## Abstract

We report an experimental and theoretical investigation of the nonlinear transmission coefficient of a heavily doped (2300 ppm) Erbium silica fiber at continuous-wave pumping at the wavelength 1560 nm. It is shown that the fiber transmission is essentially deteriorated by the nonlinear losses, which are caused by the excited-state absorption (ESA) and Erbium ion pairs (IP) presented in the fiber. These phenomena inevitably result in worsening of the amplifying and lasing potential of the heavily doped Erbium fiber. We demonstrate the latter on the example of an Erbium fiber laser (wavelength, λ = 1560 nm) under IR (wavelength, λ = 978 nm) pumping, where the heavily doped Erbium fiber is used as an active medium. The developed theory, addressing both the nonlinear transmission coefficient of the fiber at the 1560-nm pumping and the generation characteristics of the Erbium fiber laser, takes into account the additional losses and non-radiative relaxation factors stemming from the ESA- and IP-effects and allows getting a good agreement between the modeling and experimental results.

© 2005 Optical Society of America

## 1. Introduction

During the past decade, the nonlinear-optical properties of rare-earth-doped silica fibers have been under extensive investigation. The knowledge of the nonlinear-optical features of these fibers, and particularly of the Erbium-doped ones, is important for the contemporary needs of fiber-optics communication and switching, engineering with Erbium fiber amplifiers and lasers, etc. (see [1–3] and references therein). To date, there were investigated a few mechanisms explaining the nonlinear behavior of the transmission coefficient of Erbium silica fibers, apart from evident saturation of the ground-state absorption (GSA). Among those of a considerable interest are the excited-state absorption (ESA) in the system of Erbium ions and formation of Erbium ion pairs (IP) if the fiber is heavily doped with Erbium. These effects were studied both at the pumping (0.98 μm) and lasing (1.5–1.6 μm) wavelengths; however, these studies had a fragmental character [4–12], especially regarding the case of a heavily doped Erbium fiber at the lasing wavelengths. Meanwhile, both the phenomena are inevitably present in any Erbium fiber and act as un-wanted sources of additional nonlinear losses and non-radiative relaxation of excitation, that limits efficiency of Erbium fiber lasers for the 1.5–1.6-μm spectral range. Therefore, a proper account of these nonlinear effects in an Erbium-doped fiber is important for further improvements of the fiber fabrication technique for the laser needs.

In the present work, we report an experimental and theoretical study of the nonlinear
transmission coefficient of a heavily doped Erbium silica fiber SLC 110-01 (the IPHT, Jena, Germany) with concentration of Er^{3+} ions 2300 ppm (mol.), numerical aperture *NA* = 0.27, and core radius *a* = 1.35 μm at continuous-wave (cw) excitation at the wavelength *λ* = 1560 nm. The dependences of the transmission coefficient on pump power (the “bleaching” effect) and fiber length due to the GSA saturation are inspected. We also discuss in details the additional ESA- and IP- contributions in the fiber transmission nonlinearity since both the effects are the factors of additional non-radiative relaxation of the upper ^{4}
*I*
_{13/2} level, that leads to the appearance of “non-working” Erbium ions and worsening of the amplifying / lasing properties of the active fiber. We then propose a novel model for an Erbium-doped fiber laser
on the wavelength *λ* = 1.56 μm under the IR pumping (wavelength, *λ* = 0.98 μm), in which both the ESA- and IP- effects at the generation wavelength are accounted for. We finally check the model experimentally and show that a good agreement is observed between the experiment and theory.

## 2. The influence of ESA and IP effects on transmission coefficient of Erbium fiber: experiment and modeling

An experimental study of the nonlinear transmission coefficient was performed in the
arrangement where a piece of heavily doped Erbium fiber is pumped by a cw Erbium fiber
laser (wavelength, *λ* = 1560 nm) with maximal output power ${p}_{\mathit{\text{in}}}^{\text{max}}$ = 50 mW. The tested piece of Erbium fiber was spliced with the Erbium laser output; the transmission coefficient of the
former was calculated as *T* = *P*_{out}
/*P*_{in}
, where Pin and Pout are the pump powers on the input and output of the fiber. A set of dependences of the transmission coefficient versus input pump power *T*(*P*_{in}
) was obtained for different Erbium fiber lengths *L*.

