EQUIPO HUMANO

In this section, the fundamentals required to model gain in erbium-doped fibre (EDF) am- plifiers are reviewed. The dynamics of the gain and amplification process are modelled by a system of coupled atomic populations and light flux propagation equations. The simplest EDFA configuration, shown in Figure 1.11 in chapter 1, consists of an erbium-doped fibre spliced into the signal transmission link of an optical network and a source of pumped light. Different pump configurations provide EDFAs with different specifications as shown in Fig- ure 2.10, explained in the following sections in this chapter.

Figure 2.1: The three-level system used for the amplifier model. The transition rates between levels 1 and 3 are proportional to the populations in those levels and to the product of the pump flux φp and pump cross-section σp. The transition rates between levels 1 and 2 are

proportional to the populations in those levels and to the product of the signal flux φs and

signal cross-section σs. The spontaneous transition rates of the ion (including radiative and

non-radiative contributions) are given byΓ32andΓ21[6, 8, 9].

The amplification mechanism in EDFAs is stimulated emission, but in this case the pop- ulation inversion is achieved by optical pumping. During this process, pump photons raise electrons to excited states which exist at higher energy levels. This configuration is known

as the three-level atomic system [6, 8, 9, 34, 78] as shown in Figure 2.1 that provides the EDFA with the ideal characteristics for optical networks. This scheme consists of a ground state denoted by 1, an intermediate state labeled 3 and state 2. Since state 2 often has a long lifetime in the case of a good amplifier, it is sometimes referred to as the metastable level. As shown in Figure 2.1, the pump supplies energy to the erbium ions in the ground state, and then the ions are excited either to metastable level 2 if the pump wavelength is 1480 nm or to intermediate level 3 if the pump wavelength is 980 nm. The erbium ions in the latter case decay non-radiatively and quickly to the metastable level in time of the order of 1µs.

Apart from the wavelength of 980 nm, other wavelengths do not have adequate energy to pump ions to intermediate level 3 [77]. The erbium ions in the metastable level decay to the ground level either spontaneously within a time of the order of 10 ms, or through stimulated emission by an incoming signal. The radiated energy due to the ion decay between the metastable level and ground level corresponds to the photons of wavelengths between 1520 nm - 1570 nm. Light signals with wavelengths in this spectral region that traverse through the EDFA induce stimulated emission from the excited ions and then light signal amplification occurs due to this process. The rate equations for the population variation are written as [6,8]:

dN3 dt = −Γ32N3+ (N1− N3)φpσp (2.1) dN2 dt = −Γ21N2+Γ32N3− (N2− N1)φsσs (2.2) dN1 dt =Γ21N2− (N1− N3)φpσp+ (N2− N1)φsσs (2.3)

Where N1, N2, N3 are the population of ions, number of ions/volume, at energy levels

E1, E2, E3respectively. We denote the emission cross-sections for level 2 to 1 transition by

σs in units of area. We denote the absorption cross-sections for level 1 to 3 transition by

σp in units of area. Transition rate from level 2 to level 1 is denoted by Γ21 and is mostly due to radiative transitions. Transition rate from level 3 to level 2 is denoted by Γ32 and it is mostly due to non-radiative transitions. The incident pump light intensity flux at the frequency corresponding to the level 1 to level 3 transition is denoted by φp corresponds to

the pump. Theφpis in number of photons/time/area. The incident light intensity flux at the

frequency corresponding to the level 1 to level 2 transition is denoted byφsand corresponds

derivatives of the ion population in each level will be zero as shown below [6]: dN3 dt = dN2 dt = dN1 dt = 0 (2.4)

The total population of erbium ions in the fibre core is denoted by N and given by:

N= N1+ N2+ N3 (2.5)

From Equation (2.1), the population of level 3 can be written as:

N3=

1 1+Γ32/φpσp

N1 (2.6)

WhenΓ32is large (corresponding to a speedy transition from level 3 to level 2) compared to the effective pump rate into level 3, then the value of φpσp and N3 is very close to zero. Thus, the population of erbium ions is mostly in levels 1 and 2. From Equation (2.6), the population of level 2 can be written as:

N2=

(φpσp/Γ32) +φsσs

Γ21+ 2φsσs

N1 (2.7)

Then, from Equation (2.5), the populations N1, N2 and population inversion (N2− N1) can be derived as:

N2− N1=

φpσp−Γ21

Γ21+ 2φsσs+φpσp

N (2.8)

The principle of operation of an EDFA is summarized in these paragraphs. The three energy levels of erbium ions in silica are shown in Figure 2.2, and are labeled E1, E2, and

E3 in order of increasing energy. Each energy level in the figure is shown as a single line in an isolated ion of erbium, but each energy level is spread over a continuous energy band when these ions are doped or inserted into silica glass. The process of splitting is known as Stark splitting. The difference between the energy levels is denoted by the wavelength in nm of the photon corresponding to it. The process of spreading of energy levels is a useful characteristic of optical amplifiers because it increases the frequency or wavelength range of the signals that can be amplified. Moreover, an amplifier is capable of amplifying several wavelengths simultaneously. The erbium ions are distributed in the various levels within

each energy band in a non-uniform manner according to the Boltzmann distribution by a process known as thermalisation.

