We propose a method for controlling modal gain in a multimode Erbium-doped fiber amplifier (MM-EDFA) by tuning the mode content of a multimode pump. By adjusting the powers and orientation of input pump modes, modal dependent gain can be tuned over a large dynamic range. Performance impacts due to excitation of undesired pump modes, mode coupling and macro-bending loss within the erbium-doped fiber are also investigated. The MM-EDFA may potentially be a key element for long haul mode-division multiplexed transmission.
©2011 Optical Society of America
Over the past years, advances in optical coherent detection and signal processing have led to tremendous growth in the spectral efficiency achieved in optical fiber. Recently, 100-Tb/s transmission at a spectral efficiency of 11 b/s/Hz was reported in a single-mode fiber . Owing to the nonlinear refractive index of silica, it is impossible to continue increasing spectral efficiency indefinitely by merely increasing the launched power. One method to reduce fiber nonlinearity is to increase the effective area of the propagating mode, thus reducing the optical intensity and the resulting nonlinear effects . However, mode effective area is limited by bending loss and by the requirement of the waveguide to be single-mode. It is possible to transmit data in the fundamental mode of a “few-mode fiber” (FMF). Provided mode coupling is low, the signal will remain single-mode during propagation. The larger effective area of the fundamental mode in FMF in comparison with that achievable in SMF can further reduce nonlinearity . However, a nonlinear capacity limit will always exist. Even if the transmission medium was linear, Shannon’s capacity b/s per channel shows that the capacity scales only logarithmically with signal-to-noise ratio. Ultra-high spectral efficiency is therefore very power inefficient. To achieve cost-effective scaling in system capacity, new paradigms in optical transmission are required.
One promising solution is space-division multiplexing, where data is transmitted over parallel channels. Indeed, transmission over parallel orthogonal channels has been well established in wireless systems, where the achievable capacity using multiple-input multiple-output (MIMO) antennas increases with the number of independent “eigenchannels,” which under the assumption of rich multipath, scales as the minimum of the number of antennae deployed at the transmitter and receiver .
In optical fiber transmission, two space-division multiplexing (SDM) schemes have been proposed. These are (i) multicore fibers (MCF), where a single strand of glass fiber contains a number of independent single- (or multi-) mode cores each capable of communicating optical signals [5,6]; and (ii) multimode fibers (MMF), where a single strand of fiber has one core with sufficiently large cross-section area to support a number of independent guiding modes [7–9]. SDM transmission experiments have been reported for both types of fibers. Owing to the lack of available inline amplifiers, all MCF and MMF experiments to date have been single-span, with transmission distances up to 76.8 km  for MCF and 40km  for MMF.
To enable mode-division multiplexing (MDM) in MMF over long-haul distances, inline erbium-doped fiber amplifiers (EDFA) based on MMF are required . The theory of multimode EDFAs (MM-EDFA) has been studied in . Applications for MM-EDFAs have included high-powered lasers and free-space communications, where the multimode optical waveguide is essentially used in a “single-mode” manner, thus mode-dependent gain (MDG) is not critical [11,12]. In MDM transmission however, careful control over MDG is necessary to overcome mode-dependent loss (MDL) in the transmission fiber, and to ensure all signal modes are launched with optimal power maximizing the total system capacity.
Mode dependent gain (MDG) is mainly determined by three factors: (a) the concentration profile of the active dopant ions, (b) the transverse intensity profile of the pump, and (c) the transverse intensity profile of the signal. In general, a signal mode whose profile is better matched to the pump intensity profile will experience higher gain. Hence, by controlling the mode content of the pump, it is possible to control MDG. The organization of this paper is as follows. In Section 2, we review the theory of MM-EDFAs. In Section 3, we provide simulation results for a step-index “two-mode fiber,” demonstrating the feasibility of MDG control by tuning the mode content of the pump. We also explore the dependence of MDG on the excitation of unwanted modes and mode coupling within the EDF.
A schematic of a MM-EDFA is shown in Fig. 1 . To generate the desired pump intensity profile, we split the pump source into N paths, and use mode converters to transform the spatial mode of the pump source into the N spatial modes of the MMF. The variable attenuators enable N-degree control over the mode content of the pump, and thus the MDG of the device. The pump modes are spatially combined with the signal, which are injected into the erbium-doped MMF. In the paper, we assume the erbium-doped MMF has the profile shown in Fig. 2 , where the core has radius , and a region of the core for which is doped with Erbium atoms at a concentration of .
