Films minces à base de Si nanostructuré pour des cellules ...

Films minces à base de Si nanostructuré pour des cellules ... Films minces à base de Si nanostructuré pour des cellules ...

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gure 5.11. It can be seen clearly that the population in the excited state N 3 follows the pump prole with regularly alternating maxima and minima as suggested before. 5.3.3 Dynamical losses and gain In the framework of this thesis, it is assumed that the real part of refractive index is a constant, while the imaginary part may depend on the absorption and gain dynamics according to the following equation, ñ = n(λ) − ik(λ, x) + ig(λ, x) Eqn (5.31) tel-00916300, version 1 - 10 Dec 2013 where n(λ) is the real part of refractive index which we assume to be unchanged with the absorption dynamics, k(λ, x) accounts for the dynamic losses (absorption) and g(λ, x) is the dynamic gain (emission). Gain refers to an increase in the emitted electromagnetic intensity due to energy transfers within the medium as compared to the incident pump. When the gain is higher than the absorption in the thin lm, we may expect to see emission. In order to simulate the gain and losses few assumptions are made: ˆ Dynamic losses k(λ, x): The dynamic losses which depend on the population of emitters, can be expressed using the following equation, k(λ, x)= σ abs(λ).N 1 (x).λ Eqn (5.32) 4π where σ abs is the absorption cross section, and N 1 is the population of emitters in the ground state. In order to estimate the absorption cross section (σ abs ), and to give a shape to it, Lorentzian functions are used. Therefore σ abs can be represented by, σ abs = σ abs.max. 1 1 + (ω 2 ij(abs) −ω2 ) 2 ω 2 ∆ω 2 (abs) Eqn (5.33) where σ abs.max is the maximum value of absorption cross sections, ω ij(abs) is the frequency of absorption between levels i to j, and ∆ω is the width of the Lorentzian curve of absorption. The parameters in equation 5.32 are obtained by giving a shape to k(λ, x) by tting the experimentally obtained k(λ) with lorentzian function in the energy range of investigation (1.1 - 2.05 eV) of the emission behaviours (Fig. 5.12). ˆ Dynamic gain g(λ, x) : Similar to dynamic losses, dynamic gain can also be represented as a function of emission cross section (σ emis ) and the population of emitters in the excited state (N 3 ) as follows, 150

Figure 5.12: The shapes of k(λ) and k (λ, x). Lorentzian t of the k(λ) obtained through experiments, gives the shape of k(λ, x). tel-00916300, version 1 - 10 Dec 2013 g(λ, x) dynamic = σ em(λ).N 3 (x).λ 4π Eqn (5.34) For a transition between i → j, the Lorentzian function for the emission cross sections is represented as, σ em. = σ em.max 1 1 + (ω 2 ij(em.) −ω2 ) 2 ω 2 ∆ω 2 (em.) Eqn (5.35) where σ em.max is the maximum value of emission cross sections, ω ij(em.) is the frequency of emission transitions and ∆ω is the width of the Lorentzian curve of emission. Figure 5.13 shows the typical shapes of σ(λ) abs. and σ(λ) em. 5 Figure 5.13: Illustration of emission and absorption cross-sections for arbitrary inputs. 151

Figure 5.12: The shapes of k(λ) and k (λ, x). Lorentzian t of the k(λ) obtained through<br />

experiments, gives the shape of k(λ, x).<br />

tel-00916300, version 1 - 10 Dec 2013<br />

g(λ, x) dynamic = σ em(λ).N 3 (x).λ<br />

4π<br />

Eqn (5.34)<br />

For a transition between i → j, the Lorentzian function for the emission cross<br />

sections is represented as,<br />

σ em. = σ em.max<br />

1<br />

1 +<br />

(ω 2 ij(em.) −ω2 ) 2<br />

ω 2 ∆ω 2 (em.)<br />

Eqn (5.35)<br />

where σ em.max is the maximum value of emission cross sections, ω ij(em.) is the<br />

frequency of emission transitions and ∆ω is the width of the Lorentzian curve of<br />

emission. Figure 5.13 shows the typical shapes of σ(λ) abs. and σ(λ) em.<br />

5<br />

Figure 5.13: Illustration of emission and absorption cross-sections for arbitrary inputs.<br />

151

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