## Abstract

Numerical and experimental study of a Few-Mode (FM) Erbium Doped Fiber Amplifier (EDFA) suitable for mode division multiplexing (MDM) is reported. Based on numerical simulations, a Few-Mode Erbium Doped Fiber (FM-EDF) has been designed to amplify four mode groups and to equally amplify LP_{11} and LP_{21} mode groups with gains greater than 20 dB and with a differential modal gain of less than 1 dB. Experimental results confirmed the simulations with a good concordance. This modal gain equalization is obtained by tailoring the erbium spatial distribution in the fiber core with a ring-shaped profile.

© 2012 Optical Society of America

## 1. Introduction

Techniques for increasing data transmission rates in optical fibers have been widely studied during the last decades and the current technology based on single-mode fibers seems to reach its fundamental limits [1]. To go beyond these limits, new disruptive technologies based on Space Division Multiplexing (SDM) have been recently investigated, in order to be used in addition to conventional wavelength division multiplexing, polarization division multiplexing [2] and complex modulation format. Two technical possibilities for SDM transmissions are currently the subject of intense research: MultiCore Fibers (MCF) and Mode Division Multiplexing (MDM) in Few-Mode Fiber (FMF). In MCF, each core can be used as a unique channel to carry the current total data transmission capacity [3], whereas in FMF each transverse mode can be used as a path [4]. This last technique already permitted a transmission at 5x100 Gb/s over 5 transverse modes in a FMF over 40 km [5]. However, to make this technology compatible with long-haul transmissions, re-amplifying the signals that are multiplexed on the different modes will be necessary, as in single-mode transmission lines. Hence, technology of EDFA has to be revisited and upgraded in order to ensure similar gains between the different guided modes. To do so, one possibility is to tailor the transverse intensity pattern of the pump beam that is injected within the FM-EDF (by using, for example, phase plates or non-centered injection conditions) in order to decrease the Differential Modal Gain (DMG), as already proposed [6, 7]. However, this solution proves to be tricky since the pump beam profile should be carefully controlled and stable in time and could implicates strong injection losses that reduce amplifier efficiency. On top of that, exciting certain high order pump modes can generate a transverse spatial hole burning, that implicates a modal selection for signal modal gain (if they have the same azimuthal symmetry), which is an undesired effect [6, 8]. In this paper, we focus rather on the spatial distribution of the erbium doping in order to propose a FM-EDFA compatible with MDM and capable of amplifying the first four guided modes (namely LP_{01}, LP_{11}, LP_{02} and LP_{21}) and designed to exhibit similar gains for the LP_{11} and LP_{21} modes when pumped in centered injection conditions. The aim of this paper is to go deeper in the characterization of its amplifying properties.

First and foremost, the algorithm used to design the FM-EDF will be discussed and an erbium doping profile adapted to small DMG between LP_{11} and LP_{21} modes will be investigated. Then, the characteristics of the FM-EDFA will be presented and, finally, amplification of LP_{11} and LP_{21} modes alone or together will be reported and compared to numerical simulations. Results for LP_{01} and LP_{02} will also been presented.

## 2. Theoretical model and Design

The theoretical model used to obtain the modal gain in FM-EDFA is based on a modified version of the model proposed by Giles and Desurvire [9], the major difference being that this model has been generalized to FMF, as in references [8, 10, 11]. In this model, we consider a two-level system to describe the energy level configuration of erbium ions meanwhile the light is considered as *k* optical beams of frequency bandwidth Δ*λ* = 1 nm centered on the wavelength *λ _{k}* ∈ [1500–1600 nm] and propagating in the

*n*transverse fiber mode. At pump wavelength (980 nm), light is considered to be monochromatic and to propagate on different modes. By considering the effects of absorption, stimulated emission, spontaneous emission and background losses, two kinds of well-known equations are obtained, namely propagation and rate equations. We simplified the problem by using the steady state approximation. So, the algorithm makes a longitudinal, transverse, spectral and modal resolution of the equations system. In this way: pump, signal, forward and backward Amplified Spontaneous Emission (ASE) propagation (for each wavelength and each transverse mode) but also the population inversion level (at each point of the fiber) are calculated. Using this model, it becomes possible to analyze the impact of mode profiles of pump and/or signal input beam(s) and erbium doping profile on the gain performances.

