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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 25 — Dec. 12, 2005
  • pp: 9970–9981
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Refinement of Er3+-doped hole-assisted optical fiber amplifier

A. D’Orazio, M. De Sario, L. Mescia, V. Petruzzelli, and F. Prudenzano  »View Author Affiliations


Optics Express, Vol. 13, Issue 25, pp. 9970-9981 (2005)
http://dx.doi.org/10.1364/OPEX.13.009970


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Abstract

This paper deals with design and refinement criteria of erbium doped hole-assisted optical fiber amplifiers for applications in the third band of fiber optical communication. The amplifier performance is simulated via a model which takes into account the ion population rate equations and the optical power propagation. The electromagnetic field profile of the propagating modes is carried out by a finite element method solver. The effects of the number of cladding air holes on the amplifier performance are investigated. To this aim, four different erbium doped hole-assisted lightguide fiber amplifiers having a different number of cladding air holes are designed and compared. The simulated optimal gain, optimal length, and optimal noise fig. are discussed. The numerical results highlight that, by increasing the number of air holes, the gain can be improved, thus obtaining a shorter amplifier length. For the erbium concentration NEr=1.8×1024 ions/m3, the optimal gain G(Lopt) increases up to ≅ 2dB by increasing the number of the air holes from M=4 to M=10.

© 2005 Optical Society of America

1. Introduction

In recent years, the performance of Erbium Doped Fiber Amplifiers (EDFAs) operating in the third windows of optical communication has been strongly improved. Most of the nowadays communication systems include one or more EDFAs [1–2

1 . G. J. Foschini and I. M. Habbab , “ Capacity of a broadcast channels in the near-future CATV architecture ,” J. Lightwave Technol. 13 , 507 – 513 ( 1995 ). [CrossRef]

], the spreading of which is due to their intriguing characteristics such as high gain [1–5

1 . G. J. Foschini and I. M. Habbab , “ Capacity of a broadcast channels in the near-future CATV architecture ,” J. Lightwave Technol. 13 , 507 – 513 ( 1995 ). [CrossRef]

], high saturation output power, polarization independent gain [3

3 . C. R. Giles , E. Desrvire , J. R. Talman , J. R. Simpson , and P. C. Becker , “ 2-Gbit/s signal amplification at λ=1.53 μm in an erbium-doped single-mode fiber amplifier ,” J. Lightwave Technol. 7 , 651 – 656 ( 1989 ). [CrossRef]

], crosstalk absence[4

4 . E. Desrvire , C. R. Giles , and J. R. Simpson , “ Gain saturation effects in high speed, multichannel erbium doped fiber amplifiers at λ=1.53 μm ,” J. Lightwave Technol. 7 , 2095 – 2104 ( 1989 ). [CrossRef]

], low noise fig. [5

5 . O. Lumholt , J. H. Polvsen , K. Shusler , A. Bjarklev , S. D. Pedersen , T. Rasmussen , and K. Rottwitt , “ Quantum limited noise fig. operation of high gain erbium doped fiber amplifiers ,” J. Lightwave Technol. 11 , 1344 – 1352 ( 1993 ). [CrossRef]

], and low insertion loss.

Although the erbium-doped fiber technology is mature and widely employed, further research efforts are needed to obtain suitable amplifiers with higher efficiency. During the last decade the photonic crystal fibers (PCFs) have attracted a great interest of both the applied and theoretical research groups. In fact, PCFs are very suitable for the fabrication of both passive and active devices [6–11

6 . R. F. Cregan , B. J. Mangan , J. C. Knight , T. A. Birks , P. St. J. Russell , P. J. Roberts , and D. C. Allan , “ Single-mode photonic band gap guidance of light in air ,” Science 285 , 1537 – 1539 ( 1999 ). [CrossRef] [PubMed]

], since they exhibit unique optical properties which cannot be achieved via the conventional fibers. The following, for example, were obtained by employing PCFs: single mode operation over a wide range of wavelengths [7

7 . T. A. Birks , J. C. Knight , and P. St. J. Russel , “ Endlessly single-mode photonic crystal fiber ,” Opt. Lett. 22 , 961 – 963 ( 1997 ). [CrossRef] [PubMed]

], large mode area [9

9 . J. C. Knight , T. A. Birks , R. F. Cregan , P. St. J. Russel , and J. P. de Sandro , “ Large mode area photonic crystal fibre ,” Electron. Lett. 34 , 1347 – 1348 ( 1998 ). [CrossRef]

], nonlinear effects and anomalous dispersion [12–13

12 . J. Lǽgsgaard and A. Bjarklev , “ Doped photonic bandgap fibers for short-wavelength nonlinear devices ,” Opt. Lett. 28 , 783 – 785 ( 2003 ). [CrossRef] [PubMed]

], soliton formation [14

14 . W. J. Wadsworth , J. C. Knight , A. Ortisoga-Blanch , J. Arriaga , E. Silvestre , and P. St. J. Russell , “ Soliton effects in photonic crystal fiber at 850 nm ,” Electron. Lett. 36 , 53 – 55 ( 2000 ). [CrossRef]

], high birefringence [15

15 . A. Ortisoga-Blanch , J. C. Knight , W. J. Wadsworth , J. Arriaga , B. J. Mangan , T. A. Birks , and P. St. J. Russell “ Highly birefringent photonic crystal fibers ,” Opt. Lett. 25 , 1325 – 1327 ( 2000 ). [CrossRef]

], tailorable dispersion properties [16

16 . A. Ferrando , E. Silvestre , P. Andrés , J. J Miret , and M. V. Andrés , “ Designing the properties of dispersion-flattened photonic crystal fibers ,” Opt. Express 9 , 687 – 697 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687 [CrossRef] [PubMed]

] and visible continuum generation [17

17 . W. J. Wadsworth , A. Ortisoga-Blanch , C. Knight , T. A. Birks , T. P. Martin Man , and P. St. J. Russell , “ Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source ,” J. Opt. Soc. Am. B 19 , 2148 – 2155 ( 2002 ). [CrossRef]

] were obtained by employing PCFs. Many of these properties are achieved by utilizing the large index contrast between the air holes and the glass composing the PCF, via a refined design of the air hole diameter and of the hole-to-hole spacing. Moreover, rare-earth doped PCFs allow the construction of lasers and amplifiers exhibiting advantageous characteristics [11

11 . K. G. Hougaard , J. Broeng , and A. Bjarklev , “ Low pump power photonic crystal fibre amplifiers ,” Electron. Lett. 39 , 599 – 600 ( 2003 ). [CrossRef]

] [18

18 . K. Furusawa , A. N. Malinowski , J. H. V. Price , T. M. Monro , J. K. Sahu , J. Nilsson , and D. J. Richardson , “ A cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding ,“ Opt. Express 9 , 714 – 720 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714 [CrossRef] [PubMed]

].