The operation of a multimode fiber amplifier is described by coupled differential equations involving: (i) evolution of the intensities of the various signal and pump modes along the amplifying medium, and (ii) population inversion along the amplifying medium . In contrast to a single-mode EDFA, the transverse intensity distributions have to be taken into account in a multimode EDFA. We assume both signal and pump are co-propagating. Let and be the normalized intensity patterns of the i-th signal mode and j-th pump mode of the EDF, respectively; and let and be their respective power. We further assume the erbium-doped fiber (EDF) can be modeled as a quasi-three-level system at 980 nm pumping, and let and with be the population densities of Erbium atoms in the lower and upper levels at position . Loss is assumed negligible in the EDF. It can be shown that the intensity evolution equations for signal and amplified spontaneous emission (ASE) in the i-th signal mode at the wavelength are given by:13]. In the signal propagation Eq. (1), the first term on the right hand side denote net amplification due to stimulated emission, and the second term denote power coupling between the signal modes. In the ASE propagation Eq. (2), the second term on the right hand side represents spontaneous emission of the excited Erbium ions; the coefficient of ‘2’ preceding this term corresponds to two degenerate polarizations modes. The intensity evolution equation for the power in the j-th pump mode at wavelength is:Table 3 . Equations (1)–(5) can be solved by using the standard fourth-order Runge-Kutta method given initial conditions for pump and signal power . Gains and noise figures for all signal modes may similarly be calculated.
We consider a MM-EDFA where the EDF has step-index refractive profile. The parameters of the MM-EDFA are shown in Table 1 . We assume the doped region is the same size as the core (i.e., a = rc). As the normalized frequency at λs = 1.53 μm lies between 2.405 <Vs< 3.832, the EDF supports two degenerate mode groups at this wavelength. We assume a weakly guiding MMF where the modes are well approximated by linearly polarized (LP) modes . For the remainder of this paper, we use the notation LPij , s and LPxy , p to denote the LPij mode at λs and LPxy mode at λp, respectively. Figure 3 shows the intensity profiles of the modes, and their intensities viewed along the x-axis. We note that for , the LPmn,s and LPmn,p have two spatially degenerate modes. In one of these modes, referred to as the “even” mode, intensity is maximized along the x-axis at (φ = 0); in the other mode, referred to as the “odd mode”, intensity is minimized along the x-axis. All spatial modes, degenerate and non-degenerate, come with two degenerate polarization modes.
3.1 Modal Gain Control for Non-Degenerate Signal Modes
We first consider MDG using a single-mode pump. For the spatially degenerate LP11, p and LP21, p modes, we assume equal power in the even (LP11 e , p or LP21 e , p) and odd modes (LP11 o , p or LP21 o , p) so that the resulting intensity (power) patterns (e.g., ) have no azimuthal dependence (Fig. 3(a)), and hence no MDG between spatially degenerate signal modes. Figure 4 shows the gain experienced by each signal mode group when pumping in the LP01,p, LP11,p and LP21,p modes. It is assumed that the input signal to the EDF has equal power (0.05 mW) in each of its six (two LP01,s and four LP11,s) spatial and polarization degenerate modes, or 0.3 mW in total. Since the intensity profile of LP01,p is better matched to LP01,s than LP11,s, Fig. 4(a) shows higher gain for LP01,s. Conversely, pumping in LP21,p results in higher gain for LP11,s.
The quality of the match between signal and pump intensity profiles can be evaluated by the overlap integral:Table 2 shows for different pump and signal mode pairs. It is observed that and are both larger than and . Hence, higher gain is observed for LP01, s when these pump modes are used. Conversely, as is larger than , pumping in LP21, p gives higher gain for LP11, s. It is possible to control MDG by varying the relative powers of LP01, p and LP21, p. In transmission, the higher-order LP11, s mode is less confined by the core, and will experience higher bending loss than the fundamental LP01, s mode. Furthermore, the LP11, s mode has larger effective area, making the optimum power for this mode higher than the fundamental mode. Consequently, a practical MM-EDFA will pump primarily in the LP21, p. The addition of a small amount of LP01, p enables adjustment of MDG. Figure 5(a) shows modal gain versus LP21, p pump power, where the power of LP01, p was continually adjusted to maintain a 1 dB difference between the gains of LP01, s and LP11, s, which we denoted as ΔG11 s −01 s. Figure 5(b) shows the same results for ΔG11 s −01 s = 2 dB. It is observed that modal gain can be continually adjusted to values greater than 22 dB, which is sufficient to compensate the loss of a single fiber span at typical span distances. Figure 5(c) shows the sensitivity of modal gain to LP01, p power when LP21, p power is fixed at 150 mW. While the modal gain at LP11, s remain nearly constant, ΔG11 s −01 s varies by more than 4 dB as LP01, p power is changed from only 0 to 20 mW, demonstrating wide tunability of MDG in dynamic range. Thus, to establish the desired modal gain in an MM-EDFA, we first tune the power of the LP21, p pump to give the desired power for the LP11, s mode, after which, the power of LP01, p is adjusted to obtain the desired ΔG11 s −01 s.
