**3. Er3+ concentration effects in EDF**

In equation (13) the superscripts "*s*" stand for the laser wavelength (1550 nm). The model includes also the two contra-propagating waves of SE (not shown in Figure 7) spectrally

> Ω 4*π αse*0*Pse sat* <sup>Γ</sup>se *∫* 0 *a n*2

where the second term on the right side is the SE power generated by a short fiber section *dz*,

core in each direction, and *n* is the modal refractive index at *λse*. The EDF gain at *λse* is written

(*r*, *<sup>z</sup>*) *exp* -2( *<sup>r</sup>*

*wse* )2

*Pp*(*z* =0, *t*)=*Pp*<sup>0</sup> (16)

I15/2→<sup>4</sup>

(*r*, *z*)2*rdr* (14)

*rdr* - *αBG* (15)

<sup>2</sup> (17)

<sup>2</sup> (18)

is the EDFL output power. To simplify

(*z* = *L <sup>c</sup>*, *t*) =0 (19)

(*z* = *L <sup>c</sup>*, *t*)(1 - *R*2)*t* <sup>2</sup> =0 (20)

*sat* is the saturation power at 1531 nm, Γ*se* is

is the fraction of SE photons guided by the EDF

I13/2 transition),

centered at *λse*=1531 nm (this wavelength corresponds to the GSA peak of 4

264 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

(*z*)*Pse*

±(*z*) ∓

(*r*, *z*) - *n*<sup>1</sup>

powers of which obey the equation:

*gse*(*z*) <sup>=</sup> <sup>4</sup>*αse*<sup>0</sup> Γ*sew*se <sup>2</sup> *∫* 0 *a*

The boundary conditions are written as:

as

*d Pse* ±(*z*) *dz* = ± *gse*

*αse*0=0.016 cm-1 is the low-signal absorption at *λse*, *Pse*

(*ξse* - *εse*)*n*<sup>2</sup>

*Ps*

*Pse*

*Ps* -

*Ps out* (*t*)=*Ps* +

where *Pp*0 is the pump power at the EDF input and *Ps*

all the ESA transitions (curve 4).

+(*<sup>z</sup>* =0, *<sup>t</sup>*)=*Ps*

(*z* = *L <sup>c</sup>*, *t*) =*Ps*

+(*<sup>z</sup>* =0, *<sup>t</sup>*)=*Pse*

calculations, we considered that *t*1=*t*2=0 (i.e. no loss on the fiber splices).


+


The EDFL's efficiency as a function of FBG2's reflectivity *R*2, simulated using the laser model described above, is depicted in Figure 8(a). The EDFL was modeled for four different "ver‐ sions" of the energy level system: Without considering all the ESA transitions (curve 1); with considering the pump ESA only (curve 2) and the signal ESA only (curve 3); with considering

The first important observation is that the optimal reflection of output FBG2, at which EDFL demonstrates the maximal efficiency, is drastically decreased when the ESA transitions are accounted for. For instance, the optimal reflection of FBG2 is ≈66% when only the EDF background loss (3.1 dB/km, see Table 1) is present (curve 1), whereas it is ≈11% when all kinds

(*z* =0, *t*)*R*1*t*<sup>1</sup>

(*z* = *L <sup>c</sup>*, *t*)*R*2*t*<sup>2</sup>

*out*

the overlap factor for the SE waves, Ω=π*NA*/*n*<sup>2</sup>

In this part, we shall discuss the Er3+ concentration effects in EDFs resulting in a reduced efficiency of EDF based lasers and amplifiers, which is associated with the phenomenon of Er3+ ions' clustering that leads, in turn, to non-saturable absorption (NSA) through inhomo‐ geneous up-conversion (IUC).

For our experiments we selected the most representative commercial EDFs fabricated through the MCVD and direct nanoparticle deposition (DND) processes; all the fibers under scope in this section are similar in the sense of Er3+ doped core's chemical composition being the most common alumino-silicate glass (in the case of MCVD-EDFs with addition of germanium). The first series of the EDFs (MCVD-based, "M"-series) includes two fibers: M5-125-980 and M12-125-980 (*Fibercore*), hereafter M5 and M12, and the second series (DND-based, "L"-series) – three fibers: L20-4/125, L40-4/125, and L110-4/125 (*Leikki* / *nLight*), hereafter L20, L40 and L110. These fibers have very similar waveguide parameters and differ mainly in Er3+ doping level. [Notice that the EDFs employed in the whole of above experiments, see Section 2, belong to M-series.]

#### **3.1. Absorption and fluorescence spectra**

The EDFs' absorption spectra are shown in Figure 9(a) where Er3+ transitions 4 I15/2 → <sup>4</sup> I11/2 (within a 940...1020 nm range with a peak at 978 nm) and 4 I15/2 → <sup>4</sup> I13/2 (within a 1400...1600 nm range with a peak at 1.53 μm) are featured. The spectra were obtained using a white light source with fiber output and OSA with 1-nm resolution. It is seen that the absorption spectra of the EDFs of both series have a very similar shape (given by similarity of core glass chemical compositions), differing only in intensity. The ratio of the peaks' magnitudes at 1.53 μm and at 978 nm was measured to be equal to ∼1.6, for the M and L EDFs.

