**1. Introduction**

Yashkov, A. N. Gurianov, and D. V. Kuksenkov, "Single-mode all-silica photonic bandgap fiber with 20-micron mode-field diameter," Opt. Express 16, 11735–11740

.," Opt. Lett. 37, 1292–1294 (2012).

[49] M. Kashiwagi, K. Saitoh, K. Takenaga, S. Tanigawa, S. Matsuo, and M. Fujimaki, "Low bending loss and effectively single-mode all-solid photonic bandgap fiber with

[50] M. Kashiwagi, K. Saitoh, K. Takenaga, S. Tanigawa, S. Matsuo, and M. Fujimaki, "Ef‐ fectively single-mode allsolid photonic bandgap fiber with large effective area and low bending loss for compact high-power all-fiber lasers," Opt. Express 20, 15061–

[51] F. Kong, K. Saitoh, D. Mcclane,T. Hawkins, P. Foy, G.C. Gu, and L. Dong, "Mode Area Scaling with All-solid Photonic Bandgap Fibers,", Optics Express 20,

[52] G. Gu, F. Kong, T.W. Hawkins, J. Parsons, M. Jones, C. Dunn, M.T. Kalichevsky-Dong, K. Saitoh, and L. Dong, "Ytterbium-doped large-mode-area all-solid photonic

bandgap fiber lasers," Optics Express 22, 13962-13968(2014).

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

(2008).

15070 (2012).

26363-26372 (2012).

an effective area of 650μm2

Erbium-doped fiber lasers (EDFLs) are contemporary sources of coherent radiation, attractive for numerous applications requiring both continuous-wave (CW) and pulsed operations, among which telecommunications in a wide wavelength range covering C and L bands ought to be emphasized. Pulsed operation, presenting big interest for practice, is attained in EDFLs by means of standard active and passive Q-switching and mode-locking techniques, capable of enforcing a laser to generate short pulses with durations ranged from hundreds fs to hundreds ns [1]. In the meantime, transients to CW lasing and a laser's relaxation frequency are also of close attention, e.g. when targeting the sensor applications [2, 3]. The detailed knowledge of the processes involved in Erbium-doped fibers (EDFs) to be used, when pumped, as a laser or amplifying medium in each of the referred regimes cannot be overesti‐ mated. The present work is a review of some of the most featuring nonlinear-optical effects that affect the oscillation regimes of EDFLs.

In spite of certain advantages (availability at the market, low cost, and easiness of handling), EDFLs demonstrate considerably lower efficiency as compared to the lasers based on Ytterbi‐ um-doped fibers. The basic cause is the multi-level energy scheme of Er3+ ions, which makes unavoidable absorption of photons at both the pump and laser wavelengths by the ions being at upper Er3+ levels, i.e. the "excited-state absorption" (ESA) [4, 5], including the state where 1.5-μm laser transition stems from. In other words, ESA presents a kind of "up-conversion" (UC) loss inherent to EDF; see section 2.

Another, nonlinear in nature, source of UC loss in EDFs and EDFLs on their base relates to socalled "collective" (concentration-related) effects arising in Er3+ ion pairs and in more compli‐ cate clusters (percentage of which grows with Er3+ concentration) [6]; see section 3. Appearance of the latter phenomenon is also associated with the multi-level energy scheme of Er3+ ions.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Eventually, the cases of low-and heavily-doped EDFs are to be carefully segregated and properly addressed if one looks for optimization of an EDFL.

One more kind of optical nonlinearity that arises in actively Q-switched (AQS) EDFLs is stimulated Brillouin scattering (SBS). Depending of the EDF length and Q-cell's modulation frequency EDFLs may operate in one of the two QS regimes: either in the "conventional" QS (CQS) one in which QS pulses are composed of several sub-pulses separated by a photon's round-trip time (with negligible pulse jitter) [7, 8], or in the essentially stochastic SBS-induced QS (SBS-QS) one where pulse amplitude is bigger by an order of value as compared with CQS but where pulses suffer severe jitter [9]. The results of an experimental study of the basins the CQS and SBS-QS regimes belong to and the basic spectral features of these regimes are discussed in section 4.

In section 5, the review's conclusions are formulated.

### **2. ESA in EDF at the pump and laser wavelengths**

In this section, we report a study aiming at determination of the ESA's spectral depend‐ ence in EDF, covering the most important for applications spectral range, 1.48...1.59 μm, and at ~978 nm, the wavelength that usually is used for pumping EDFs by laser diodes (LDs). In the experiments discussed hereafter a low-doped silica EDF (*Thorlabs M5-970-125*, ∼300 ppm of Er3+ concentration, Al-Ge-silicate host composition, *NA*=0.24, cut-off wave‐ length – 0.94 μm) was chosen to avoid possible effect of UC (observed in heavily-doped EDFs; see section 3). [This fiber belongs to M-series (fabricated through the modified chemical vapor deposition (MCVD) process), to be under scope in section 3 where Er3+ concentration effects are treated; "M5" signifies that small-signal absorption peaked at 978 nm is 5 dB/m.] The measurements were completed by modeling, a necessary chain in interpretation of the experimental results. The developed modeling also provides an estimate for the so-called ESA parameter, obtained at both pump and laser (signal) wavelengths, and thereafter allows determination of the EDF's net-gain coefficient, "deliberated" from the ESA interfere.

Figure 1 shows the fife level Er3+ ion energy scheme upon excitation at the pump (*λp* ≈980 nm) and signal (*λ<sup>s</sup>* ≈1.5 μm) wavelengths, useful for the discussion presented in this section.

The equations that describe functioning of the Er3+ system, in accord to the scheme shown in Figure 1, at excitation at sole or at both excitation wavelengths (*λs* and/or *λp*) in the steady-state are written as follows:

$$\frac{\sigma\_{12}I\_s}{h\nu\_s}N\_1 - \frac{\sigma\_{21}I\_s}{h\nu\_s}N\_2 - \frac{\sigma\_{24}I\_s}{h\nu\_s}N\_2 - \frac{N\_2}{\tau\_{21}} + \frac{N\_3}{\tau\_{32}} = 0\tag{1}$$

Eventually, the cases of low-and heavily-doped EDFs are to be carefully segregated and

One more kind of optical nonlinearity that arises in actively Q-switched (AQS) EDFLs is stimulated Brillouin scattering (SBS). Depending of the EDF length and Q-cell's modulation frequency EDFLs may operate in one of the two QS regimes: either in the "conventional" QS (CQS) one in which QS pulses are composed of several sub-pulses separated by a photon's round-trip time (with negligible pulse jitter) [7, 8], or in the essentially stochastic SBS-induced QS (SBS-QS) one where pulse amplitude is bigger by an order of value as compared with CQS but where pulses suffer severe jitter [9]. The results of an experimental study of the basins the CQS and SBS-QS regimes belong to and the basic spectral features of these regimes are

