2. Numerical simulation

We start our analysis considering first a tapered fiber section formed in a singlemode step index fiber surrounded by air. This scheme has been extensively studied in un-doped fibers showing that no changes of the transmitted power by temperature are obtained when only air is considered as surrounding media [11–14]. These modifications on the transmitted power are reached when a thermochromic material surrounds the tapered fiber section. However, this situation is not considered in our study because our principal interest is to analyze the transmitted power variations caused by the temperature response of the amplified spontaneous emission in the tapered doped core.

Once air is considered as the surrounding media, we proceed to analyze the mode field expression of the fundamental core mode within the tapered fiber. The mode field expression is an important parameter in doped fiber amplifiers because it determines the core mode fraction of the pump and signal, which affects the power conversion in the doped core [23–28]. In this study, Gaussian shapes for the pump

and signal fundamental core modes were considered. Thus, we can write the transverse intensity pattern using a Gaussian envelope approximation given by [29]:

$$f\_{p,s}(r) = \frac{1}{\pi \Omega^2} e^{-r/\Omega^2} \tag{1}$$

where subscripts p and s refer to pump and signal radiations and Ω is determined by the characteristic of the fiber: the refractive indexes and radius of the core and cladding, respectively. The multiplying factor in Eq. (1) is chosen to normalize f rð Þ as follows:

$$2\pi \int f\_{p,s}(r) dr = 1\tag{2}$$

For a step index fiber, Ω is approximately given by [29]:

$$
\Omega = a f\_0(U) \frac{V}{U} \frac{K\_1(W)}{K\_2(W)} \tag{3}
$$

where a is the core radius and U ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 n2 <sup>1</sup> � <sup>β</sup><sup>2</sup> q , W ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> n2 1 q , <sup>V</sup> <sup>¼</sup> ka ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 <sup>1</sup> � n<sup>2</sup> 2 <sup>p</sup> , <sup>k</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> <sup>λ</sup> , n<sup>1</sup> and n<sup>2</sup> are the refractive indices of the active core and cladding, respectively, and β is the propagation constant of the pump or signal mode. For a given V, the value of W can be obtained using the empirical relationship

$$W = 1.1428V - 0.996,\tag{4}$$

It is valid only for step index fibers. In this way, according to the pump and signal wavelengths, one can obtain the respective values of U, W, and V using the numerical aperture (NA) of the fiber where

$$\text{NA} = \sqrt{n\_1^2 - n\_2^2},\tag{5}$$

and subsequently the parameter Ω. This procedure is accurate for 1:5<V <2:5.

Now, we analyze the thermal effects on the tapered doped fiber considering the changes of the absorption and emission cross-section with temperature. Previous works in un-tapered fibers have reported changes in population of the energy levels in Yb-doped glasses, and the broadening of the homogeneous linewidth as temperature is increased [18–21]. It results in modifications of the absorption and emission cross-sections for the signal and pump radiations. These effects may vary for each individual doped fiber due to the different glass composition, concentration of dopants and co-dopants, and the degree of structural disorder on the glass network used in different active fibers. In particular, we study the impact of variations on the cross-sections due to temperature in tapered doped fiber amplifiers. Without loss of generality, we use the absorption and emission cross-section changes with temperature reported in [18], which is a representative of several Yb-doped fibers. These changes are expressed in the following equations:

$$
\sigma(T) = \sigma(\text{20}^\circ \text{C}) + \frac{d\sigma}{dT} \tag{6}
$$

$$\frac{d\sigma\_{abs}^{1064nm}}{dT} = 7.78 \ast 10^{-29} \frac{m^2}{^\circ K} \tag{7}$$

$$\frac{d\sigma\_{em}^{1064nm}}{dT} = -2.44 \ast 10^{-28} \frac{m^2}{^\circ K} \tag{8}$$

$$\frac{d\sigma\_{abs}^{976um}}{dT} = \frac{d\sigma\_{em}^{976um}}{dT} = -1.63 \ast 10^{-27} \frac{m^2}{^\circ K} \tag{9}$$

where σ1064nm abs and σ1064nm em are the absorption and emission cross-sections for the signal wavelength and σ976nm abs and σ976nm em are the absorption and emission crosssections for the pump wavelength, respectively.

In order to model a temperature-dependent tapered Yb-doped fiber amplifier, we numerically analyze the following coupled equations:

$$\frac{dI\_p(r,z)}{dz} = \left[\sigma\_{em}^p(T)n\_2(T) - \sigma\_{abs}^p(T)n\_1(T)\right]N\_{tol}I\_p(r,z)\tag{10}$$

$$\frac{dI\_s(r,z)}{dz} = \left[\sigma\_{em}^{\prime}(T)n\_2(T) - \sigma\_{abs}^{\prime}(T)n\_1(T)\right]N\_{tot}I\_s(r,z) \tag{11}$$