Figure 9(b) demonstrates the normalized fluorescence spectra for L fibers (to simplify the picture the spectra for M fibers are not shown), measured at the maximal pump power at 978 nm, *Pp* ∼400 mW, within the 450...1650 spectral range (the area nearby the pump wavelength is cut out). In this experiment the lateral geometry, when fluorescence is captured by a multimode fiber patch cord from the lateral surface of the short fiber samples, was arranged. In spite of Er3+ concentration increases in the row of fibers L20 → L40 → L110, the Er3+ fluorescence band, centered at ∼1.53 μm, is indistinguishable in shape. Although the 1.53-μm band dominates in the EDFs' fluorescence spectra, there also exist the spectral lines at its anti-Stokes side (∼450...1100 nm), which evidences the presence of UC. Note that, in contrast to the ∼1.53-μm band's stability against Er3+ concentration, the higher Er3+ content, the more intense the UCE (compare curves 1 – 3 in Figure 9(b)).

To understand the origin of UCE and the dependence of UC intensity upon Er3+ concentration in the EDFs, let's refer to Figure 10 in which the scheme of Er3+ energy levels and a sketch of the processes involved at the excitation at *λp*=978 nm are presented. UCE (shown in the figure by grey arrows) seems to be mostly associated to Er3+ ion clusters being in states 4 I11/2 and 4 I13/2, because the ESA process, equally acting for single and clustered Er3+ ions, is ineffective at 978 nm excitation.

The Influence of Nonlinear Effects Upon Oscillation Regimes of Erbium-Doped Fiber Lasers http://dx.doi.org/10.5772/59146 267

**Figure 9.** (a) Absorption spectra of the EDFs of L-(blue curves) and M-(red curves) series in the near-IR. (b) Fluores‐ cence spectra of the EDFs of L-series in the VIS...near-IR spectral range at 978-nm pumping.

**Figure 10.** Scheme of Er3+ energy levels, applicable for the EDFs with high Erbium content. Functioning of Er3+ clusters (shown for simplicity as ion pairs) is sketched by the blue and red arrows for long-living manifolds 4 I11/2 and 4 I13/2; the black dotted arrows depict non-radiative relaxations; the grey arrows show the UC and "fundamental" (1.53-μm) emissions; the short-living levels are shown by dashed lines.

#### **3.2. Fluorescence decay kinetics**

Er3+ ions' clustering that leads, in turn, to non-saturable absorption (NSA) through inhomo‐

266 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

For our experiments we selected the most representative commercial EDFs fabricated through the MCVD and direct nanoparticle deposition (DND) processes; all the fibers under scope in this section are similar in the sense of Er3+ doped core's chemical composition being the most common alumino-silicate glass (in the case of MCVD-EDFs with addition of germanium). The first series of the EDFs (MCVD-based, "M"-series) includes two fibers: M5-125-980 and M12-125-980 (*Fibercore*), hereafter M5 and M12, and the second series (DND-based, "L"-series) – three fibers: L20-4/125, L40-4/125, and L110-4/125 (*Leikki* / *nLight*), hereafter L20, L40 and L110. These fibers have very similar waveguide parameters and differ mainly in Er3+ doping level. [Notice that the EDFs employed in the whole of above experiments, see Section 2, belong

The EDFs' absorption spectra are shown in Figure 9(a) where Er3+ transitions 4

range with a peak at 1.53 μm) are featured. The spectra were obtained using a white light source with fiber output and OSA with 1-nm resolution. It is seen that the absorption spectra of the EDFs of both series have a very similar shape (given by similarity of core glass chemical compositions), differing only in intensity. The ratio of the peaks' magnitudes at 1.53 μm and

Figure 9(b) demonstrates the normalized fluorescence spectra for L fibers (to simplify the picture the spectra for M fibers are not shown), measured at the maximal pump power at 978 nm, *Pp* ∼400 mW, within the 450...1650 spectral range (the area nearby the pump wavelength is cut out). In this experiment the lateral geometry, when fluorescence is captured by a multimode fiber patch cord from the lateral surface of the short fiber samples, was arranged. In spite of Er3+ concentration increases in the row of fibers L20 → L40 → L110, the Er3+ fluorescence band, centered at ∼1.53 μm, is indistinguishable in shape. Although the 1.53-μm band dominates in the EDFs' fluorescence spectra, there also exist the spectral lines at its anti-Stokes side (∼450...1100 nm), which evidences the presence of UC. Note that, in contrast to the ∼1.53-μm band's stability against Er3+ concentration, the higher Er3+ content, the more intense

To understand the origin of UCE and the dependence of UC intensity upon Er3+ concentration in the EDFs, let's refer to Figure 10 in which the scheme of Er3+ energy levels and a sketch of the processes involved at the excitation at *λp*=978 nm are presented. UCE (shown in the figure

because the ESA process, equally acting for single and clustered Er3+ ions, is ineffective at 978-

by grey arrows) seems to be mostly associated to Er3+ ion clusters being in states 4

I15/2 → <sup>4</sup>

I15/2 → <sup>4</sup>

I11/2 and 4

I13/2,

I13/2 (within a 1400...1600 nm

I11/2

geneous up-conversion (IUC).

**3.1. Absorption and fluorescence spectra**

the UCE (compare curves 1 – 3 in Figure 9(b)).

(within a 940...1020 nm range with a peak at 978 nm) and 4

at 978 nm was measured to be equal to ∼1.6, for the M and L EDFs.

to M-series.]

nm excitation.

