## The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation

Optics Express, Vol. 13, Issue 13, pp. 5179-5185 (2005)

http://dx.doi.org/10.1364/OPEX.13.005179

Acrobat PDF (180 KB)

### 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

*nm*). In addition, the temperature dependence of the gain characteristics of EDFAs has also of great importance for WDM systems [1

1. J. Kemtchou, M. Duhamel, and P. Lecoy, “Gain Temperature Dependence of Erbium-Doped silica and Fluoride Fiber Amplifiers in Multichannel Wavelength-Multiplexed Transmission Systems,” IEEE J. Lightwave Tech. **15 (11)**, 2083–2090 (1997). [CrossRef]

2. M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,” Opt. Lett. **15**, 842–844 (1990). [CrossRef] [PubMed]

3. N. Kagi, A. Oyobe, and K. Nakamura, “Temperature Dependence of the Gain in Erbium-Doped Fibers,” IEEE J. Lightwave Tech. **9 (2)**, 261–265 (1991). [CrossRef]

4. H. Wei, Z. Tong, and S. Jian, “Use of a genetic algorithm to optimize multistage erbium-doped amplifier systems with complex structures,” Opt. Express **12 (4)**, 531–544 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-531. [CrossRef]

5. K. Furusawa, T. M. Monro, and D. J. Richardson, “High gain efficiency amplifier based on an erbium doped aluminosilicate holey fiber,” Opt. Express **12 (15)**, 3452–3458 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3452. [CrossRef]

6. M. Yamada, M. Shimizu, M. Horiguchi, and M. Okayasu, “Temperature Dependence of Signal Gain in *Er*^{3+}-Doped Optical Fiber Amplifiers,” IEEE J. Quantum Electron. **28 (3)**, 640–649 (1992). [CrossRef]

7. Q. Mao, J. Wang, X. Sun, and M. Zhang, “A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifier with single erbium-doped fiber,” Opt. Commun. **159**, 149–157 (1999). [CrossRef]

8. F. Prudenzano, “Erbium-Doped Hole-Assisted optical Fiber Amplifier: Design and Optimization,” IEEE J. Light-wave Tech. **23 (1)**, 330–340 (2005). [CrossRef]

*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

*nm*.

^{-1}),

*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:

*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].

*dN*

_{i}/

*dt*=0 (

*i*=1, 2). In the last two equations,

*hν*

_{p}/

*hν*

_{s}/

*, ν*

_{p}and

*ν*

_{s}are the pump and the signal frequencies, respectively;

*I*

_{p}and

*I*

_{s}are the pump and signal intensities and

^{3+}ions is

*N*,

*N*=

*N*

_{1}+

*N*

_{2-}+

*N*

_{2+}or in terms of β,

*N*=

*N*

_{1}+(1+

*β*)

*N*

_{2-}.

*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:

*α*

_{s}=2

*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:

*A*

*N*

_{2-}

*f*(

*r*)

*rdr*/

*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*)~

*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

*z*=

*L*and hence establish a relationship between the amplifier gain and length:

*G*=

*P*

_{s}(

*L*)/

*P*

_{s}(0) can be calculated from the following equation:

*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

2. M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,” Opt. Lett. **15**, 842–844 (1990). [CrossRef] [PubMed]

12. M. C. Lin and S. Chi, “The Gain and Optimal Length in the Erbium-Doped Fiber Amplifiers with 1480 nm Pumping,” IEEE Photonics Tech. Lett. **4 (4)**, 354–356 (1992). [CrossRef]

*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

*R*=

*N*(

*r*)

*f*(

*r*)

*rdr*/

*N*(

*r*)

*rdr*. It is notes that the output pump power is a function of the temperature.

## 3. Results and discussion

*f*(

*r*) in Gaussian form,

*f*(

*r*)=

*exp*(-

*r*

^{2}/

*ω*

_{0}is the spot size and the effective core area is

*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

11. H. Zech, “Measurment Technique for the Quotient of Cross Section σ_{e}(λ_{S})/σ_{a}(λ_{S}) of Erbium-Doped Fibers,” IEEE Photonics Tech. Lett. **7 (9)**, 986–988 (1995). [CrossRef]

*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

12. M. C. Lin and S. Chi, “The Gain and Optimal Length in the Erbium-Doped Fiber Amplifiers with 1480 nm Pumping,” IEEE Photonics Tech. Lett. **4 (4)**, 354–356 (1992). [CrossRef]

*OptiAmplifier 4.0*for generating

*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.

*β*and

*η*are given in Table 2.

*C*, 20 °

*C*and 60 °

*C*in Fig. 3.