The hole-assisted lightguide fibers (HALFs) constitute an intriguing subclass of the PCFs. They are composed of a high-index core, a low index cladding and a suitable number of air holes surrounding the core. The air holes assist and improve the propagation mode confinement. Moreover, HALFs exhibiting losses comparable to those of the conventional fibers have been recently constructed [19–20

19 . Z. Zhu and T. G. Brown , “ Multipole analysis of hole-assisted optical fibers ,” Opt. Commun. 206 , 333 – 339 ( 2002 ). [CrossRef]

]. These fibers can be activated by rare earth doping and their section can be designed to opportunely change the guided mode area. In particular, it is possible to optimize the intensity of the mode electromagnetic field in the region closer to the doped core.

Most papers investigate the dispersion features of passive microstructured fibers or pertain to the fabrication and characterization of PCF lasers; the number of articles reporting the design of microstructured rare earth doped fibers is rather small.

In this paper, the design and refinement criteria pertaining to an Er3+-doped hole-assisted optical fiber amplifier in the third band of fiber optic communication are investigated. The HALF electromagnetic investigation is carried-out by employing a finite element method (FEM) solver. The mode electromagnetic field, simulated at different wavelengths, constitutes the input data for a home-made computer code which solves power propagation and population rate equations via a Runge-Kutta iterative algorithm.

The model takes into account the effects of the amplified spontaneous emission (ASE) in the band around the signal wavelenght, the ion-ion energy transfer, the propagation loss and wavelength refractive index dispersion. Object of research is also to investigate the influence of the number of air holes on the performance of the Er3+ doped HALF amplifier. Numerous simulations are performed in order to evaluate the actual feasibility of the silica/germania erbium doped PCF amplifier.

2. Theory

N1t=W13N1+W21N2+N2τ21W12N1C14N1N4+CupN22+CupN32
(1)
N2t=W12N1W21N2+N3τ32N2τ21+2C14N1N42CupN22
(2)
N3t=W13N1N3τ32+N4τ432CupN32
(3)
NEr=N1+N2+N3+N4
(4)

where the 4I15/2, 4I13/2, 4I11/2, and 4I9/2 levels of the erbium ion are indicated by the labels 1, 2, 3, 4, and their population density with N1, N2, N3, and N4, respectively. Moreover, 1/τ21 is the Er3+ spontaneous emission rate; 1/τ32 and 1/τ43 are the non-radiative relaxation rates; Cup and C14 are the up-conversion and cross relaxation coefficients; whereas NEr is the total erbium concentration. It should be noted that, due the quadratic terms in equation (1–3), the coupling of the rate equations complicates the numerical solution of the problem. The Wij absorption and emission rate are expressed by:

W13(r,ϑ,z)=σ13(νp)hνpPp(z)Ψ(r,ϑ,νp)
(5)
W12(r,ϑ,z)=σ12(νs)hνsPs(z)Ψ(r,ϑ,νs)+0+σ12(ν)hν[SASE+(z,ν)+SASE(z,ν)]Ψ(r,ϑ,ν)dν
(6)
W21(r,ϑ,z)=σ21(νs)hνsPs(z)Ψ(r,ϑ,νs)+0+σ21(ν)hν[SASE+(z,ν)+SASE(z,ν)]Ψ(r,ϑ,ν)dν
(7)

02π0+Ψ(r,ϑ,ν)rdrdϑ=Re{02π0+E(r,ϑ,ν)×H*(r,ϑ,ν)ẑrdrdϑ}=1
(8)

In Eq. (8), E and H are the electric and magnetic fields of the propagation modes, is the versor of the z propagation direction. The integrals in the equation (6–7) pertain to the amplified spontaneous emission (ASE) occurring via the 4I13/24I15/2 transition and absorbed via the 4I15/24I13/2 transition; SASE+(z,ν) and SASE(z,ν) are the forward and backward propagating power spectral densities of the amplified spontaneous emission.

The Er3+ stimulated emission in the pump band is neglected, the nonradiative decay of the 4I11/2 level being very fast (of the order of 1 ns).

The rate of change of the pump, the signal and the ASE power is given by the equations:

dPp(z)dz=Pp(z)[Aσ13(νp)N1(r,ϑ,z)Ψ(r,ϑ,νp)rdrdϑ+α(νp)]
(9)
dPs(z)dz=Ps(z)[A[σ21(νs)N2(r,ϑ,z)σ12(νs)N1(r,ϑ,z)]Ψ(r,ϑ,νs)rdrdϑ+α(νs)]
(10)
dPASE±(z,ν)dz=±PASE±(z,ν)A[σ21(ν)N2(r,ϑ,z)σ12(ν)N1(r,ϑ,z)]Ψ(r,ϑ,ν)rdrdϑ
±mP0(ν)σ21(ν)Aσ21(ν)N2(r,ϑ,z)rdrdϑα(ν)PASE±(z,ν)
(11)

where P0(ν)=hνdν is the contribution of the spontaneous emission to the guided mode power; m is the number of guided modes propagating at signal wavelength; α(ν) is the propagation mode loss. The ASE noise spreads in a continuum wavelength range and in the calculation, it is discretized in K wavelength slots. Thus both the forward and backward ASE noise are modelled as K optical beams having λK wavelength and ΔλK bandwidth centered around λK. The rate equations (1–4) and the 2(K+1) equations of pump, signal and ASE powers (9–11

9 . J. C. Knight , T. A. Birks , R. F. Cregan , P. St. J. Russel , and J. P. de Sandro , “ Large mode area photonic crystal fibre ,” Electron. Lett. 34 , 1347 – 1348 ( 1998 ). [CrossRef]

) are solved, in the steady state condition (∂/∂t=0), via a Runge-Kutta iterative method. The iterative procedure is stopped when the change of the signal gain between two consecutive integrations is less than 10-7. This high precision is necessary when the amplifier optimal length, for which gain is maximized, has to be evaluated.