Like at the fluorescence spectra' measurements discussed above, the kinetics of near-IR fluorescence at ∼1.53 μm was detected using the lateral geometry. However, the pump light at 978 nm was in this case switched on / off by applying a rectangular modulation of LD current at Hz-repetition rate. The launched into the EDF samples pump power was varied between zero and ∼400 mW. The fluorescence signal was detected either using an InGaAs PD with a Si filter placed between the fiber and a multimode patch cord delivering fluorescence to PD (being so the measurements above ~1.1 μm where the use of Si filter allows cutting off the pump light's spectral component), or using a fast Si-PD with no spectral filtering applied (being in fact the measurements below ~1.1 μm), placed directly above a slit segregating a portion of fluorescence from the EDF's surface. To diminish ASE and re-absorption on the results, we used short (∼0.5 cm) EDF pieces.

Typical kinetics of the fluorescence signal, recorded after switching pump light at 978 nm off, are presented in Figure 11 for the heavier doped EDFs M12 (a) and L110 (b); the data were acquired using InGaAs-PD with Si filtering (transmission band above 1.1 μm). We don't present here the results for other, lower doped, EDFs as these showed similar but less featured trends in the decay kinetics.

**Figure 11.** Fluorescence decay kinetics obtained for the EDFs M12 (a) and L110 (b). Curves 1 to 6 are captured for dif‐ ferent pump levels (see the insets in the right upper corners). Zero-time corresponds to the moment when the pump light is switched off.

It is seen from Figure 11 that for these two EDFs fluorescence power, corresponding to 1.53 μm spectral band, is saturated (as is saturated GSA of Er3+ ions) yet at a few mW of pump power. However the key feature is deviation from the exponential law in the fluorescence kinetics in EDF L110 (see Figure 11(b)). A similar trend occurs but is less expressed in the rest of L and M fibers with lower Er3+ concentration; see e.g. Figure 11(a). Another fact deserving attention is the presence of a sharp drop in the fluorescence signal in fiber L110 at high pump powers, which happens after switching pump off (refer to curves 4–6 in Figure 11(b)). Such a feature is present but in a smaller degree also in fibers L40, L20, and M12 (having substantially lower Er3+ contents) and almost vanishes in fiber M5 (having the lowest Er3+ content). Note that similar fluorescence kinetics were observed in some of the earlier reports, see e.g. [4, 10].

Overview of the fluorescence decays for all the EDFs under scope is provided in Figure 12 (points). These data were obtained at maximal pump power, *Pp* at 978 nm (400 mW; the high pump power was found to be the right choice for minimizing spatial diffusion of excitation; see e.g. [10]). The dependences in the figure demonstrate the fluorescence decay "tails" obtained after cutting off the short initial segments just after switching pump off (~30 μs), which permits elimination of the influence of non-instant LD power decay (~8 μs).

at 978 nm was in this case switched on / off by applying a rectangular modulation of LD current at Hz-repetition rate. The launched into the EDF samples pump power was varied between zero and ∼400 mW. The fluorescence signal was detected either using an InGaAs PD with a Si filter placed between the fiber and a multimode patch cord delivering fluorescence to PD (being so the measurements above ~1.1 μm where the use of Si filter allows cutting off the pump light's spectral component), or using a fast Si-PD with no spectral filtering applied (being in fact the measurements below ~1.1 μm), placed directly above a slit segregating a portion of fluorescence from the EDF's surface. To diminish ASE and re-absorption on the results, we

268 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

Typical kinetics of the fluorescence signal, recorded after switching pump light at 978 nm off, are presented in Figure 11 for the heavier doped EDFs M12 (a) and L110 (b); the data were acquired using InGaAs-PD with Si filtering (transmission band above 1.1 μm). We don't present here the results for other, lower doped, EDFs as these showed similar but less featured

**Figure 11.** Fluorescence decay kinetics obtained for the EDFs M12 (a) and L110 (b). Curves 1 to 6 are captured for dif‐ ferent pump levels (see the insets in the right upper corners). Zero-time corresponds to the moment when the pump

It is seen from Figure 11 that for these two EDFs fluorescence power, corresponding to 1.53 μm spectral band, is saturated (as is saturated GSA of Er3+ ions) yet at a few mW of pump power. However the key feature is deviation from the exponential law in the fluorescence kinetics in EDF L110 (see Figure 11(b)). A similar trend occurs but is less expressed in the rest of L and M fibers with lower Er3+ concentration; see e.g. Figure 11(a). Another fact deserving attention is the presence of a sharp drop in the fluorescence signal in fiber L110 at high pump powers, which happens after switching pump off (refer to curves 4–6 in Figure 11(b)). Such a feature is present but in a smaller degree also in fibers L40, L20, and M12 (having substantially lower Er3+ contents) and almost vanishes in fiber M5 (having the lowest Er3+ content). Note that similar fluorescence kinetics were observed in some of the earlier reports, see e.g. [4, 10].

used short (∼0.5 cm) EDF pieces.

trends in the decay kinetics.

light is switched off.

As seen from Figure 12, 1.53-μm fluorescence decays get more and more deviated from the single exponential law when Er3+ concentration increases: The fibers with smaller contents of Er3+ ions (M5, M12, and L20) demonstrate decays nearly a single-exponent law whereas fibers L40 and L110 – the decays, apparently different from this law. These features, associated to the Er3+ concentration effect, can be addressed in terms of the IUC process – see Section 3.3 where the results of modeling of Er3+ fluorescence kinetics are presented. The modeling of the fluorescence kinetics allowed us to get, for each EDF, lifetime *τ*<sup>0</sup> and constant *C*HUP\* (charac‐ terizing the homogeneous UC process, HUC; see modeling below) and thereafter to build their dependences upon small-signal absorption value *α*0 at 1.53 μm (and hereby upon Er3+ ions concentration, proportional to *α*0).

