## Abstract

We experimentally validate a numerical model to study multimode erbium-doped fiber amplifiers (MM-EDFAs). Using this model, we demonstrate the improved performance achievable in a step index MM-EDFA incorporating a localized erbium doped ring and its potential for Space Division Multiplexed (SDM) transmission. Using a pure LP_{01} pump beam, which greatly simplifies amplifier construction, accurate modal gain control can be achieved by carefully tuning the thickness of the ring-doped layer in the active fiber and the pump power. In particular, by optimizing the erbium-ring-doped structure and the length of active fiber used, over 20dB gain for both LP_{01} and LP_{11} signals with a maximum gain difference of around 2 dB across the C band are predicted for a pure LP_{01} pump beam delivering 250mW power at 980nm.

©2012 Optical Society of America

## 1. Introduction

With the information carrying capacity of conventional single-mode optical fiber close to fundamental limits in the laboratory there is increasing interest in the use of SDM as means to increase single fiber transmission capacity. Mode-Division-Multiplexing (MDM) transmission over “few-mode fibers” (FMFs) represents one potential SDM approach and encouraging early results have been achieved [1–4]. If such an approach is ever to be applied to long haul networks then the development of practical, high performance in-line FMF optical amplifiers will be essential and initial results in this direction have also recently been reported [5–9]. FM-EDFAs represent a particularly attractive approach, promising high-efficiency, high-gain devices with low differential modal gain (DMG) between all supported modes [6–9]. To date DMG in FM-EDFAs has been managed through control of the modal pump distribution [6] (making for a complex and sensitive pump arrangement) with significant further improvements provided by simultaneous control of the fiber refractive-index profile (FRIP) and of the erbium ion distribution profile [8,9]. Whilst encouraging results have been demonstrated (e.g. gains of >20 dB across the full C-band, with DMG values of <3dB for all 6 modes in a two-mode fiber (TMF) supporting LP_{01} and LP_{11} modes [9]) significant scope exists for new fiber concepts and designs providing simplified means of DMG control and improved overall gain performance.

In this paper we report the development and experimental validation of a modeling tool that allows the accurate prediction of FM-EDFA performance subject to a user-specified complex pump field. Since DMG depends strongly on the overlap between signal modes, pump modes and the rare earth dopant distribution we propose a fiber design concept based on ring-doping where the erbium ions are confined in a ring inside the fiber core. Previously ring-doping designs have been used in fiber laser applications, for example to increase slope efficiency at short emission waveband [10] of a 3-level system, or to selectively amplify certain modes in high-order-mode fiber [11]. Here we numerically demonstrate a MM-EDFA incorporating ring doping that allows for accurate DMG control whilst incorporating a greatly simplified pumping arrangement that uses just the fundamental LP_{01} pump mode.

## 2. Theoretical model

The schematic diagram of the simulation model is shown in Fig. 1
. In this model, the following assumptions are made: (1) the EDFA is considered as a two-level system; (2) The few-moded active fiber is uniform along the longitudinal direction and inter-modal coupling effects are negligible; (3) All fibers are weakly guiding and the modes are well approximated by linearly polarized (LP) modes [12]. For the remainder of this paper, the notation LP_{ijs} and LP_{ijp} will be used to denote the LP_{ij} modes at signal wavelength (i.e. within the C-band) and at the pump wavelength (i.e. 980nm), respectively.

In practice, it is well known that the slightly different propagation constants between the vector modes which form the corresponding approximated LP mode result in the apparent rotation of the cross sectional intensity pattern of the LP mode (i.e. LP_{11}) as the mode propagates along the fiber [13]. For the EDFs considered in this work, all weakly guiding and with maximum index difference between core and cladding below 0.004, the full beat length of the LP_{11a} mode (formed by a combination of TE_{01} and HE_{21} vector modes) is below 1m at all wavelengths within the C-band, which implies that the LP_{11a} apparently evolves into LP_{11b} every few tens of centimeters. However, the impact on the overall modal gain caused by the rotation of the propagating LP_{11s} modes is in general averaged out by the use of EDF lengths (i.e. 3-5m) much longer than the beating period. Moreover, in the numerical studies presented in this work, the pump beams are mostly in LP_{01p} and LP_{21p} modes, which minimize the effects on overall gain performance caused by the rotation of the propagating LP_{11s}, as has been proven numerically in a simplified model presented in [14]. Hence, although a more rigorous amplifier model based on vector modes would in principle provide a more accurate representation; our scalar-mode based model already provides fairly accurate results in terms of the gross amplifier performance as confirmed by the experimental validation reported in Section 3.1. We also numerically verified that the impact of bending of our fiber can be neglected for both pump and signal modes providing that the bend radius is kept larger than ~7.5cm. This bend radius could be further reduced if required by adding additional features in the fiber cladding (i.e. refractive index trenches).

