1. Introduction

Thanks to the recent advance in giant-micro photonics [1], it has become clear that the extremely high power optical output can be extracted even from microchip laser configurations by means of thoroughly investigated optimal cavity designs [2]. The near infrared giant pulse generated from high power microchip lasers can offer high power visible, ultraviolet, and terahertz-wave by wavelength conversion [3–5], which are very useful not only for science fields but also for various industrial fields [6,7].

It is important for generating intense laser pulses with high repetition rates to manage the temperature distribution inside the solid-state laser gain media, because thermal problems often cause the deterioration of laser performances. Since the pump energy concentrates on the small laser volume inside laser media, thermal effects are easily enhanced even under the low power operation in the case of microchip lasers [8]. Especially thermal lensing and thermal birefringence of laser gain media have been well studied among various thermal problems in high power laser cavity [9]. However, only few attentions have been paid for the temperature dependencies of the spectroscopic characteristics in laser gain media traditionally. If these untraditional thermal problems exist, they must cause the deterioration of laser performances immediately.

The existence of temperature-dependent spectroscopic characteristics in Nd:YVO₄ was proved by laser performances where output pulse energies of high power Q-switched Nd:YVO₄ lasers were severely dependent on temperature [10]. Therefore, we must consider the variation of spectroscopic properties due to temperature change in laser gain media besides its thermal quenching of luminescence. On the contrary, the input-output characteristics of high power Q-switched Nd:YAG lasers were hardly dependent on their operation temperature [11]. Because the energy of Q-switched pulse depends not on fluorescent lifetime but the stimulated emission cross section (σ_em), the precise numerical models of σ_em depending on temperature T for Nd-doped laser gain media have been strongly desired.

There are two important evaluations about the temperature dependencies of spectroscopic characteristics for Nd:YAG [12]. One is the temperature independent fluorescent lifetime. This fact indicates the invariant decay rate of the spontaneous emission emitted from Nd-ions, which gives the basic assumption that the temperature dependency of σ_em is caused by the variation in the spectral profile in fluorescence I(ν) at frequency ν. Another shows that σ_em of Nd:YAG has a temperature dependence of −0.12%/K. Although σ_em of Nd:YAG reported in [12] is independent on Nd-doping concentration (C_Nd), we recently found that σ_em of Nd:YAG depended on C_Nd and its fabricating process from the measurement of the fluorescence with high spectral resolution [13]. Compared to our finer spectral resolution of 0.05 nm, the resolution of 0.4 nm in [12] could bring less sensitivity for the variation of the emission intensity with narrow bandwidth from rare-earth trivalent. From these situations, we consider that the −0.12%/K-dependence of σ_em in Nd:YAG is doubtful.

About the problem of the accuracy in the evaluation of the temperature-dependent stimulated emission cross section σ_em(T), very pregnant results are shown in [14] and [15], where reported dependences of σ_em in Nd:YVO₄ were −0.18%/K and −0.68%/K, respectively. While finer resolution of 0.07 nm in [15] compared to 0.5 nm in [14] can gives more accurate result in general, we must take care that an amplified spontaneous emission occurs easily in the Nd:YVO₄ with high population inversion. Since high pump density over 2.5 kW/cm² was comparable to 4.4 kW/cm² of pump saturation intensity of Nd:YVO₄ [16], −0.68%/K in [15] could be an enhanced result affected by the amplified spontaneous emission. Thus, comparative researches for various laser media and fine evaluations with the high spectral resolution under the weak pump field are desired in order to establish the numerical model for σ_em(T) in solid-state laser media.

In this work, σ_em(T) of Nd:YAG, Nd:YVO₄, and Nd:GdVO₄ were comparatively evaluated. By using measured data, we developed the experimentally proved general model for the spectroscopic properties of Nd-doped solid-state laser media. We also presented the numerical approximation for σ_em(T) in Nd:YAG, Nd:YVO₄, and Nd:GdVO₄, where differences between σ_em(T) of Nd:YAG and Nd:orthovanadates were discussed.

