LOS ESTILOS DE LIDERAZGO EN LA GENERACIÓN DE UN BUEN DESEMPEÑO

Absorption cross-section can be established by direct light transmission measurements. To obtain an absorption spectrum, a broadband light source is usually used in conjunction with a monochromator to provide wavelength discrimination. When a sample is illuminated with beam of intensity I0(λ), the output power intensity I(L,λ) can

be written according to the Beer-Lambert law:

𝐼(𝐿, 𝜆) = 𝐼0(𝜆)𝑒𝑥𝑝(−𝛼(𝜆)𝐿) (2.7)

where α(λ) is the absorption coefficient and L is the optical path length of the sample along the propagation direction. The absorption cross-section σa(λ) is then defined as the

absorption coefficient, α(λ), normalised by the ion concentration, N:

𝜎_𝑎(𝜆) =𝛼(𝜆) 𝑁 = 1 𝑁𝐿𝑙𝑛 𝐼0(𝜆) 𝐼(𝐿,𝜆) (2.8)

(b) Emission cross-section

The estimation of the emission cross-section is more complicated than the absorption cross-section. When it is not straightforward to measure both the absorption and stimulated emission cross-sections for a rare-earth ion doped material, the Einstein A and B coefficients for a two-level system are often employed to calculate one from the other. In a two level system, both lower state (level 1) and upper state (level 2) are split into multiple components, and the following relationship holds [161]:

𝑔₁∫ 𝑣2𝜎_𝑎(𝑣)𝑑𝑣 = 𝑔2∫ 𝑣2𝜎𝑒(𝑣)𝑑𝑣 (2.9)

where 𝑔𝑖 is the degeneracy of level i, v is the photon frequency, σa and σe are absorption

and emission cross-sections, respectively. This equation is a more general form of the Ladenburg-Fuchtbauer relationship [162], and the validity condition in a rare-earth ion system is that either the two levels must be equally populated, or all the transitions must have the same rate. Unfortunately, erbium ion doped glasses cannot meet either of these conditions. The manifold width of the 4I15/2 and the 4I13/2 states of erbium ion doped

glass are typically a few meV (300-400 cm-1), which is larger than kT=200 cm-1 at room temperature. In addition, low temperature absorption and emission measurements indicate the transition strength is sensitive to the Stark levels involved, making it impossible for all transitions to have the same rate. If an emission spectrum is captured and the metastable state radiative lifetime is accurately measured, then a formula derived from the Ladenburg-Fuchtbauer relationship can be applied to calculate the emission cross-section [97, 161]: 1 𝜏𝑒 = 8𝜋𝑛2 𝑐2 ∫ 𝑣 2_𝜎 𝑒(𝑣) 𝑑𝑣 = 8𝜋𝑛2𝑐 ∫ 𝜎𝑒(𝜆) 𝜆4 𝑑𝜆 (2.10)

where τe is the radiative lifetime of metastable state, n is the refractive index of the host

material, c is the light velocity in vacuum, and σe is emission cross-section. This method

is known to work well for erbium ion doped fluorophosphate glass [161].

On the other hand, a relationship between absorption and emission cross-sections was proposed by McCumber in 1964 [156], as follows:

𝜎_𝑒(𝑣) = 𝜎𝑎(𝑣)𝑒𝑥𝑝 (

𝜀−ℎ𝑣

𝑘𝑇 ) (2.11)

where σe is the stimulated emission cross-section, v is the photon frequency, h is Plank’s

constant, k is the Boltzmann constant, and ε is the effective energy required to excite one Er3+ from the lower state of interest to the upper state of interest at temperature T. The only assumption required by this theory is the time needed to establish a thermal

distribution within each manifold must be shorter than the lifetime of the manifold. Thus, an accurate value of ε is essential to calculate emission cross-sections.

Miniscalco and Quimby [161] suggested a simple method for approximation of ε by using the room temperature absorption and emission spectrum half-width. The ion distribution between the ground level and excited level can be described as follows [161]:

𝑁1

𝑁2 = 𝑒𝑥𝑝⁡(

𝜀

𝑘𝑇) (2.12)

where N1 and N2 are the equilibrium population of ground level and excited level at

temperature T. For Er3+, if all eight Stark components of the ground state 4I15/2 and

seven Stark components of the excited state 4I13/2 are known, the ion distribution can

also be expressed as [161]: 𝑁1 𝑁2 = 1 𝑒𝑥𝑝(−𝐸0⁄𝑘𝑇)× 1+∑8𝑗=2𝑒𝑥𝑝(−𝐸1𝑗⁄𝑘𝑇) 1+∑7𝑗=2𝑒𝑥𝑝(−𝐸2𝑗⁄𝑘𝑇) (2.13) where E0 is the separation between the lowest components of each manifold, Eij is the

energy difference between the jth and the lowest component of level i. The electronic structure could be simplified by assuming the Stark levels for a given manifold are equally spaced, which means Eij=(j-1)Ei. This reduces the unknown parameters from 14

to 3: E0 and the manifold spacing ΔE1 and ΔE2. From low temperature measurement

results, the absorption and emission peaks corresponding to the transition between the lowest components of each manifold, therefore, the average value of absorption and emission peaks are taken as the value of E0. Also the seven ΔE1 spaced levels are chosen

based on the low-energy half-width of the room temperature emission spectrum, while the six ΔE2 spaced levels are chosen from the high-energy half-width of the room

temperature absorption spectrum accordingly. With these three parameters (E0, ΔE1 and

ΔE2), the N1/N2 value is calculated, then according to equation 2.12, the parameter ε,

which is essential to emission cross-section calculation, could be extracted. The accuracy of the calculated emission cross-section using this method was verified by Miniscalco and Quimby in Al/P-silica fibre and fluorophosphate glasses [161]. This method is also widely applied on rare-earth ion doped chalcogenide glass hosts with reasonable accuracy [48, 120, 163, 164].

(c) Maximum pump efficiency

Knowing the absorption and emission cross-sections, the fraction of ions in the 4I13/2

metastable state can be estimated as a function of excitation wavelength. Under high- power excitation, the expression for the maximum achievable excited state fraction is [97]: 𝜂𝑚𝑎𝑥 = 𝑛2 𝑛 = 𝜎𝑎(𝜆𝑝) 𝜎𝑎(𝜆𝑝)+𝜎𝑒(𝜆𝑝) (2.14)

where n is the total Er3+ concentration, n2 is the population of the 4I13/2 state, σa is the

absorption cross-section, σe is the emission cross-section and λp is the excitation

wavelength.