1. Introduction

Energy transfer upconversion (ETU) process is a well known mechanism responsible by heating in laser crystals under high intensity pump conditions [1, 2]. This deleterious effect enhances thermal lensing and decreases the population inversion, thus limiting the performance of solid state lasers. Therefore, the knowledge of ETU is very important in the design of solid-state lasers. For instance, Chen demonstrated that ETU has a critical effect in the choice of pump-to-mode size ratio of a laser cavity [3]. Some methods has been used in order to characterize the ETU process such as fluorescence decay behavior [1, 2], different thermal lens techniques [4, 5] and analysis of the laser behavior. However, the emission dynamics of Nd:YAG is quite complex and departures from single exponential decay even at quite low concentrations and low pump intensity (negligible ETU effect) [5]. Consequently, the ETU effect can only be detected at very high excitation levels, higher than those usually found in operating lasers. Therefore, taking advantage on the dependence of the transmitted light through the sample with the ETU process, the Z-scan technique can be applied as an alternative approach in the study of the energy transfer mechanisms. Z-scan is the most popular technique for the characterization of nonlinear refraction and absorption behavior in several classes of materials [6]. The time resolved Z-scan technique was introduced to study materials presenting slow nonlinearities (response times > 10⁻⁴ s) using cw lasers [7]. This technique is able to monitor the transient behavior of the excited state population [8]. It is an attractive method to quantitatively study the ETU effect due to its simplicity, sensitivity and accuracy [9, 10]. Although the ETU effect is known to be important in Nd:YAG, the literature data is scarce with conflicting results [1–3]. The aim of this article is to study the ETU process in Nd:YAG crystals and in its excited state population dynamics. Moreover, the nonlinear properties related to refractive index change and absorption changes were also determined.

2. Theoretical background

In ion doped laser materials (transparent crystals and glasses), it is well known that a refractive index change appears with the variation of the excited state population or, consequently, the gain. This effect is due to the fact that the ion excited state polarizability, α_pex, is different from that the ground state one, α_pg. For Nd³⁺ ion, the ground, ⁴I_9/2, and excited, ⁴F_3/2, states of Nd³⁺ ions will be hereafter referred as g and ex, respectively. The contribution of the ion to the total susceptibility is given by${\chi }_{ion}={\alpha }_{pg}{N}_g+{\alpha }_{pex}{N}_{ex}$ [11], with N_g (N_ex) the ground (excited) ion population density. Here, the usual case where the system has a main excited metastable state that traps all excited population is considered. Therefore, in the stationary regime the total ion concentration is given by N_t ~N_ex + N_g and consequently ${\chi }_{ion}={\alpha }_{pg}{N}_t+\Delta {\alpha }_p{N}_{ex}$ where Δα_p = α_pex - α_pg . Under cw excitation, and in the stationary regime, the excited state population is given by N_ex ~N_tI/I_s for I/I_s<<1, where $I_S=h\nu /{\tau }_0{\eta }_p{\sigma }_g$ is the pump saturation intensity, hν is the photon energy, τ_ο is the excited state lifetime, σ_g is the ground state absorption cross section and η_p is the pump efficiency (i.e. the fraction of absorbed pump photons that result in excitation of the emitting level). Lower than unity η_p may be caused by the presence of impurities or “dead sites”. However, there is no clear evidence of these effects, so it is usually assumed η_p ~1 [12]. Therefore, in first order the electronic athermal refractive index change is proportional to the intensity as$\Delta n=n_{2}I$, so that [11]:

On the other hand, it is already known that the ETU in Nd³⁺ systems involves the interaction of two excited ions in the ⁴F_3/2 metastable laser level. Here one ion returns to the ⁴I_11/2 (and/or ⁴I_13/2) level while the other is promoted to the higher-lying excited level followed by a multiphonon decay back to the ⁴F_3/2 level [17, 18]. The ETU process in doped solids is satisfactorily described through standard population rate equation using the ETU rate given by W_up = γN_ex, where γ is the upconversion parameter [1]:

3. Experimental

In this work the used sample was a 2.1 mm thick Nd³⁺:YAG crystal, pumped in resonance with ⁴I_9/2 →⁴F_5/2 transition (808.7 nm), from a cw Ti:Sapphire laser. The linear absorption coefficient was found as α_abs = (7.15 ± 0.02) cm⁻¹. The Z-scan experiments were performed using a 10 cm focal length lens to focalize the laser beam, with a w = (35 ± 1) μm beam waist. The chopper frequency was adjusted to its maximum value (820 Hz) in order to minimize the aperture time (~10 µs). The time-resolved Z-scan measurements were performed at t_i = 20 μs and t_f = 600 μs (initial and final times for data acquisition). More details of the experimental set-up can be found if Refs [7, 14, 16]. For the transient measurements the sample was positioned at z = 5 mm, which corresponds to the peak of the purely refractive curve (closed/open ratio).