2. Experimental setup

We evaluated a 1.0at.% Nd:YAG single crystal (Scientific materials Co.), a 1.0at.% Nd:YVO₄ single crystal (ITI Electro Optics Co.), and a 1.0at.% Nd:GdVO₄ single crystal (Shandong Newphotons Science and Technology Co.). These samples had 1-mm thickness and were mirror polished on both surfaces.

During the measurement of fluorescence spectra samples were pumped by 808-nm radiation from a fiber coupled LD (LIMO40-F400-DL808, LIMO GmbH) with 100Hz repetition rate and 10% duty ratio. The averaged pump power was limited below 4 mW in order to prevent the amplified spontaneous emission. Emitted pump beam was collimated and focused onto sample surfaces by lenses with focal length of 100 mm and 80 mm, respectively. Fluorescence from samples was analyzed by a monochrometer (TRIAX-550, JOBIN YVON) with less than 0.05-nm resolution and detected by linear-InGaAs-array detector (IGA512-1x1, JOBIN YVON).

Samples were sandwiched by the copper plates of which temperature were controlled between 15°C and 65°C by a pertier device and a thermo-electric controller (LDT-5948, ILX Lightwave Co.). Similarly temperature of samples was tuned within the range from 25°C to 145°C by using of heater for the purpose of examining the developed numerical model for the spectroscopic properties of Nd-doped solid-state laser media.

The calibration of the wavelength was done within an error of 0.2 cm⁻¹ by Ne lamp (Pencil style calibration lamp 6032, Oriel), Xe lamp (Pencil style calibration lamp 6033, Oriel) and Hg lamp (Pen-Ray Mercury lamp, UVP). We referred 80 lines in these lamps within the range from 1033nm to 1150nm.

3. Results

Figure 1 shows fluorescent intensities emitted from Nd:YAG, Nd:YVO₄ and Nd:GdVO₄ at 20°C accompanied with transitions from ⁴F_3/2 to ⁴I_11/2. These intensities were normalized, and factors for normalization was proportional to η defined by Eq. (B8) where the maximum intensity at 20°C was set to 1.0. While the line-bandwidth of the main peak in fluorescence from Nd:YAG in Fig. 1 was 0.78 nm that was comparable to 0.80 nm of π-polarized fluorescence from Nd:YVO₄, π-polarized fluorescence from Nd:GdVO₄ had wider line-width of 0.94 mn. Temperature dependence of these maximum emission intensity of Nd:YAG, Nd:YVO₄ and Nd:GdVO₄ are shown in Fig. 2 .Temperature dependence of σ_em is almost the same as that of the intensity in fluorescence as discussed in Appendix B. From the dependence of peak intensities on T shown in Fig. 3 , we evaluated thermal reductions of σ_em for Nd:YAG, Nd:YVO₄ (π- and σ-polarization), and Nd:GdVO₄ (π- and σ-polarization) to be −0.20%/°C, −0.50%/°C, −0.37%/°C, −0.48%/°C, and −0.37%/°C within the temperature range between 15°C and 65°C, respectively.

 

Fig. 1 Normalized emission intensities in fluorescence at 1.06 μm under 25°C: Nd:YAG (a), Nd:YVO₄ π- and σ-polarization (b) and (c), and Nd:GdVO₄ π- and σ-polarization (d) and (e).

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Fig. 2 Temperature dependence of fluorescences from Nd-doped laser crystals: Nd:YAG (a), Nd:YVO₄ π- and σ-polarization (b) and (c), and Nd:GdVO₄ π- and σ-polarization (d) and (e).

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Fig. 3 Temperature dependence of the normalized intensities at the emission peak.