MDG depends not only on pump power, but also the length of the EDF. Figure 6 shows modal gain vs. EDF length for two pump configurations. At short fiber lengths, modal gain increases with increasing EDF length. Eventually, pump depletion and high signal power depletes population inversion, so modal gain decreases for further increases in EDF length. For any given EDF length, ΔG11 s −01 s is maximized when only the LP21, p mode is pumped. Figure 6(a) shows that a longer EDF gives larger ΔG11 s −01 s, and thus larger range of achievable MDG. We can select an EDF length which approximately maximizes modal gain, while ensuring relatively large difference between the gains of LP11, s and LP01, s. In Fig. 6(b), it is observed that adding only 8 mW of pump power in LP01, p enables flattening of the modal gains responses. Even as device length is swept from 20 to 40 meters, less than ± 1 dB change in the modal gains of LP11, s and LP01, s, and less than ± 0.5 dB variation in MDG are observed.
3.2 Modal Gain Control for Spatially Degenerate Signal Modes
In 3.1 section, it was assumed that odd and even modes of the spatially degenerate LP11, p and LP21, p modes are pumped with equal power, resulting in no MDG between spatially degenerate signal modes such as LP11, s. In practical MDM systems, however, spatially degenerate signal modes may have different losses. One mechanism for MDL is fiber bending, where the loss experienced by each spatially degenerate mode depends on its orientation relative to the plane containing the bend . For our MM-EDFA to overcome MDL, the gains for LP11e, s and LP11 o , s need to be adjustable. It was previously observed in Fig. 4 that modal gain is related to the intensity overlap between the pump mode and the signal mode in question. The assumption of equal pump power in even & odd modes for LP11, p or LP21, p caused the overlap integral to be independent of azimuthal angle. We now consider a pump with azimuthal intensity dependence. Consider a rotated LP11 e , p pump as shown in Fig. 7 , which we denote as LP11 θ , p. It can be shown that the overlap intensity integral between a rotated pump mode LPmkθ , p (with m azimuthal nulls) and an even signal mode LPnje , s with n azimuthal nulls is proportional to:Figure 8(a) shows the θ dependence of modal gain assuming we pump only in the LP11 θ ,p mode. As the rotation angle θ is scanned from 0 to 2π, the modal gains for LP11 e , s and LP11 o , s oscillates with a period of π, reflecting the rotational symmetry of the LP11 θ , p mode. When the intensity peaks of the LP11 θ , p pump align with LP11 e , s, its gain is maximized while the gain of LP11 o , s is minimized, and vice-versa. We observed that MDG between these modes can be as much as 20 dB. Meanwhile, the fluctuation in gain for the LP01, s mode is a second order effect caused by gain competition between the signal modes; when the pump is at 45° relative to both the LP11 e , s and LP11 o , s modes, maximum gain occurs for LP01, s. In Fig. 8(b), we assume pumping in only LP21 θ ,p. In this pumping regime, the rotation angle θ has no impact on MDG, as expected from the overlap integral in Eq. (7), which is constant when . Finally, Fig. 8(c) shows MDG for a pump regime similar to one in a real implementation, where the pump is mostly in LP21,p (as defined in Fig. 3(a)), with small amounts of LP01,p and LP11e,p added to adjust the MDG between the three modes. The results show that the gain difference between LP11e,s and LP11o,s can be tuned by proper rotation of the LP11θ,p, while the variation in gain for LP01,s is as small as ± 0.05dB.