Our FMF amplifier simulator is based on the rate and propagation equations dealing with multi-transverse-spatial modes [14,15]:

With the specified boundary conditions at z = 0 and z = L (e.g. input signal and pump power, backward and forward ASE power), Eq. (3-4) can be solved numerically by using the standard fourth-order Runge-Kutta method. Throughout this work, the fluorescence lifetime as well as the emission/absorption cross section are derived from [17] which concerned a fiber with a similar core composition (Er^{3+} doped aluminosilicate host) as the EDF used in [8,9] (and whose structure is shown in Fig. 2(a)
(Fiber 1)). The fluorescence spectrum of the Er-doped aluminosilicate glass has a peak at λ_{s} = 1531nm (7nm wide) and a flat region (20nm wide) centered at 1545nm [18]. For this reason, 1530nm and 1550nm are used as the representative signal wavelengths for the two gain regions of interest. The key emission/absorption parameters used in the model are: σ_{a,980nm} = 1.879 × 10^{−25}m^{2}, σ_{e,980nm} = 0m^{2}, σ_{a,1530nm} = 5.5 × 10^{−25}m^{2}, σ_{e,1530nm} = 5.674 × 10^{−25}m^{2}, σ_{a,1550nm} = 2.46 × 10^{−25}m^{2}, σ_{e,1550nm} = 3.667 × 10^{−25}m^{2}. The Er^{3+} doping concentration is set to be 15 × 10^{24}m^{−3} as estimated from the measured absorption (10.1dB/m) at the pump wavelength of 980nm. The background loss is neglected due to the short length of fibers required (3-5m).

To illustrate use of the model we first show general results from modeling Fiber 1 which has a core diameter of 18 µm with an effective NA of 0.101 and guides 3 distinct spatial modes, LP_{01}, LP_{11a} and LP_{11b} within the C-band. In the simulations, we use LP_{11a} to represent the LP_{11s} mode group of the signal, and LP_{21a} to represent the LP_{21p} mode group of the pump. In Fig. 2(b), we show the simulated DMG (G(LP_{01s})-G(LP_{11s})) as a function of input signal power per mode for both Fiber 1 and a conventional Step Index (SI) TM-EDF (of equivalent effective NA and dopant concentration) when pumped by pure LP_{01p} and LP_{21p} modes with a fixed power of 250mW. The EDFAs were forward pumped and a fiber length of 3.5m was used. As shown in Fig. 2(b), the gain differences between the two signal modes of Fiber 1 are significantly less than those of the step index TM-EDF, which confirms that the EDF with a “batman” profile improves the amplifier performance in terms of reducing DMG. However, the simulations also indicate that a complex pump field distribution (i.e. a combination of LP_{01p} and LP_{21p} pump modes) is required for both fiber designs to minimize the DMG.

## 3. Simulation results and discussion

#### 3.1 Fitting experimental data

To validate the model predictions above we undertook detailed two mode group gain measurements using the experimental setup previously reported [9] and tried to fit these with our model. The fiber length was set to be 3.5m and a counter-propagating pump was used. The wavelengths of both the signal modes were set to 1550nm while the average input signal power per mode was changed from −10dBm to 0dBm. Results for both central and offset pump launch arrangement, which were described in [9], were considered in the modeling.