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4. Discussions

4.1 Population inversion generated under measured condition

The effect of gain narrowing causes the underestimation of measured line-bandwidth [17], therefore it can bring fatal estimation errors in σ_em(T). In order to reduce these estimation errors, the optical gain in samples is required to be low. Optical gain coefficient g in pumped laser media is given by [16]

4.2 General model for σ_em(ν) and lorentzian decomposition of I(ν)

We propose a general model for σ_em(T) in Appendix B, where σ_em(T) is given by

In order to use above model, it is necessary to determine a_i, b_if, ν_if, Δν_if experimentally. These parameters can be evaluated from measured I(ν) at various temperature by the fitting to following expression:

Russell-Saunders terms relating to optical transition around 1-μm region in Nd-doped laser gain media are explained at Appendix A. While the emitting manifold is ⁴F_3/2 term consists of Stark-levels of R₁ and R₂, the terminating manifold is ⁴I_11/2 term made of Stark-levels of Y_n (n is from 1 to 6). Therefore, our fluorescence contains 12 emission peaks. Figure 4 shows experimentally obtained b_if, ν_if, Δν_if in Nd:YAG, and least square fitting to these values. Detected temperature dependences in a_i and b_if are less than error level, thus these values are treated as a constant in following discussions. ν_if and Δν_if were well fitted by using of Debye temperatures of YAG, YVO₄, GdVO₄ that were reported in [20] and were 795K, 718K, and 677K, respectively. The fitting results in Nd:YAG and Nd:orthovanadates are summarized in Tables 1 , 2 and 3 .

 

Fig. 4 Temperature dependence of b_if (a), ν_if (b), and Δν_if (c) in Nd:YAG. In these figures markers show experimental results and lines show the fitting.

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Table 1. Spectral parameters of representative transitions in Nd:YAG

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Table 2. Spectral parameters of representative transitions in Nd:YVO₄ in π-polarization

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Table 3. Spectral parameters of representative transitions in Nd:GdVO₄ in π-polarization

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4.3 Simulation of the fluorescence dependent on temperature

Equations (4) and (5) and parameters shown in Tables 1-3 can give the prediction for I(ν) and σ_em(ν) at any required temperature. Figure 5 shows simulations of I(ν) emitted from Nd:YAG at 25°C, 85°C, and 145°C. Although these simulations are calculated by using of parameters derived from experiments carried within the range under 65°C, they coincide to additional experiments with tuned temperature from 25°C to 145°C by heater. Therefore, this model was experimentally proved within the range from 15°C to 145°C. Since the electron-phonon interaction assumes the simple density of state defined by Debye model in order to develop this model, this model should be correct within the temperature range without phase transition under Debye temperature.Figure 6 shows the prediction for I(ν) emitted from Nd:YAG, Nd:YVO₄, and Nd:GdVO₄ by means of Eq. (5) and Tables 1-3. These predictions directly indicate the difference of the temperature-stability in fluorescence between Nd:YAG and Nd:orthovanadates, which is discussed in detail at section 4.5.

 

Fig. 5 I(ν) emitted from Nd:YAG at various temperatures. Solid lines are simulations calculated from Eq. (5) with parameters in Table 1, and dashed lines are experimentally measured under temperature tuning by heater.

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Fig. 6 Predictions for the temperature-dependent I(ν) emitted from Nd:YAG (a), Nd:YVO₄ and Nd:GdVO₄ (b) by means of Eq. (5) and Tables 1-3.

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4.4 Numerical models for σ_em

It is not convenient for laser cavity design involving thermal problems to use Eq. (4), because there are too many spectral parameters to be managed. Since σ_em at only the maximum emission peak is important for laser designs, Eq. (4) can be approximated by following polynomial:

Table 4. e_i for Nd:YAG, Nd:YVO₄, and Nd:GdVO₄ under T₀ is 20°C

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Simulations calculated by Eq. (5) and approximations expressed by Eq. (8) with parameters in Table 4 within the range from 15°C to 300°C compared to experimental peak intensity are shown in Fig. 7 . Figure 7 indicates that Eq. (8) is enough applicable for the cavity design of high-power microchip lasers.

 

Fig. 7 Temperature dependence of emission intensity of various Nd-doped laser media.

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4.5 Difference between Nd:YAG and Nd:orthovanadates

Similarly to previous reports [12,14,15], obtained temperature dependencies in Nd:YAG and Nd:orthovanadates seem to be linear and non-linear, respectively. However, it is not correct to consider that either will not show a behavior interpreted by the model such as Appendix B in Nd:YAG and Nd:orthovanadates. It is because the behavior of other emission peaks of Nd:YAG except the main peak is equivalent to Nd:orthovanadates.

