## Abstract

We compare the amplification of various modes in large mode area fibers when the pump is coupled into the fundamental core mode versus the standard cladding-pump scheme. Our simulations show that pumping in the LP_{01} core mode results in differential gain for the fundamental signal mode which suppresses the higher order modes and amplified spontaneous emission compared to the cladding pump scheme. This differential gain effect is predicted to increase with core size and may provide a path to scale fiber mode area to several thousand square micron.

©2009 Optical Society of America

## 1. Introduction

Novel fiber designs and amplifier schemes have been used to scale the mode area of large-modearea (LMA) gain fibers while still maintaining single mode operation in multi-mode fibers [1]–[11]. Most fiber amplifiers are pumped using low brightness diodes coupled into their inner cladding. This pumping scheme results in a near uniform distribution of pump energy across the inner cladding and hence core modes see gain proportional to their overlap with the gain profile, which is near identical for a large number of modes.

We recently proposed and demonstrated that by coupling a high brightness 1480 nm Raman fiber laser as pump into the fundamental mode of the core, we were able to obtain stable and robust amplification of the fundamental mode in a 1800*µ*m^{2} mode area Er doped fiber amplifier [12, 13]. This amplification scheme is useful when one is willing to trade efficiency for reduced nonlinear signal distortion, such as for the amplification of ultrashort pulses to high peak intensities [14, 15, 16], and stimulated Brillouin scattering (SBS) limited amplification of narrow line width signals.

In this paper numerical simulations are performed to investigate the effect of the ion-inversion profile on the gain dynamics of various modes of a core-pumped, step-index, uniformly doped LMA amplifier. The results are compared to the case where the pump is launched in the cladding. The model proposed by Jiang and Marciante in Ref. [17] accounts for the spatial gain dependence, and we use it to explore the relative amplification of modes in the two pump schemes.

## 2. Theory

A model describing the amplification of multiple spectral components in single-mode fiber amplifiers was presented in Ref. [18]. This model has been used successfully to describe propagation of multiple spatial modes in LMA amplifiers [17] where it was shown that the spatial distribution of the propagating fields have a significant impact resulting in transverse spatial-hole burning (TSHB). In Ref. [12] we had introduced a model that used approximations to calculate the overlap between the modes and dopant. Since then we have found that simulations with the model in Ref. [17] match better with experimental results, especially for calculations relating to the change in HOM power fraction with pump power (presented in Section 3.3). Hence we use this model for our simulations.

The modes of the fiber were solved for using the approximate solutions to the scalar Helmholtz-eigenvalue equation described in Refs. [17, 19], which are accurate when radial symmetry and an infinite cladding are assumed.

The following propagation equation is solved for the signal and pump mode powers using the fourth order Runge-Kutta method:

where *G _{mn}*(

*z*) is the gain while

*α*is the background loss (scattering loss) of the fiber. The background loss is assumed to be constant for all propagating modes and it dominates confinement loss in straight LMA fibers. The subscripts

*m*and

*n*refer to the azimuthal and radial mode numbers, respectively. The pump and signal are assumed to be continuous-wave (CW).

The gain depends on the ion inversion distribution and the absorption and emission cross-sections of the doping material [20] and is given by

where *r*, and *ϕ* are the radial and azimuthal coordinates, and *i _{mn}*(

*r*,

*ϕ*) is the normalized mode intensity distribution of a particular mode given by

where *I _{mn}* is the intensity profile. In Eq. (2),

*n*

_{2}and

*n*

_{1}are the number of ions in the excited and ground states respectively, and

*n*

_{1}+

*n*

_{2}=

*n*the total ion concentration. σ

_{t}*, and σ*

_{a}*are the absorption and emission cross-sections at the wavelength of the propagating light, respectively. This formalism can treat arbitrary doping profiles in the fiber [21].*

_{e}Mode coupling between different transverse modes due to scattering can be modeled by adding the term ∑* _{m}*(

*κ*

_{mn,pq}

*P*-

_{mn}*κ*

_{pq,mn}

*P*) to the power evolution Eq. (1), where

_{pq}*κ*

_{mn,pq}is the coupling coefficient for power transfer from the mode given by the azimuthal and radial mode number

*mn*to the mode

*pq*. This effect was found to be negligible in Ref. [17] and so is neglected in our paper.