In ion doped materials, it is well known that both thermal and the electronic effects should contribute to the Z-scan the signal. The magnitude of the TL effect (in the cw regime) is proportional to phase parameter θ [14]:

The characteristic Thermal Lens response time is related to the radial heat diffusion, and is given by t_c = w_o²/4D, where D is the thermal diffusivity. In our experiment t_c ~70 μs, so the response time of the thermal lens is typically smaller than the electronic population lens (τ_o = 240 μs).

4. Results and discussion

Firstly, the results obtained in the low intensity regime, I<<I_s, where saturation effects are negligible, will be analyzed. This corresponds to the regime of validity of the standard Z-scan theory and of Eq. (1) since it assumes that N_ex ~N_tI/I_s. Fig. 1 shows typical Z-scan curves obtained at 808.7 nm.

 

Fig. 1 – Time resolved Z-scan measurement for (a) closed aperture, (b) open aperture and (c) the ratio closed/open at λ = 808.7 nm, chopper frequency f = 820 Hz, P = 120 mW and ω₀ = 35μm.

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Since for Nd³⁺:YAG the magnitude of the nonlinear absorption is important, we normalized the closed-aperture signal (Fig. 1(a)), dividing it by the open aperture data (Fig. 1(b)), as is typically done in the Z-scan technique [6]. Therefore, Fig. 1(c) presents the purely refractive Z-scan curve. The open aperture curve presented an increase of the transmitted intensity at the focus position which indicates that absorption saturation is the predominant effect, Fig. 1(b).

Transient measurements were also performed in the low intensity regime in order to further investigate the origin of the measured refractive index changes. In these experiments, the sample was placed at the peak of the Z-scan curve and the transmittance transient signals in both detectors were recorded, Fig. 2.

 

Fig. 2 – Transient transmittance obtained from both closed, 50% transmittance, and opened, 100% transmittance, Z-scan apertures. The solid lines are exponential fits.

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Both curves where fit by single exponentials, in good agreement with the fluorescence lifetime of this sample, τ_o = (240 ± 10) μs. In three level systems like ruby, this behavior would indicate a predominance of the electronic contribution compared to the thermal contribution. However, in Nd³⁺ doped materials, the heat released has a fast and a slow component. The fast one is due to the nonradiative transition ⁴F_5/2 → ⁴F_3/2, so that its dynamics is given by the typical heat diffusion time (t_c = w²/4D), where D is the thermal diffusivity. The slow component is due to the transitions after the radiative decay, the cascade decay from the terminal levels (⁴I_15/2, ⁴I_13/2 ⁴I_11/2) to the ground state (⁴I_9/2). In our experiment t_c << τ_o, so the slow heat component has a dynamics similar to the one of the excited state population. In Nd:YAG, ~67% of the heat is released in the slow process and 33% in the fast one. Using the value (φK⁻¹ds/dT) = 2.7 × 10⁻⁶ cm/W from Ref [20]. in Eq. (6) we estimated θ/ΔΦ_ο ∼10%. Moreover, Fig. 1(c) presents a peak-to- valley distance ΔZ_p–v ~1.7z_c (with z_c = πw²/λ) corroborating our assumption that the electronic contribution is the dominant one. In the cw regime, the thermal lens effect is known to present ΔZ_p–v, ~3.4z_c, as a consequence of heat diffusion [19, 21]. Considering the superposition of thermal and electronic effects [19] we estimate that ~14% of the Z-scan signal is due to the thermal effect. Therefore, the electronic component of $n_2^{,}=(4.4\pm 0.2)\times {10}^{-9}$ cm²/W. The imaginary part, $n_2^{,}$, is not affected by the thermal effect, and is obtained from the theoretical fitting of the Fig. 1(b) as $n_2^{,,}=(2.4\pm 0.2) {10}^{-9}$ cm²/W.