In this section, we report a study aiming at determination of the ESA's spectral depend‐ ence in EDF, covering the most important for applications spectral range, 1.48...1.59 μm, and at ~978 nm, the wavelength that usually is used for pumping EDFs by laser diodes (LDs). In the experiments discussed hereafter a low-doped silica EDF (*Thorlabs M5-970-125*, ∼300 ppm of Er3+ concentration, Al-Ge-silicate host composition, *NA*=0.24, cut-off wave‐ length – 0.94 μm) was chosen to avoid possible effect of UC (observed in heavily-doped EDFs; see section 3). [This fiber belongs to M-series (fabricated through the modified chemical vapor deposition (MCVD) process), to be under scope in section 3 where Er3+ concentration effects are treated; "M5" signifies that small-signal absorption peaked at 978 nm is 5 dB/m.] The measurements were completed by modeling, a necessary chain in interpretation of the experimental results. The developed modeling also provides an estimate for the so-called ESA parameter, obtained at both pump and laser (signal) wavelengths, and thereafter allows determination of the EDF's net-gain coefficient,

Figure 1 shows the fife level Er3+ ion energy scheme upon excitation at the pump (*λp* ≈980 nm) and signal (*λ<sup>s</sup>* ≈1.5 μm) wavelengths, useful for the discussion presented in this section. The equations that describe functioning of the Er3+ system, in accord to the scheme shown in Figure 1, at excitation at sole or at both excitation wavelengths (*λs* and/or *λp*) in the steady-state

21 32

 tt

*hh h* (1)

0

12 21 24 2 3 12 2

 s

*II I N <sup>N</sup> NN N*

 n


properly addressed if one looks for optimization of an EDFL.

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

In section 5, the review's conclusions are formulated.

"deliberated" from the ESA interfere.

ss

 n

*ss s*

n

are written as follows:

**2. ESA in EDF at the pump and laser wavelengths**

discussed in section 4.

**Figure 1.** Fife-level scheme of Er3+ ion used in modeling. GSA and ESA indicate the ground-state and excited-state ab‐ sorptions, UCE marks the up-conversion emission transitions (weak but detectable), σ*ij* and τ*ij* are, respectively, the cross-sections and decay times for the transitions between the levels *i* and *j*. Three closely-spaced energy levels 4 S3/2, 2 H11/2, and 4 F7/2 are regarded as an effective level "5".

$$\frac{\sigma\_{13}I\_p}{h\nu\_p}N\_1 - \frac{\sigma\_{31}I\_p}{h\nu\_p}N\_3 - \frac{N\_3}{\tau\_{32}} - \frac{\sigma\_{35}I\_p}{h\nu\_p}N\_3 + \frac{N\_4}{\tau\_{43}} = 0\tag{2}$$

$$\frac{\sigma\_{24} I\_s}{h\nu\_s} N\_2 - \frac{N\_4}{\tau\_{43}} + \frac{N\_5}{\tau\_{54}} = 0 \tag{3}$$

$$\frac{\sigma\_{\rm 35} I\_p}{h \nu\_p} N\_{\rm 3} - \frac{N\_{\rm 5}}{\tau\_{\rm 5}} = 0 \tag{4}$$

$$N\_1 + N\_2 + N\_3 + N\_4 + N\_5 = N\_0 \tag{5}$$

where *N*0 is Er3+ concentration in the EDF core, *Ni* are the populations of the correspondent Er3+ levels (*i=*1 to 5), *h* is Plank constant, *νp* and *ν<sup>s</sup>* are the frequencies of the pump and signal waves, and *Is* and *Ip* are the pump and signal waves' intensities. The Er3+ parameters' values used in modeling are listed in Table 1.


**Table 1.** EDF "M5" / Er3+ parameters used in modeling

#### **2.1. Spectral features of ESA in low-doped EDF within the 1.48 to 1.59-µm range**

The first experiment serves to reveal the existence of the ESA process in the EDF at pumping through 4 I15/2→<sup>4</sup> I13/2 transition, refer to Figure 1. The UC emission (UCE) spectra were recorded in the wavelengths range near 980 nm (4 I11/2→<sup>4</sup> I15/2 transition). The pump source was a 12-mW narrow-line LD (*Anritsu Tunics Plus SC*), tunable through the spectral interval, *λs*=1.48...1.59 μm. Experimentally, light from the LD was launched into the EDF (length, 1 m) through a standard 980-nm / 1550-nm wavelength division multiplexer (WDM) while de-multiplexed backward emission from the EDF was registered by an optical spectrum analyzer (OSA, *Ando AQ6315A*). The recorded UCE spectra, at 10-mW in-fiber power and for four different excita‐ tion wavelengths *λs*, are shown in Figure 2(a). It is seen that UCE power depends on *λ<sup>s</sup>* and that the maximal UCE signal spectrally matches the peak wavelength of the Er3+ ground-state absorption (GSA) contour. The appearance of the UCE spectra at *λ<sup>p</sup>* ~980 nm is very similar to that of 4 I11/2→<sup>4</sup> I15/2 emission, thus testifying for effective populating of Er3+ 4 I11/2 state at pumping the EDF at *λs*=1.48...1.59 μm and so for the presence of the ESA process (4 I13/2→<sup>4</sup> I9/2), followed by fast non-radiative (4 I9/2→<sup>4</sup> I11/2) relaxation. Note that no UCE spectral components in the range below 960 nm were detected.

The second experiment, allowing us to reveal the spectral dependence of ESA, was arranged through measurement of the integral lateral emission power collected from the EDF's lateral surface, using a Si photo-detector directly placed above the fiber. Given Si is sensitive from the visible to near-IR (Si band-gap wavelength is ∼1.1 μm), the Si photo-detector (PD) does register the UCE signal (centered at ∼980 nm) and does not the spontaneous emission (SE) from level 4 I13/2 (∼1.5 μm); a scattered pump-light component was found to be extremely weak. The results obtained at various pump powers are shown in Figure 2(b) (see curves 1–3). For comparison, we also plot in this figure (by curve 4) the integrated backward UCE power measured using OSA (from a 10-cm EDF sample). It is seen that the spectral dependence of the UCE signal is similar to the one of ESA power on the excitation wavelength. In fact, the shape of the presented dependencies is established by a convolution of the known GSA and yet unknown ESA spectra of Er3+ (notice that the latter depends on Er3+ ions population inversion).

**Parameter Value Units** Low-signal absorption at 1531 nm (experimental data) *αs0* = 0.016 cm-1 Low-signal absorption at 977 nm (experimental data) *αp0* = 0.012 cm-1

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

**2.1. Spectral features of ESA in low-doped EDF within the 1.48 to 1.59-µm range**

I11/2→<sup>4</sup>

the EDF at *λs*=1.48...1.59 μm and so for the presence of the ESA process (4

of Er3+ (notice that the latter depends on Er3+ ions population inversion).