In a steady-state approximation (*dn*
_{2}/*dt*=0), the inverted population can be expressed analytically as follows

where *τ* is the fluorescence lifetime of the dopant, *f* is the frequency, and *h* is the Planck constant. The summation over the propagating mode powers enables us to account for the gain competition in the LMA amplifier. The TSHB effect due to saturation is accounted for by the summation in the denominator while the small signal gain is represented by the numerator. The two-level rate equation (4) is valid under the assumption that multiple modes are incoherent with respect to each other and therefore do not interfere.

## 3. Simulation results

We model an erbium fiber amplifier with a uniformly doped step index core of radius 35 *µ*m and fundamental mode effective area of approximately 1800 *µ*m^{2}. The signal input power is 17 mW and the system is pumped by a 7 W continuous wave Raman fiber laser at 1480 nm wavelength. These parameters are similar to the experimental amplification results reported in [12, 13], where both signal and pump light were coupled from a standard single mode fiber into the fundamental mode of the LMA fiber using a fiber based mode converter that matched the mode field between the two fibers. Greater than 18 dB coupling into the fundamental mode of the LMA was reported. Simulations are performed under conditions where the pump is launched (i) in the fundamental mode of the core (as in the experiments), and (ii) in the inner cladding. The fiber parameters are chosen so that in both cases the fundamental signal mode is amplified by 22 dB. In the cladding pump case, since the pump intensity is very weak, the doping concentration, and cladding radius are adjusted to achieve the required 22 dB gain with the same amount of pump power. The inner cladding radius is set to 50 *µ*m instead of 100 *µ*m used for simulating the core pump case. The pump is assumed to have a uniform distribution across the inner cladding cross-section. The parameters used in the simulations are listed in Table 1.

## 3.1. Ion inversion profile

In order to understand the gain dynamics of the various modes it is important to understand the inversion profile created by the two pump schemes. The theory above is used to simulate the ion inversion as a function of radius at various gain fiber positions when the core, and cladding pump schemes are employed. Figure 1 shows the radial distribution of the inversion level when the LP_{01} (first row), LP_{03} (second row), and LP_{06} (third row) signal modes are launched one at a time. The case where the pump is in the fundamental core mode is shown in the left panel, and the case of cladding pump is shown in the right panel. The input power of each mode was 17 mW. At the start (*z*=0), in the core pump case the inversion is nearly uniform for *r* < 0.8*R*
_{co} after which it declines rapidly, whereas for the cladding pump case the inversion is uniform throughout the core. With propagation down the fiber the inversion decreases due to depletion of pump intensity and due to gain saturation across the fiber cross section due to increasing signal strength. The decrease due to gain saturation is maximum at radial positions where the signal has the highest intensity leading to a highly modulated inversion profile for higher order modes (HOM).

We first compare the inversion for the LP_{01} mode for the core and cladding pump cases (top row of Fig. 1). In the core pump case, the pump intensity distribution maintains a high inversion at radial positions of high signal intensity where the saturation effect is maximum, and at the same time it reduces inversion where the signal is weak. This overlap between the pump and signal intensity profiles leads to a nearly constant inversion for *r*<0.8*R _{co}* ensuring uniform gain across the signal mode which is mainly confined within this radius. The good overlap between the signal mode and inversion results in efficient extraction of energy by the signal. The inversion decreases rapidly for

*r*>0.8

*R*in accordance with the pump mode distribution. As a result very few HOM are supported which reduces amplified spontaneous emission (ASE). The inversion profile is nearly flat near the center instead of the expected distribution similar to the LP

_{co}_{01}mode. This is due to pump saturation which occurs for pump powers

*P*>0.8-1 W, for smaller

_{p}*P*the shape of

_{p}*n*

_{2}(

*r*) approaches the LP

_{01}distribution. In our case the pump power exceeds the saturation threshold at all fiber lengths and therefore the shape of the inversion profile is nearly preserved.

For the cladding pump case with signal in the LP_{01} mode the inversion varies strongly with radial position due to gain saturation. At all lengths (*z*>0) the inversion is lower at the core center than at its edge due to gain saturation at the mode peak. This results in non-uniform gain across the mode cross-section. The weakness of the fundamental mode near the core edge means that most of the inverted population near the edge remains undepleted and can seed unwanted ASE.