As shown in Fig. 3(a), (b)
, the DMG exhibits a significant dependence on the pump mode configuration. For the central launch condition outlined in Fig. 3(a), it was found that the best agreement between theory and experiment was achieved by assuming that 65% of the 280mW total pump was split into LP_{01p} mode with the remaining 35% in the LP_{21p} mode. For offset pump launch conditions the best fit was achieved by assuming 40% of the pump in LP_{01p} and 60% in the LP_{21p}, as shown in Fig. 3(b). Although the experimental data points agree very well with the simulations at high input signal powers, a significant (3dB) discrepancy appears to occur at low powers until account is taken of the 4% Fresnel reflection from the flat input end face that is currently required in order to avoid significant mode-mixing at the signal launch. With this included, the apparent discrepancy observed at low powers is reduced from 3dB to 1.5dB and the general performance of the amplifier is well described over the full operating range. We thus conclude the scalar mode approximation and the various other assumptions we have listed hold well in modeling our FM-EDFA and can be reliably used to investigate similar amplifier concepts and designs.

#### 3.2 Modal gain control by further tailoring FRIP incorporating complex pumping configuration

Previously, we have validated that for the TM-EDF, the central dip in its FRIP gives a lower differential gain between the LP_{01} and LP_{11} modes than the conventional step index design [8,9]. In this section, we investigate whether even better equalized modal gain (or maybe even higher gain for the LP_{11} mode) might be achieved by optimizing the depth of the central dip of Fiber 1. Figure 4(a)
above shows various idealized modified index profiles similar to Fiber 1 but with a Normalized Depth (ND) varying from 0.3 to 1.0 while the width (as shown in Fig. 4(a)) is fixed at 40% of the core diameter. The ND is defined as the ratio between the index depth (as shown in Fig. 4(a)) and the maximum difference between the core and cladding indices. The corresponding intensity distributions (normalized to the same power) of the LP_{01s} and LP_{11s} signal modes are given in Fig. 4(b) and 4(c) respectively. Here, we still assume that the erbium doping distribution closely follows the FRIP.

It can be seen from Fig. 4(c) that the shape of the mode field profile of LP_{11s} remains essentially unchanged for the entire range of NDs. However, Fig. 4(b) clearly shows that the LP_{01s} mode profile deforms significantly from a Gaussian shape when the normalized depth increases beyond 0.5. This level of modal distortion will have serious consequences in terms of coupling losses at the interface of TM-EDF and any passive transmission fiber to which it is spliced. To illustrate the impact of this we consider splicing the TM-EDF with various values of ND to Fiber 1 (chosen as a representative transmission fiber FRIP). The coupling efficiencies ${\eta}_{LP01}$ and ${\eta}_{LP11}$ for the LP_{01s} and LP_{11s} modes can be calculated through the use of Eq. (5) from ref [19],

_{11s}mode remains relatively insensitive to changes in ND, this is not so for the LP

_{01s}mode, as the coupling efficiency drops down to 75% (i.e. ~1.25dB loss) for the extreme central dip depth under the ideal central launch condition. We have also calculated the dependence of the differential modal gain (ΔG = G(LP

_{01s})-(GLP

_{11s})) on the ND as well as on the pump field distribution and the result is illustrated in the contour map shown in Fig. 5 . Once again we have used backward pumping with a total pump power of 250mW. The signal power of each mode is assumed to be −10dBm at 1550nm whilst the EDF length is 3.5m.

It can be seen from Fig. 5 that |ΔG|< 1.5dB can be obtained for a wide range of normalized depths. Smaller depths require a higher ratio of LP_{21p} than LP_{01p}, which is difficult to control in practice due to the complexity of reliably exciting higher order pump modes. Although larger normalized depths require increasingly smaller levels of LP_{21p}, the resulting intensity distribution of the LP_{01} signal mode is however significantly distorted relative to the well-known Gaussian distribution and is far from ideal in terms of splicing loss. Considering both the modal gain requirement as well as the tolerable coupling loss for the LP_{01s} mode, the most desirable region of operation lies in the upper left hand corner of the contour map as indicated by the dotted ring in Fig. 5, where the power ratio between LP_{21p} and LP_{01p} is around 8:2 and the ND is below 0.5 corresponding to an LP_{01s} coupling loss of less than 5%. Moreover, the modal gains for both the LP_{01s} and LP_{11s} modes in the region enclosed by the dotted ring are well above 20dB.

