Modeling population and thermal lenses in the presence of Auger Upconversion for Nd³⁺ doped materials

1. Introduction

Thermal lens (TL) spectrometry is a highly sensitive photothermal technique that is used for the thermal and optical analysis of materials [1,2 ]. It relies on the measurement of the laser-induced local heating in a sample by probing the change in the optical path, mainly by the radial refractive index gradient, known as the TL effect. The study of thermo-optical properties is generally done using the time resolved TL method. However, it is well known that some non-linear effects may also be present in the illuminated volume. One example is the photon-induced population change in the ground and excited states that can alter the refractive index according the excitation intensity profile [1,3,4 ]. This effect is referred to as population lens (PL), and it may mainly be important in rare-earth doped optical solids [1,3–6 ]. Indeed, the study of nonlinear properties of photonics materials is particularly important because standing waves, principally in laser cavities, produce self-focusing, temporal and spatial self-phase modulation, and light-induced gratings that cause effects like hole-burning [1,7,8 ].

The laser-induced optical path change produces lens-like optical elements into the sample. Each of these optical elements introduces an additional phase shift that can be probed by a laser beam that is passed through the illuminated volume of the sample and exhibits a corresponding on-axis intensity change. Both the PL and TL effects are generally present, appearing as a complex transient signal. Theoretical and experimental efforts have been made to describe properly and to discriminate these effects [1,8–11 ]. When the lifetime () of the excited state is so shorter than the TL formation time ($t_c$), the transient, in principle, contains only TL effect. This occurs, for instance, with the Nd-doped NAB crystal that has a lifetime of $~20 \mu s$ [12]. The opposite occurs to Ruby crystal because $t_c\ll \tau$ [3,13 ]. However, when $t_c$ is of the same order of magnitude of $\tau$ both effects will be present in the transient. This situation occurs for many $N{d}^{3+}$ and mainly $Y{b}^{3+}$ doped materials and when the PL and TL transients have the same sign (both either positive or negative), the interpretation could be difficult and mistakes can be made easily. Indeed, we have recently shown that, for $Y{b}^{3+}$ doped systems, the population lens can be accounted in the framework of the TL transient [11].

Among the intrinsic processes that can relatively impact laser action is the energy transfer upconversion, also known as Auger upconversion (AU) [12,14–18 ]. It occurs when two neighbor ions, both at the same metastable level, interact non-radiatively such that one of them is transferred to a higher excited state while the other goes to a lower state. Since this process leads to a reduction of the excited state population and lifetime, and, therefore, changes the PL and the generated heat (TL), it has been evaluated using the PL and TL techniques [12,14,16–18 ]. In this way, it is also desirable to study the influence of the AU on the transient signal of the TL + PL spectrometry. In this letter, we present a theoretical model describing the cw laser induced time-resolved lens spectrometry with both TL and PL effects, and including the AU process, for Nd³⁺ doped systems.

2. Results

The modeling is based on the rate equation solution for the ion excited-state population $N_{ex}(g,z,t)$. In a conventional mode-mismatched TL experiment, a cw $TE{M}_{00}$ Gaussian beam excites a sample of thickness L, producing a TL signal. A weak $TE{M}_{00}$ Gaussian beam, collinear with the excitation beam, is used to probe the TL. The temporal dependency of the temperature gradient, $T(r,z,t)$, is given by the diffusion equation [2,11 ]:

For $N{d}^{3+}$ doped systems, the excited-state population taking into account AU is given by the following rate equation [14]:

The solution of Eq. (2), considering that $N_e\left(r,z,t\right)=N_e\left(r,t\right) e^{-A_{e}z}$, is given by:

As mentioned before, if the change of $\eta$ due to the additional decay rate is taken into account, using Eq. (3) we can now study the AU process effect on the temperature distribution. As the results presented in Fig. 1(a) , the exact numerical temperature profile can be well described by using in the diffusion equation the steady solution for $N_e(r,z,t\to \infty )$, and, in addition, by replacing, $S\left(r\right)\to S_0/2$. This procedure gives the resulting approximated solution of the diffusion equation:

 

Fig. 1 (a) Radial dependence of the laser induced temperature profile, $T(r,0,t)$, for different exposure times. Dashed and solid lines are analytical solutions, Eq. (4), without and considering AU process, respectively. Open circle are the exact numerical solution from the diffusion Eq. (1) with AU process. $\beta =10$ and the other parameters listed in Table 1 (Part 1) were used in simulations. I(t) dependence on ${\theta }_{el}$ for (b) diverging TL (ZBLAN:Nd fluoride glass) and (c) converging TL (Q-98:Nd phosphate glass). ${\theta }_{el}$ was artificially varied from 0 to 4. All these results of the right figure are for β = 0 (without AU).