Next we focus on amplification of the higher order modes LP_{03} and LP_{06}. When the amplifier is cladding pumped the gain of higher order modes is governed by their overlap with the inversion level and therefore a larger number of modes are supported. For both pump situations the inversion shows several peaks where the mode has a null (due to undepleted gain), and minima where the mode has peaks (due to gain saturation). The modulation in gain increases with propagation as the mode gains in intensity. This causes strong variation in local gain across the mode cross-section. The regions around the nulls of the mode have large amount of undepleted gain which can contribute to ASE. For the core pump case the outer rings of the HOMs extend beyond the inverted region and this poor overlap reduces mode gain compared to the cladding pump case. Thus the core pump scheme discriminates against HOMs.

Saturation depths in the plots of the inversion level are significantly larger for HOMs in the cladding pump case than in the core pump case. This is because HOMs see higher gain in the cladding pump scheme (as indicated in Fig. 1).

The large radial variation of inversion in the cladding pump case for all modes implies a significant variation in gain across the mode cross-section. As the modes propagate and are amplified in each slice of the fiber, the intensity profile is expected to deviate from the ideal mode shape, leading to scattering and redistribution of energy into other modes. This should lead to increased interference between modes (also called multi-path interference (MPI)) at the output of the fiber. Our current model is limited and it does not capture this effect.

## 3.2. Amplification of modes

For both the cladding and core pump schemes we simulate the amplification of twenty modes in the LMA fiber accounting for gain competition between them (the modes are assumed to

be incoherent with respect to each other). We assume that at the fiber input most of the signal is coupled into the fundamental mode and the strength of the higher order modes (HOM) is weaker by 21 dB. This is considered a high quality fundamental mode launch in experiments. Each HOM has an input power of 132 *µ*W while the LP_{01} power is 17 mW. The gain per unit length as a function of propagation distance is plotted for six modes in Fig. 2. The total gain for all modes is listed in Table 2 in order of decreasing effective refractive index and is plotted in Fig. 3.

Comparing Figs. 2(a), and 2(b), the variation of gain with fiber length is higher for the clad-pump case than for the core-pump case. The larger dopant concentration for the clad-pump case is chosen so as to provide the required gain for the fundamental mode in the face of strong gain competition from higher order modes. In the clad-pump case significant amount of pump is absorbed by the ions near the core-clad interface, which do not provide much gain for the fundamental mode. Therefore amplification of the fundamental mode is less efficient resulting in lower residual pump compared to the core-pump case (as shown by the curve labeled “pump” in Figs. 2(a) and 2(b)). The higher dopant concentration results in larger gain over the initial fiber length when compared with the core-pumped case. At longer fiber lengths the gain decreases in both cases due to pump depletion and gain saturation, with the drop in gain being more rapid for the clad-pump case.

In the core-pump case (Fig. 2(a)) the LP_{01} mode sees the highest gain through the length of the amplifier. In the cladding-pump scheme the gain for most HOMs exceeds the gain for the LP_{01} mode - this is due to the larger overlap of HOMs with the gain profile. Only modes with significant extent beyond the core (such as the LP_{06}) see lower gain.

The total gain plotted in Fig. 3 shows that for the core pump scheme the LP_{01} mode achieves the highest total gain which is ~1.4 dB higher than the LP_{11} mode. Whereas for the cladding pump case the LP_{01} mode has lower total gain than all other modes (except the LP06 mode). These results are expected from our discussions in the previous section on overlap of the modes with the ion inversion. The gain of HOMs has a significant effect on beam quality. In the core pump case the fraction of HOM power relative to the fundamental mode power is lower at the amplifier output than at the input. Whereas for the cladding pump case the fraction of HOM power increases on amplification. A high fraction of HOM is undesirable because it makes the beam non-diffraction limited and leads to instability in beam pointing due to MPI [22].

Differential gain effects can also be induced in cladding pumped fiber amplifiers by confining the dopant ions to a radius smaller than that of the core or to a Gaussian-shaped profile [21] so that the fundamental mode has a larger overlap with the gain profile than the HOM. However in this scheme when the fiber is bent the overlap of the modes with the doped region changes which modifies the differential gain effect. In our architecture both pump and signal beams suffer the same distortion and therefore their overlap remains unchanged to first order.