Fig. 1. Sketch of the HALF sections

Being an important characteristic of the amplifiers to be used in optical communication systems, the noise fig. is accurately evaluated. The computation is performed by using the formula reported in literature [21–22

21 . A. D’Orazio , M. De Sario , L. Mescia , V. Petruzzelli , F. Prudenzano , A. Chiasera , M. Montagna , C. Tosello , and M. Ferrari , “ Design of Er 3+ doped SiO 2 -TiO 2 planar waveguide amplifier ,” J. Non-Crystalline Solids 322 , 278 – 283 ( 2003 ). [CrossRef]

]:

F=1G+PASE+(L,νs)GhνsΔνs
(12)

where Δν s is the slot width centered at the signal frequency, ν s, and PASE+(L,νs ) = SASE+(L,νsνs .

3. Numerical results and discussion

The HALF consists of a silica/germania (SiO2/GeO2) glass core, a silica SiO2 cladding and a suitable number of air holes surrounding the core. Figure 1 shows four HALF sections including different air hole numbers: M=4 (a), M=6 (b), M=8 (c), M=10 (d). The other geometrical parameters for all the HALF sections reported in Fig. 1 (a)-(d) are the core diameter dcore, the hole diameter dhole and the distance between the core center and the air hole center R.

The mode electromagnetic field profiles and the dispersion properties have been evaluated via a solver based on the Finite-Element Method (FEM). The HALF has been preliminary designed in order to obtain single mode operation at both the signal, λs=1536 nm, and the pump, λp=980 nm, wavelength. More precisely, in the design the hole diameters dhole=0.8 μm, the distance R=4 μm and the refractive index change between the core and the cladding Δn=8×10-3 have been roughly chosen in order to obtain the mode guiding and a strong dependence of the amplifier performance on the air hole number. Too high a refractive index change between the core and the cladding can make the fiber multimode and the device performance insensitive to the change of the air holes, owing to the strong bounding of the electromagnetic field in the core. The core diameter dcore=5.2 μm has been identified in this first design step for all four sections in order to obtain the single mode operation. No refractive index variation due to the Er3+ ions has been considered, the variation being negligible because the concentration levels used in the design are rather low. Moreover, the wavelength dispersion of both SiO2/GeO2 and SiO2 refractive indices has been expressed via a Sellmeier equation [21–23

21 . A. D’Orazio , M. De Sario , L. Mescia , V. Petruzzelli , F. Prudenzano , A. Chiasera , M. Montagna , C. Tosello , and M. Ferrari , “ Design of Er 3+ doped SiO 2 -TiO 2 planar waveguide amplifier ,” J. Non-Crystalline Solids 322 , 278 – 283 ( 2003 ). [CrossRef]

].

The emission and absorption cross-section spectra of the erbium in GeO2–SiO2 glass employed in the simulation are those reported in [24

24 . W. L Barnes , R. I. Laming , E. J. Tarbox , and P. R. Morkel , “ Absorption and emission cross section of Er 3+ doped silica fiber ”, IEEE J. Quantum Electron. 27 , 1004 – 1010 ( 1991 ). [CrossRef]

]; the wavelength range from λ=1450 nm to λ=1599 nm is discretized in K=150 wavelength slots. The propagation mode losses are αs=0.41 dB/km at signal wavelength λs= 1536 nm and αp=2 dB/km at pump wavelength λp= 980 nm [20

20 . T. Hasegawa , E. Sasaoka , M. Onishi , M. Nishimura , Y. Tsuji , and M. Koshiba , “ Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss ,” Opt. Express 9 , 681 – 686 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681 [CrossRef] [PubMed]

]. According to the literature data, the ion-ion interaction coefficients are Cup=10-22 m3/s, C3=10-22 m3/s, C14=3.5×10-23 m3/s [25–26

25 . P. Blixt , J. Nilsson , T. Carlnas , and B. Jaskorzynska , “ Concentration dependent upconversion in Er 3+ -doped fiber amplifiers: experiments and modelling ,” IEEE Photon. Techonol. Lett. 3 , 996 – 998 ( 1991 ). [CrossRef]

], the lifetime of the metastable level is τ21=10 ms [24

24 . W. L Barnes , R. I. Laming , E. J. Tarbox , and P. R. Morkel , “ Absorption and emission cross section of Er 3+ doped silica fiber ”, IEEE J. Quantum Electron. 27 , 1004 – 1010 ( 1991 ). [CrossRef]

], the non-radiative lifetimes are τ32 = 1 ns and τ43 = 1 ns [21

21 . A. D’Orazio , M. De Sario , L. Mescia , V. Petruzzelli , F. Prudenzano , A. Chiasera , M. Montagna , C. Tosello , and M. Ferrari , “ Design of Er 3+ doped SiO 2 -TiO 2 planar waveguide amplifier ,” J. Non-Crystalline Solids 322 , 278 – 283 ( 2003 ). [CrossRef]

].

The hole pattern has been designed in order to optimize the electromagnetic field bounding and to enhance the power distribution over the erbium doped core. A parameter pertaining to the electromagnetic field bounding is the effective mode area [27

27 . K. Saitoh , M. Koshiba , T. Hasegawa , and E. Sasaoka “ Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion ,” Opt. Express 11 , 843 – 852 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843 [CrossRef] [PubMed]

]:

Aeff(λ)=(02π0+E(r,ϑ,λ)2rdrdϑ)202π0+E(r,ϑ,λ)4rdrdϑ
(13)

where |E| is the electromagnetic field modulus of the propagation mode.