Download Full Size | PPT Slide | PDF

The probe beam propagating through the illuminated volume of the sample has its wave front slightly distorted. This distortion can be expressed in terms of phase shifts. The total phase shift is the superposition of the individual phase shifts caused by the PL and TL effects:$\varphi \left(g,t\right)={\varphi }_{PL}\left(g,t\right)+{\varphi }_{TL}(g,t)$ . These phase shifts are expressed by:

The phase shifts due to the TL and PL effects can then be numerically integrated and the intensity on the center of the probe beam spot at the detector plane, which is positioned at the far field, can be written as $I\left(t\right)=I\left(0\right){\left|\underset{0}{\overset{\infty }{{\displaystyle \int }}}\text{exp}\{-\left(1+iV\right)g-i{\text{Φ}}_{\text{th}}\left(g,t\right)-i{\text{Φ}}_{\text{PL}}\left(g,t\right)\} dg\right|}^2$, with V = $Z_1/Z_c$ and I(0) as the intensity signal at t = 0. $Z_c$ is the confocal distance of the probe beam and $Z_1$ the distance between the waist of the probe beam and the sample. The parameters from the setup are m = 13, V = 1.73, ${\omega }_{0e}=40 \mu m$, and ${\lambda }_p=632.8nm$. Table 1 (Part 1) presents the values of the physical properties for ZBLAN:Nd and Q-98:Nd glasses from the literature, which are fixed in this work.

Table 1. Part 1: Physical properties of the samples. Parameters for ZBLAN and Q-98 are from references [1,17–22]. Part 2: Parameters obtained from the computational fits for ZBLAN:Nd and Q-98:Nd samples.

View Table

The contributions of the PL and TL effects on the transient, without AU (β = 0), can be seen in Fig. 1(b) and 1(c) for respectively a representative diverging TL (ZBLAN:Nd) and a converging TL (Q-98:Nd). All other parameters are kept fixed with exception of the strength of PL, that could be modified by increasing ${\theta }_{el}$. Note that ${\theta }_{el}=0$ represents a pure thermal contribution. Similarly to what we have found for the Yb³⁺ doped systems [24], the additional PL contributions appear as an overshoot of the initial signal at short times, tending to follow the TL transient afterwards in the converging TL effect. On the other hand, when a diverging TL is present, it appears more intensively modifying inclusive the PL response (at the short time). Indeed, it can be observed that both TL and PL effects could affect significantly the signal along all transient. However, the effect is more marked when one is positive and the other is negative. In order to study the influence of the AU on the TL + PL transient we consider both Eqs. (7) and (8) and compare the transients. Furthermore, we also investigate how the thermal diffusivity affects the transient. The notable influence of the AU shown in Fig. 2(a), 2(b), 2(c), and 2(d) only confirms what we expected, since, according to this Figure, the AU plays an important role in the temperature distribution. However, a detailed analysis is necessary to exactly evaluate the AU process on both effects, mainly when both are present in the same transient. Naturally the AU contribution to PL is more easily noted than to the TL when observing the Eq. (2), but it also makes itself present in the TL signal changing φ by means of η that now appears as a function of $N_e$. The question is: what are the AU contributions on the TL and PL transients?

 

Fig. 2 [(a), (b), (c), and (d)] - Normalized TL signal transients for a typical value of ${\theta }_{el}=4$, Auger upconversion parameter varying from β = 0 up to β = 20, and thermal diffusivity of [(a) and (b)] $D=3\times {10}^{-7}{m}^2/s$ and [(c) and (d)] $D=3\times {10}^{-7}{m}^2/s$, typics of glasses and crystals, respectively. In [(a) and (c)] and [(b) and (d)] were used other representative parameters given in Table 1 (Part 1), which lead to positive and negative TL signals. [(e) and (f)] - Typical normalized experimental lensing signals for (e) ZBLAN:Nd (1.0 mol%) and (f) Q-98:Nd (1.0 wt.%) samples. Continuous lines denotes least-squares curves fit using I(t) equation. The excitation power for these transients was fixed at $P_{in}$ = 200 mW. The obtained parameters are given in Table 1 (Part 2).