## 3.3. Relative mode amplification at different pump intensities

We now calculate the ratio of power in the HOM versus the fundamental mode - this quantity is directly related to the level of MPI observed in the output beam. Our calculations show that the ratio changes depending upon the pump power and its distribution. The growth in modal power versus pump power is presented for various modes in Figs. 4(a) and 4(b) for the core and cladding pump schemes. For the calculations the HOMs were launched with 21 dB lower power than LP_{01} as before.

Figures 4(c) and 4(d) show the ratio of power in various HOMs relative to the power in the fundamental mode for the two pump schemes. In the core-pump case at low pump powers only the center of the core inverted so that HOMs suffer differential absorption which drops their relative strength to much below the starting value of 21 dB. This absorption effect is relatively minor in the cladding pump case. As the pump is increased, amplification of HOMs occurs more efficiently leading to a growth in HOM power relative to the fundamental (except for the LP_{06} which sees low gain because of its poor overlap with the gain profile and therefore its fraction keeps decreasing). Hence MPI increases with pump power. However this growth in MPI with pump power saturates in both pumping schemes. As expected from the previous section, at high pump values the HOM power fraction is reduced below the input value for the core pump case while it increases for the cladding pump case.

This behavior of MPI saturation in the core-pump scheme has also been measured experimentally by us using the 1800 *µ*m^{2} mode area erbium amplifier detailed in Ref. [12, 13]. The S^{2}-imaging technique [23] was used to measure the LP_{11}, LP_{02}, and LP_{03} mode content of our amplifier output [13] as a function of output signal power. The experimental results are shown in Fig. 5.

The results of this and the previous section imply that with the commonly used cladding-pump scheme, the HOMs see greater gain than the fundamental mode leading to a deterioration of beam quality at the output relative to the input. Even small amounts of HOM excitation at the input can lead to significant MPI build up at the output. Whereas with the pump in the fundamental core mode the growth of the MPI is curtailed by the differential gain effect.

## 3.4. Effect of increasing mode area

Next we examine how the modal gain changes for fibers as their mode area is increased by increasing the core size while keeping the Δ*n* constant. For cladding pump simulations the ratio of core to inner cladding area was fixed so that *R*
_{clad}=*R*
_{core}/√0.49, the same as in Table 1.

All other parameters were also left unchanged from those listed in Table 1. Calculations were performed accounting for the gain competition.

Figure 6 shows the total gain experienced by various modes for the core and cladding pump schemes as the mode area is increased. In both cases the gain decreases as a function of core size due to the decrease in pump intensity. We also note that the gain length used was optimum for an 1800 *µ*m^{2} fundamental mode area (ideally calculations should be performed for optimized gain lengths that vary with mode area, however it has a second-order effect on the differential gain between modes).

For the core-pump scheme (Fig. 6(a)) the gain of the fundamental mode remains higher than for the HOMs, independent of area. The difference in gain between the fundamental and HOMs increases with area, i.e. the larger the mode area, the higher the differential gain effect induced by our core pump scheme. For the cladding pump scheme the fundamental mode has lower gain than the HOMs for relatively small core areas, and at larger areas the gain for all modes is approximately equal (Fig. 6(b)). These results therefore show that the core pumping scheme is scalable to larger mode areas and should be preferred over the conventional cladding pump scheme when scaling the mode area to several thousand square microns.

## 4. Conclusion

Very large mode area Er fiber amplifiers were modeled by including spatially dependent gain with a two level rate equation for the cases where the pump was in the fundamental core mode, and the inner cladding. The effects of gain competition were included. Our simulations showed that the commonly used cladding pump geometry provides higher gain for HOMs relative to the fundamental mode and therefore even with a high quality launch into the fundamental mode at the input, the MPI increases at the amplifier output. On the other hand launching the pump in the fundamental core mode provides differential gain for the fundamental mode thereby improving MPI at the output. In this architecture the gain for the fundamental mode is uniform across its cross section leading to mode stability and efficient energy extraction. This differential gain effect is predicted to increase with increase in mode area and the scheme should enable significant scaling of the mode area.

## Acknowledgement

The authors thank Dr. J. W. Nicholson for analysis of the S2 measurement data.

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