Figure. 2 shows the effective mode area versus the wavelength for M=4 (full curve), M=6 (broken curve), M=8 (dot curve) and M=10 (dash-dot curve). The number of air holes strongly affects the overlapping between the dopant concentration in the core and the electromagnetic field distribution. This occurs especially for high wavelengths, where the electromagnetic field well surrounds the air holes. By increasing the number of air holes, the effective mode area decreases, the electromagnetic field of the propagation modes being more bounded.

The performance of the optical amplifier is also affected by the overlap factor Γp at the pump and Γs at the signal wavelength. The overlap factors are defined as the overlapping integrals between the normalized optical intensity distribution and the fiber core [28–29

28 . P. C. Becker , N. A. Olsson , and J. R. Simpson , Erbium-doped fiber amplifiers: fundamentals and technology ( Academic Press , 1999 ) pp. 140 – 144 .

]:

Γp=02π0RcoreΨ(r,ϑ,νp)rdrdϑ
(14)
Γs=02π0RcoreΨ(r,ϑ,νs)rdrdϑ
(15)

Moreover, the overlap factor between pump and signal modes is described by the integral of the product of the normalized optical intensity distributions of pump and signal modes [29

29 . B. P. Petreski , P. M. Farrell , and S. F. Collins , “ Optical amplification on the 3P0→3F2 transition in praseodymium-doped fluorozirconate fiber ,” Fiber Integr. Opt. 18 , 21 – 32 ( 1999 ). [CrossRef]

]:

Γp,s=02π0RcoreΨ(r,ϑ,νp)Ψ(r,ϑ,νs)rdrdϑ
(16)
Fig. 2. Effective mode area Aeff versus wavelength λ for different number of air holes: M=4 (full curve), M=6 (broken curve), M=8 (dot curve), M=10 (dash-dot curve). Hole diameter dhole=0-8 μm, core diameter dcore= 5.2 μm, core to hole distance R=4 μ m.

Table 1. Overlap integrals and effective mode area for different numbers of air holes.

table-icon
View This Table

Table 1 reports the overlap factors Γp, Γs and Γp,s and the effective mode area Aeff calculated at both pump and signal wavelengths. The overlap factors Γp, Γs and Γp,s increase as the number of the air holes is increased.

By varying the air holes, the overlap factors change. The effects on the amplifier performance have been also investigated. A number of simulations have been performed to identify the optimal fiber length and optimal gain by varying the erbium concentration, the fiber length, the pump power, the signal power and the number M of the air holes. The electromagnetic field of propagation modes, calculated via the FEM solver, are the input data for the home made computer code which integrate the equation (1–11) by employing a Runge-Kutta algorithm.

The optimal fiber length Lopt, which maximizes the signal gain, and the corresponding optimal gain G(Lopt) depend on the erbium concentration NEr. The optimal gain curves are reported in Fig. 3(a) and the optimal length in Fig. 3(b) for M=4 (full curve), M=6 (broken curve), M=8 (point curve), M=10 (dash-dot curve), the input pump power being Pp(0)=30 mW and the input signal power Ps(0)=-40 dBm. The maximum erbium concentration considered in the simulation is NErmax= 2×1024 ions/m3 because a higher Er3+ concentration worsens the amplifier performance due to pair-induced quenching (PIQ) [26

26 . P. Myslinsky , D. Nguyen , and J. Chrostowski , “ Effects of concentration on the performance of erbium-doped fiber amplifiers ,” J. Lightwave Technol. 15 , 112 – 120 ( 1997 ). [CrossRef]

]. The optimal gain, in all four cases, exhibits a peak Gmax(Lopt)=41.43 dB (full curve), Gmax(Lopt)=41.53 dB (broken curve), Gmax(Lopt)=41.60 dB (point curve), Gmax(Lopt)=41.64 dB (dash-dot curve) close to the erbium concentration NEr=0.16×1024 ions/m3. Therefore this is the optimal erbium concentration NEropt. For higher erbium concentration the optical gain decreases because the effects of the ion-ion interactions become stronger. The ion-ion interactions partially deplete the metastable level, causing a decrease in the amplifier performance. It is worthwhile to notice that by increasing the number of air holes the gain increases, the overlap factors being larger. The relative gain variation is defined by the formula ΔG=[Gmax(Lopt)-Gmin(Lopt)]/Gmax(Lopt) where the minima of the optical gain are calculated for the concentration NEr=2×1024 ions/m3 while the maxima are calculated for NEropt. The calculated values are: ΔG=0.740 (full curve), ΔG=0.690 (broken curve), ΔG=0.654 (point curve), and ΔG=0.630 (dash-dot curve). These numerical values suggest that a high number of air holes minimizes the gain degradation due to high erbium concentration. The optimal fiber length (see Fig. 3(b)) slightly decreases by increasing the number of the air holes. This effect can be easily explained: by increasing the number of air holes the overlap factors at pump, Γp, and signal, Γs, wavelength increase. Therefore, both the pump absorption and signal stimulated emission rate are enhanced. Figure 3(c) depicts the behaviour of the noise fig. F(Lopt), calculated for the optimal length, versus the erbium concentration. The noise fig. decreases by increasing the erbium concentration. Note that a large number of air holes causes a slight worsening of the noise fig. Figures 3 (a), (b), and (c) yield that, for all the four HALFs and the erbium concentration changing within the range from NEr≅0.1×1024 to NEr≅2×1024 ions/m3: i) the optimal gain varies G(Lopt) in a reduced range, from about 41.6 dB to about 35.6 dB; ii) also the optimal noise fig. slightly changes F (Lopt) in the range from about 4.53 dB to 4.16 dB; iii) the optimal length Lopt strongly varies it changing from about 296 m to 13.7 m. Thus, according with the aforesaid remarks, the erbium concentration NEr=1.8×1024 ions/m3 will be fixed in the following simulations, this allowing the design of short HALF amplifier with appreciable performance.