**Figure 12.** Normalized fluorescence decay kinetics obtained for the EDFs of L-(a) and M-(b) series; points – experimen‐ tal data (using InGaAs-PD with Si filtering); plain curves are the theoretical fits made using formula (22).

Furthermore, Figure 13 demonstrates the results of the fluorescence kinetics' measurements in the optical band below 1.1 μm within a short (tens of μs) interval just after switching pump light off, for EDFs M12 (a), L40 (b), and L110 (c). The measurements were fulfilled using Si-PD without optical filtering at *Pp*=400 mW. As is seen from the figure, there are two "fast" components in the PD signal's decay: The shortest (≈8 μs) one, being in fact the setup's technical resolution and originating from the scattered pump light, and the longer one, measured by 21 μs to 26 μs for fibers M12, L40, and L110 (for fibers M5 and L20 this component was not resolved). A similar component was also detected in the 1.53-μm fluorescence kinetics; see Figure 11, which evidences its non-radiative nature. Note that there are known the processes in Er3+ -doped materials attributed by similar times [19, 20].

**Figure 13.** Fluorescence decay kinetics in the EDFs measured using Si-PD: M12 (a), L40 (b), and L110 (c); the short-time components of the fluorescence signals are specified in each plot.

We suggest that the found feature stems from a partial excitation relaxation in Er3+ clusters since it is present in the heavier doped EDFs but almost vanishes in the lower doped ones. The magnitude of the short-living component is a function of Er3+ concentration (and therefore of *α*0), as seen when comparing the plots (a), (b), and (c) in Figure13: The higher Er3+ concentration the larger is relative (to the technical, i.e. originated from pump-light scattering) magnitude of this component.

#### **3.3. Nonlinear absorption coefficient**

The nonlinear absorption coefficient of a rare-earth doped fiber (e.g. EDF) as a function of pump power *α*(*Pp*) contains the useful information about GSA saturation and, consequently, about the fiber's potential as a laser medium. On the other hand, such effects deteriorating laser 'quality' of the fiber as ESA and concentration-related HUC / IUC (lifetime quenching and non-saturable absorption) ought to affect the behavior of *α*(*Pp*), too [18].

In the study to be reported hereafter, pump light was delivered to an EDF sample from the same LD operated at 978 nm (used in the measurements of fluorescence spectra and lifetimes); pump power was varied from ≈0.5 to 400 mW. We measured first the nonlinear transmission coefficient of the EDF sample with length *L*0, which is defined as *T*<sup>978</sup> = *PP out* / *PP in* where *PP in* and *PP out* are the pump powers at the EDF's input and output. Then we made a formal re-calculation of the experimental transmission coefficient *T*978(*PP in*) in the absorption coefficient, applying formula: *α*(*PP in*) =-*ln*(*T*978) / *<sup>L</sup>* <sup>0</sup>. The EDFs' lengths were chosen such that overall trends in the dependences *α*(*Pp in*) can be viewed within the whole range of pump powers. The ratio of the EDFs' lengths was such that the optical density (the product *α*0*L*0) is almost the same for all the samples, which is worth for estimation of the nonlinearity (saturation) of absorption at increasing Er3+ concentration in the fibers. Using the OSA, we checked the ratio of pump to ASE powers at the EDFs' outputs; it was found that the ASE contribution is negligible in all samples at *Pp in*> 0.5 mW.

**Figure 14.** (a) Nonlinear absorption coefficients of the EDFs of L-and M-types vs. pump power at 978 nm. Symbols: experimental points; plain curves: theoretical fits obtained using Eqs. (25-27). Fiber lengths used in experiments and at modeling were, correspondingly: 188.6 (M5, curve 1), 59.4 (M12, curve 2), 43.5 (L20, curve 3), 22.4 (L40, curve 4), and 9.5 (L110, curve 5) cm. (b) Non-saturable absorption *β* vs. small signal absorption *α*0, measured for the entire EDFs' set; the fitting curve is for guiding the eye.

The results obtained by applying the drawn procedure are shown in Figure 14(a) by symbols. Coefficients *α*0 and *β* (saturated absorption at pump wavelength) marked in the upper left and right corners of the figure correspond to the limits of small-signal and saturated pump absorptions. First notice that absorption is "bleached" (in other words, transmission is "satu‐ rated") by a more or less similar manner for either fiber. However, as is also seen from the figure, the "residual" absorption (*β*) rises drastically with increasing Er3+ concentration in the fibers' sequences M5→M12 and L20→L40→L110. This trend points out that the ratio between the residual (*β*) and small-signal (*α*0) absorptions is much bigger for the heavier doped fibers (M12, L40 and L110). In fact, pump-induced (looking as residual) absorption *β* is the measure of nonlinear absorption loss in EDF, as it stems from the modeling results (see Section 3.4 below). Furthermore, the dependence *β*(*α*0) plotted in Figure 14(b) allows one to reveal that this pump-induced excessive loss in the EDFs appear as one of the most important Er3+ concentration effects.