3.3 Impact on Performance Due to Inexact Excitation and Mode Coupling
Previously, it was assumed that (a) the pump can be excited in particular modes of the EDF without leakage into undesired mode, and (b) no mode-coupling between different pump modes during propagation inside the EDF. In practice, either of these assumptions may be difficult to satisfy. Firstly, to generate the desired pump mode (Fig. 1), phase plates are likely to be used – as was demonstrated in an MDM transmission experiment in . A phase plate alone (without modulation of transverse amplitude) will result in excitation of other unwanted pump modes. Additionally, spatial misalignment between pump and signal at the beam combiner (shown as the dichroic mirror in Fig. 1) will cause further energy leakage into unwanted modes and excess loss, leading to performance degradation. We consider only leakage from LP21, p to the other modes, as LP21, p has the most power. Figure 9 shows the modal gains in LP01, s and LP11, s versus the percentage of LP21, p power leaked into the unwanted modes. Note the power in LP01, p is chosen to give an MDG of ΔG11 s −01 s = 2dB when 0% of LP21, p is coupled to unwanted modes. As previously shown in Fig. 4, pumping in the other modes gives higher gain for LP01, s than LP11, s. Hence we observe a reduction in ΔG11 s −01 s that as leakage increases. At sufficiently high leakage, ΔG11 s −01 s become negative, indicating higher modal gain for LP01, s. It is also observed that ΔG11 s −01 s has greater sensitivity to leakage into LP01, p and LP02, p, as these modes have larger overlap integrals with LP01, s than LP11, p (Table 2). Hence, the pump generation mechanism should minimize leakage into these modes.
Mode coupling during propagation inside the EDF will have similar impact as excitation of unwanted modes, since both result in power transferred into undesired modes. The strength of mode coupling is inversely proportional to the difference in effective refractive index between the modes (Δneff). Hence, we focus on coupling between LP21, p and LP02, p, as the effective refractive index difference Δneff ,( 21p−02p ) between this pair of modes is the smallest among all the mode pairs in our FMF-based EDF. Assuming the power coupling coefficient between LP21, p and LP02, p, which was defined as in Eq. (3), is constant throughout the EDF, Fig. 10 shows MDG as a function of . The value of coupling coefficient refers to . As expected, ΔG11s–01s decreases with increasing strength of mode coupling. This favors using a shorter length EDF, and a refractive index profile that maximizes effective refractive index difference between the modes to reduce mode coupling.
3.4 Impact on Performance Due to Macro-Bending Loss
In the results thus far, the loss of EDF is assumed to be negligible. In practice, however, the EDF has to be spooled to create a module, which may introduce macro-bending loss. As the higher order LP11, s and LP21, p modes are less confined, they are more likely to couple into cladding modes when the fiber is bent, resulting in higher macro-bending loss than the fundamental modes LP01, s and LP01, p. This must either be taken into account by increasing the power of the higher order pump, or the bending radius has to be large enough to render bending loss negligible. The theoretical macro-bending loss can be calculated using Marcuse’s curvature loss formula . Figure 11 shows the macro-bending losses for the various signal and pump modes as functions of bend radius.
It is observed that macro-bending loss of all modes grows exponentially as bending radius decreases. In particular, the LP11, s mode is the least spatially confined. To reduce its macro-bending loss to less than 0.01 dB/m, the bending radius must be at least 9.46 cm. Figure 12 shows modal gain as a function of bend radius assuming only LP21 p is pumped at a power of 150 mW. It is observed that the gain of LP11, s is reduced significantly as bend radius falls below 9 cm. In particular, MDG between the modes is reduced to zero at a bend radius of 8 cm, making it impossible to equalize MDL in the transmission by pumping in LP01 p as outlined in Section 3.1. It is observed that at low bend radius, the gain of LP01, s is increased due to reduced mode competition. However, the increase in gain saturates when LP11, s is completely stripped out. At very small bend radius, macro-bending loss will again reduce the gain of LP01, s. The results indicate that a practical EDF should be spooled with a bend radius greater than 9 cm for a step-index fiber design. If device size is an issue, it is also possible to create fibers more complex refractive index profiles, such as using refractive index trenches, to better confine all signal and pump modes.
A multimode EDFA with modal gain control is proposed. By adjusting relative amount of LP01, p and LP21, p, the gains of the LP01, s and LP11, s signal modes can be tuned over a wide dynamic range. The relative gain between the two spatially degenerate LP11, s signal modes can also be adjusted by adding a small amount of LP11 θ , p which is the even LP11 e , p mode rotated by angle θ. Performance impact due to excitation of unwanted pump modes at the input of the EDF, mode coupling and macro-bending loss in the fiber was also investigated. The proposed modal gain control scheme can be generalized for an N-mode MM-EDFA by varying the powers of N well-chosen pump modes.
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