## Abstract

An analytically expression for the temperature dependence of the signal gain of an erbium-doped fiber amplifier (EDFA) pumped at 1480 *nm* are theoretically obtained by solving the propagation equations with the amplified spontaneous emission (ASE). It is seen that the temperature dependence of the gain strongly depends on the distribution of population of Er^{3+}-ions in the second level. In addition, the output pump power and the intrinsic saturation power of the signal beam are obtained as a function of the temperature. Numerical calculations are carried out for the temperature range from -20 to +60 °*C* and the various fiber lengths. But the other gain parameters, such as the pump and signal wavelengths and their powers, are taken as constants. It is shown that the gain decreases with increasing temperature within the range of *L*≤27 *m*.

© 2005 Optical Society of America

## 1. Introduction

EDFAs have been presenting many advantages such as high gain and low noise in the optical communication networks and providing a broadband amplification of radiation whose wavelength is in the so-called third window for fiber-optic communication (~1530 *nm*). In addition, the temperature dependence of the gain characteristics of EDFAs has also of great importance for WDM systems [1]. An analytical solution of the rate equation has been derived and the gains at optimum amplifier lengths have been experimentally found for the various temperature values in the previous works [2, 3, 4, 5]. The theoretical and experimental results for the temperature dependence of the gain with the various lengths of EDFAs have been reported [6], but the temperature dependent analytic expressions has not been given in terms of the Boltzmann factor. Afterwards, the theoretical analysis of amplification characteristics of EDFAs has been developed to calculate the signal gain, using the rate equation model, [7], and this rate equation has been modified by including the temperature and cross section factors to understand the dependence of the gain on the temperature [8, 9] for EDFAs pumped at 1480 *nm*. In this article, we present an analytical expression for the signal gain in EDFAs, using the propagation equations improved by including the temperature effects, and the numerical results for the temperature ranges of -20 °*C* to +60 °*C*. We took into account the amplified spontaneous emission (ASE), but neglected the excited state absorption (ESA) effect for the simplicity.

## 2. Theory

The simplest treatment of EDFA considers the two-level amplification system with energy levels as shown in Fig. 1, when it is pumped at 1480 *nm*.

In this figure, level 1 is the ground level and level 2 is the metastable level characterized by a long lifetime τ (=γ^{-1}), ${R}_{p}^{a\mathit{,}e}$
is the pump absorption and stimulated emission rates, *S*
_{12}
_{,21} is the signal stimulated absorption and emission rates, respectively. *N*
_{2+} and *N*
_{2-} are the populations of Er^{3+} ions within the sub-levels of the second energy state and it is possible to consider each of them as a single energy level. Actually, this system contains many sub-levels where the erbium-ions reside and they are unequally populated due to the thermal distribution of the ions. Thus, the relative occupation of the sub-levels in the thermal equilibrium must be arranged as a function of the temperature. This arrangement is governed by Boltzmann’s distribution law:

where *T* is the temperature in degrees Kelvin, kB is Boltzmann’s constant. *E*
_{2+} and *E*
_{2-} are the higher and lower sub-levels energies of the second level, respectively, and Δ*E*
_{2}=*E*
_{2+}-*E*
_{2-}[10]. ${C}_{\mathit{\text{nr}}}^{+}$
and ${C}_{\mathit{\text{nr}}}^{-}$
are the nonradiative rates which correspond to the thermalization process occurring within each manifold of the second level. The rate equations corresponding to the two levels 1 and 2 can be given as follow

Thus, at stationary conditions we now obtain

or

where the populations are time invariant, i.e., *dN*_{i}
/*dt*=0 (*i*=1, 2). In the last two equations, ${b}_{p}^{a\mathit{,}e}$
=*hν*_{p}
/${\tau \sigma}_{p}^{a\mathit{,}e}$${\mathit{,}b}_{s}^{a\mathit{,}e}$
=*hν*_{s}
/${\tau \sigma}_{s}^{a\mathit{,}e}$*, ν*_{p}
and *ν*_{s}
are the pump and the signal frequencies, respectively; ${\sigma}_{p}^{a\mathit{,}e}$
is the stimulated absorption and emission cross sections of the pump beam while ${\sigma}_{s}^{a\mathit{,}e}$
is the the stimulated absorption and emission cross sections of the signal beam, respectively; *I*_{p}
and *I*_{s}
are the pump and signal intensities and ${I}_{\mathit{\text{ASE}}}^{\pm}$
is the forward (+ sign) and backward (- sign) propagating optical intensities, respectively. η is the ratio between the signal emission and absorption cross sections, and the total concentration distribution of Er^{3+} ions is *N*,*N*=*N*
_{1}+*N*
_{2-}+*N*
_{2+} or in terms of β,*N*=*N*
_{1}+(1+*β*)*N*
_{2-}.

The differential equations for propagation of the signal, pump and ASE powers are given, respectively, as follows

where ${f}_{\mathit{\text{ASE}}}^{\pm}$ is the normalized ASE intensity profile, ${P}_{\mathit{\text{ASE}}}^{\pm}$ is the amplified spontaneous emission power at the position z and has to be determined from a forward as well as a backward travelling ASE spectrum,