#### 3.3 Modal gain control in TM-EDFA using ring doping

The physical origin of the DMG results from differences in the overlap of the pump modes, signal modes and the distribution of the rare earth dopant. As illustrated in section 3.2, we can substantially change the DMG by tailoring the Refractive Index (RI) profile and additionally fine tuning the pump mode content. Here we relax the condition that the dopant profile follows the FRIP (a specific feature of the fabrication process adopted for Fiber 1, but not fundamental). For example, it is possible to envisage incorporating the dopant in an annular ring surrounding an undoped central core region of a step index fiber and which, as is clear from examination of Fig. 6
(Fiber 2), provides great scope for engineering the relative overlap of the LP_{01} and LP_{11} signal modes with the Er-ions.

To illustrate the benefits to be derived from this design we plot the variation of DMG between LP_{01s} and LP_{11s} signal beams in a length of Fiber 2 forward pumped by a pure LP_{01p} 980nm pump beam for various values of normalized ring thickness (defined by the ratio t/a as shown in Fig. 6). The signal wavelength was chosen to be 1530nm while the pump power was fixed at 250mW. It should be noted that the pump absorption per unit length increases with increasing t/a. Thus for a fair comparison, the Fiber Length (FL) is adjusted to ensure the total output signal power (the sum of LP_{01s} and LP_{11s}) reaches a maximum for each t/a value.

The DMG values for a series of t/a values and the corresponding FL used are shown in Fig. 7(a)
. As can be seen, for values of t/a around 0.52, very low levels DMG can be obtained for a wide range of input powers. Moreover, through fine tuning of t/a, both negative and positive values of DMG can be achieved. For comparison, we have plotted the DMG for the conventional uniformly doped SI-EDF (with t/a = 1), which remains large and positive at all input powers. Furthermore, the trend of modal gain dependence on LP_{01p} pump power with input signal power of −10dBm per mode based on the ratio of t/a = 0.52 shown in Fig. 7(b), indicates that the minor adjustment of DMG is possible through the changing of the pump power. From the above discussion, it can be concluded that large gain differences can be addressed by tailoring the value of t/a, while fine tuning of the DMG can be achieved by adjusting the LP_{01p} pump power. In addition, further calculations reveal that the Er ring-doping fiber is relatively tolerant to variations of the order ± 10% in the refractive index of the central undoped region compared with that of the Er-doped region.

In practice, we need to consider modal gain for the entire C-band. To investigate this aspect we selected signal wavelengths at 1530nm and 1550nm and chose a ratio for t/a = 0.52 so as to have higher gain for the higher order modes when the LP_{01p} pump power was set at 250mW. An input power of −10dBm was used for both LP_{01s} and LP_{11s}. The corresponding signal power evolution along the fiber is shown in Fig. 8(a)
. At a fiber length of ~3.5m, the gains for signals at 1530nm are already saturated whilst those at 1550nm are still increasing, which indicates that the position within the fiber where the minimum absolute DMG value appears for signals at 1530nm and 1550nm are different. However, it is possible to minimize the gain difference between LP_{01s} and LP_{11s} to within 2dB across the full C-band by choosing an EDF length of 4.3 m, as depicted by black lines in Fig. 8(b). Similarly, by using a ratio of t/a = 0.6 and the EDF length to 3.5m, we are able to obtain higher gain for the LP_{01s} with minimal gain difference (~2dB) across the C-band (red lines in Fig. 8(b)). This combination of high gain, low DMG and spectral flatness is achieved by fiber design (i.e. ring-doping with an appropriate ratio of t/a) and a proper choice of LP_{01p} pump power as well as fiber length. Thus by carefully engineering the fiber design we are able to achieve low values of DMG for a greatly simplified and far more practical pump configuration than previous FM-EDFA designs.

## 4. Conclusion

We report an experimentally validated tool for the modeling of FM-EDFAs. Using this tool we propose a novel ring-doped fiber design capable of providing accurate control of differential modal gain in a TM-EDFA for a significantly simplified (and much more practical) LP_{01}-only based pumping configuration. We believe this ring-doped fiber design offers significant practical benefits relative to other TM-EDF designs proposed to date. In future SDM transmission systems, more than two transverse mode groups may be used as independent signal channels, thus multi-ring dopant distributions may be considered for gain equalization in more heavily-MM EDFAs.

## Acknowledgment

This work was supported by the European Communities 7th Framework Program under grant agreement 228033 (MODE-GAP).

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