The un-saturated absorption spectrum of the tested Erbium fiber in the spectral range
corresponding to the ^{4}
*I*
_{15/2} - ^{4}
*I*
_{13/2} transition is shown in Fig. 1(a). It is seen that at the pump (*λ* = 1560 nm) wavelength the un-saturated absorption and full-saturated gain coefficients are, respectively, *α*
_{0} = 0.06 cm^{-1} and *g*
_{0} = 0.08 cm^{-1} (Fig. 1(b)). The propagation of the pumping wave in the Erbium fiber was single-mode (the cut-off wavelength is about 1040 nm, see Fig.1 (c), where the near-IR absorption band ^{4}
*I*
_{15/2} - ^{4}
*I*
_{11/2} is shown).

The experimentally measured dependences *T*(*P*_{in}
, *L*) are shown in Figs. 2(a) and (b) by open symbols. It is seen that the dependence of transmission coefficient versus input power (Fig. 2(a)) demonstrates the “bleaching” character that is characteristic for the GSA saturation effect. On the other hand, the dependence of transmission coefficient on fiber length has also a well-known behavior that is the appearance of an “interplay” between the resonant absorption and gain and depleting of the pump power through the fiber length.

Another feature seen from Fig. 2 (a) is that the full-saturated transmission coefficient of the fiber is considerably less than 100% that is an indication of the presence of strong nonlinear losses. Our attempts to fit the experimental dependences shown in Fig. 2 by the theoretical ones, accounting for the GSA saturation effect only, have been failed. So, we have arrived to the conclusion that an account of the ESA- and IP- contributions in the fiber transmission nonlinearity is crucial to get a matching between the experiment and theory. Our idea based on the fact that the ESA-process at 1.5-μm excitation is inevitably present in any Erbium-doped material, and in particular, in Erbium-doped silica fiber. Meanwhile, the formation of Erbium ion-pairs in this fiber, as it will be shown below, is also the case.

We have derived a system of equations that addresses the propagation of a pump wave at
the wavelength falling into the resonant ^{4}
*I*
_{15/2} - ^{4}
*I*
^{13/2} band of Erbium. We considered the scheme of Erbium energy levels to have the simplified structure shown in Fig. 3, where the GSA and gain relate to the transitions between the states “1” (^{4}
*I*
_{15/2}) and “2” (^{4}
*I*
_{13/2}) and the ESA - to the transition from the upper state “2” to the higher excited state(s) “3”. Additionally, we supposed that a return of an Erbium ion from the state “3” to the state “2” is instantaneous, i.e., there is no change in the state “2” population owing to the ESA process.

In this case, the balance equations for the pump power *P* (in s^{-1}) and dimensionless (normalized on the total concentration *N*_{0}
of Erbium ions in the fiber) population *n*
_{2} (0 ≤ *n*
_{2} ≤ 1) of the metastable state “2” are as follows:

where *α*
_{0} = *N*_{0}*σ*_{12}
is the un-saturated (small-signal) absorption coefficient of Erbium fiber at
the pump wavelength; *γ0* is the linear loss coefficient; *ξ* = (*σ*_{12}
+ *σ*_{21}
)/*σ*_{12}
and *η* = *σ*_{23}
/*σ*_{12}
are
the coefficients that stand for the ratios between the GSA, ESA and gain cross-sections, where *σ*_{12}
and *σ*_{21}
are the cross-sections of the GSA and gain transitions between the states “1” and
“2” and *σ*_{23}
is the ESA cross-section; *τ*_{0}
is the lifetime of Erbium ions in the excited state “2”; *τ*_{p}
is the effective time parameter which addresses efficiency of up- and down- conversion at non-radiative excitation relaxation in Erbium ion-pairs; *Γ* = 1 - *exp*[-2(*a*/*w*
_{0})^{2}] = 1-*exp*(-*S*_{a}
/*S*_{w}
) is the overlap factor for the pump wave and fiber core, where *S*_{w}
= ${\pi w}_{0}^{2}$/2 and *S*_{a}
= *πa*
^{2} are, respectively, the geometrical cross-sections of the beam and fiber core, with *w*
_{0} being the beam radius. It is seen from Eqs. (1) and (2) that the ESA- and IP- effects are accounted for in our model by the coefficients *η* = *σ*_{23}
/*σ*_{12}
and *τ*_{p}
= *β*/*N*_{0}
. Notice that *τ*_{p}
itself is dependent of Erbium concentration *N*_{0}
in the fiber, whilst the coefficient *β* is the “microscopic” parameter characterizing the Erbium IP binding strength.

