (1) non-diagonal and non-unitary matrix. If there are no separate eq amp fiber (k) k=1 equalizers in the system, Meq is the identity matrix. The overall MDG operator of the system at a frequency ω (k) (k) (k) where Mfiber(ω), Λamp and Meq are the propagation opera- is given by M(ω)MH (ω).
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