#### **3.4. Modeling**

**Figure 13.** Fluorescence decay kinetics in the EDFs measured using Si-PD: M12 (a), L40 (b), and L110 (c); the short-time

We suggest that the found feature stems from a partial excitation relaxation in Er3+ clusters since it is present in the heavier doped EDFs but almost vanishes in the lower doped ones. The magnitude of the short-living component is a function of Er3+ concentration (and therefore of *α*0), as seen when comparing the plots (a), (b), and (c) in Figure13: The higher Er3+ concentration the larger is relative (to the technical, i.e. originated from pump-light scattering) magnitude

The nonlinear absorption coefficient of a rare-earth doped fiber (e.g. EDF) as a function of pump power *α*(*Pp*) contains the useful information about GSA saturation and, consequently, about the fiber's potential as a laser medium. On the other hand, such effects deteriorating laser 'quality' of the fiber as ESA and concentration-related HUC / IUC (lifetime quenching

In the study to be reported hereafter, pump light was delivered to an EDF sample from the same LD operated at 978 nm (used in the measurements of fluorescence spectra and lifetimes); pump power was varied from ≈0.5 to 400 mW. We measured first the nonlinear transmission

are the pump powers at the EDF's input and output. Then we made a formal re-calculation

EDFs' lengths was such that the optical density (the product *α*0*L*0) is almost the same for all the samples, which is worth for estimation of the nonlinearity (saturation) of absorption at increasing Er3+ concentration in the fibers. Using the OSA, we checked the ratio of pump to ASE powers at the EDFs' outputs; it was found that the ASE contribution is negligible in all

*in*) =-*ln*(*T*978) / *<sup>L</sup>* <sup>0</sup>. The EDFs' lengths were chosen such that overall trends in the

*in*) can be viewed within the whole range of pump powers. The ratio of the

*out* / *PP*

*in*) in the absorption coefficient, applying

*in* where *PP*

*in* and

and non-saturable absorption) ought to affect the behavior of *α*(*Pp*), too [18].

coefficient of the EDF sample with length *L*0, which is defined as *T*<sup>978</sup> = *PP*

of the experimental transmission coefficient *T*978(*PP*

components of the fluorescence signals are specified in each plot.

270 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

of this component.

*PP out*

formula: *α*(*PP*

samples at *Pp*

*in*> 0.5 mW.

dependences *α*(*Pp*

**3.3. Nonlinear absorption coefficient**

Firstly, the kinetics of near-IR (∼1.53 μm) fluorescence decays obtained for the entire set of the EDF samples were modeled, which allows us to find Er3+ fluorescence lifetimes *τ*<sup>0</sup> and HUC coefficient *C*HUP. The model implicitly implies, in accord to the definition of the HUC process, the only interactions between rather distant single Er3+ ions, not forming "chemical", tightly coupled, clusters, which are in turn supposed to interact via the IUC mechanism, discussed further. At this step of modeling we disregard the short-time features in the fluorescence decays, which are assumed to originate from Er3+ ions gathered into quenched (weakly fluorescing) clusters. That fact that in the experiments on fluorescence kinetics we used comparatively high pump powers (400 mW) allows us to neglect the intensity dependent HUC contribution and excitation migration effects.

For the normalized population density *n*<sup>2</sup> *s* of single (index "s") Er3+ ions being in the first excited (laser) state 4 I13/2, the following rate equation holds [21]:

$$\frac{dn\_2^\*}{dt} = -\frac{n\_2^\*}{\tau\_0} - C\_{HILP} \{ n\_2^\* \}^2 \tag{21}$$

where *n*<sup>2</sup> *<sup>s</sup>* <sup>=</sup> *<sup>N</sup>*<sup>2</sup> *s* / *N*<sup>0</sup> *s* ; *N*<sup>0</sup> *<sup>s</sup>* is the population density of single Er3+ ions in the excited state 2 (4 I13/2), *N*0 *s* is their concentration, and *C*HUP [s−1] is the UC parameter, being a product of the "volu‐ metric" HUC constant *CHUP \** [s−1cm3 ] and concentration *N*<sup>0</sup> *s* : *CHUP* = *N*<sup>0</sup> *s CHUP \** .

Assuming that pump power at 978 nm is high enough to achieve maximal populating of the excited state 4 I13/2, i.e. at "infinite" pump power, the part of Er3+ ions being in the excited state is *k*=*σ*12/(*σ*12+*σ*21). Furthermore, since in our experimental circumstances (where near-IR fluorescence is detected at ∼1.53 μm while excitation is at *λ*p=978 nm) the SE process can be disregarded by means of formal zeroing cross-section *σ*<sup>21</sup> in the dominator of this ratio (k=1). Implying that *n*<sup>2</sup> *<sup>s</sup>*(*t* =0)=1 and that pump is switched off at *t*=0, equation (21) is solved analyti‐ cally, giving:

$$m\_2^\*(t) = \frac{e^{-\frac{t}{\tau\_0}}}{1 + \pi\_0 C\_{HIP} \left(1 - e^{-\frac{t}{\tau\_0}}\right)}\tag{22}$$

Formula (21) is a worthy approximation for fitting of the whole of experimental near-IR fluorescence decay kinetics, reported above for *Pp* ∼400 mW (providing maximal population of manifold 4 I13/2). The modeling results obtained by using formula (22) are shown by plain curves in Figure 12 (the fitting procedure has been fulfilled until the residual sum *R*<sup>2</sup> exceeded 0.99) and are seen to be in good agreement with the experimental decay kinetics (points in the figure). The values of constants *τ*0 (lifetime of single Er3+ ions, found to be ~10.8 ms for all the fibers under study), and *C*HUP (an attribute of the HUC process, determined as the result of fitting) are plotted in Figure 15 in function of the small signal absorption *α*0 at 978 nm.