Let us briefly explain the physical origin of the terms in Eqs. (1) and (2) regarding to the ESA- and IP- effects. Evidently, the parameter *η* enters Eq. (1) only as it addresses the ESA-process from level “2” to level “3” (see Fig.3). Its absence in Eq. (2), which describes the population of the excited state “2”, is explained by taking as instantaneous the relaxation process of the higher level “3” [21]. Meanwhile, the dependence of the last term of Eq. (2) on ${n}_{2}^{2}$ stems from the accepted model of Erbium IP, where undertaken are the mechanisms of instantaneous up- and down- conversion of excitation of both the ions composing the pair. When both ions in the pair are excited, i.e., they are in the state “2”, one of them (donor) gives its energy to the other (acceptor), which is promoted to a higher energy level “3”, while the donor relaxes to the ground (“1”) state. In turn, the acceptor from the state “3” relaxes non-radiatively back to the metastable level “2” [10,11]. From here, the quadratic dependence of relaxation time of the pair composed by two excited Erbium ions stems immediately. By the way notice that if only a single ion in an IP is excited, no changes occur neither in effective relaxation decay of Erbium, nor in effective bleaching of the fiber.

It is also necessary to point out that Eqs. (1) and (2) are obtained from the more general Eqs. (1’) and (2’) (see Attachment) provided that one neglects by a deformation of the radial distribution of the excited state population owing to the saturation effects. As our estimates show such a simplification is acceptable indeed.

The numerical calculations were performed for various values of the Erbium fiber length
*L*. All the parameters, apart from the coefficients *η* and *β*, were given by the experimental arrangement. The overlap factor for fundamental propagation mode at 1560 nm is calculated to be *Γ* = 0.42, and the beam radius at the same wavelength is *w*
_{0} = 2.6×10^{-4} cm. The linear loss coefficient *γ*_{0}
in the fiber under study is rather small, about 2×10^{-4} cm^{-1}. The coefficients characterizing the resonant-absorption properties of the Erbium fiber at the pump (*λ* = 1560 nm) wavelength are: *α*_{0}
= 0.06 cm^{-1} (see Fig. 1); the cross-sections *σ*_{12}
= 3×10^{-21} cm^{2} and *σ*_{21}
= 1.25*σ*_{12}
(that stems from the measured ratio of the small-signal absorption, 0.06 cm^{-1}, and full-saturated gain, 0.08 cm^{-1}, coefficients of the Erbium fiber, see Figs. 1(a) and (b)). This results in *ξ* = 2.25. The concentration of Erbium ions in the fiber is *N*_{0}
= *α*_{0}
/*σ*_{12}
= 2×10^{19} cm^{-3} and their relaxation time is *τ*_{0}
= 10^{-2} s. Concerning the value of the ESA cross-section, there
are some estimations in the literature for this coefficient at the wavelength λ̣ = 980 nm: *σ*_{23}
= 1.5×10^{-21} cm^{2} [12], that leads to the estimate η = 0.43. However, the value of the ESA parameter η at the wavelength 1560 nm as well as the value of the parameter β (or *τ*_{p}
), characterizing the IP-effect in the heavily-doped Erbium fiber, are to be found from the best fitting of the experimental data (see Fig. 2, symbols) by the theory (see Eqs. (1) and (2)).

The modeling results are shown in Figs. 2(a) and (b) by lines, which are the theoretical counter-parts for the experimental data (symbols), and also in Fig. 4, where either the ESA- or the IP- mechanism is treated separately, implying that another one is neglected (see Figs. 4(a) and (b), respectively). The latter modeling has been done for showing that only at a simultaneous account of both the ESA-effect (as a source of the “resonant” nonlinear loss) and the IP-effect (as a source of non-radiative relaxation of the “working” state “2” of Erbium ions via the up- and down- conversion of excitation), one is able to obtain a true agreement between the experiment and theory (compare Fig. 4 and Fig. 2).