Fig. 3. (a) Optimal gain G(Lopt), (b) optimal length Lopt, (c) optimal noise fig. F(Lopt) versus erbium concentration NEr for different number of holes M=4 (full curve), M=6 (broken curve), M=8 (dot curve), M=10 (dash-dot curve). Input signal power Ps(0)=-40 dBm and input pump power Pp(0)=30 mW.

By changing the nominal pump power a further amplifier optimization should be calculated. Fig 4(a) shows the calculated gain coefficient αp = G(Lopt)/Pp(0), i.e. the ratio between the optimal signal gain and the pump power launched into the active optical fiber versus the input pump power Pp(0), for M=4 (full curve), M=6 (broken curve), M=8 (point curve), M=10 (dash-dot curve). The input signal power is Ps(0)=-40 dBm, the erbium concentration is the optimal one, NEr=1.8×1024 ions/m3. In all four cases the threshold pump power is Pth ≅ 2.8 mW. By increasing pump power, the gain coefficient steeply increases showing a peak. In particular, the optimal gain coefficient is αp=1.33 dB/mW (full curve), αp=1.40 dB/mW (broken curve), αp=1.46 dB/mW (point curve), αp=1.5 dB/mW (dash-dot curve) for the input pump power Pp(0)=21.93 mW, Pp(0)=20.91 mW, Pp(0)= 19.48 mW and Pp(0)=18.98 mW, respectively. Thereafter the gain coefficient decreases, tending to αp ≅ 0.8 dB/mW for all the air hole configurations. The last occurrence can be explained by considering that, for high input pump power, the erbium ions are inverted almost completely along the whole fiber. For this reason, further increase of the input pump power does not allow the gain coefficient improvement. The optimal length, Lopt, versus input pump power Pp(0) is shown in Fig. 4(b). The optimal length increases as the input pump power is increased until Pp(0) ≅30 mW. Then the curves become flat, the almost constant optimal fiber length being Lopt≅ 17.78 m (full curve), Lopt≅ 16.83 m (broken curve), Lopt≅ 16.18 m (point curve) and Lopt≅ 15.76 m (dash-dot curve) for input pump power changing in the range from 30 to 50 mW.

Fig. 4. (a) Gain coefficient αp, (b) optimal length Lopt, (c) optimal noise fig. F(Lopt) versus input pump power Pp(0) for different number of holes M=4 (full curve), M=6 (broken curve), M=8 (dot curve), M=10 (dash-dot curve). Input signal power Ps(0)=-40 dBm and erbium concentration NEr=1.8×1024 ions/m3.

Note that the gain coefficient calculated for M=10 is 12.8 % higher than that calculated for M=4 and the optimal length is 12.8 % shorter.

Figure 4(c) illustrates the variation of the noise fig. calculated for the optimal length F(Lopt) versus the input pump power Pp(0). By increasing the input pump power the noise fig. steeply increases until Pp(0) ≅ 7 mW, where a maximum occurs. For higher input pump power, i.e. for changes in the range from 30 to 50 mW, the noise fig. at first decreases and then it is almost constant, exhibiting the following values F(Lopt) ≅ 4.34 dB (full curve), F(Lopt) ≅ 4.41 dB (broken curve), F(Lopt) ≅ 4.47 dB (point curve), F(Lopt) ≅ 4.51 dB (dash-dot curve).

Summing up, even by the inspection of Figs. 4(a), (b), (c), the effect of a better field confinement due to a large number of holes is apparent, allowing higher gain coefficient and reduced optimal length but causing a slight worsening of the noise fig.

The amplifier characteristics versus the output signal power have been simulated, too. In fact, the signal output power affects the feasibility of transmission and repetition distances as well as the number of output ports utilized for service distribution. Fig 5(a) shows the optimal signal gain G(Lopt) versus the output signal power Ps(Lopt)=Ps(0)G(Lopt) for different number of the air holes: M=4 (full curve), M=6 (broken curve), M=8 (point curve), M=10 (dash-dot curve). The input pump power and erbium concentration are Pp(0)=30 mW and NEr=1.8×1024 ions/m3 respectively. The gain remains almost constant for low signal output power. A significant gain reduction is observed for output signal power higher than Ps(Lopt)=4 dBm (corresponding to input signal power higher than Ps(0)≅-30 dBm). The optimal gain, as expected, decreases; this occurrence is due to the reduction of the population inversion which is induced by the high signal power producing a strong stimulated emission. Moreover, the saturation output power Psat (Lopt), i.e. the output power at which the signal gain is 3dB below its small signal value (calculated for Ps(0)=-40dBm), increases by increasing the number of air holes. The calculated values are Psat(Lopt)=2.78 dBm, Psat(Lopt)=3.15 dBm, Psat(Lopt)=3.39 dBm and Psat(Lopt)=3.60 dBm for M=4, M=6, M=8, and M=10, respectively.

Fig 5(b) illustrates the optimal length Lopt versus the output signal power. In all cases it remains almost constant close to the value Lopt≅17 m (full curve), Lopt≅16.2 m (broken curve), Lopt≅15.6 m (point curve) and Lopt≅15.2 m (dash-dot curve) for output signal power below Ps(Lopt)=5 dBm. For higher signal power Lopt strongly decreases. Moreover, for higher number of air holes M=8 and M=10 the optimal length very slightly increases in the range from 0 to 5 dBm. This occurs because a high number of holes increases the gain and the forward-traveling signal can pull away power from the backward-travelling ASE [28

28 . P. C. Becker , N. A. Olsson , and J. R. Simpson , Erbium-doped fiber amplifiers: fundamentals and technology ( Academic Press , 1999 ) pp. 140 – 144 .

]. For low number of air holes, M=4 and M=6, this effect is not observed.

Fig. 5(c) depicts the noise fig. calculated at the optimal length, F(Lopt), as a function of the output signal power. For the input signal power Ps(0)≅-45 dBm, i.e. for the output signal power Ps(Lopt)=-10 dBm, the calculated values are: F(Lopt)=4.21 dB (full curve), F(Lopt)=4.28 dB (broken curve), F(Lopt)=4.34 dB (point curve), and F(Lopt)=4.38 dB (dash-dot curve).