As seen from Figure 15, the parameter *C*HUP is proportional to Er3+ concentration (GSA *α*<sup>0</sup> at 978 nm). From this figure we found the value of HUC constant: *CHUP \** =2.7×10−18 s−1cm3 . This value agrees well with the published data for EDFs of similar types; see e.g. Refs. [22-24]. Note that the quantity attributing the HUC phenomenon ( *CHUP \** constant) should be proportional to the ESA cross-section (see e.g. [25]). As the latter does not depend on Er3+ concentration, *CHUP \** should be concentration-independent. Indeed, the dependence *C*HUP vs. *α*<sup>0</sup> is seen from Figure 15 to be almost linear.

The next step in modeling is simulation of Er3+ clusters' contribution on the base of the experimental dependences of nonlinear absorption vs. pump power (see Figure 14(a)). A method to model nonlinear absorption of an EDF is based on the idea that ensemble of Er3+ ions in a fiber consists of two independent subsystems, assumed to be single ("s") and clustered

For the normalized population density *n*<sup>2</sup>

(laser) state 4

where *n*<sup>2</sup>

excited state 4

Implying that *n*<sup>2</sup>

cally, giving:

of manifold 4

*CHUP*

Figure 15 to be almost linear.

*N*0 *s* *<sup>s</sup>* <sup>=</sup> *<sup>N</sup>*<sup>2</sup> *s* / *N*<sup>0</sup> *s* ; *N*<sup>0</sup>

metric" HUC constant *CHUP*

*s*

*<sup>τ</sup>*<sup>0</sup> - *CHUP*(*n*<sup>2</sup>

is their concentration, and *C*HUP [s−1] is the UC parameter, being a product of the "volu‐

] and concentration *N*<sup>0</sup>

Assuming that pump power at 978 nm is high enough to achieve maximal populating of the

is *k*=*σ*12/(*σ*12+*σ*21). Furthermore, since in our experimental circumstances (where near-IR fluorescence is detected at ∼1.53 μm while excitation is at *λ*p=978 nm) the SE process can be disregarded by means of formal zeroing cross-section *σ*<sup>21</sup> in the dominator of this ratio (k=1).

> *t τ* 0

Formula (21) is a worthy approximation for fitting of the whole of experimental near-IR fluorescence decay kinetics, reported above for *Pp* ∼400 mW (providing maximal population

curves in Figure 12 (the fitting procedure has been fulfilled until the residual sum *R*<sup>2</sup> exceeded 0.99) and are seen to be in good agreement with the experimental decay kinetics (points in the figure). The values of constants *τ*0 (lifetime of single Er3+ ions, found to be ~10.8 ms for all the fibers under study), and *C*HUP (an attribute of the HUC process, determined as the result of fitting) are plotted in Figure 15 in function of the small signal absorption *α*0 at 978 nm.

As seen from Figure 15, the parameter *C*HUP is proportional to Er3+ concentration (GSA *α*<sup>0</sup> at

value agrees well with the published data for EDFs of similar types; see e.g. Refs. [22-24]. Note that the quantity attributing the HUC phenomenon ( *CHUP \** constant) should be proportional to the ESA cross-section (see e.g. [25]). As the latter does not depend on Er3+ concentration,

*\** should be concentration-independent. Indeed, the dependence *C*HUP vs. *α*<sup>0</sup> is seen from

The next step in modeling is simulation of Er3+ clusters' contribution on the base of the experimental dependences of nonlinear absorption vs. pump power (see Figure 14(a)). A method to model nonlinear absorption of an EDF is based on the idea that ensemble of Er3+ ions in a fiber consists of two independent subsystems, assumed to be single ("s") and clustered

(1 - *<sup>e</sup> t τ*

*s*

*<sup>s</sup>* is the population density of single Er3+ ions in the excited state 2 (4

I13/2, i.e. at "infinite" pump power, the part of Er3+ ions being in the excited state

*<sup>s</sup>*(*t* =0)=1 and that pump is switched off at *t*=0, equation (21) is solved analyti‐

I13/2). The modeling results obtained by using formula (22) are shown by plain

*s*

: *CHUP* = *N*<sup>0</sup>

*s CHUP \** .

I13/2, the following rate equation holds [21]:

272 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

*dn*<sup>2</sup> *s dt* = *n*2 *s*

*\** [s−1cm3

*n*2

978 nm). From this figure we found the value of HUC constant: *CHUP*

*<sup>s</sup>*(*t*)= *<sup>e</sup>*

1 + *τ*0*CHUP*

of single (index "s") Er3+ ions being in the first excited

)<sup>2</sup> (21)

<sup>0</sup> ) (22)

*\** =2.7×10−18 s−1cm3

. This

I13/2),

**Figure 15.** The values of HUC constant *C*HUP vs. Er3+ concentration (in terms of *α*<sup>0</sup> at 978 nm), obtained for the entire set of EDFs as the result of modeling of the experimental fluorescence kinetic using formula (22); the fitting line in for guiding the eye.

("c") ions. Considering this hypothesis, we generalize the model developed in [21] for propagation of a pump wave through the system of single and paired resonantly absorbing and fluorescing centers (pairs are the simplest case of clusters). The model's generalization signifies here that the clusters' subsystem is meant to comprise an arbitrary number of centers (Er3+ ions in our case) whereas the other subsystem – to consist of single species [6, 26].

We assume that a cluster of Er3+ ions can occupy only one of the two permitted states – the state <11>, where all ions forming the cluster are in the ground state, and the state <12>, where one excepted ion (an acceptor of energy transferred from the adjacent donor ions within the cluster) is in the excited state. The latter holds because if other cluster's constituents absorb pump photons and move to the excited state, all them except one leave state 2 (down to state 1) immediately, whereas only this excepted one can stay in state 2.