^{th}In an attempt to design a FM-EDFA, our primary goal was to minimize the DMG between the LP_{11} and LP_{21} mode groups. These modes are attractive because they can only couple with themselves when they experience a splice between two different fibers, as long as the two fibers both guide the same number of modes and the splice is centered (because of azimuthal symmetry considerations, as explained later in this paper). This effect limits cross-talk and ensures modal control. Moreover, considering both polarization (noted *x* and *y*) and spatial degeneracies (noted *a* and *b*), these two mode families open the possibility to multiplex data over 8 modes (namely LP_{11ax}, LP_{11ay}, LP_{11bx}, LP_{11by}, LP_{21ax}, LP_{21ay}, LP_{21bx} and LP_{21by} modes). The study of LP_{01} and LP_{02} mode was only a secondary goal, but they have also been numerically and experimentally studied.

Since the intensity profile of LP_{11} and LP_{21} modes remains mainly off-centered in the fiber core, ring-shaped profile for erbium doping appears as a natural geometry to allow gain equalization of these two mode groups. Such a doping profile has already been used in our previous reference [12] and has also been recently theoretically investigated to obtain small DMG between LP_{01} and LP_{11} modes [11]. So, in order to propose the optimum ring dimension, the following assumptions have been made: i) the shape of the erbium profile is assumed to be a flat ring and ii) this ring is delimited by two radii, namely the internal one (R_{di}), and the external one (R_{de}) (Fig. 1(a)).

By only changing the ring dimensions, the gain of both LP_{11} and LP_{21} mode groups has been computed, considering that these four modes (LP_{11a}, LP_{11b}, LP_{21a} and LP_{21b}) are simultaneously injected in the fiber and simultaneously amplified. The fiber length has been defined as the one that gives the optimal gain for LP_{21} modes. Results of these simulations are shown in Fig. 1. Modal gain for LP_{11} and LP_{21} mode groups are reported on Figs. 1(c) and 1(d) as a function of R_{di} and R_{de}. The differential modal gain between the two mode groups (|*G*_{LP21} − *G*_{LP11}|) is reported on Fig. 1(b). It can be seen on these figures that there exists an optimal area, i.e. some erbium doping profiles, for which the modal gains are equal (25 dB) and close to each maximum gain (27 dB). According to these numerical results, the selected profile corresponding to R_{di}=0.4×R_{de} and R_{de}=R_{core} (radius of the fiber core) has been chosen.

## 3. Fiber realization

Based on these results, a preform has been made by Modified Chemical Vapor Deposition (MCVD) process. The core has been realized in a two step process: first, a porous layer has been deposited, soaked with erbium/aluminum salts and sintered and, second, several erbium-free germanium-doped silica layers have been deposited so as to reach the suited R_{di}/R_{de} ratio. The preform has then been collapsed. The refractive index profile was measured on the preform as well as the erbium profile that was measured by Electron Probe Micro-Analysis (EPMA). A R_{di}/R_{de} ratio of 0.44 has been measured (at 1/*e*^{2}), which is close to the target. On Fig. 2, it can be observed that the index profile is not an ideal step, however the erbium profile is close to what we prospected for. Then, the preform has been drawn into fiber, so that the FM-EDF effectively guides four mode groups (LP_{01}, LP_{11}, LP_{21} and LP_{02}) at 1550 nm. It has to be pointed out that the core composition of the erbium-doped region makes this fiber compatible with WDM applications due to gain flatness authorized by aluminosilicate composition. Erbium-related absorption is about 16 dB/m at 1530 nm when the FM-EDF is tested, spliced to a standard single mode fiber (SSMF). This relatively high absorption should permit to limit mode coupling and background losses by making it possible to use a short piece of fiber.