Download Full Size | PPT Slide | PDF

It can also be noticed that as D increases, both the thermal and population lens formation times decrease (which is evident by the different scales used on the left and right sides of Fig. 2 [(a), (b)] and [(c), (d)]. Also, when $\beta$ increases the peaks of the graph occur faster (see the diverging lens case in the inferior part). This tell us that, if $\beta$ and D are large enough, thus we can make the population effects take action under a very short period of time. However, the AU effect is so strong that it could not be neglected in the transient analysis of both TL and PL.

In order to validate the theoretical model here developed, experimental results were obtained to Nd³⁺ doped glass samples of fluoride (ZBLAN) and phosphates (Q-98) doped respectively with 1.0 mol% and 1.0 wt.%. Two representative experimental transients are displayed in Fig. 2(e) and 2(f). As excitation laser, it was used a Ti:sapphire laser tuned at 792 nm and 802 nm for ZBLAN and Q-98 samples, respectively. A low power HeNe laser at 632.8 nm was used as probe beam. Note that the signal of the PL effect appears in both cases, following the same behavior presented in the simulations. As one can see, the non-linear adjusts based on the theoretical model presented here fits very well with the experimental data.

To estimate the magnitude of PL and TL contributions, including the AU effect, several transients with different pump powers were obtained and adjusted with the proposed model. As we can see in Eqs. (3)-(6) , the model allows the access to a large number of physical properties. However, in a multiparameter regression analysis the output could present large fluctuation, which can induces error in the results. To minimize this fluctuation, we kept fixed some parameters by using values obtained from the literature (see Table 1 – Part 1). The parameters obtained by the regression analysis were ${\theta }_{th}$, ${\theta }_{el}$, D and $\beta$, and by means of these $k_T$ and $\text{Δ}{\alpha }_p$. As we can see from Eq. (7), the effects of the absorbed power and AU process appear as the product ${\theta }_{th}\phi (S_0,\beta )$ in the TL phase shift. This induces an additional difficult in the data fitting. As showed in Fig. 3 , the fitting of the experimental data with inappropriate values for $\beta$ leads to a wrong nonlinear behavior of the curve ${\theta }_{th}$ versus $P_e$. The strategy used was thus to change the parameter $\beta$ in order to obtain the best linear fit to the curve ${\theta }_{th}$ versus excitation power. Note that ${\theta }_{el}$ does not depend on $P_e$.

 

Fig. 3 θ_th versus the excitation power P_e for ZBLAN:Nd sample obtained from the experimental data fitting with β = 0, 2, and 5. Dashed lines are guides for the eye. The fitting parameters obtained for β = 2 were θ_th/P_e = (810 ± 50) × 10⁶ K/W, $〈{\theta }_{el}〉=(1.5\pm 0.1)$, and D = (2.7 ± 0.3) × 10⁻⁷ m²/s.

Download Full Size | PPT Slide | PDF

Figure 3 shows the results to θ_th versus P_e obtained with ZBLAN:Nd by means of experimental data fitting with different values of AU parameter $\beta$. The $\beta =2$ for ZBLAN:Nd glass, corresponds to the linear behavior of ${\theta }_{th}$ ($R^2=0.999$. In this case, the constant values of ${\theta }_{th}/P_e$ proves its definition $(\frac{{\theta }_{th}}{P_e}=\frac{A_e{L}_{eff}}{k_T{\lambda }_P})$. Neglecting the AU process ($\beta =0$) or an overestimation of the effect ($\beta =5$) induces nonlinearity in the curve. For the best value of $\beta$, the corresponding parameters D, ${\theta }_{el}$, and ${\theta }_{th}$ were determined. Equivalent procedure was performed in the Q-98:Nd glass giving the best value of $\beta \simeq 1.2$. Several tests were made, leading to a confidence of around 70% in the values of AU parameter. A small discrepancy with the literature, in which $\beta =3.66$ and 1.54 for ZBLAN:Nd and Q-98:Nd, respectively, could be justified by the fact that the PL contribution on the phase shift was neglected in previous works [1, 9, 14, 18 ]. The values for diffusivity are in good agreement with those reported in the literature [1,19–22 ]. Table 1 (Part 2) summarizes all the values obtained in the present work.

3. Conclusions

In this letter we present a theoretical model to distinguish between population and thermal contributions to the laser-induced optical path change, including the influence of the Auger Upconversion (AU) effect. From the results, it is evident that the AU cannot be so easily neglected and that it plays an important role on the temperature distribution and, therefore on the TL spectroscopy. Considering that the studied effects are usually present in ions doped optical materials and the combination of wide use of materials and lasers, the model presented here may be useful in the materials´ characterization as well as to avoid misleading analysis of lenses´ transients.