The first experiment serves to reveal the existence of the ESA process in the EDF at pumping

narrow-line LD (*Anritsu Tunics Plus SC*), tunable through the spectral interval, *λs*=1.48...1.59 μm. Experimentally, light from the LD was launched into the EDF (length, 1 m) through a standard 980-nm / 1550-nm wavelength division multiplexer (WDM) while de-multiplexed backward emission from the EDF was registered by an optical spectrum analyzer (OSA, *Ando AQ6315A*). The recorded UCE spectra, at 10-mW in-fiber power and for four different excita‐ tion wavelengths *λs*, are shown in Figure 2(a). It is seen that UCE power depends on *λ<sup>s</sup>* and that the maximal UCE signal spectrally matches the peak wavelength of the Er3+ ground-state absorption (GSA) contour. The appearance of the UCE spectra at *λ<sup>p</sup>* ~980 nm is very similar to

The second experiment, allowing us to reveal the spectral dependence of ESA, was arranged through measurement of the integral lateral emission power collected from the EDF's lateral surface, using a Si photo-detector directly placed above the fiber. Given Si is sensitive from the visible to near-IR (Si band-gap wavelength is ∼1.1 μm), the Si photo-detector (PD) does register the UCE signal (centered at ∼980 nm) and does not the spontaneous emission (SE) from level

I13/2 (∼1.5 μm); a scattered pump-light component was found to be extremely weak. The results obtained at various pump powers are shown in Figure 2(b) (see curves 1–3). For comparison, we also plot in this figure (by curve 4) the integrated backward UCE power measured using OSA (from a 10-cm EDF sample). It is seen that the spectral dependence of the UCE signal is similar to the one of ESA power on the excitation wavelength. In fact, the shape of the presented dependencies is established by a convolution of the known GSA and yet unknown ESA spectra

I15/2 transition [10] *τ*21 = 10 ms

I13/2 transition[11] *τ*32 = 5.2 μs

I11/2 transition [12] *τ*43 = 5 ns

I13/2 transition @ 1531 nm [14] *σ*12 = 5.1×10-21 cm2

I11/2 transition @ 977 nm [14] *σ*13 = 1.7×10-21 cm2

I9/2 transition [13] *τ*54 = 1 μs

I13/2 transition, refer to Figure 1. The UC emission (UCE) spectra were recorded

I15/2 emission, thus testifying for effective populating of Er3+ 4 I11/2 state at pumping

I11/2) relaxation. Note that no UCE spectral components in the range

I15/2 transition). The pump source was a 12-mW

I13/2→<sup>4</sup>

I9/2), followed

Relaxation time for 4

Relaxation time for 4

Relaxation time for 4

Relaxation time for (4

Cross-section of 4

Cross-section of 4

through 4

that of 4

4

I11/2→<sup>4</sup>

by fast non-radiative (4

below 960 nm were detected.

I15/2→<sup>4</sup>

I13/2 → <sup>4</sup>

I11/2 → <sup>4</sup>

I9/2 → <sup>4</sup>

F7/2/ 2 H11/2/ 4 S3/2) → <sup>4</sup>

**Table 1.** EDF "M5" / Er3+ parameters used in modeling

in the wavelengths range near 980 nm (4

I9/2→<sup>4</sup>

I15/2 → <sup>4</sup>

I15/2 → <sup>4</sup>

**Figure 2.** (a) UCE spectra at *λ<sup>s</sup>* ∼1.5-μm excitation. (b) Dependencies of PD signal (circles, left scale, curves 1 to 3) and normalized frontal UCE power (stars, right scale, curve 4) on excitation wavelength. Curves 1, 2, and 3 correspond to pump powers of 1, 5, and 10 mW; curve 4 is for 10-mW pump.

Considering the simplified Er3+ ion's model and the processes involved at excitation at *λs* ~1.5 μm (see Figure 1 and formulas (1-5)), we obtain simple formula for the normalized population of 4 I11/2 state, *n*3=*N*3/*N*0 [11]:

$$m\_{\rm s} = \frac{\left\| \mathbf{s}\_s \mathbf{y}\_s \mathbf{s}\_s \right\|^2}{1 + \left\| \mathbf{s}\_s \mathbf{s}\_s + \mathbf{s}\_s \mathbf{y}\_s \mathbf{s}\_s^2 \right\|},\tag{6}$$

where *εs*=*σ*24/*σ*12 is the ESA parameter at signal wavelength, *γs*=*τ*42/*τ*<sup>21</sup> ≈ *τ*32/*τ*<sup>21</sup> (*τ*<sup>43</sup> ≈5 ns is neglected in simulations given its smallness respectively the other decay times), *ξs*=1+*σ21*/*σ12* is the spontaneous emission (SE) parameter at the signal wavelength, and *ss*=*Is* / *Is sat* <sup>=</sup> *Ps* / *Ps sat* is the saturation parameter for signal wave (with *Is* being the signal wave intensity, *Is sat* =*hνs*/*σ*12*τ*<sup>21</sup> the saturation intensity, *Ps* = *AsIs*, *Ps sat* <sup>=</sup> *AsIs sat* , *Ps* and *Ps sat* the signal wave power and saturation power, respectively, and *As* the area of the Gaussian distribution of the signal wave in the EDF core). Notice that since *εsγsss* << *ξs* (*γs* ≈7×10-4 and *εs* ≤ 0.6, see below), the term with a second power on *s* in the denominator of (6) is omitted in further calculus. From formula (6), we find the normalized population *n*<sup>3</sup> averaged over the fiber core area if the pump wave intensity is taken to obey the Gaussian law, *ss*(*r*)=*ss*0exp[–2(*r*/*ws*) 2 ]:

$$\overline{n}\_{3} = \frac{1}{\pi a^{2}} \int\_{0}^{a} n\_{3}(r) \times 2\pi r dr = \frac{\mathfrak{s}\_{s} \mathfrak{z}\_{s} \mathfrak{s}\_{s0}}{2\mathfrak{z}\_{s}} \left(\frac{\mathfrak{w}\_{s}}{a}\right)^{2} \ln\left[\frac{1 + \mathfrak{f}\_{s}\mathfrak{s}\_{s0}}{1 + \mathfrak{f}\_{s}\mathfrak{s}\_{s0}\left(1 - \Gamma\_{s}\right)}\right],\tag{7}$$

where *ss*0 is the on-axis saturation parameter that depends on the pump power, *Ps*, as *Ps*=*Is sat ss*0(*πw<sup>s</sup>* 2 /2), *a* is the EDF core radius, *w<sup>s</sup>* is the modal radius of the Gaussian wave, and Γ*s* is the mode's to fiber core's radii overlap factor at *λs*. The parameter *ξ<sup>s</sup>* comprises the ESA parameter *εs*, the measured EDF low-signal absorption coefficient *α<sup>s</sup>*0=Γ*sσ*12*N*0, and the full-saturated fiber gain *gs*=Γ*sσ*21*N*0 – Γ*sσ*24*N*0=*gs*0 – *εsαs*0 (*gs*0 is the EDF net gain at *λs*):

$$\mathcal{L}\_s = 1 + \frac{\sigma\_{21}}{\sigma\_{12}} = 1 + \frac{\mathbf{g}\_{s0}}{\alpha\_{s0}} = 1 + \frac{\mathbf{g}\_s}{\alpha\_{s0}} + \mathcal{e}\_s. \tag{8}$$

Apparently, UCE power depends on *n*<sup>3</sup> that in turns depends on *εs.* It is worth wising that the latter can be easily found from the UCE lateral signal, registered using PD (see Figure 2(b)). The results of calculation of the ESA parameters are plotted in Figure 3(a) by filled symbols (left scale), where the *εs*-spectrum is normalized on the GSA peak value (at *λs*=1.531 μm). It is seen that the ESA parameter weakly depends on *λs* within the spectral interval 1.48 μm to 1.55 μm whilst it sharply grows as *λs* approaches ∼1.6 μm; this behavior reminiscences the trend reported in [15].