Fig. 5. (a) Optimal signal gain G(Lopt), (b) optimal length Lopt, (c) optimal noise fig. F(Lopt) versus the output signal power Ps(Lopt) for different number of holes M=4 (full curve), M=6 (broken curve), M=8 (dot curve), M=10 (dash-dot curve). Input pump power Pp(0)=30 mW and erbium concentration NEr=1.8×1024 ions/m3.

For Ps(Lopt) >0 dBm, the noise fig. is slightly decreasing. The curves exhibit a smooth behavior because low signal powers have a non-relevant effect on the backward-travelling ASE [28

28 . P. C. Becker , N. A. Olsson , and J. R. Simpson , Erbium-doped fiber amplifiers: fundamentals and technology ( Academic Press , 1999 ) pp. 140 – 144 .

].

Moreover, for low signal power the noise fig. increases by increasing the number of air holes. In fact, a high hole number causes an increase of the signal gain and thus a higher backward ASE at the input section of the optical fiber. When the signal output is larger than the saturation value, the noise fig. strongly increases because the depletion of the inverted population become stronger. A striking feature is the dip, ΔF [30

30 . E. Desurvire , Erbium doped fiber amplifiers ( Wiley-Interscience Inc. , 1993 ) pp. 354 – 382 .

], defined as the difference between the noise fig. F(Lopt) calculated at the output signal power Ps(Lopt)=-10 dBm and the minimum noise fig. value, calculated at output signal power close to Ps(Lopt) = 7 dBm. More precisely, the calculated values of ΔF are: ΔF=0.29 dB (full curve), ΔF=0.37 dB (broken curve), ΔF=0.43 dB (point curve), and ΔF=0.48 dB (dash-dot curve).

The effect of the variation of the doped area on the amplifier characteristics has been investigated. In the following simulations the erbium ions concentration NEr=1.8×1024 ions/m3 have been supposed to be uniformly distributed on a circular region having diameter dd. Figure 6(a) shows the dependence of the optimal gain G(Lopt) as a function of the dopant confinement ratio dd/dcore, for M=4 (full curve), M=6 (broken curve), M=8 (point curve), and M=10 (dash-dot curve). The pump and signal input power are Pp(0)=30 mW, Ps(0)=-40 dBm, respectively.

Fig. 6. (a) Optimal signal gain G(Lopt), (b) optimal length Lopt, (c) optimal noise fig. F(Lopt) versus the dopant confinement ratio dd/dcore for different number of holes M=4 (full curve), M=6 (broken curve), M=8 (dot curve), M=10 (dash-dot curve). Input pump power Pp(0)=30 mW, input signal power Ps(0)=-40 dBm, erbium concentration NEr=1.8×1024 ions/m3.

The signal gain decreases by increasing the ratio dd/dcore. This occurs because the erbium ions are more efficiently inverted in the region where the pump intensity of the fundamental mode is stronger, i.e. in the central part of the fiber core. Nevertheless, the gain increases by increasing the air hole number N. However, a low dopant confinement ratio requires a longer active fiber. This is illustrated in Fig. 6(b) where the variation of the optimal length Lopt is plotted as a function of the confinement ratio. In particular, the optimal length decreases as the confinement ratio increases while its numerical value remains almost constant (Lopt≅14 m) for confinement ratios larger than 1.4. In fact, the input pump power is not further absorbed in the region where it is too weak for large dd/dcore values. In Fig. 6(c) the noise fig. calculated at the optimal length F(Lopt) versus the confinement ratio is reported. An interesting point is that by increasing the number of holes, the noise fig. curve becomes smoother. This is due to better pump and signal field confinement.

4. Conclusion

Acknowledgments

This work has been partially supported within the following plants: CNR (National Council Research) and MIUR (Ministry of Instruction, University, and Research) plan (16/10/2000 FISR Funds no. CU 03.00204); MIUR plan PRIN 2004 prot. 2004025738_004; MIUR plan D.M 593 8/8/00 prot. 5910 10/07/2003 (FIBLAS).

References and links

1 .

G. J. Foschini and I. M. Habbab , “ Capacity of a broadcast channels in the near-future CATV architecture ,” J. Lightwave Technol. 13 , 507 – 513 ( 1995 ). [CrossRef]

2 .

T. Otami , K. Goto , T. Kawazawa , H. Abe , and M. Tanaka , “ Effect of span loss increase on the optically amplified communication system ,” J. Lightwave Technol. 15 , 737 – 742 ( 1997 ). [CrossRef]

3 .

C. R. Giles , E. Desrvire , J. R. Talman , J. R. Simpson , and P. C. Becker , “ 2-Gbit/s signal amplification at λ=1.53 μm in an erbium-doped single-mode fiber amplifier ,” J. Lightwave Technol. 7 , 651 – 656 ( 1989 ). [CrossRef]

4 .

E. Desrvire , C. R. Giles , and J. R. Simpson , “ Gain saturation effects in high speed, multichannel erbium doped fiber amplifiers at λ=1.53 μm ,” J. Lightwave Technol. 7 , 2095 – 2104 ( 1989 ). [CrossRef]

5 .

O. Lumholt , J. H. Polvsen , K. Shusler , A. Bjarklev , S. D. Pedersen , T. Rasmussen , and K. Rottwitt , “ Quantum limited noise fig. operation of high gain erbium doped fiber amplifiers ,” J. Lightwave Technol. 11 , 1344 – 1352 ( 1993 ). [CrossRef]

6 .

R. F. Cregan , B. J. Mangan , J. C. Knight , T. A. Birks , P. St. J. Russell , P. J. Roberts , and D. C. Allan , “ Single-mode photonic band gap guidance of light in air ,” Science 285 , 1537 – 1539 ( 1999 ). [CrossRef] [PubMed]

7 .

T. A. Birks , J. C. Knight , and P. St. J. Russel , “ Endlessly single-mode photonic crystal fiber ,” Opt. Lett. 22 , 961 – 963 ( 1997 ). [CrossRef] [PubMed]

8 .