Another result of our modeling is that the best fitting of the experimental data by the
theory is obtained at the following values of the parameters *η* and *β*: *η* = 0.32 and *β* = 1.7×10^{16} cm^{3} s (*τ*_{p}
= 8.5×10^{-4} s). Note here that the value of the ESA-parameter *η* is universal for any Erbium-doped fiber at the 1560-nm excitation since it is independent of concentration of Erbium ions in a fiber. On the other hand, the value of the IP-parameter *β* is valid only for the treated fiber with Erbium concentration of 2300 ppm since at other concentrations the effective interaction radius of the adjacent ions in a pair can be dependent of Erbium concentration [13].

## 3. The impact of ESA and IP effects in Erbium fiber laser: experiment and modeling

The laser (see Fig. 5(a)) was composed by splicing the heavily doped Erbium fiber (the same as in above experiments on studying the nonlinear transmission, Section 2) of the length *L* = 150 cm with two fiber Bragg gratings (FBG1 and FBG2) as the output couplers, having maximum reflection coefficients of 94% (wavelength, 1560.8 nm) and 90% (wavelength, 1559.7 nm) - see Fig. 5(b), where the product of the reflection coefficients of the FBGs 1 and 2 is shown. The active Erbium fiber was pumped through a WDM multiplexer by a commercial low-power (300 mW) pigtailed diode laser (pump wavelength, 978 nm).

The experimentally measured output power of the laser (as a sum of generation powers on its outputs 1 and 2) versus pump power is shown in Fig. 6 (filled symbols). It is seen that this dependence has a linear character. One can notice that the laser efficiency is very low; this is caused by the following main factors: (i) non-optimized reflectivity of FBGs, (ii) high losses on splices between the Erbium active fiber and fibered output FBG-couplers, and (iii) nonlinear losses stemming from the treated above ESA- and IP- effects. Also notice that at the pump powers exceeding 150 mW the Erbium fiber laser transits from cw operation to the regime of giant pulses. The last feature is out of the present study’s scope and has been investigated in details recently (see Ref. [14]).

At the present time, there are known a few models for an Erbium fiber laser (see, e.g.,
Refs. [15–21] and references therein); however to the best of our knowledge, the ESA- and IP effects in Erbium have not been addressed in ever of them. One of the aims of the present work is to fill this gap. Let us use in a subsequent analysis the approach for the Erbium fiber laser modeling developed in Ref. [21]. The balance equations (see Eqs. (3) and (4) below) for the generation intra-cavity power *P*_{g}
(that is a sum of the contra-propagating waves’ powers inside the cavity, in s^{-1}) at the wavelength *λ*_{g}
= 1560 nm (lying within the transition ^{4}
*I*
_{13/2} - ^{4}
*I*
_{15/2},) and the averaged over the active fiber length population *y* of the upper “2” level (that is a dimensionless variable, 0 ≤ *y* ≤ 1) under the IR pumping at the wavelength *λ*_{p}
= 978 nm (within the transition ^{4}
*I*
_{15/2} - ^{4}
*I*
_{11/2}) are as follows:

In Eqs. (3) and (4), $y=\underset{0}{\overset{L}{\int}}\frac{{N}_{2}\left(z\right)dz}{{N}_{0}L}$, where *N2( z)* is the local population of the level “2”;

*T*

_{r}= 2

*n*(

*L*+

*l*

_{0})/

*c*is the photon intra-cavity round-trip time, where

*L*is the active fiber length,

*l*

_{0}is the total length of FBG-couplers’ tails inside the cavity, and

*n*is the refractive index of silica;

*α*

_{th}=

*γ*+

*l*

_{n}(1/

*R*)/2

*L*is the intra-cavity overall losses on the threshold, where

*γ*=

*γ*

_{0}+

*γ*

_{1}is the total non-resonant losses which is a sum of the loss

*γ*

_{0}inside the Erbium active fiber and losses

*γ*

_{1}on splices of this fiber with the fibered FBGs, and

*R*is the effective reflection coefficient of FBG-couplers; ${P}_{sp}\left(y\right)=\frac{y{10}^{-3}}{{\tau}_{0}{T}_{r}}{\left(\frac{{\lambda}_{g}}{{w}_{0}}\right)}^{2}\frac{{a}^{2}{\alpha}_{0}L}{4{\pi}^{2}{\sigma}_{12}}$ is the spontaneous emission power into the fundamental laser mode of the radius