## 4. Amplifier set-up

To experimentally characterize the gain performances of the FM-EDF, pump (974 nm laser diode from Oclaro) and signal beams (81640A tunable laser source from Agilent) are multiplexed in free space, and then, injected into 6 m of FMF especially designed for weakly-coupled MDM transmissions. This FMF is a step index fiber with core radius equal to 7.5 *μ*m and core/cladding refractive index difference equal to 9.7×10^{−3}. A complete characterization of this fiber is reported in reference [13]. As shown on Fig. 3, the FMF is spliced to a 3 m-long piece of FM-EDF, which is also spliced to another 6 m-long piece of FMF. The output end was angle-cleaved so as to prevent laser effect and it is connected to the input port of an Optical Spectrum Analyzer (6370 from Yokogawa) or an IR camera (C10633 InGaAs camera from Hamamatsu). Phase plates are used to shape the desired signal field profile: either the LP_{11} (2-quadrant phase plate) and/or the LP_{21} modes (4-quadrant phase plate) are then injected in the FMF. LP_{01} and LP_{02} modes cannot be tested alone with this set-up (as it is explained later in this paper) and no demultiplexing set-up is used to separate modes at the output. Coupling losses of the pump beam are about 1.1 dB. At signal wavelength, coupling losses depend on which mode is excited: without phase plate and in centered injection conditions, these losses are equal to 2.1 dB. With a 2-quadrant phase plate, LP_{11} mode is excited with 10 dB coupling losses and with a 4-quadrant phase plate, LP_{21} is excited with 16 dB coupling losses. These high coupling losses can be explained by the fact that it was not possible to conserve a 4f setup (meaning that the phase plate is not at the focal length of the two lenses, due to the bulky micropositioner and dichroic miror obstruction). Nevertheless, such coupling losses are not a limiting factor in the context of this experiment where the goal is to study the characteristics of the FM-EDFA. Note that potential alternatives exist to overcome the drawbacks of phase plates such as mode-selective couplers [14] or asymetric Y-junction [15].

The FMF has a step index profile, whereas FM-EDF has not (Fig. 2). So as to be sure that the signal modes are well injected in the FMF, and also conserved when they experience a splice between FMF and FM-EDF, they have been individually imaged on the camera, successively before the splice (FMF) and after the splice (FM-EDF). On Fig. 4, it can be seen that LP_{01}, LP_{11} and LP_{21} are well injected in the FMF, and that the LP_{11} and LP_{21} modes are conserved when they undergo a splice between two different fibers. This can be explained by overlap integrals: the LP_{11} and LP_{21} modes can only couple with themselves if the splice is perfectly centered. We report the coupling efficiency between FMF and FM-EDF modes in Tab. 1 and Tab. 2. Mode coupling efficiency factors (Γ* _{ij}*)

^{2}were calculated as the square of the overlap integrals Γ

*between FM-EDF and FMF transverse mode field profiles (Eq. (1)), for pump and signal wavelengths. Then, these factors were reported (in percent) in Tab. 1 and Tab. 2, for both wavelengths respectively.*

_{ij}Where *E _{i}* and Ψ

*are the mode field profiles of the FMF and FM-EDF respectively. Mode profiles at pump and signal wavelengths of the FM-EDF have been computed, using a Finite Element Method (FEM), based on the refractive index profile of the FM-EDF presented on Fig. 2. Note that the mode profiles don’t have exactly the same shape after the splice: it is due to the difference of refractive index structure between the two fibers that impose a new shape for the modes. Checking of the modal purity in the FMF and in the FM-EDF has been made possible by controlling either that intensity profiles return to zero between the different lobes or tuning the signal wavelength (or slightly move the fiber) in order to eventually observe mode beating on the camera. This kind of tests guarantee a relative good control of mode purity. The case of the LP*

_{j}_{01}signal mode is slightly different, compared to LP

_{11}and LP

_{21}, because when LP

_{01}signal mode of the FMF undergoes the splice, it excites both LP

_{01}and LP

_{02}modes of the FM-EDF (Fig. 4), due to mode mismatch (Tab. 1). This is confirmed by the observation of beating between this two modes on the camera as the wavelength is changed. So it isn’t possible to clearly measure the individual modal gain of these two modes with our set-up.