where Ipð Þ r, z and Isð Þ r, z are the pump and signal intensities; Ntot is the total ytterbium population; σ p abs, σ p em, σ<sup>s</sup> abs, σ<sup>s</sup> em, are the temperature dependent absorption and emission cross-sections of the pump and signal at 976 nm and 1064 nm, respectively; and n1ð Þ T , n2ð Þ T are the temperature-dependent upper- and lower-state populations of Yb, which are given at a steady state by the following equations:

$$m\_2 = \frac{R\_{abs} + W\_{abs}}{R\_{abs} + R\_{em} + W\_{abs} + W\_{em} + A\_{ep}} \tag{12}$$

$$n\_1 = 1 - n\_2 \tag{13}$$

where Rabs <sup>¼</sup> <sup>σ</sup><sup>p</sup> absIphvp, Rem ¼ σ p emIphvp, Wabs <sup>¼</sup> <sup>σ</sup><sup>s</sup> absIshvs, and Wem <sup>¼</sup> <sup>σ</sup><sup>s</sup> emIshvs. In these equations, the ASE generation is not considered, and only effects of the taper and temperature modifications are investigated.

In order to consider the taper effects and the overlap of the pump and signal fundamental mode with the active core, we can use [29]:

$$I\_{p,s} = P\_{p,s}(z) f\_{p,s}(r) \tag{14}$$

where subscripts p and s refer to pump and signal radiations, Pp,<sup>s</sup>ð Þz are the zdependent powers at the pump and signal wavelengths, and f rð Þ is given by Eq. (1). It is clear at this point that the effect of taper and temperature in parameter Ω defined in Eq. (3) directly modifies the evolution of pump and signal intensities described in Eqs. (10) and (11).

If we consider the pump power at any value of z, we have [30]

$$P\_{p, \boldsymbol{\varphi}}(\boldsymbol{z}) = \int\_0^\infty \int\_0^{2\pi} I\_{p, \boldsymbol{\varphi}}(r, \boldsymbol{z}) r dr d\boldsymbol{\phi} = 2\pi \int\_0^\infty I\_{p, \boldsymbol{\varphi}}(r, \boldsymbol{z}) r dr \tag{15}$$

Then,

$$\frac{dP\_{p,\boldsymbol{\varsigma}}(\boldsymbol{z})}{dz} = 2\pi \int\_0^\infty \frac{dI\_{p,\boldsymbol{\varsigma}}(r,\boldsymbol{z})r dr}{dz} \tag{16}$$

Using Eqs. (14) and (16), we can rewrite Eqs. (10) and (11) as follows:

Temperature Sensing Characteristics of Tapered Doped Fiber Amplifiers DOI: http://dx.doi.org/10.5772/intechopen.89894

Figure 1.

Modeling scheme using tapered Yb-doped amplifiers with different tapered core shapes and with co-propagating single pump at different initial taper ends: (a) pump at the end with wider radius and (b) pump at the end with lower radius.

$$\frac{dP\_p(z)}{dz} = 2\pi \int\_0^{a(z)} \left[\sigma\_{em}^p(T)n\_2(T) - \sigma\_{abs}^p(T)n\_1(T)\right] \mathcal{N}\_{tot} f\_p(r) r dr \tag{17}$$

$$\frac{dP\_s(\mathbf{z})}{dz} = 2\pi \int\_0^{\mathbf{a}(\mathbf{z})} \left[ \sigma\_{em}^{\epsilon}(T) n\_2(T) - \sigma\_{abs}^{\epsilon}(T) n\_1(T) \right] \mathbf{N}\_{tot} f\_s(r) r dr \tag{18}$$

On these equations, we assume that the fiber is doped with uniform Yb concentration up to the core radius "a" which depends on z. Besides, it is worth to note that n1ð Þ T and n2ð Þ T depend on f <sup>p</sup>,<sup>s</sup> ð Þr due to their relation with Ip and Is intensities.

Once the temperature-dependent coupled equations are defined, we proceed to model a tapered Yb-doped fiber amplifier with different longitudinal tapered core shapes and different pump schemes [31] as is shown in Figure 1.

$$r(z) = \frac{1}{C} \left(-D - Bz - z^2\right) \tag{19}$$

$$r(z) = \frac{1}{C} \left( -D - B(L - z) - (L - z)^2 \right) \tag{20}$$

This model can be applied to a different doped material as the thulium; in this case Eqs. (6) to (9) must be replaced in agreement with absorption and emission cross-section changes with temperature reported in [32–35]. These changes are expressed in the following equations:

$$
\sigma\_{abs}^{1600nm}(T) = \mathbf{15.56x10^{-25}} - \mathbf{38x10^{-28}T} \tag{21}
$$

$$
\sigma\_{em}^{1600nm}(T) = \mathbf{1.8x10}^{-26} + \mathbf{1.99x10}^{-28}T \tag{22}
$$

$$
\sigma\_{abs}^{1841nm}(T) = -\mathbf{1}.96\mathbf{x}\mathbf{10}^{-26} + \mathbf{3}.53\mathbf{x}\mathbf{10}^{-28}T \tag{23}
$$

$$
\sigma\_{em}^{1841mm}(T) = \text{3.75x10}^{-25} - \text{1.57x10}^{-29}T \tag{24}
$$