We also take into account the presence of the short time *τ*1 (together with much longer *τ*0) for Er3+ clusters as an essential element of the model, coming from the experimental data.

We consider that the populations of single ( *N*1, <sup>2</sup> *<sup>S</sup>* ) and clustered ( *N*1, <sup>2</sup> *<sup>c</sup>* ) Er3+ ions, satisfy the following relations:

$$\mathbf{N}\_1^s + \mathbf{N}\_2^s = \mathbf{N}\_0^s = \{\mathbf{1} - \chi\mathbf{x}\} \mathbf{N}\_0 \tag{23}$$

$$\mathbf{N}\_1^c + \mathbf{N}\_2^c = \mathbf{N}\_0^c = \chi \mathbf{x} \mathbf{N}\_0 \tag{24}$$

where *κ* is the partial weight of clustered ions in ensemble, *χ* is the effective (averaged) number of Er3+ ions in a cluster, and *N*0 is the total Er3+ ions concentration. The correspondent normal‐ ized population densities are defined as *n*1, <sup>2</sup> *s*, *c* = *N*1, <sup>2</sup> *s*, *c* / *N*0, where the lower indices assign, correspondingly, the ground (1 or <11>) and the excited (2 or <12>) states, and *N*<sup>0</sup> *s*, *c* are the concentrations of single ions and clusters, respectively.

The balance equations for pump power (*Pp*) and normalized dimensionless population densities of single and clustered ions being in metastable states 2 and <12>, *n*<sup>2</sup> *s*, *c* (0 ≤ *n*<sup>2</sup> *s*, *c* ≤ 1), are as follows [27]:

$$\frac{dP\_p(z)}{dz} = -\alpha\_{p0} \left[ 1 - \left( 1 + \xi\_s - \varepsilon\_s \right) \left( n\_2^s(z) + n\_2^c(z) \right) \right] P\_p(z) - \alpha\_{BG} P\_p\left(z\right) \tag{25}$$

$$\frac{a\_0}{\hbar \left[ \nu\_p N\_0 \Gamma\_p A\_c \right]} \Big[ \chi \kappa \cdot \left( 1 + \xi\_s \right) \nu\_2^c(z) \Big] P\_p(z) \cdot \left( \frac{1}{\tau\_0} + \frac{1}{\tau\_1} \right) \nu\_2^c(z) = 0 \tag{26}$$

$$\frac{a\_{p0}}{2\pi\nu\_p N\_0 \Gamma\_p A\_c} \mathbf{[1 - \chi\chi - (1 + \xi\_s)n\_2^s(\mathbf{z})]} \mathbf{P}\_p(\mathbf{z}) - \frac{n\_2^s}{\tau\_0} - \mathbf{C}\_{H \text{II} \mathbf{P}} \left(n\_2^s\right)^2 = 0 \tag{27}$$

where majority of the quantities have been designated above, parameter *A*c=π*a*<sup>2</sup> is the EDF core area (*a*=1.5 μm is the core radius), and *α*BG=0.03...0.1 dB/m, depending weakly upon the EDF type). In formulas (25-27) we omit the ASE contribution as negligible in our experiments at pump powers exceeding 0.5 mW (see above). Note that these formulas are written in a general form, applicable not only to the Er3+ ion but also to any other resonantly absorbing center, having a three equivalent level system and subjected to the aforementioned concentration effects.

The modeling results are plotted by plain curves 1 to 5 in Figure 16(a). It is seen that they fit well the whole of the experimental data for the EDFs of both (M and L) types. Thus, the IUC process, treated by us as mostly non-radiative relaxation within Er3+ ion clusters, is justified as the key mechanism responsible for nonlinear absorption (attributed by coefficient *β*). The dependence of *β* upon Er3+ concentration in terms of small signal absorption *α*p0 is shown in Figure 16(b) (see curve 1) as the result of modeling; the errors' bars in the curve show uncer‐ tainties of fitting the experimental data by the theory.

When making the numerical calculations, we found that, once searching for the best fit of the experiment by the theory, any *χ*-value (*χ*=2, 3, and so on) can be used, with *κ*-value being varied accordingly. Thus, we have concluded that the product *χκ* (a relative number of clustered Er3+ ions in the system) serves an adjusting parameter at fitting rather than quantities *χ* and *κ* separately. Given by the modeling results, useful insight can be made to interrelation between the relative number of clustered Er3+ ions *χκ* (modeling: Figure 16(a)) and the measured nonlinear (saturated) absorption *β* (experiment: Figure 14(a)). Placing on the same plot (at double logarithmic scaling) the dependences of these two quantities vs. small-signal absorption *α*p0, we found that they have the slopes related as (∼1.44/∼0.63) ≈ 2.3; see Figure 16(b). The found slope's value signifies the average number of Er3+ ions in clusters, whose

where *κ* is the partial weight of clustered ions in ensemble, *χ* is the effective (averaged) number of Er3+ ions in a cluster, and *N*0 is the total Er3+ ions concentration. The correspondent normal‐

> *s*, *c* = *N*1, <sup>2</sup> *s*, *c*

The balance equations for pump power (*Pp*) and normalized dimensionless population