The method to find the absolute values of the ESA parameter *εs* is based on the measurements of the EDF's transmission coefficient *T(λs, Ps)*, i.e. in function of excitation power, and subse‐ quent comparison of the experimental dependencies with the simulated ones, obtained after integrating, through the fiber length, of the saturation parameter *ss*0(*z*), with the latter being obtained from the equation for the excitation wave propagating in the fiber (*z* is the direction of light propagation) (see [11]):

$$\frac{ds\_{s0}}{dz} = -\alpha\_{s0} \frac{\mathcal{E}\_s}{\tilde{\xi}\_s} \Gamma\_s \mathbf{s}\_{s0} - \alpha\_{s0} \frac{\tilde{\xi}\_s - \mathcal{E}\_s}{\tilde{\xi}\_s^2} \ln \frac{1 + \tilde{\xi}\_s \mathbf{s}\_{s0}}{1 + \tilde{\xi}\_s \mathbf{s}\_{s0} \left(1 - \Gamma\_s\right)}.\tag{9}$$

Note that the contribution of amplified SE (ASE) to the output spectra was as low as 0.1–1.9% respectively to the overall transmitted power, provided a short (50 cm) EDF piece has been used.

The results of fitting of the experimental EDF transmission curves within the interval 1.48...1.59 μm are shown in Figure 3(a) by asterisks (see right scale). It is evident that these results, as the ones regarding the UCE experiments, almost coincide. The solid curve shown in this figure is the best fit of the experimental data by the polynomial regression.

The knowledge of the *εs-*spectrum for the 1.48...1.59 μm band allows one to find the spectral dependence of the net-gain coefficient *gs*0 in the EDF, which is the real gain in the fiber, in contrast to the gain coefficient *gs*, the quantity commonly but uncritically dealt with in experiments and simulations with EDF and EDFL13: In fact *gs*0 is diminished by the ESA parameter's value. Figure 3(b) demonstrates the spectral dependences of the three coefficients (*gs0*, *gs*, and *αs0*), where the spectra for *gs* and *αs0* were measured using standard methods (see

**Figure 3.** (a) ESA parameter *ε<sup>s</sup>* calculated from the measurements of PD signal (filled symbols) and the ones of the EDF transmission coefficient (stars); solid line is the best polynomial fit. (b) Spectral dependencies of the EDF's small-signal absorption coefficient *αs*0, full-saturated gain coefficient *gs*, and net-gain coefficient *gs*0.

*e.g.* Ref. 7) and the spectrum for *gs*<sup>0</sup> was calculated using the *gs*-and *αs0*-spectra and the fitted *εs*-spectrum, taken from Figure 3(b). Worth noticing, the data obtained above are universal for any silica-based EDF weakly doped with Er3+, where the "concentration" effects (to be discussed in section 3) are negligible.

#### **2.2. ESA in low-doped EDF at 977 nm**

where *ss*0 is the on-axis saturation parameter that depends on the pump power, *Ps*, as *Ps*=*Is*

mode's to fiber core's radii overlap factor at *λs*. The parameter *ξ<sup>s</sup>* comprises the ESA parameter *εs*, the measured EDF low-signal absorption coefficient *α<sup>s</sup>*0=Γ*sσ*12*N*0, and the full-saturated fiber

> 12 0 0 111 .

Apparently, UCE power depends on *n*<sup>3</sup> that in turns depends on *εs.* It is worth wising that the latter can be easily found from the UCE lateral signal, registered using PD (see Figure 2(b)). The results of calculation of the ESA parameters are plotted in Figure 3(a) by filled symbols (left scale), where the *εs*-spectrum is normalized on the GSA peak value (at *λs*=1.531 μm). It is seen that the ESA parameter weakly depends on *λs* within the spectral interval 1.48 μm to 1.55 μm whilst it sharply grows as *λs* approaches ∼1.6 μm; this behavior reminiscences the trend

The method to find the absolute values of the ESA parameter *εs* is based on the measurements of the EDF's transmission coefficient *T(λs, Ps)*, i.e. in function of excitation power, and subse‐ quent comparison of the experimental dependencies with the simulated ones, obtained after integrating, through the fiber length, of the saturation parameter *ss*0(*z*), with the latter being obtained from the equation for the excitation wave propagating in the fiber (*z* is the direction

0 0

*s s s s s s*


*ds s*

 xe

Note that the contribution of amplified SE (ASE) to the output spectra was as low as 0.1–1.9% respectively to the overall transmitted power, provided a short (50 cm) EDF piece has been

The results of fitting of the experimental EDF transmission curves within the interval 1.48...1.59 μm are shown in Figure 3(a) by asterisks (see right scale). It is evident that these results, as the ones regarding the UCE experiments, almost coincide. The solid curve shown in this figure is

The knowledge of the *εs-*spectrum for the 1.48...1.59 μm band allows one to find the spectral dependence of the net-gain coefficient *gs*0 in the EDF, which is the real gain in the fiber, in contrast to the gain coefficient *gs*, the quantity commonly but uncritically dealt with in experiments and simulations with EDF and EDFL13: In fact *gs*0 is diminished by the ESA parameter's value. Figure 3(b) demonstrates the spectral dependences of the three coefficients (*gs0*, *gs*, and *αs0*), where the spectra for *gs* and *αs0* were measured using standard methods (see

xx

*dz <sup>s</sup>* (9)

*s s ss s*

0 00 2

*s ss s*

the best fit of the experimental data by the polynomial regression.

*s*

 a

e

x

a

*s s*

saa

=+ =+ =+ + *s s s s*

21 0

gain *gs*=Γ*sσ*21*N*0 – Γ*sσ*24*N*0=*gs*0 – *εsαs*0 (*gs*0 is the EDF net gain at *λs*):

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

x

s

/2), *a* is the EDF core radius, *w<sup>s</sup>* is the modal radius of the Gaussian wave, and Γ*s* is the

 e

*g g* (8)

( )

0 <sup>1</sup> ln . 1 1

+ -G

 x

*ss*0(*πw<sup>s</sup>* 2

reported in [15].

used.

of light propagation) (see [11]):

*sat*

We report here the results of a study of the other ESA process, 4 I11/2→<sup>4</sup> F7/2 (see Figure 1), which takes the place when EDF is excited simultaneously at the two GSA wavelengths: *λp* ∼977 nm (through 4 I15/2→<sup>4</sup> I11/2 transition) and *λ<sup>s</sup>* ∼1531 nm (through 4 I15/2→<sup>4</sup> I13/2 transition), a situation normally encountered in a diode-pumped (at *λp*∼980 nm) EDFL or EDF-based amplifier (EDFA). Upon simultaneous excitation of EDF at these two wavelengths, population of 4 I11/2 state increases, resulting in significant growth of the pump-induced ESA loss (when 4 I11/2→<sup>4</sup> F7/2 transition gets "switched on", see Figure 1). On the contrary, without the presence of a signal wave in the fiber, almost all ions will be at the upper laser level 4 I13/2, which leads to a lower absorption at *λp*.