Figure 2.2: Three energy levels E1, E2, and E3 of erbium ions in silica glass used in the amplification process. The fourth energy level, E4, is present in fluoride glass not in silica glass. The difference between the energy levels is denoted by the wavelength in nm of the photon corresponding to it. The upward arrows indicate wavelengths at which the amplifier can be pumped to excite the ions into the higher energy level. The 980 nm transition corre- sponds to the band gap between the E1 and E3 levels. The 1480 nm transition corresponds to the band gap between the bottom of the E1 to the top of the E2 levels. The downward arrows indicate the wavelength of the photons emitted owing to stimulated and spontaneous emission [1, 6, 8].

We explained that the three energy level atomic system can be reduced into a two energy level atomic system. In the two energy level system, an optical signal at frequency fm could

be amplified only if h fm is equal to the energy difference (E2− E1). Here, h is Planck’s constant (6.63 x 10−34J s). As we noted, these energy levels are spread into bands, thus all frequencies that correspond to the energy difference between level E2 and level E1 can be amplified. As a result, in the case of erbium ions in silica glass, all wavelengths in the range 1525 nm to 1570 nm can be amplified by stimulated emission from the E2 band to the E1. The band of wavelengths is 50 nm wide with a peak around 1532 nm. This is in the low attenuation region of SMF that is applied in optical WDM networks.

In thermal equilibrium condition, N1> N2> N3. The condition for stimulated emission from E2to E1is that the N2 > N1, this can be achieved both by absorption and spontaneous emission. When the amplifier is pumped by an optical signal at wavelength 980 nm, it will cause transitions of ions from E1to E3, because N1> N3. The ions that have been moved to level E3by the pumping process will rapidly transit to level E2by the spontaneous emission. As we mentioned earlier, the lifetime for this process,τ32, is about 1 µs. The ions that have

moved to level E2by spontaneous emission will transit to level E1 by spontaneous process. The lifetime for this process,τ21, is about 10 ms.

The erbium has pump bands at 514.5, 532, 670, 800, 980, and 1480 nm. These wave- lengths correspond to the energy differences between E1 (4I15/2) and the first six excited

states of the erbium ion. When an amplifier is injected at any of these wavelengths, absorp- tion of pump photon raises erbium ions to an excited state at the corresponding energy. The erbium ion decays non-radiatively to the energy level E2(4I13/2), metastable state. However,

the pump efficiency is one of the factors that decides which of these wavelengths of the pump band could be used for pumping an EDFA. Pump efficiency is degraded by Excited State Ab- sorption (ESA) transitions, the erbium ions in level E2can be raised to a higher excited state by absorbing pump light photons. It has been demonstrated that the ESA at E2does not take place at the wavelengths 980 and 1480 nm. Thus, the pumping process is more efficient, and the EDFA uses less pump power for a given gain at 980 nm than other wavelengths. However, higher power pump lasers exist at 1480 nm, in comparison to 980 nm, and so 1480 nm pumps are applied in amplifiers developed to provide high output powers.

Chemical compound, length and ion populations are the properties which affect the EDFA gain at each wavelength. These also form the basis of the design of the different types of EDFA, such as L-band or L+C band EDFAs [55, 81].

To develop a model for the EDFA, it is necessary to realize that even in a simple one- dimensional model of the fibre amplifier, the transverse shape of the optical mode and its overlap with the transverse erbium ion distribution profile are important, this is parameter- ized by a term denoted by the overlap factor [82]. Only that part of the optical mode that overlaps with the erbium ion distribution will stimulate absorption or emission from the Er3+ transitions. The whole mode, however, will experience gain or attenuation as an outcome of this interaction. For instance, for a step index fibre, if the erbium is doped only in the fibre core, since a portion of the optical mode extends into the cladding, only that portion of the optical mode that is in the core will experience the effects of the Er3+ present. This can be used advantageously. By doping the erbium only in the very centre of the fibre core, the erbium ions will only face the very high intensity part of the optical mode and the pump will more easily invert the maximum number of Er3+ ions [83]. In general, the erbium density is not necessarily constant and can vary significantly across the fibre core and cladding regions.

By considering the light fields in EDFAs to be confined in a core of very small dimensions, it is assumed the pump and the signal intensities as well as the erbium ion distribution are constant in transverse dimension and the situation simplifies to a one dimensional problem which is easier to solve.

The energy levels of rare earth ions consist of relatively well separated groups of closely spaced energy levels. The pumping wavelength required for transition to energy level 3 is different than that of energy level 2, and the assumption is made that rapid relaxation occurs from level 3 to level 2. For all practical purposes, the population in level 3 is then effectively zero, and the three level system reduces to a two level system. The whole behaviour of the EDFA can be predicted from the absorption and emission cross-sections which are shown in Figure 2.3. The thermal distribution of energy within the closely spaced energy levels causes the difference in spectral shape between the emission and absorption spectra [6, 84, 85].

Figure 2.3: Experimentally obtained emission and absorption cross-sections of an erbium doped Ge/Al/P silica fibre [10–12].

Due to the reduction of the three-level system to a two-level system, the rate equation can be written in terms of the total population densities of level 1 and 2 as shown in more detail in Appendix B.2. The two level rate equation illustrated in Appendix B.2 was ana- lytically investigated by Saleh et al [75]. The ASE power effect on the EDFA saturation was ignored, this was an essential assumption for the derivation of the model. Then, another

model was developed including the ASE power effect on saturation by the same research group as illustrated in [86].