Though temperature dependence of the emission bandwidth in the main emission peak of Nd:YAG seems to be quite small, the emission bandwidth of two emission peaks that composed the main peak shows similar dependencies to other emission peaks in Nd:YAG, and Nd:orthovanadete as shown in Fig. 8 . However, the splitting between these two peaks becomes narrower under higher temperature. This is the reason why the main emission peak in Nd:YAG shows the different temperature-dependences from other emission peaks.

 

Fig. 8 Temperature dependence of the line-bandwidth of emission peaks in Nd-doped materials.

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Our experimental results indicated the novel design rule for the selection of the laser medium: Nd:YAG is suitable for highly stable operations, and Nd:orthovanadate is useful for temperature tuning. In both case our experimentally proved numerical model of σ_em(T) in Eq. (8) will be eminently useful for developing high power lasers by use of Nd-doped laser gain media.

5. Conclusion

We evaluated σ_em(T) of Nd:YAG, Nd:YVO₄, and Nd:GdVO₄ from temperature dependent I(ν). Under our evaluation the population inversion was induced into samples by a weak pumping field, and fluorescence was detected with fine spectral resolution. At the temperature range from 15°C to 65°C, the variation of σ_em in Nd:YAG was estimated to be −0.20%/°C at 1.06 μm, while those in Nd:YVO₄ for π- and σ-polarization were 2.5 times and 1.8 times compared to Nd:YAG, respectively. Temperature dependency of fluorescence emitted from Nd:GdVO₄ is quite similar to that of Nd:YVO₄.

By using of temperature dependence in the electron-phonon interaction, we developed a general numerical model of σ_em(T) for Nd-doped solid-state laser materials. Since this model is based on the density of state defined by Debye model, it should be correct in the temperature range without phase transition under Debye temperature. We have also presented numerical approximations of this model for our samples by a simple polynomial. This approximation should be applicable from 15°C to 350°C, and was experimentally confirmed at the temperature range from 25°C to 145°C.

Appendix A: Stark levels in Nd-doped laser media

Energy levels of the lower terms of Nd trivalent are composed of three 4f electrons. They are electrically shielded from the external field by the octet of perfectly filled 5s- and 5p-orbital. Therefore, these lower terms of Nd trivalent form Russell-Saunders coupling due to Coulomb interaction between three 4f electrons. This Russell-Saunders term can be described by ^2S+1L, where S and L are spin angular momentum quantum number and aztimuthal quantum number of three 4f electrons. Spin-orbit interaction inside 4f electron makes ^2S+1L split to the terms ^2S+1L_J that have different total angular momentum quantum number J.

When ^2S+1L_J terms in Nd:YAG are arranged in low-energy order, they are ⁴I_9/2, ⁴I_11/2, ⁴I_13/2, ⁴I_15/2, ⁴F_3/2, ⁴F_5/2, ²H_9/2, ⁴F_7/2, ⁴S_3/2, ⁴F_9/2, .... . Therefore the energy level of ground state in Nd:YAG is ⁴I_9/2. Excitation by 808 nm pump source excite Nd trivalent from ⁴I_9/2 to ⁴F_5/2, then they relaxed to the metastable state of ⁴F_3/2, which is the emitting level of fluorescence. Via 1-μm fluorescence Nd trivalent in ⁴F_3/2 decays to ⁴I_11/2, which is the terminating level of optical transition in 1-μm region. Finally Nd trivalent in ⁴I_11/2 non-radiatively relaxes to ⁴I_9/2.

Although ^2S+1L_J has multiplicity of 2J+1, ^2S+1L_J of Nd trivalent shows splitting to utmost J+1/2 levels due to Kramers' degeneration under zero magnetic field condition. In the case of the emitting level ⁴F_3/2, it splits into 2 levels named R₁ and R₂. Similarly, the terminating level ⁴I_11/2, it splits into 6 levels named Y_i (i is an integer within the range from 1 to 6) by crystal field splitting. The schematic diagram of energy levels in Nd trivalent is shown in Fig. 9 .

