## Abstract

We present a theoretical model to thermal (TL) and population (PL) lenses effects in the presence of Auger upconversion (AU) for analysis of Nd^{3+} doped materials. The model distinguishes and quantifies the contributions from TL and PL. From the experimental and theoretical results, the AU cannot be neglected because it plays an important role on the excited state population and therefore on the temperature and polarizability difference between excited and ground states. Considering the extensive use of these techniques, the model presented here could be useful for the investigation of materials and also to avoid misleading analysis of lenses transients.

© 2015 Optical Society of America

## 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 Nd^{3+} 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 ]:

*k*are the mass density, specific heat, and thermal conductivity, respectively. For $N{d}^{3+}$ ions, one can write $\phi =1-{\eta}_{0}({\lambda}_{ex}/\u3008{\lambda}_{em}\u3009)$, with ${\eta}_{0}$ being the fluorescence quantum efficiency, ${\lambda}_{ex}\approx 800nm$ and ${\lambda}_{em}\approx 1060nm$ are excitation and average emission wavelengths, respectively [1]. In the presence of AU an additional decay rate $({W}_{AU})$ appears changing ${\eta}_{0}$ of the emitter level to $\eta (r,t)={\eta}_{0}/[1+\beta {N}_{e}(r,t)/{N}_{T}]$, i.e., $\eta $ presents spatial and temporal dependences [16], $\beta $ is a dimensionless parameter that determines the strength of the AU process as shown below. Therefore, before we study the thermal contribution to the transient signal, we must first investigate the excited state population.

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

^{4}F

_{3/2}excited-state lifetime in absence of AU, ${R}_{P}={\sigma}_{abs}I(r,z)/E$ is the pumping rate, ${\sigma}_{abs}$ is the absorption cross-section, $I(r,z)$ is the beam intensity profile, E is the energy of the excitation photon, and $\gamma $ is the AU parameter.

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:

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:

^{3+}ions [23]. Note that the

*r*dependence of

*N*[Eq. (3)] is seen in

_{e}(r,z,t)*S*because $I\left(r,z=0\right)={I}_{o}{e}^{-2{r}^{2}/{w}_{oe}^{2}}$. With the help of Eq. (3) and (4) , the TL and PL phase shifts can then be given respectively by [19]:

*Y*is the Young's modulus, ${\alpha}_{T}$ is the thermal expansion coefficient, ${q}_{\parallel}$ and ${q}_{\perp}$ refer to the piezo-optic coefficients for stresses applied parallel and perpendicular to the polarization axis, respectively. In the limit of $L\to 0$, $\chi (\alpha ,L)$ recovers the form for the temperature coefficient of the optical path length change in the plane-stress approximation, usually denoted by $ds/dT$ [1,19 ].

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 $I\left(t\right)=I\left(0\right){\left|\underset{0}{\overset{\infty}{{\displaystyle \int}}}\text{exp}\{-\left(1+iV\right)g-i{\text{\Phi}}_{\text{th}}\left(g,t\right)-i{\text{\Phi}}_{\text{PL}}\left(g,t\right)\}dg\right|}^{2}$, 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.

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 Yb^{3+} 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?

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 Nd^{3+} 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 $\text{\Delta}{\alpha}_{p}$. 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 ${P}_{e}$.

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.

## Acknowledgments

The authors gratefully acknowledge the financial support from Brazilian agencies: PRONEX/FAPEAL (Project 2009-09-006), FINEP (Financiadora de Estudos e Projetos), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico through the grants INCT NANO(BIO)SIMES and Project Universal n^{o} 483238/2013-9), CAPES (Coordenadoria de Aperfeiçoamento de Pessoal de Ensino Superior) – Project PVE A077/2013, and Fundação Araucária. The research of E. C. Ximendes is supported by CNPq. The Q-98:Nd phosphate sample was provided by the company Kingre Inc.

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