*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

## Acknowledgments

## References and links

1. | J. Kemtchou, M. Duhamel, and P. Lecoy, “Gain Temperature Dependence of Erbium-Doped silica and Fluoride Fiber Amplifiers in Multichannel Wavelength-Multiplexed Transmission Systems,” IEEE J. Lightwave Tech. |

2. | M. Peroni and M. Tamburrini, “Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,” Opt. Lett. |

3. | N. Kagi, A. Oyobe, and K. Nakamura, “Temperature Dependence of the Gain in Erbium-Doped Fibers,” IEEE J. Lightwave Tech. |

4. | H. Wei, Z. Tong, and S. Jian, “Use of a genetic algorithm to optimize multistage erbium-doped amplifier systems with complex structures,” Opt. Express |

5. | K. Furusawa, T. M. Monro, and D. J. Richardson, “High gain efficiency amplifier based on an erbium doped aluminosilicate holey fiber,” Opt. Express |

6. | M. Yamada, M. Shimizu, M. Horiguchi, and M. Okayasu, “Temperature Dependence of Signal Gain in |

7. | Q. Mao, J. Wang, X. Sun, and M. Zhang, “A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifier with single erbium-doped fiber,” Opt. Commun. |

8. | F. Prudenzano, “Erbium-Doped Hole-Assisted optical Fiber Amplifier: Design and Optimization,” IEEE J. Light-wave Tech. |

9. | C. Berkdemir and S. Özsoy, “An investigation on the temperature dependence of the relative population inversion and the gain in EDFAs by the modified rate equations,” accepted for publication in Opt. Commun. (2005). |

10. | E. Desurvire, |

11. | H. Zech, “Measurment Technique for the Quotient of Cross Section σ |

12. | M. C. Lin and S. Chi, “The Gain and Optimal Length in the Erbium-Doped Fiber Amplifiers with 1480 nm Pumping,” IEEE Photonics Tech. Lett. |

13. | OptiAmplifier Version 4.0; |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.2360) Fiber optics and optical communications : Fiber optics links and subsystems

(060.2410) Fiber optics and optical communications : Fibers, erbium

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 3, 2005

Revised Manuscript: June 21, 2005

Published: June 27, 2005

**Citation**

Cüneyt Berkdemir and Sedat �?zsoy, "The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation," Opt. Express **13**, 5179-5185 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-5179

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### References

- J. Kemtchou, M. Duhamel, and P. Lecoy, �??Gain Temperature Dependence of Erbium-Doped silica and Fluoride Fiber Amplifiers in Multichannel Wavelength-Multiplexed Transmission Systems,�?? IEEE J. Lightwave Tech. 15 (11), 2083�??2090 (1997). [CrossRef]
- M. Peroni and M. Tamburrini, �??Gain in erbium-doped fiber amplifiers: a simple analytical solution for the rate equations,�?? Opt. Lett. 15, 842�??844 (1990). [CrossRef] [PubMed]
- N. Kagi, A. Oyobe, and K. Nakamura, �??Temperature Dependence of the Gain in Erbium-Doped Fibers,�?? IEEE J. Lightwave Tech. 9 (2), 261�??265 (1991). [CrossRef]
- H. Wei, Z. Tong, and S. Jian, �??Use of a genetic algorithm to optimize multistage erbium-doped amplifier systems with complex structures,�?? Opt. Express 12 (4), 531�??544 (2004), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-531">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-531</a>. [CrossRef]
- K. Furusawa, T. M. Monro, and D. J. Richardson, �??High gain efficiency amplifier based on an erbium doped aluminosilicate holey fiber,�?? Opt. Express 12 (15), 3452�??3458 (2004), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3452">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3452</a>. [CrossRef]
- M. Yamada, M. Shimizu, M. Horiguchi, and M. Okayasu, �??Temperature Dependence of Signal Gain in Er3+- Doped Optical Fiber Amplifiers,�?? IEEE J. Quantum Electron. 28 (3), 640�??649 (1992). [CrossRef]
- Q. Mao, J. Wang, X. Sun, M. Zhang, �??A theoretical analysis of amplification characteristics of bi-directional erbium-doped fiber amplifier with single erbium-doped fiber,�?? Opt. Commun. 159, 149�??157 (1999). [CrossRef]
- F. Prudenzano, �??Erbium-Doped Hole-Assisted optical Fiber Amplifier: Design and Optimization,�?? IEEE J. Lightwave Tech. 23 (1), 330�??340 (2005). [CrossRef]
- C. Berkdemir and S. �?zsoy, �??An investigation on the temperature dependence of the relative population inversion and the gain in EDFAs by the modified rate equations,�?? accepted for publication in Opt. Commun. (2005).
- E. Desurvire, Erbium-Doped fiber Amplifiers; Principle and Applications (John Wiley and Sons. Inc, New York, 1994).
- H. Zech, �??Measurment Technique for the Quotient of Cross Section �?e(λs)/�?a(λs) of Erbium-Doped Fibers,�?? IEEE Photonics Tech. Lett. 7 (9), 986�??988 (1995). [CrossRef]
- M. C. Lin, and S. Chi, �??The Gain and Optimal Length in the Erbium-Doped Fiber Amplifiers with 1480 nm Pumping,�?? IEEE Photonics Tech. Lett. 4 (4), 354�??356 (1992). [CrossRef]
- OptiAmplifier Version 4.0; Optical Fiber Amplifier and Laser Design Software (Copyright © 2002 Optiwave Corporation, 2002).

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