J. C. Knight , T. A. Birks , P. St. J. Russel , and D. M. Aktin , “ All silica single mode optical fiber with photonic crystal cladding ,” Opt. Lett. 21 , 1547 – 1549 ( 1996 ). [CrossRef] [PubMed]

9 .

J. C. Knight , T. A. Birks , R. F. Cregan , P. St. J. Russel , and J. P. de Sandro , “ Large mode area photonic crystal fibre ,” Electron. Lett. 34 , 1347 – 1348 ( 1998 ). [CrossRef]

10 .

J. Broeng , D. Mongilevstev , S. E. Barkou , and A. Bjarklev , “ Photonic crystal fibers: a new class of optical waveguide ,” Opt. Fiber Technol. 4 , 305 – 330 ( 1999 ). [CrossRef]

11 .

K. G. Hougaard , J. Broeng , and A. Bjarklev , “ Low pump power photonic crystal fibre amplifiers ,” Electron. Lett. 39 , 599 – 600 ( 2003 ). [CrossRef]

12 .

J. Lǽgsgaard and A. Bjarklev , “ Doped photonic bandgap fibers for short-wavelength nonlinear devices ,” Opt. Lett. 28 , 783 – 785 ( 2003 ). [CrossRef] [PubMed]

13 .

J. C. Knight , J. Arriaga , T. A. Birks , A. Ortisoga-Blanch , W. J. Wadsworth , and P. St. J. Russell , “ Anomalous dispersion in photonic crystal fiber ,” IEEE Photon. Techonol. Lett. 12 , 807 – 809 ( 2000 ). [CrossRef]

14 .

W. J. Wadsworth , J. C. Knight , A. Ortisoga-Blanch , J. Arriaga , E. Silvestre , and P. St. J. Russell , “ Soliton effects in photonic crystal fiber at 850 nm ,” Electron. Lett. 36 , 53 – 55 ( 2000 ). [CrossRef]

15 .

A. Ortisoga-Blanch , J. C. Knight , W. J. Wadsworth , J. Arriaga , B. J. Mangan , T. A. Birks , and P. St. J. Russell “ Highly birefringent photonic crystal fibers ,” Opt. Lett. 25 , 1325 – 1327 ( 2000 ). [CrossRef]

16 .

A. Ferrando , E. Silvestre , P. Andrés , J. J Miret , and M. V. Andrés , “ Designing the properties of dispersion-flattened photonic crystal fibers ,” Opt. Express 9 , 687 – 697 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687 [CrossRef] [PubMed]

17 .

W. J. Wadsworth , A. Ortisoga-Blanch , C. Knight , T. A. Birks , T. P. Martin Man , and P. St. J. Russell , “ Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source ,” J. Opt. Soc. Am. B 19 , 2148 – 2155 ( 2002 ). [CrossRef]

18 .

K. Furusawa , A. N. Malinowski , J. H. V. Price , T. M. Monro , J. K. Sahu , J. Nilsson , and D. J. Richardson , “ A cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding ,“ Opt. Express 9 , 714 – 720 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714 [CrossRef] [PubMed]

19 .

Z. Zhu and T. G. Brown , “ Multipole analysis of hole-assisted optical fibers ,” Opt. Commun. 206 , 333 – 339 ( 2002 ). [CrossRef]

20 .

T. Hasegawa , E. Sasaoka , M. Onishi , M. Nishimura , Y. Tsuji , and M. Koshiba , “ Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss ,” Opt. Express 9 , 681 – 686 ( 2001 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681 [CrossRef] [PubMed]

21 .

A. D’Orazio , M. De Sario , L. Mescia , V. Petruzzelli , F. Prudenzano , A. Chiasera , M. Montagna , C. Tosello , and M. Ferrari , “ Design of Er 3+ doped SiO 2 -TiO 2 planar waveguide amplifier ,” J. Non-Crystalline Solids 322 , 278 – 283 ( 2003 ). [CrossRef]

22 .

F. Prudenzano , “ Erbium-doped hole-assisted optical fiber amplifier: design and optimization ,” J. Lightwave Technol. 23 , 330 – 340 ( 2005 ). [CrossRef]

23 .

J. W. Fleming , “ Dispersion in GeO 2 -SiO 2 glasses ,” Appl. Opt. 23 , 4486 – 4493 ( 1984 ). [CrossRef] [PubMed]

24 .

W. L Barnes , R. I. Laming , E. J. Tarbox , and P. R. Morkel , “ Absorption and emission cross section of Er 3+ doped silica fiber ”, IEEE J. Quantum Electron. 27 , 1004 – 1010 ( 1991 ). [CrossRef]

25 .

P. Blixt , J. Nilsson , T. Carlnas , and B. Jaskorzynska , “ Concentration dependent upconversion in Er 3+ -doped fiber amplifiers: experiments and modelling ,” IEEE Photon. Techonol. Lett. 3 , 996 – 998 ( 1991 ). [CrossRef]

26 .

P. Myslinsky , D. Nguyen , and J. Chrostowski , “ Effects of concentration on the performance of erbium-doped fiber amplifiers ,” J. Lightwave Technol. 15 , 112 – 120 ( 1997 ). [CrossRef]

27 .

K. Saitoh , M. Koshiba , T. Hasegawa , and E. Sasaoka “ Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion ,” Opt. Express 11 , 843 – 852 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843 [CrossRef] [PubMed]

28 .

P. C. Becker , N. A. Olsson , and J. R. Simpson , Erbium-doped fiber amplifiers: fundamentals and technology ( Academic Press , 1999 ) pp. 140 – 144 .

29 .

B. P. Petreski , P. M. Farrell , and S. F. Collins , “ Optical amplification on the 3P0→3F2 transition in praseodymium-doped fluorozirconate fiber ,” Fiber Integr. Opt. 18 , 21 – 32 ( 1999 ). [CrossRef]

30 .