We can decompose the intensity as *I*_{s,p}
*(z,r*)=*P*_{s,p}
(*z*)*f*_{s,p}
(*r*) where *P*_{s,p}
(*z*) is z-dependent signal od pump powers and *f*_{s,p}
(*r*) is the normalized signal and pump transverse intensity profiles, respectively. At this point, by substituting *N*=*N*
_{1}+(1+*β*)*N*
_{2}- into Eq. (7), we have the propagation equation for the signal power:

where *α*_{s}
=2${\pi \sigma}_{s}^{a}$
${\int}_{0}^{\infty}$
*N*(*r*)*f*(*r*)*rdr* is the absorption constant of the signal beam. To evaluate the integral at the right-hand side of Eq. (11), we make use of Eq. (5). In this case, multiplying both-hand side of Eq. (5) with *rdr* and then integrating between 0 and ∞, we obtain the following equations:

$$+{\int}_{0}^{\infty}\frac{\tau {I}_{\mathit{ASE}}^{+}}{h{v}_{s}}\left({\sigma}_{s}^{a}{N}_{1}-{\sigma}_{s}^{e}{N}_{2-}\right)\mathit{rdr},$$

where we define the confinement factor Γ=*A*${\int}_{0}^{\infty}$
*N*
_{2-}
*f*(*r*)*rdr*/${\int}_{0}^{\infty}$
*N*
_{2}-*rdr* and A is the effective doped area. We can put the equations into more practical form supposing the pump, signal and ASE profiles to be approximately equal, so that the transverse profiles *f*_{p}
(*r*)~*f*_{s}
(*r*)~${f}_{\mathit{\text{ASE}}}^{+}$
(*r*)=*f*(*r*) and considering the co-propagating scheme in the positive z direction for the simplicity. Inserting Eq. (14) into Eq. (11), we have

where the intrinsic saturation power of the signal beam is introduced as follow

Therefore, we define the intrinsic saturation power as a function of the temperature. Integrating Eq. (15), we obtain the output signal power at *z*=*L* and hence establish a relationship between the amplifier gain and length:

The amplifier gain *G*=*P*_{s}
(*L*)/*P*_{s}
(0) can be calculated from the following equation:

with boundary condition ${P}_{\mathit{\text{ASE}}}^{+}$
(0)=0. If one neglects the effect of the *β* parameter and the ASE power in the gain equation, it can be easily seen that the relevant equation is reduced to the previous works [2, 12]. Thus, Eq. (18) is a more accurate solution for the propagation equations. In order to obtain the output pump power *P*_{p}
(*L*) in Eq. (18) for the maximal pumping efficiency, it should be substituted Eq. (6) into Eq. (7) and Eq. (8), and then Eq. (7) divided by Eq. (8). If the obtained result makes equal to zero, we have

where *R*=${\int}_{0}^{\infty}$
*N*(*r*)*f*(*r*)*rdr*/${\int}_{0}^{\infty}$
*N*(*r*)*rdr*. It is notes that the output pump power is a function of the temperature.

## 3. Results and discussion

The gain against fiber lengths is calculated in the following way. Firstly, we take *f*(*r*) in Gaussian form, *f*(*r*)=*exp*(-*r*
^{2}/${\omega}_{0}^{2}$)/${\pi \omega}_{0}^{2}$ where *ω*
_{0} is the spot size and the effective core area is ${\pi \omega}_{0}^{2}$=33 *µm*
^{2}. Dopant distribution *N*(*r*) is also assumed to be Gaussian, *N*(*r*)≃*exp*(-*r*
^{2}/*ω*
^{2})/*πω*
^{2}. In addition, the ratio (*ω*/*ω*
_{0}) between Gaussian dopant distribution and transverse intensity profiles is selected as 0.3. Secondly, we obtain *R* and *α*_{s}
by using *N*(*r*) and *f*(*r*) for the relevant fiber parameters. Thus, the output pump power in Eq. (19) is calculated with the different temperature values for the fiber length of 45 *m*. In this case, it is bear in mind that the ratio of cross-sections, which are belong to the signal beam, depends on the temperature. To calculate the parameter *η* as a function of the temperature, we benefit by McCumber’s theory, which gives a highly accurate relation between emission and absorption cross sections [11].

In the numerical calculations, we select the Al/P-silica erbium-doped fiber as an amplifier operated at the pump wavelength *λ*_{p}
=1480 *nm* and the input pump power *P*_{p}
(0) is fixed at 30 mW. The signal wavelength *λ*_{s}
and the signal power *P*_{s}
(0) are taken as 1530 *nm* and 10 *µW*, respectively. The other parameters assigned to the fiber are given in Table 1 [12]. Moreover,

we used the simulation programme *OptiAmplifier 4.0* for generating ${P}_{\mathit{\text{ASE}}}^{+}$
(*L*) only, and we set up the basic system seen in Fig. 2 [13]. The energy difference between the sublevels of the metastable level (level 2) is assumed as 300 *cm*
^{-1} in the room temperature for the simplicity.

The results calculated for the various output pump powers and the intrinsic saturation powers as well as the parameters *β* and *η* are given in Table 2.

The variation of the signal gain against the fiber length is illustrated for the temperatures -20 °*C*, 20 °*C* and 60 °*C* in Fig. 3.

For a given pump and signal powers, the gain decreases with increasing temperature within the range of* L*≤27 *m*. The difference between the maximum gains for -20 and 60 °*C* is 0.67 *dB*. There is a temperature insensitivity for the length about *L*≈30 *m* for the relevant pump and signal powers. On the other hand, this temperature insensitive length is equivalent to the length at which the gain curves intersect each other.

## 4. Conclusion

We have introduced a more accurate model including the temperature effect for the signal gain of the erbium-doped fiber amplifier. In addition, we have shown the possibility of deriving an analytical solution of the propagation equations for some practical temperature ranges. The temperature dependence of the output pump power is smaller than that of the intrinsic saturation power. Thus, in terms of the practical applications we can neglect the dependence of the output pump power on the temperature. However, it is taken into consideration that the gain performance of EDFAs strongly depends on the temperature.

## Acknowledgments

This study is supported by Scientific Research Projects Council (SRPC) of Erciyes University under Grant No FBT-04-17. The authors are grateful to A. ALTUNCU for his useful comments and discussions on the original version of the paper.

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