The pump beam is injected in centered conditions, so that it only excites LP_{01}, LP_{02} and LP_{03} modes. Modal content at pump wavelength was evaluated in the FMF, using the experimental waist of the injected pump beam. A waist of 3.25 *μ*m has been measured by using a flat cleaved SMF fiber as a probe to scan the focalized pump beam. Then, the overlap integrals between the injected Gaussian beam and the transverse modes of the FMF have been calculated (in centered injection conditions). These calculations give us an idea of modal excitation in the first piece of FMF: 71% of pump power in LP_{01} mode, 28% in LP_{02} mode and 1% in LP_{03} mode. Knowing the coupling efficiency factors between FMF modes and FM-EDF modes at pump wavelength (Tab. 2), it is quite simple to have an idea of the theoretical modal composition of pump beam in the FM-EDF, i.e. the pump profile that excites the erbium ions. When it experiences the splice, each excited mode of the FMF excites the different modes of the FM-EDF with different proportions. So, by taking the modal composition in the FMF as a vector **v** and by using Tab. 2 as a mode transition matrix *M*, a new vector is obtained **v’** = *M* * **v** that represents the modal composition of the pump in the FM-EDF. After normalization of this new vector **v’** (so all its components sum to one) the outcome of all these reciprocal modes can be calculated. It is finally found that the pump power is distributed as follows in the FM-EDF: 57% on LP_{01} mode, 23 % on LP_{02} mode and 20 % on LP_{03} mode. In order to validate these theoretical results, the pump profile has been captured on the camera, both in FMF (before splice) and in FM-EDF (after splice). The results are shown on Fig. 5. A good accordance between predictions and experimental observations can be noted, and modal composition of the pump is approximately known. This modal composition has been used for the simulations reported throughout the following of this paper.

## 5. Experimental characterization

The FM-EDF has been characterized by measuring the gain of each signal mode (LP_{11}, LP_{21} but also LP_{01} and LP_{02}), individually injected at 1550 nm, as a function of the total pump power coupled in the FM-EDF (Fig. 6). Gain values were obtained by comparing output (i.e. full length) and input (after a few centimeters) spectra in the EDF. Injection and splice were checked at the beginning and at the end of each experiment in order to validate the results. Then, these results have been compared to simulations performed using the experimental parameters: experimental refractive index/erbium profiles of the FM-EDF, signal/pump mode profiles obtained by FEM, modal composition of the pump beam deduced from Tab. 2. Pump and signal powers were measured after a few centimeters of FM-EDF with a powermeter, at the end of each experiment. The simulations reported in the following of this article were performed using these measured pump/signal powers. Note that no degree of freedom was left in order to fit the experimental data.

It is seen on Fig. 6(a) and 6(b) that the gains are nearly equal for LP_{11} and LP_{21} modes, with a maximum DMG of 1 dB for all gain values greater than 10 dB, and a maximum gain close to 20 dB. The threshold of the amplifier is about 40 mW. A good accordance between experiments and simulations has to be noticed, with a maximum difference between experimental and theoretical gains of 1.5 dB for all gain values greater than 0 dB. Situation is slightly different for LP_{01} mode because, experimentally, both LP_{01} and LP_{02} modes were excited in the FM-EDF. It can be seen on Fig. 6 (c) that an average gain of 15 dB is experimentally observed. Simulations for both LP_{01} and LP_{02} modes are reported in Fig. 6 (c) and show that these two modes can be amplified with gains larger than 10 dB.