E. Desurvire , Erbium doped fiber amplifiers ( Wiley-Interscience Inc. , 1993 ) pp. 354 – 382 .

OCIS Codes
(060.2410) Fiber optics and optical communications : Fibers, erbium
(140.4480) Lasers and laser optics : Optical amplifiers
(160.5690) Materials : Rare-earth-doped materials

ToC Category:
Research Papers

Citation
A. D'Orazio, M. De Sario, L. Mescia, V. Petruzzelli, and F. Prudenzano, "Refinement of Er3+-doped hole-assisted optical fiber amplifier," Opt. Express 13, 9970-9981 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-25-9970


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References

  1. G. J. Foschini, I. M. Habbab, "Capacity of a broadcast channels in the near-future CATV architecture," J. Lightwave Technol. 13, 507-513 (1995). [CrossRef]
  2. T. Otami, K. Goto, T. Kawazawa, H. Abe, and M. Tanaka, "Effect of span loss increase on the optically amplified communication system," J. Lightwave Technol. 15, 737-742 (1997). [CrossRef]
  3. C. R. Giles, E. Desrvire, J. R. Talman, J. R. Simpson, and P. C. Becker, "2-Gbit/s signal amplification at λ=1.53 µm in an erbium-doped single-mode fiber amplifier," J. Lightwave Technol. 7, 651-656 (1989). [CrossRef]
  4. E. Desrvire, C. R. Giles, and J. R. Simpson, "Gain saturation effects in high speed, multichannel erbium doped fiber amplifiers at λ=1.53 µm," J. Lightwave Technol. 7, 2095-2104 (1989). [CrossRef]
  5. O. Lumholt, J. H. Polvsen, K. Shusler, A. Bjarklev, S. D. Pedersen, T. Rasmussen, and K. Rottwitt, "Quantum limited noise fig. operation of high gain erbium doped fiber amplifiers," J. Lightwave Technol. 11, 1344-1352 (1993). [CrossRef]
  6. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  7. T. A. Birks, J. C. Knight, P. St. J. Russel, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  8. J. C. Knight, T. A. Birks, P. St. J. Russel, and D. M. Aktin, "All silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
  9. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, J. P. de Sandro, "Large mode area photonic crystal fibre," Electron. Lett. 34, 1347-1348 (1998). [CrossRef]
  10. J. Broeng, D. Mongilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: a new class of optical waveguide," Opt. Fiber Technol. 4, 305-330 (1999). [CrossRef]
  11. K. G. Hougaard, J. Broeng, and A. Bjarklev, "Low pump power photonic crystal fibre amplifiers," Electron. Lett. 39, 599-600 (2003). [CrossRef]
  12. J. Láegsgaard, and A. Bjarklev, "Doped photonic bandgap fibers for short-wavelength nonlinear devices," Opt. Lett. 28, 783-785 (2003). [CrossRef] [PubMed]
  13. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortisoga-Blanch, W. J. Wadsworth, and P. St. J. Russell, "Anomalous dispersion in photonic crystal fiber," IEEE Photon. Techonol. Lett. 12, 807 - 809 (2000). [CrossRef]
  14. W. J. Wadsworth, J. C. Knight, A. Ortisoga-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, "Soliton effects in photonic crystal fiber at 850 nm," Electron. Lett. 36, 53 - 55 (2000). [CrossRef]
  15. A. Ortisoga-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks and P. St. J. Russell "Highly birefringent photonic crystal fibers," Opt. Lett. 25, 1325 - 1327 (2000). [CrossRef]
  16. A. Ferrando, E. Silvestre, P. Andrés, J. J Miret, and M. V. Andrés, "Designing the properties of dispersion-flattened photonic crystal fibers," Opt. Express 9, 687 - 697 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687</a> [CrossRef] [PubMed]
  17. W. J. Wadsworth, A. Ortisoga-Blanch, C. Knight, T. A. Birks, T. P. Martin Man, and P. St. J. Russell, "Supercontinuum generation in photonic crystal fibers and optical fiber tapers: a novel light source," J. Opt. Soc. Am. B 19, 2148 - 2155 (2002). [CrossRef]
  18. K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, "A cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding," Opt. Express 9, 714- 720 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714</a> [CrossRef] [PubMed]
  19. Z. Zhu, T. G. Brown, "Multipole analysis of hole-assisted optical fibers," Opt. Commun. 206, 333-339 (2002). [CrossRef]
  20. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, "Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss," Opt. Express 9, 681-686 (2001). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681</a> [CrossRef] [PubMed]
  21. A. D'Orazio, M. De Sario, L. Mescia, V. Petruzzelli, F. Prudenzano, A. Chiasera, M. Montagna, C. Tosello, and M. Ferrari, "Design of Er3+ doped SiO2-TiO2 planar waveguide amplifier," J. Non-Crystalline Solids 322, 278-283 (2003). [CrossRef]
  22. F. Prudenzano, "Erbium-doped hole-assisted optical fiber amplifier: design and optimization," J. Lightwave Technol. 23, 330-340 (2005). [CrossRef]
  23. J. W. Fleming, "Dispersion in GeO2-SiO2 glasses," Appl. Opt. 23, 4486-4493 (1984). [CrossRef] [PubMed]
  24. W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, "Absorption and emission cross section of Er3+ doped silica fiber", IEEE J. Quantum Electron. 27, 1004-1010 (1991). [CrossRef]
  25. P. Blixt, J. Nilsson, T. Carlnas, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+ - doped fiber amplifiers: experiments and modelling," IEEE Photon. Techonol. Lett. 3, 996 - 998 (1991). [CrossRef]
  26. P. Myslinsky, D. Nguyen, and J. Chrostowski, "Effects of concentration on the performance of erbium-doped fiber amplifiers," J. Lightwave Technol. 15, 112-120 (1997). [CrossRef]
  27. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka "Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion," Opt. Express 11, 843 - 852 (2003). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843</a> [CrossRef] [PubMed]
  28. P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-doped fiber amplifiers: fundamentals and technology (Academic Press, 1999) pp. 140-144.
  29. B. P. Petreski, P. M. Farrell, S. F. Collins, "Optical amplification on the 3P0→3F2 transition in praseodymium-doped fluorozirconate fiber," Fiber Integr. Opt. 18, 21-32 (1999). [CrossRef]
  30. E. Desurvire, Erbium doped fiber amplifiers (Wiley-Interscience Inc., 1993) pp. 354-382.

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