*w*

_{0}(we assume here that the laser spectrum width is 10

^{-3}of the Erbium luminescence spectral bandwidth); ${P}_{\mathit{pump}}\left(y\right)={P}_{p}\frac{1-\mathrm{exp}\left[-\delta {\alpha}_{0}L\left(1-y\right)\right]}{{N}_{0}h{v}_{p}L\pi {w}_{p}^{2}}$ is the pump power (in s

^{-1}), with

*P*

_{p}(in W) being the pump power at the fiber entrance, δ the dimensionless coefficient that accounts for the ratio of absorption coefficients of Erbium fiber at the pump wavelength

*λ*

_{p}to that at the laser wavelength

*λ*

_{g}, and

*hυ*

_{p}= 2.037×10

^{-19}J the pump energy quanta. Notice that we take further for simplicity that the geometrical cross-section ${\pi w}_{p}^{2}$ of the pump beam (which is multi-mode at the wavelength 978 nm, see Fig. 1(c)) coincides with the Erbium-doped fiber core

*S*

_{a}=

*πa*

^{2}. Other parameters of the model are the same as in the modeling of the nonlinear transmission coefficient of the Erbium fiber (see Section 2).

The system of Eqs. (3) and (4) was numerically calculated for obtaining the dependence of laser output power ${P}_{g}^{\mathit{\text{out}}}$
at the generation wavelength, *λ*_{g}
= 1560 nm versus pump power *P*_{p}
at the pumping wavelength, *λ*_{p}
= 978 nm. The parameters *NA*, *a*, *w*_{0}
, *Γ*, *α*_{0}
, *σ*_{12}
, *σ*_{21}
, *σ*_{23}
, *ξ*, *N*_{0}
, *γ*_{0}
, and *τ*_{0}
, which were used in the routine, are the same as in the modeling in Section 2, and the parameters *η* = 0.32 and *τ*_{p}
= 8.5×10^{-4} s (*β* = 1.7×10^{16} cm^{3}s) were the resultant ones obtained from the above modeling of the nonlinear transmission coefficient of the Erbium fiber. Other parameters used in the calculations are as follows: *L* = 150 cm, *n* = 1.45 and l0 = 20 cm, giving *T*_{r}
= 16.4 ns; *δ* = 0.5 (compare the spectra in Figs. 1(a) and (c)); *R* = 0.78 (see Fig. 5(b)); *γ* = 2.61. Such a large value of the intra-cavity passive losses, about 84%, is mainly due to the losses on splices *γ*_{1}
since in our arrangement the beam diameters in the Erbium-doped active fiber (2.6×10^{-4} cm, see above) and fibered FBG-couplers (5×10^{-4} cm^{-1}) are strongly different. Thus, we get the following values of the threshold loss and population of the upper level “2”: *α*_{th}
= 8.7×10^{-3} cm^{-1}, ${y}_{th}=\left(1+\frac{{\alpha}_{th}}{\Gamma {\alpha}_{0}}\right)\frac{1}{\left(\xi -\eta \right)}=0.69$. Alone variable parameter in the modeling was the excess over the laser threshold *ϵ* (defined as *P*_{p}
= ϵ*P*_{th}
). Note that the calculated threshold pump power ${P}_{th}={y}_{th}\frac{{N}_{0}h{v}_{p}L\pi {w}_{p}^{2}\lfloor {{y}_{th}}^{2}{\tau}_{0}+{y}_{th}{\tau}_{p}\rfloor}{{\tau}_{0}{\tau}_{p}\left\{1-\mathrm{exp}\left[-{\alpha}_{0}L\delta \left(1-{y}_{th}\right)\right]\right\}}=28\phantom{\rule{.2em}{0ex}}\mathrm{mW}$ is very closed to the experimentally measured one, about 23 mW (see Fig. 6). Finally, the generation output power is calculated as

where *hv*_{g}
= 1.274×10^{-19} J is the generation energy quanta.

The results of modeling are shown in Fig. 6(a) by a solid line, which is the dependence of laser output power ${P}_{g}^{\mathit{\text{out}}}$
versus pump power *P*_{p}
. It is seen that the theoretical results agree with a high precision with the experimental ones (Fig. 6(a), symbols), thus showing the necessity to take into account both the ESA- and IP- contributions at the laser dynamics modeling.

