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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 13 — Jun. 27, 2005
  • pp: 5179–5185
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The temperature dependent performance analysis of EDFAs pumped at 1480 nm: A more accurate propagation equation

Cüneyt Berkdemir and Sedat Özsoy  »View Author Affiliations


Optics Express, Vol. 13, Issue 13, pp. 5179-5185 (2005)
http://dx.doi.org/10.1364/OPEX.13.005179


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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 Er3+-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

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.

Fig. 1. Two level amplification system and main transitions of erbium ion.

In this figure, level 1 is the ground level and level 2 is the metastable level characterized by a long lifetime τ (=γ-1), Rpa,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 Er3+ 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:

β=N2+N2=Cnr+Cnr=exp(ΔE2kBT)
(1)

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 ΔE 2=E 2+-E 2-[10

10. E. Desurvire, Erbium-Doped fiber Amplifiers; Principle and Applications (John Wiley and Sons. Inc, New York, 1994).

]. Cnr+ and Cnr 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

dN2+dt=RpaN1RpeN2++CnrN2Cnr+N2+,
(2)
dN2dt=S12N1S21N2N2γCnrN2+Cnr+N2+,
(3)
dN1dt=RpeN2+RpaN1+S21N2S12N1+N2γ.
(4)

Thus, at stationary conditions we now obtain

N2=τ[(σpaN1βσpeN2−)Iphvp+(σsaN1σseN2)(Is+IASE±)hvs],
(5)

or

N2N=Ipbpa+(Is+IASE±)bsa(1+β)Ipbpa+βIpbpe+(1+β+η)(Is+IASE±)bsa+1
(6)

where the populations are time invariant, i.e., dNi /dt=0 (i=1, 2). In the last two equations, bpa,e =p /τσpa,e,bsa,e =s /τσsa,e, νp and νs are the pump and the signal frequencies, respectively; σpa,e is the stimulated absorption and emission cross sections of the pump beam while σsa,e is the the stimulated absorption and emission cross sections of the signal beam, respectively; Ip and Is are the pump and signal intensities and IASE± 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 Er3+ 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

dPsdz=2π0Is[σseN2(r)σsaN1(r)]rdr,
(7)
dPpdz=±2π0Ip[βσpeN2(r)σpaN1(r)]rdr,
(8)
dPASE±dz=±2hvs02πσseN2fASE±(r)rdr±2π0[σseN2−(r)σsaN1(r)]PASE±fASE±rdr,
(9)

where fASE± is the normalized ASE intensity profile, PASE± 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,

PASE±=PASE++PASE.
(10)

We can decompose the intensity as Is,p (z,r)=Ps,p (z)fs,p (r) where Ps,p (z) is z-dependent signal od pump powers and fs,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:

dPsdz=2πσsaPs(1+β+η)0N2f2(r)rdrPsαs,
(11)

where αs =2πσsa 0 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:

0N2rdr=0τIphvp(σpaN1βσpeN2)rdr+0τIshvs(σsaN1σseN2)rdr
+0τIASE+hvs(σsaN1σseN2)rdr,
(12)
0N2rdr=τ2πhvpdPpdzτ2πhvsdPsdzτ2πhvsdPASE+dz+2τσse0N2f(r)rdr,
(13)
0N2f(r)rdr=τ2π(AΓ2τσse)[1hvpdPpdz+1hvs(dPsdz+dPASE+dz)],
(14)

where we define the confinement factor Γ=A0 N 2- f(r)rdr/0 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 fp (r)~fs (r)~fASE+ (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

dPsdz=Ps(αs+hvsPsint[1hvpdPpdz+1hvs(dPsdz+dPASE+dz)]),
(15)

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

Psint=hvs(A2τσseΓ)τσsaΓ(1+β+η).
(16)

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:

Ps(L)Ps(0)=exp(αsL)exp(hvsPsint[Pp(0)Pp(L)hvp+(Ps(0)+PASE+(0))(Ps(L)+PASE+(L))hvs]).
(17)

The amplifier gain G=Ps (L)/Ps (0) can be calculated from the following equation:

G=exp(αsL)exp(hvsPsint[Pp(0)Pp(L)hvpPs(0)hvs(1G)PASE+(L)hvs]),
(18)

Pp(L)=1R(ηbpaβbpe),
(19)

where R=0 N(r)f(r)rdr/0 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/ω02)/πω02 where ω 0 is the spot size and the effective core area is πω02=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 αs 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 σeS)/σaS) of Erbium-Doped Fibers,” IEEE Photonics Tech. Lett. 7 (9), 986–988 (1995). [CrossRef]

].

In the numerical calculations, we select the Al/P-silica erbium-doped fiber as an amplifier operated at the pump wavelength λp =1480 nm and the input pump power Pp (0) is fixed at 30 mW. The signal wavelength λs and the signal power Ps (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]

]. Moreover,

Table 1. Typical fiber parameters for an Al/P-silica erbium-doped fiber (from Ref.[12]).

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we used the simulation programme OptiAmplifier 4.0 for generating PASE+ (L) only, and we set up the basic system seen in Fig. 2 [13

13. OptiAmplifier Version 4.0; Optical Fiber Amplifier and Laser Design Software (Copyright © 2002 Optiwave Corporation, 2002).

]. 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.

Fig. 2. Simulation setup for measurement of the co-propagating ASE power in an Er3+-doped optical fiber amplifier (from OptiAmplifier 4.0).

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.

Table 2. The relevant fiber parameters as a function of temperature.

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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.

Fig. 3. Gain as a function of fiber length. Pp (0)=30 mW and Ps (0)=10 µW.

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.

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. 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 Er3+-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]

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, Erbium-Doped fiber Amplifiers; Principle and Applications (John Wiley and Sons. Inc, New York, 1994).

11.

H. Zech, “Measurment Technique for the Quotient of Cross Section σeS)/σaS) of Erbium-Doped Fibers,” IEEE Photonics Tech. Lett. 7 (9), 986–988 (1995). [CrossRef]

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]

13.

OptiAmplifier Version 4.0; Optical Fiber Amplifier and Laser Design Software (Copyright © 2002 Optiwave Corporation, 2002).

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

  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), <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]
  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), <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]
  6. 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]
  7. 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]
  8. F. Prudenzano, �??Erbium-Doped Hole-Assisted optical Fiber Amplifier: Design and Optimization,�?? IEEE J. Lightwave Tech. 23 (1), 330�??340 (2005). [CrossRef]
  9. 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).
  10. E. Desurvire, Erbium-Doped fiber Amplifiers; Principle and Applications (John Wiley and Sons. Inc, New York, 1994).
  11. 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]
  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]
  13. OptiAmplifier Version 4.0; Optical Fiber Amplifier and Laser Design Software (Copyright © 2002 Optiwave Corporation, 2002).

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