In principle, the values of N_t and I_s should be known in order to obtain Δα_p and Δσ from n₂ through Eq. (1). However, the reported σ_g values in the literature varies in the range of (4 – 10) × 10⁻²⁰ cm² (at 808nm) [2, 22–24], probably due to the uncertainty of concentration determination, since α_abs = N_tσ_g. This difficulty can be eliminated by noticing that N_t/I_s = α_absτ₀/hν, where both α_abs and τ₀ can be obtained with relative good precision. Here it was assumed unity pump efficiency (η_p ~1), for our Nd:YAG sample. This assumption is based on previous photothermal measurements performed on this sample that indicates no presence of defects that could affect the pump efficiency [12, 24]. The Z-scan allow the determination of both Δα_p and Δσ by Eq. (1). Thus, for λ = 808.7nm, Δα_p = (5.5 ± 1) × 10⁻²⁶ cm³ and Δσ = (−5.1 ± 0.5) × 10⁻²⁰ cm² were found. This Δα_p value is comparable with 4.9 × 10⁻²⁶ cm³ obtained from four-wave-mixing experiments [25], 4.0 × 10⁻²⁶ cm³ from interferometric measurements [26] and Δα_p = (3.0 ± 0.2) × 10⁻²⁶ cm³ by both interferometry and transient grating (λ = 633nm) [27]. In principle, we could expect some dispersion in Δα_p(ω) due to the superposition of resonant and nonresonant contributions. In our experiment, the laser is tuned at line-center of an absorption transition (808.7nm), so the resonant contribution vanishes as demonstrated in pump-probe n₂ spectroscopy measurements [15]. For instance, in the case of ruby (Al₂O₃:Cr³⁺), is was observed by Z-scan measurements that in the 543 – 457 nm range, Δα_p(ω) varies only between 1.7 – 1.9 (10⁻²⁵ cm³) and this variation is within the experimental uncertainty [28]. Similarly, the nonresonant contribution to Δα_p in Nd:YAG is expected to be the most important one, consequently Δα_p(ω) should be nearly constant in the 808 – 515 nm range [23].

The Δσ data is related to Nd:YAG excited state absorption (ESA) spectra studied in detail by Guyot et al. [2] and Kuck et al. [22]. Both papers indicate that at 808 nm, σ_ex is at least one order of magnitude smaller than σ_g. From the spectra of Guyot, we estimated σ_ex ~0.5 × 10⁻²⁰ cm² and Δσ = −5.5 × 10⁻²⁰ cm², in very good agreement with the value we obtained from n₂”.

Let us now consider the results for the pump saturation regime, where I is comparable to I_s so both saturation and upconversion processes should be taken into account. Firstly, applying the Beer law to the beam crossing a sample, of thickness L, the transmittance can be written as T~T₀(1- ΔσLN_ex), for N_exΔσ <<1. In this expression T₀ = exp(-α_abs L) is the linear transmittance, found for N_ex = 0. Therefore, the amplitude of the open aperture Z-scan signal is given by:

Figure 3 shows ΔT, obtained for different incident intensities, as a function of average radial intensity, <I(r)> = I_o/2, where I₀ is the on-axis intensity of the Gaussian beam profile I(r) = I_o.exp(−2r²/w²).

 

Fig. 3 – Open aperture transmittance variation as a function of average intensity, <I> = I₀/2. For comparison it is also depicted the β = 0 case.