*<sup>s</sup>*(*z*) <sup>+</sup> *<sup>n</sup>*<sup>2</sup>

*<sup>c</sup>*(*z*) *Pp*(*z*) - ( <sup>1</sup>

*<sup>s</sup>*(*z*) *Pp*(*z*) -

area (*a*=1.5 μm is the core radius), and *α*BG=0.03...0.1 dB/m, depending weakly upon the EDF type). In formulas (25-27) we omit the ASE contribution as negligible in our experiments at pump powers exceeding 0.5 mW (see above). Note that these formulas are written in a general form, applicable not only to the Er3+ ion but also to any other resonantly absorbing center, having a three equivalent level system and subjected to the aforementioned concentration

The modeling results are plotted by plain curves 1 to 5 in Figure 16(a). It is seen that they fit well the whole of the experimental data for the EDFs of both (M and L) types. Thus, the IUC process, treated by us as mostly non-radiative relaxation within Er3+ ion clusters, is justified as the key mechanism responsible for nonlinear absorption (attributed by coefficient *β*). The dependence of *β* upon Er3+ concentration in terms of small signal absorption *α*p0 is shown in Figure 16(b) (see curve 1) as the result of modeling; the errors' bars in the curve show uncer‐

When making the numerical calculations, we found that, once searching for the best fit of the experiment by the theory, any *χ*-value (*χ*=2, 3, and so on) can be used, with *κ*-value being varied accordingly. Thus, we have concluded that the product *χκ* (a relative number of clustered Er3+ ions in the system) serves an adjusting parameter at fitting rather than quantities *χ* and *κ* separately. Given by the modeling results, useful insight can be made to interrelation between the relative number of clustered Er3+ ions *χκ* (modeling: Figure 16(a)) and the measured nonlinear (saturated) absorption *β* (experiment: Figure 14(a)). Placing on the same plot (at double logarithmic scaling) the dependences of these two quantities vs. small-signal absorption *α*p0, we found that they have the slopes related as (∼1.44/∼0.63) ≈ 2.3; see Figure 16(b). The found slope's value signifies the average number of Er3+ ions in clusters, whose

*<sup>c</sup>*(*z*)) *Pp*(*z*) - *<sup>α</sup>BGPp*

*<sup>τ</sup>*<sup>0</sup> - *CHUP*(*n*<sup>2</sup>

*s*

*<sup>τ</sup>*<sup>0</sup> + 1 *τ*1 )*n*2

*n*2 *s*

correspondingly, the ground (1 or <11>) and the excited (2 or <12>) states, and *N*<sup>0</sup>

densities of single and clustered ions being in metastable states 2 and <12>, *n*<sup>2</sup>

where majority of the quantities have been designated above, parameter *A*c=π*a*<sup>2</sup>

/ *N*0, where the lower indices assign,

*s*, *c* (0 ≤ *n*<sup>2</sup> *s*, *c* ≤ 1),

*<sup>c</sup>*(*z*)=0 (26)

)<sup>2</sup> =0 (27)

is the EDF core

(*z*) (25)

*s*, *c*

are the

ized population densities are defined as *n*1, <sup>2</sup>

are as follows [27]:

effects.

*dP <sup>p</sup>*(*z*)

*α*0

*αp*<sup>0</sup>

concentrations of single ions and clusters, respectively.

274 Advances in Optical Fiber Technology: Fundamental Optical Phenomena and Applications

*dz* = - *αp*<sup>0</sup> 1 - (1 + *ξ<sup>s</sup>* - *εs*)(*n*<sup>2</sup>

*<sup>h</sup> <sup>ν</sup> <sup>p</sup>N*0<sup>Γ</sup> *<sup>p</sup>Ac χκ* - (1 + *ξs*)*n*<sup>2</sup>

*<sup>h</sup> <sup>ν</sup> <sup>p</sup>N*0<sup>Γ</sup> *<sup>p</sup>Ac* 1 - *χκ* - (1 + *ξs*)*n*<sup>2</sup>

tainties of fitting the experimental data by the theory.

**Figure 16.** (a) Er3+ clusters' contribution *χκ* vs. Er3+ concentration (in terms of *α*p0) obtained for the entire set of EDFs of L-and M-types as the result of modeling the experimental dependences of nonlinear absorption coefficient using for‐ mulas (25-27); the fitting curve is for guiding the eye. (b) Er3+ clusters' contribution *χκ* (left scale) and non-saturable absorption loss *β* (right scale) vs. Er3+ concentration (in terms of *α*p0); the ratio of the slopes attributing the dependences reveals an effective number of Er3+ ions in clusters.

presence, according to the model, is responsible for NSA. This result deserves attention since it shows that NSA in the EDFs of both types mostly originates from paired Er3+ ions rather than from more complicate aggregates. Thus, in contrast to [28] where the role of "heavier" clusters in the IUC phenomenon is discussed, our results evidence for negligible contribution of Er3+ clusters, "heavier" than simple ion pairs.

Notice that the presence in EDFs of NSA at increasing Er3+ concentration affects net gain in heavily-doped Er3+ fibers, which becomes more and more limited (saturated) as it stems from the presence of single Er3+ ions being in the excited state, whereas the clustered ions, in their big part, i.e. (*χκ*–1) ions in each cluster, are always in the ground state. As a consequence, efficiency of an EDFL or EDFA is expected to drop down with increasing concentrations of Er3+ ions in the active fiber. Some recent studies confirm a severe character of the problem [29-30]. It seems that another problem could be encountered at the use of heavily-doped EDFs for pulsed operation where extensive heating via excitation relaxation within Er3+ clusters would affect the dispersive properties of the fibers and deteriorate the regime.