The first experiment was focused on the demonstration of the presence of "pump-ESA", when EDF is pumped simultaneously at *λp*=977 nm and at *λs*=1531 nm. It was aimed to reveal of whether UCE is observed (through transitions 2 H11/2→<sup>4</sup> I15/2 and 4 S3/2→<sup>4</sup> I15/2 "5"→"3", see Figure 1), as following the ESA process at *λp*. Experimentally, an EDF piece was pumped, using 980 nm / 1550 nm WDM supporting up to 1 W of optical power, by the pump and signal beams from the same EDF's side. These beams were delivered from a standard fibered LD (*λp*=977 nm) and from another semiconductor laser with wavelength *λs*=1531 nm, followed by an EDFA. Figure 4 presents two photographs of lateral emission from the EDF sample when the fiber was pumped, respectively, (*i*) solely at wavelength *λ<sup>p</sup>* (Figure 4(a)) and (*ii*) simultaneously at wavelengths *λp* and *λs* (Figure 4(b)); the incidence pump and signal powers were *Pp*=*Ps*=260 and very weak UCE when it is pumped at

I11/2<sup>4</sup>

coefficients (*gs0*, *gs*, and

977 nm (through <sup>4</sup>

population of <sup>4</sup>

loss (when <sup>4</sup>

4

*s0*-spectra and the fitted

**2.2. ESA in low-doped EDF at 977 nm** 

I15/2<sup>4</sup>

I13/2, which leads to a lower absorption at

when EDF is pumped simultaneously at

and  situation normally encountered in a diode-pumped (at

"concentration" effects (to be discussed in section 3) are negligible.

We report here the results of a study of the other ESA process, <sup>4</sup>

I11/2 transition) and

*s0*), where the spectra for *gs* and

standard methods (see *e.g.* Ref. 7) and the spectrum for *gs*0 was calculated using the *gs*-

obtained above are universal for any silica-based EDF weakly doped with Er3+, where the

which takes the place when EDF is excited simultaneously at the two GSA wavelengths:

amplifier (EDFA). Upon simultaneous excitation of EDF at these two wavelengths,

the presence of a signal wave in the fiber, almost all ions will be at the upper laser level

was pumped, using 980 nm / 1550 nm WDM supporting up to 1 W of optical power, by the pump and signal beams from the same EDF's side. These beams were delivered from a

*<sup>s</sup>*=1531 nm, followed by an EDFA. Figure 4 presents two photographs of lateral emission from the EDF sample when the fiber was pumped, respectively, (*i*) solely at wavelength

**Figure 4**. (a) and (b) Photographs of ~520/~545 nm UCE obtained from the EDF's surface at in-core

*<sup>p</sup>* only.

*p*. The first experiment was focused on the demonstration of the presence of "pump-ESA",

I11/2<sup>4</sup>

I15/2<sup>4</sup>

*<sup>p</sup>*980 nm) EDFL or EDF-based

*<sup>s</sup>* = 1531 nm. It was aimed to

I15/2 and <sup>4</sup>

*<sup>p</sup>*. Experimentally, an EDF piece

*<sup>s</sup>*-spectrum, taken from Figure 3(b). Worth noticing, the data

*<sup>s</sup>* 1531 nm (through <sup>4</sup>

I11/2 state increases, resulting in significant growth of the pump-induced ESA

*p* = 977 nm and at

F7/2 transition gets "switched on", see Figure 1). On the contrary, without

H11/2<sup>4</sup>

*s0* were measured using

F7/2 (see Figure 1),

I13/2 transition), a

S3/2<sup>4</sup>

I15/2

*p*

*p*

mW. One can readily see bright UCE (in the green spectral range, ~520–560 nm) when the EDF is pumped at both the wavelengths, and very weak UCE when it is pumped at *λp* only. (Figure 4(a)) and (*ii*) simultaneously at wavelengths *p* and *<sup>s</sup>* (Figure 4(b)); the incidence pump and signal powers were *Pp* = *Ps* = 260 mW. One can readily see bright UCE (in the green spectral range, ~520–560 nm) when the EDF is pumped at both the wavelengths,

reveal of whether UCE is observed (through transitions <sup>2</sup>

"5""3", see Figure 1), as following the ESA process at

excitation (a) only at *<sup>p</sup>* = 977 nm and (b) simultaneously at *<sup>p</sup>* = 977 nm and *<sup>s</sup>* = 1531 nm. (c) Frontal UCE spectra recorded from a 5-cm EDF sample at in-core excitation at pump wavelength (*Pp* = 260 mW) and variable signal powers (*Ps* = 0, 25, and 260 mW). **Figure 4.** (a) and (b) Photographs of ~520/~545 nm UCE obtained from the EDF's surface at in-core excitation (a) only at *λp*=977 nm and (b) simultaneously at *λp*=977 nm and *λs*=1531 nm. (c) Frontal UCE spectra recorded from a 5-cm EDF sample at in-core excitation at pump wavelength (*Pp*=260 mW) and variable signal powers (*Ps*=0,25, and 260 mW).

The UCE spectra, obtained using a 5-cm EDF sample, are shown in Figure 4(c). The spectra were recorded for three signal powers (0, 25, and 260 mW) while fixed pump power (260 mW). One can see from the figure that even low-power (25 mW) signal radiation tremendously enhances UCE (compare curves 1 and 2) and that at further increasing signal The UCE spectra, obtained using a 5-cm EDF sample, are shown in Figure 4(c). The spectra were recorded for three signal powers (0, 25, and 260 mW) while fixed pump power (260 mW). One can see from the figure that even low-power (25 mW) signal radiation tremendously enhances UCE (compare curves 1 and 2) and that at further increasing signal power, up to *Ps*=260 mW, growth of the UCE power becomes slower, demonstrating a saturating behavior (compare curves 2 and 3). Thus, the signal at 1531 nm "ignites" the ESA process at *λ<sup>p</sup>* and thereby increases the number of Er3+ ions in the ground state, resulting in an increase of the pump-light absorption. Consequently, there will be a non-negligible population of Er3+ ions on 4 I11/2 level and, as the fact of matter, a more effective ESA process at the pump wavelength, seen as growth of pump-induced loss.