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In principle, ΔT should be calculated integrating over this profile using N_ex given by Eq. (3). Fortunately, numerical results of this integral are in very good agreement with calculations using just the average radial intensity in Eq. (7). For the intensity range used in this paper, the difference between these two procedures is better than 8%. So, the full line in Fig. 3 was calculated using the average intensity. In the same Figure one can see the case β = 0 for comparison. The difference between them is due to ETU mechanism that decreases the excited state population, weakening the population lens, therefore reducing ΔT. The experimental points were fitted with Eq. (7) using Δσ obtained in the low intensity Z-scan measurement and leaving β and I_s as fit parameters. Therefore, β = (6.4 ± 0.8) and I_s = (18.1 ± 0.9) kW/cm² were obtained. Reminding that I_s = hν/η_pσ_gτ_o, with η_p = 1, τ_o = 240 μs results in σ_g ~(5.6 ± 0.4) × 10⁻²⁰ cm², in agreement with σ_g ~5.8 × 10⁻²⁰ cm² of Ref [23]. Moreover, it also agrees with Δσ = −5.1 × 10⁻²⁰ (obtained from n₂”) since no significant ESA absorption is expected at 808 nm [22, 23]. Since α_abs = N_tσ_g it was possible to obtain the Nd concentration as N_t = (1.28 ± 0.09) × 10²⁰ ions/cm³ or 0.92 at. %. Finally, the ETU effect could be evaluated from the dimensionless parameter β = N_tγτ_o as γ = (2.0 ± 0.3) × 10⁻¹⁶ cm³/s. This value is in good agreement with γ = (2.8 ± 1) × 10⁻¹⁶ cm³/s and γ = (1.8 ± 0.2) × 10⁻¹⁶ cm³/s, found by Guyot et al [2] and Chen et al [3], while it is slightly higher than γ = 0.5 × 10⁻¹⁶ cm³/s found by Guy et al [1]. It should be noticed that these γ values were evaluated in samples with ~1 at. %-Nd, since γ increases linearly with Nd concentration [9, 10, 14]. However, the literature results cited above were obtained via the fluorescence decay behavior, which is sensitive to ETU process only in the initial decay curve. This can increase the uncertainty in γ determination since it requires high pump intensities and/or high doping. On the other hand, time resolved Z-scan transmittance is defined as the ratio between the transmitted intensity before any nonlinear effect and the intensity at the stationary state. Therefore, it is unnecessary to know the system dynamics (with its initial conditions, e. g., N_ex(0)) after pumping. It can also be noted that the value obtained for γ presents a good accuracy when compared with the literature.

The Z-scan theory was obtained with the assumption that the refractive index, and consequently, the phase profile due to the refractive index variation, have a Gaussian profile. However, in the saturation regime, N_ex(r) is no longer proportional to a Gaussian profile. It is well known that the standard Z-scan theory cannot be applied in the saturation regime, where the refractive index is proportional to S/(1 + S) [7]. In the present paper, both saturation and upconversion effects should contribute to a non-Gaussian refractive index profile. To approach this problem, we used the Fresnel-Kirchhoff diffraction integral considering the Gaussian beam propagation through a radial phase profile, Δϕ(r) proportional to N_ex(r) given by

Figure 4 shows the power dependence of ΔT_pv (the difference between peak and valley transmittance) for β between 4 and 8. A good agreement between experimental and calculated values was obtained using β = 6, n₂’ = 4.4 x 10⁻⁹ cm²/W and I_s = 18 kW/cm², corroborating the values previously obtained.

 

Fig. 4 – Peak-and-valley transmittance variation, of the closed Z-scan curve, as a function of the saturation parameter at the peak position, for λ = 808.7 nm.

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5. Conclusions

In this paper time resolved Z-scan measurements on a Nd:YAG crystal, on saturated and unsaturated regimes, were performed. The nonlinear refractive index, n_2, was determined from the linear dependence of the Z-scan signals, from both closed an open aperture, with intensity, leading to Δα_p = (5.5 ± 1) × 10⁻²⁶ cm³and Δσ = (−5.1 ± 0.5) × 10⁻²⁰ cm². However, a nonlinear behavior was observed which could be attributed to an excited state population N_ex I/I_s - (1 + β)I²/I_s² + …, which is equivalent to consider Δn = n₂I + n₄I² + ... where the real and imaginary parts of Δn are both in good agreement with the values β = 6 and I_s = 18 kW/cm². In this way, the Z-scan method was important to corroborate that both, the nonlinear refraction and absorption, are proportional to N_ex(I). From the open aperture transmittance variation the ETU parameter, γ, could also be obtained. The saturation intensity was also obtained in the same measurement. It was shown that on the presence of ETU an effective saturation intensity (1 + β/2) times greater than the usual one appears. These parameters are particularly useful for laser designers in order to optimize the Nd:YAG gain transitions.

The results also show that, while the optical electronic nonlinearity and the energy transfer upconversion can be determined from several techniques, the Z-scan technique is confirmed as a good option because it combines sensitivity, simple analysis and simple experimental apparatus.