Notice, that otherwise, i.e., if these effects are not incorporated in the modeling, significant deviations in the laser threshold and its output power appear between the theoretical and experimental data - see inset (b). In this figure, equally with the theoretical data shown in Fig.6 (a) (see curve 1), are demonstrated the modeling results, obtained for the situations where either ESA-, or IP- contributions, or both these contributions are omitted (see, respectively, curves 3, 4, and 2).

## 4. Final remarks

Note that in both the presented models, addressing the nonlinear transmission coefficient of a heavily doped Erbium fiber and Erbium fiber laser on its base, we didn’t take into account the effects of (i) excitation migration through the Erbium ions’ system [22] and (ii) clustering-induced non-saturable absorption in a heavily doped fiber [23–25].

The first of them (the excitation migration among the Erbium ions, see Ref. [22]) seems to be responsible only for an effective change in lifetime of the excited state “2” of Erbium ions and not to result in additional nonlinear losses in the fiber. Meanwhile, an account of this effect can seriously complicate the model because of an appearance of other free parameters and thus uncertainties in the results interpretation. An accurate comparison of the excitation migration effect [22] with the ESA- and IP- effects might be a point of a separate further investigation.

Concerning the second effect (the clustering-induced non-saturable absorption in a heavily doped Erbium silica fiber, see Refs. [23,24]), we omit it in our modeling because of the following facts. (i) The manufacturer of this Erbium fiber (SLC 110-01, IPHT, Jena, Germany) manifests that their fiber is free from clustering of Erbium ions. (ii) Our own comparison of the ratios of absorption-to-gain spectra of the used heavily doped fiber (concentration, 2300 ppm) and an analogous low doped fiber (concentration, 300 ppm) has showed that their amplitudes are nearby the same (0.85 and 0.91, respectively). Therefore, no pronounceable non-saturable absorption due to clusters was observed indeed in our heavily doped Erbium fiber. So, we can point out that the above modeling, which implies as the dominant in the fiber under study the ESA- and IP-effects (the last of them requires interaction between only two excited ions of Erbium), fully describes the transmitting and lasing characteristics of our active fiber.

## 5. Conclusions

We have reported the results of an experimental and theoretical analysis of the effects of excited-state absorption (ESA) and Erbium ion pairs’ (IP) formation in heavily doped Erbium silica fiber on the fiber nonlinear transmission coefficient at the cw 1560-nm pumping and the characteristics of the correspondent 1560-nm Erbium fiber laser pumped at the wavelength 978 nm. It has been shown that the ESA- and IP- effects result in the appearance of nonlinear losses and non-radiative relaxation of the excited state via up- and down- conversion of excitation, that seriously changes the fiber transmission coefficient bleaching under the 1560- nm excitation and thus worsens the amplifying and lasing potential of the Erbium-doped fiber. An account of these effects has allowed us to get a whole agreement between the experimentally measured and modeled dependences of the nonlinear transmission coefficient on pump power and Erbium-doped fiber length and, as the result, to obtain the values of the microscopic parameters, addressing the ESA- and IP- effects in the heavily doped Erbium fiber. We have also proposed a simple system of equations for the Erbium-doped fiber laser operating at the wavelength 1560 nm under the 978-nm pumping, where both the additional loss and non-radiative relaxation factors stemming from the ESA and IP effects are taken into consideration. We have demonstrated that these equations precisely model the experimentally measured output parameters of the laser.

## Appendix

The general form of equations addressing the propagation of a pump wave in an Erbium-doped fiber is as follows:

where the additional parameter $\mu =\frac{{S}_{a}}{{S}_{w}}\mathrm{exp}\left(-\vartheta \frac{{S}_{a}}{{S}_{w}}\right)$ characterizes the deformation of the radial population distribution of the excited state “2” due to the saturation effects, where *θ* = 0.65. It is easily to show that in the conditions of our experiment *μ* ≈ *Γ* with a high accuracy, thus Eqs. (1’) and (2’) are transformed in Eqs. (1) and (2). However, at other relative sizes of the fundamental mode propagating in Erbium fiber *S*_{w}
= *πw*_{0}
/2 and core-area doped with Erbium *S*_{a}
= *πa*
^{2}, one may meet the situation where Eqs. (1’) and (2’) (nor Eqs. (1) and (2)) are valid.

An analogous generalized form of the equations describing an Erbium fiber laser (see Eqs. (3) and (4) above) is as follows:

## Acknowledgments

The present research has been partly supported through the Project # 47029-F (CONACyT, Mexico).

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