Figure 5 shows the dependence of "green" UCE signal (transition "5"→"3", see Figure 1), detected from the EDF, on powers of pump and signal radiations launched into the fiber (in this case, again, the pump power was kept fixed, *Pp*=260 mW, and the signal power was varied, *Ps*=0...290 mW). Experimentally, lateral UCE power *PUCE*(*Ps*) was measured using a photomultiplier with a cesium cathode from a short section (∼1 mm) of the EDF near its splice with WDM. It is seen that the UCE's behavior demonstrates the already noticed aspects: At very low signal power UCE is extremely weak, whilst upon its increase UCE first strongly enhances and then gets saturated.

Since UCE power is proportional to population of the 5th level of Er3+ ions, the experimental data can be fitted well by a simulated curve of normalized population inversion *n*5=*N*5/*N*0 (*N*<sup>5</sup> is the population of the 5th Er3+ ion's level) averaged over the fiber core area, which confirms the theory we built. Considering that both the excitation waves (at 977 nm and at 1531 nm) have Gaussian spatial distributions, one can obtain from the steady-state rate equations for the Er3+ ion (formulas 1-5) the following expression for the averaged population *n*¯ 5:

mW. One can readily see bright UCE (in the green spectral range, ~520–560 nm) when the EDF is pumped at both the wavelengths, and very weak UCE when it is pumped at *λp* only.

(Figure 4(a)) and (*ii*) simultaneously at wavelengths

reveal of whether UCE is observed (through transitions <sup>2</sup>

"5""3", see Figure 1), as following the ESA process at

coefficients (*gs0*, *gs*, and

977 nm (through <sup>4</sup>

standard fibered LD (

excitation (a) only at

seen as growth of pump-induced loss.

and then gets saturated.

on 4

I11/2<sup>4</sup>

population of <sup>4</sup>

loss (when <sup>4</sup>

4

*s0*-spectra and the fitted

**2.2. ESA in low-doped EDF at 977 nm** 

I15/2<sup>4</sup>

I13/2, which leads to a lower absorption at

when EDF is pumped simultaneously at

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

and very weak UCE when it is pumped at

mW) and variable signal powers (*Ps* = 0, 25, and 260 mW).

**Figure 4.** (a) and (b) Photographs of ~520/~545 nm UCE obtained from the EDF's surface at in-core excitation (a) only at *λp*=977 nm and (b) simultaneously at *λp*=977 nm and *λs*=1531 nm. (c) Frontal UCE spectra recorded from a 5-cm EDF sample at in-core excitation at pump wavelength (*Pp*=260 mW) and variable signal powers (*Ps*=0,25, and 260 mW).

The UCE spectra, obtained using a 5-cm EDF sample, are shown in Figure 4(c). The spectra were recorded for three signal powers (0, 25, and 260 mW) while fixed pump power (260 mW). One can see from the figure that even low-power (25 mW) signal radiation tremendously enhances UCE (compare curves 1 and 2) and that at further increasing signal power, up to *Ps*=260 mW, growth of the UCE power becomes slower, demonstrating a saturating behavior (compare curves 2 and 3). Thus, the signal at 1531 nm "ignites" the ESA process at *λ<sup>p</sup>* and thereby increases the number of Er3+ ions in the ground state, resulting in an increase of the pump-light absorption. Consequently, there will be a non-negligible population of Er3+ ions

I11/2 level and, as the fact of matter, a more effective ESA process at the pump wavelength,

Figure 5 shows the dependence of "green" UCE signal (transition "5"→"3", see Figure 1), detected from the EDF, on powers of pump and signal radiations launched into the fiber (in this case, again, the pump power was kept fixed, *Pp*=260 mW, and the signal power was varied, *Ps*=0...290 mW). Experimentally, lateral UCE power *PUCE*(*Ps*) was measured using a photomultiplier with a cesium cathode from a short section (∼1 mm) of the EDF near its splice with WDM. It is seen that the UCE's behavior demonstrates the already noticed aspects: At very low signal power UCE is extremely weak, whilst upon its increase UCE first strongly enhances

Since UCE power is proportional to population of the 5th level of Er3+ ions, the experimental data can be fitted well by a simulated curve of normalized population inversion *n*5=*N*5/*N*0 (*N*<sup>5</sup> is the population of the 5th Er3+ ion's level) averaged over the fiber core area, which confirms the theory we built. Considering that both the excitation waves (at 977 nm and at 1531 nm) have Gaussian spatial distributions, one can obtain from the steady-state rate equations for the

Er3+ ion (formulas 1-5) the following expression for the averaged population *n*¯ 5:

and  situation normally encountered in a diode-pumped (at

"concentration" effects (to be discussed in section 3) are negligible.

We report here the results of a study of the other ESA process, <sup>4</sup>

I11/2 transition) and

*s0*), where the spectra for *gs* and

standard methods (see *e.g.* Ref. 7) and the spectrum for *gs*0 was calculated using the *gs*-

obtained above are universal for any silica-based EDF weakly doped with Er3+, where the

which takes the place when EDF is excited simultaneously at the two GSA wavelengths:

amplifier (EDFA). Upon simultaneous excitation of EDF at these two wavelengths,

the presence of a signal wave in the fiber, almost all ions will be at the upper laser level

was pumped, using 980 nm / 1550 nm WDM supporting up to 1 W of optical power, by the pump and signal beams from the same EDF's side. These beams were delivered from a

*<sup>s</sup>*=1531 nm, followed by an EDFA. Figure 4 presents two photographs of lateral emission from the EDF sample when the fiber was pumped, respectively, (*i*) solely at wavelength

pump and signal powers were *Pp* = *Ps* = 260 mW. One can readily see bright UCE (in the green spectral range, ~520–560 nm) when the EDF is pumped at both the wavelengths,

**Figure 4**. (a) and (b) Photographs of ~520/~545 nm UCE obtained from the EDF's surface at in-core

UCE spectra recorded from a 5-cm EDF sample at in-core excitation at pump wavelength (*Pp* = 260

The UCE spectra, obtained using a 5-cm EDF sample, are shown in Figure 4(c). The spectra were recorded for three signal powers (0, 25, and 260 mW) while fixed pump power (260 mW). One can see from the figure that even low-power (25 mW) signal radiation tremendously enhances UCE (compare curves 1 and 2) and that at further increasing signal

*<sup>p</sup>* = 977 nm and

*<sup>s</sup>* = 1531 nm. (c) Frontal

*<sup>p</sup>* = 977 nm and (b) simultaneously at

*<sup>p</sup>* only.