Signal-to-Noise Ratio (SNR) deterioration has also been measured, comparing both SNR at the input of the FM-EDF and SNR at the output of the FM-EDF. It is reported on Fig. 7. SNR deterioration is not an exact Noise-Figure (NF) measurement, the difference being that NF measurement needs a mode demultiplexer in order to separate the different modes at the output of the amplifier and measure the SNR of each ”modal-channel”. As there is no mode demultiplexer in our set-up, the output spectrum measured is a sum of optical power over all the modes. So, the SNR deterioration measured in our set-up is overestimated compared to the actual NF. The SNR deterioration values are rather high (up to 10 dB) because each mode has its own ASE, and so the total noise power increases as a function of the number of modes. This phenomenon is already known from previous works [16, 17]. SNR deterioration values should decrease by adding a mode demultiplexer in the set-up, and NF values should be of the same order of magnitude than in a single mode EDFA, as in ref. [18].

Finally, simultaneous amplification of LP_{11} and LP_{21} modes has been tested by adding a second signal channel on set-up of Fig. 2. So as to facilitate the measurement, the two modes have been used at two different wavelengths, namely 1554 nm for LP_{11} and 1550 nm for LP_{21}. The results are presented on Fig. 8.

Once again, besides the fairly good agreement between theory and experiment (for both modal gain values and ASE shape, respectively in Fig. 8(a) and 8(b)), it appears that more than 20 dB gain can be obtained for both modes and that a DMG of less than 3 dB is observed for all averaged gains between 13 dB and 27 dB. Moreover, a DMG of 0.4 dB is experimentally obtained for about 100 mW pump power. The experimental and theoretical input/output spectra obtained for this pump power are reported on Fig. 8(b). Since the two signal wavelengths are close from each other (4 nm) and located in a spectral region where the gain is flat, spectral dependence of modal gain can be neglected, so that the two measured modal gains can be properly compared. This hypothesis has been validated by numerical results: the same simulation as the one reported on Fig. 8(a) has been performed considering that the two signal were injected at the same wavelength (1552 nm). A maximum difference of 0.5 dB gain has been found between the two simulations, for all the gain values between 10 and 30 dB. This proves that spectral dependence can be neglected in this experiment.

Even if this primary design of FM-EDFA was to obtain similar gains for the LP_{11} and LP_{21} modes, it can be used for MDM transmissions using LP_{21}, LP_{11}, LP_{01} and LP_{02} modes, since all these modes are amplified. LP_{01} and LP_{02} gains were not measured in our set-up due to mode profile mismatching. However, note that this FM-EDF has been employed in a 6 modes amplifier set-up where the ring-doped FM-EDF was concatenated with an other FM-EDF (which has a flat erbium doping profile), so that the first part of the amplifier amplifies LP_{11} and LP_{21} modes and the second part re-amplifies LP_{01} and LP_{02} modes that suffer from lower gains [18].

## 6. Conclusion

In this work, an algorithm able to simulate the modal gain in a FM-EDFA has been used to design a four mode groups FM-EDFA and a FM-EDF has been synthesized and characterized in amplification regime. Experimental results are in good accordance with simulations and it is demonstrated that the proposed FM-EDFA is able to amplify both LP_{11} and LP_{21} mode groups with more than 20 dB gain and less than 1 dB DMG between these modes. In these conditions, LP_{01} and LP_{02} modes also benefit of about 12 dB gain. These results could make it possible to use the algorithm to develop an improved design of FM-EDF, able to equalize modal gains on more than two mode groups (for example LP_{11}, LP_{21} LP_{01} and LP_{02}).

## Acknowledgment

This work has been supported by the French government, in the frame of STRADE research project (ANR-09-VERS-010). We also acknowledge financial support from the Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and the FEDER through the “Contrat de Projets Etat Region (CPER) 2007-2013”.

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