*p*. The first experiment was focused on the demonstration of the presence of "pump-ESA",

I11/2<sup>4</sup>

I15/2<sup>4</sup>

*<sup>p</sup>*980 nm) EDFL or EDF-based

*<sup>s</sup>* = 1531 nm. It was aimed to

*<sup>s</sup>* (Figure 4(b)); the incidence

I15/2 and <sup>4</sup>

*<sup>p</sup>*. Experimentally, an EDF piece

*<sup>s</sup>*-spectrum, taken from Figure 3(b). Worth noticing, the data

*<sup>s</sup>* 1531 nm (through <sup>4</sup>

I11/2 state increases, resulting in significant growth of the pump-induced ESA

*p* = 977 nm and at

F7/2 transition gets "switched on", see Figure 1). On the contrary, without

*<sup>p</sup>*=977 nm) and from another semiconductor laser with wavelength

*p* and 

H11/2<sup>4</sup>

*s0* were measured using

F7/2 (see Figure 1),

I13/2 transition), a

S3/2<sup>4</sup>

I15/2

*p*

*p*

**Figure 5.** Dependence of UCE power on signal power (left scale): Circles and rhombs correspond to two different ex‐ perimental series; plain curve is a theoretical fit of normalized inversion of the 5th Er3+ level (right scale).

$$\bar{\mathbf{m}}\_{\mathfrak{F}} = \frac{e\_p \boldsymbol{\gamma}\_2}{a^2} \mathbf{J} \frac{1 - n\_i(r) - n\_2(r)}{1 + e\_p \boldsymbol{\gamma}\_2 s\_p(r)} \mathbf{s}\_p(r) \mathbf{2} r dr \tag{10}$$

where the normalized populations *n*1(*r*) and *n*2(*r*) are taken from [16] (found from the same set of rate equations for the Er3+ ion); *εp*=*σ*35/*σ*<sup>13</sup> is the ESA parameter at *λp*, *γp*=*τ*53/*τ*31 (with *τ*53 ≈ *τ*<sup>54</sup> ≈1 μs and *τ*31 ≈ *τ*21 ≈10 ms, see Table 1), and *sp*= *I <sup>p</sup>* / *I <sup>p</sup> sat* is the saturation parameter at the pump wavelength (*Ip* is the pump intensity, *I <sup>p</sup> sat* =*hνp*/*σ*13*τ*21, and *hνp* is the quanta energy at *λp*). As seen from Figure 5, the simulated curve for population of the 5th level ( *n*¯ 5) fits well the experimental data, thus confirming correctness of the theory.

We found that the best way to find the ESA parameter at the pump wavelength, *λp*=977 nm, is to measure the EDF's nonlinear transmission coefficient at the pump wavelength in function of signal power, *Tp(Ps)*, and then compare this data with the simulated ones, obtained from the steady-state rate equations (formulas 1-5) for the fife level Er3+ energy diagram (see Figure 1). Considering the Gaussian radial intensity distributions for the pump and signal waves, the equations describing *Tp(Ps)* at fixed pump power (*Pp*=260 mW in our case) take the form [16]:

$$\frac{d P\_p(z)}{dz} = -\frac{4\alpha\_{p0}}{\Gamma\_p w\_p^2} \left[ \oint\_l n\_1(r, \ z) \cdot \{\xi\_p \cdot \varepsilon\_p\} n\_3(r, \ z) \right] \exp\left[-2\left(\frac{r}{w\_p}\right)^2\right] r dr + \alpha\_{BG} \right] P\_p(z) \tag{11}$$

$$\frac{d^2 P\_s(z)}{dz} = \frac{4a\_{s0}}{\Gamma\_s w\_s^2} \Bigg[ \prod (\xi\_s - \varepsilon\_s) n\_2(r\_s \ z) - n\_1(r\_s \ z) \Bigg] \exp \Bigg[ -2 \left( \frac{r}{w\_s} \right)^2 \Bigg] r dr - \alpha\_{BG} \Bigg] P\_s(z) \tag{12}$$

where *αp*0 is the small-signal EDF absorption and Γ*p* is the pump wave to fiber core overlap factor, both at the pump wavelength, *αBG* is the small background EDF loss (~3 dB/km for the EDF used), *wp* is the radius of the Gaussian distribution of pump wave, and *ξp*=1+*σ*31/*σ*13 is the SE parameter at the pump wavelength. The normalized populations of Er3+ levels *n*1, *n*2, and *n*<sup>3</sup> are found from the rate equations for the Er3+ ion (the routine is not discussed here because of its completeness; refer for details to [16]).

Actually, the coefficient adjacent to *Pp* from the left side (Equation (11)) is the EDF absorption coefficient at the pump wavelength, whereas the one adjacent to *Ps* from the left side (Equation (12)) is the EDF gain at the signal wavelength; both the coefficients depend on the pump and the signal powers. It is worth of mentioning that the set of equations (11) and (12), added by the corresponding boundary conditions, can be also used for modeling a CW EDFL.

Figure 6 shows the EDF transmissions measured for three different EDF lengths along with the best fits obtained using equations (7) at varying *εp* and *ξp* (and as the result their most relevant values found to be: *εs*=0.17 and *ξs*=1.08, both at 1531 nm). As seen from the figure, the best fitting of the experimental EDF transmission at the pump wavelength is provided at *εp*=0.95 and *ξp*=1.08. Thus, the ESA cross-section at the pump wavelength is strong enough (being less than the GSA one only slightly), which ought to affect (decrease) an EDFL's efficiency via the pump-induced loss distributed along the active fiber.

**Figure 6.** Experimental (symbols) and theoretical (curves) dependencies of pump (*λp*=977 nm) wave's transmission co‐ efficient on signal (*λs*=1531 nm) wave's power; the data are obtained for three different EDF lengths (indicated in the inset). Theoretical curves are the best fits to the experimental data obtained at *εp*=0.95 and *ξp*=1.08.

#### **2.3. Effect of ESA in EDF upon efficiency of CW EDFL.**

where *αp*0 is the small-signal EDF absorption and Γ*p* is the pump wave to fiber core overlap factor, both at the pump wavelength, *αBG* is the small background EDF loss (~3 dB/km for the EDF used), *wp* is the radius of the Gaussian distribution of pump wave, and *ξp*=1+*σ*31/*σ*13 is the SE parameter at the pump wavelength. The normalized populations of Er3+ levels *n*1, *n*2, and *n*<sup>3</sup> are found from the rate equations for the Er3+ ion (the routine is not discussed here because

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

Actually, the coefficient adjacent to *Pp* from the left side (Equation (11)) is the EDF absorption coefficient at the pump wavelength, whereas the one adjacent to *Ps* from the left side (Equation (12)) is the EDF gain at the signal wavelength; both the coefficients depend on the pump and the signal powers. It is worth of mentioning that the set of equations (11) and (12), added by

Figure 6 shows the EDF transmissions measured for three different EDF lengths along with the best fits obtained using equations (7) at varying *εp* and *ξp* (and as the result their most relevant values found to be: *εs*=0.17 and *ξs*=1.08, both at 1531 nm). As seen from the figure, the best fitting of the experimental EDF transmission at the pump wavelength is provided at *εp*=0.95 and *ξp*=1.08. Thus, the ESA cross-section at the pump wavelength is strong enough (being less than the GSA one only slightly), which ought to affect (decrease) an EDFL's

**Figure 6.** Experimental (symbols) and theoretical (curves) dependencies of pump (*λp*=977 nm) wave's transmission co‐ efficient on signal (*λs*=1531 nm) wave's power; the data are obtained for three different EDF lengths (indicated in the

inset). Theoretical curves are the best fits to the experimental data obtained at *εp*=0.95 and *ξp*=1.08.

the corresponding boundary conditions, can be also used for modeling a CW EDFL.

efficiency via the pump-induced loss distributed along the active fiber.

of its completeness; refer for details to [16]).

We discuss hereafter the results of a theoretical study of influence of ESA inherent in Er3+ ions upon efficiency of an EDFL assembled in the linear (Fabry-Perot) configuration. The laser setup we shall deal with at modeling is shown in Figure 7. The low-doped EDF discussed above was considered to be an active medium and two fiber Bragg Gratings (FBGs) – to form selective couplers of the cavity, both centered at *λs*=1550 nm, the laser (signal) wavelength. The EDF length was chosen to be 4 m whereas the EDFL cavity length *Lc*, including the EDF piece and FBGs' tails, to be 6 m. Pump power on the EDF input was fixed in modeling at 200 mW.

**Figure 7.** Sketch of the EDFL's geometry: *Ps* + and *Ps* are the laser (signal) waves' powers propagating in the positive (+) and negative (-) z direction, respectively; Pp is the pump power propagating in the positive z direction; t1 and t2 are the transmissions on the fibers' splices.

We imply (see Figure 7) that the pump wave at 977 nm (absorption peak of 4 I15/2→<sup>4</sup> I11/2 transition, see Figure 1) propagates rightward while the laser waves – rightward (marked by "+" super‐ script) and leftward (marked by "–" superscript), respectively. In the scheme, FBG1 plays the role of a rear 100% reflector (reflection *R*1=1), transparent for the pump wave, whereas FBG2 forms an output coupler with reflection *R*2 varied for optimizing the laser efficiency. To decrease loss on the fiber splices, both FBGs were considered to be written in a photosensitive fiber with waveguide parameters similar to the EDF's ones. At the laser wavelength the lowsignal EDF absorption *αs*0 was taken to be 0.0069 cm-1, according to [17].

The laser is simulated by applying the two contra-propagating laser waves' model discussed in details in [17] with taking into account Gaussian distributions of the laser and pump waves. In this model, the pump wave is described by equation (11) and the signal waves – by equation (12) with a small modification being that powers of the two contra-propagating signal waves are assumed to be governed by the equation:

$$\frac{d\,P\_s^{\pm}(z)}{dz} = \pm \frac{4\alpha\_{s0}}{\Gamma\_s w\_s^2} \left[ \iint (\xi\_s - \varepsilon\_s) n\_2(r\_s, z) \cdot n\_1(r\_s, z) \mathbf{I} \exp\left[ -2\left(\frac{r}{w\_s}\right)^2 \right] r dr \mp \alpha\_{\rm BG} \right] \mathbf{P}\_s^{\pm}(z) \tag{13}$$

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 centered at *λse*=1531 nm (this wavelength corresponds to the GSA peak of 4 I15/2→<sup>4</sup> I13/2 transition), powers of which obey the equation:

$$\frac{d P\_{\rm sc}^{\pm}(z)}{dz} = \pm \left[ g\_{\rm sc}(z) P\_{\rm sc}^{\pm}(z) \mp \frac{\Omega}{4\pi} \frac{a\_{\rm sc0} P\_{\rm sc}^{\rm sat}}{\Gamma\_{\rm sw}} f\_0^a n\_2(r, \, \, z) 2r dr \right] \tag{14}$$

where the second term on the right side is the SE power generated by a short fiber section *dz*, *αse*0=0.016 cm-1 is the low-signal absorption at *λse*, *Pse sat* is the saturation power at 1531 nm, Γ*se* is the overlap factor for the SE waves, Ω=π*NA*/*n*<sup>2</sup> is the fraction of SE photons guided by the EDF core in each direction, and *n* is the modal refractive index at *λse*. The EDF gain at *λse* is written as

$$\mathcal{G}\_{sc}\begin{pmatrix} z \\ \end{pmatrix} = \frac{4a\_{sc0}}{\Gamma\_{sc} w\_{sc}^2} \left[ \prod (\xi\_{sc} - \varepsilon\_{sc}) n\_2(r\_s \ z) - n\_1(r\_s \ z) \right] \exp\left[ -2 \left( \frac{r}{w\_{sc}} \right)^2 \right] r dr - \alpha\_{BG} \right] \tag{15}$$

The boundary conditions are written as:

$$P\_p\begin{pmatrix}\mathcal{Z}=\mathcal{0}, & t\end{pmatrix} = P\_{p\,0} \tag{16}$$

$$P\_s^+(z=0,\ \ t) = P\_s^-(z=0,\ \ t)R\_1t\_1^2\tag{17}$$

$$P\_s\{z=L\_{\bf \cdot \varepsilon'}\; t\} = P\_s\{z=L\_{\bf \cdot \varepsilon'}\; t\} R\_2 t\_2^{\bf \cdot 2} \tag{18}$$

$$P\_{se}^{+}(z=0,\ \ t) = P\_{se}^{-}(z=L\_{\ c'}\ \ t) = 0\tag{19}$$

$$P\_s^{\
out}(t) = P\_s^{\
\ast}(z = L\_{-c'} \ t)(1 - R\_2)t\_2 = 0\tag{20}$$

where *Pp*0 is the pump power at the EDF input and *Ps out* is the EDFL output power. To simplify 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 all the ESA transitions (curve 4).

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

**Figure 8.** (a) EDFL output power and (b) fractions of the absorbed pump photons spent on the laser output (quantum efficiency) as functions of output (FBG2) reflectivity, *R*2. The designations of curves 1 to 4 are given in the text. The small fraction of pump photons spent at the background loss and the ASE contribution are not shown.

of the ESA loss are accounted for (curve 4). The other important fact is that the range of FBG2's reflections, in which the output laser power varies within 10% (with respect to its maximum value if all the ESA transitions are considered), is relatively broad: ~2.5% to ~34%. At the optimal FBG2's reflectivity (R2=11%) the laser efficiency reaches ~34% (implying all fiber splices are made lossless).

The fractions of pump and signal photons spent on the ESA transitions with respect to the absorbed by EDF pump photons are shown in Figure 8(b). It is seen that the contribution of the signal ESA loss is bigger when FBG2's reflection is bigger (curve 2). Furthermore, if reflection of the output coupler approaches 100%, the absorbed pump power is spent entirely on the ESA transitions (see curves 2 and 3) and no photons at the laser output are produced (see curve 1). At optimal FBG2's reflectivity (≈11%), about 23% of absorbed pump photons are spent on ESA at the laser wavelength and about 9.5% on ESA at the pump wavelength. Note that the sum of the relative photon numbers, as can be revealed from curves 1, 2 and 3, is approximately equal to 1 in the whole range of FBG2's reflections.

The reader is advised here to refer to [18] for comparison of the developed theory with some of the experimental data on laser efficiency of EDFLs based on the EDF of M-type with relatively high Er3+ concentrations.

Finally, we conclude that the ESA processes at the laser and pump wavelengths strongly affect an EDFL's efficiency and output coupler's optimal reflectivity, at which the laser output power is maximal.