 

Fig. 9 Energy levels of the lowest terms of Nd trivalent. Only ⁴F_3/2 become the emitting level under 808-nm pumping, and only ⁴I_11/2 become the terminating level for 1-μm fluorescence.

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Appendix B: Numerical model for σ_em(T)

In this work the fluorescence from Nd-doped materials is a composition of the emission between Stark level-i and level-f that are belonging to different Russell-Saunders manifolds as described in Appendix A. In ordered crystalline materials each emission peak can be assumed to have a lorentzian spectral profile, where the line-shape function g_if(ν) of the transition from the Stark level-i to Stark level-f is expressed by

(B1)$$g_{if}\left(\nu \right)=\frac{\Delta {\nu }_{if}}{2\pi }\frac{1}{{\left(\Delta {\nu }_{if}/2\right)}^2+{\left(\nu -{\nu }_{if}\right)}^2},$$

Thus the line-shape function g_i^k(ν) of the transition from i to Russell-Saunders manifold k is expressed by

(B2)$$g_i^k\left(\nu \right)={\displaystyle \sum _{f\in k}{b}_{if}{g}_{if}\left(\nu \right)},$$

and b_if should be normalized by

(B3)$${\displaystyle \sum _{f\in k}{b}_{if}}=1.$$

Here we can obtain the line-shape function g^jk(ν) of the transition from manifold-j to manifold-k that is given by

(B4)$$g^{jk}\left(\nu \right)={\displaystyle \sum _{i\in j}{a}_i{f}_i{g}_i^k\left(\nu \right)},$$

where f_i is the fractional population of level-i in the manifold-j, and a_i is defined from the decay rate of level-i (A_i) by

(B5)$$a_i=A_i/{\displaystyle \sum _{i\in j}{A}_i{f}_i}.$$

By using of g^jk(ν), we can obtain σ^jk(ν) and the spectral profile of fluorescent intensity I ^jk(ν) as

(B6)$${\sigma }^{jk}\left(\nu \right)=\frac{{\lambda }^2}{8\pi n^2}{g}^{jk}\left(\nu \right){\displaystyle \sum _{i\in j}{A}_i},$$

(B7)$$I^{jk}\left(\nu \right)=\frac{\eta {\nu }^3}{c^2}{g}^{jk}\left(\nu \right)=\frac{\eta {\nu }^3}{2\pi c^2}{\displaystyle \sum _{\begin{array}{l}i\in j\\ f\in k\end{array}}\frac{a_i{b}_{if}{f}_i\Delta {\nu }_{if}}{{\left(\Delta {\nu }_{if}/2\right)}^2+{\left(\nu -{\nu }_{if}\right)}^2}},$$

where c and η are the photon speed, and a normalization factor that is defined by

(B8)$$\eta ={\displaystyle \int \lambda I^{jk}\left(\nu \right)d\lambda },$$

and proportional to a total intensity of fluorescence, respectively. Equation (B6) is well-known as Füchtbauer-Ladenburg relation.

By using of radiative lifetime of manifold j τ_j that is equal to the inverse of the total decay ratio of manifold j, σ^jk(ν) is given by

(B9)$${\sigma }^{jk}\left(\nu \right)=\frac{{\lambda }^2}{16{\pi }^2{n}^2{\tau }_j}{\displaystyle \sum _{\begin{array}{l}i\in j\\ f\in k\end{array}}\frac{a_i{b}_{if}{f}_i\Delta {\nu }_{if}}{{\left(\Delta {\nu }_{if}/2\right)}^2+{\left(\nu -{\nu }_{if}\right)}^2}}.$$

From Eqs. (B7) and (B9) it is shown that temperature dependence of σ^jk(ν) is almost equal to I ^jk(ν) at each emission peak, because line-shift of emission peaks due to temperature change are negligible compared to wavenumber itself. Temperature dependent components in the expression of σ^jk_em(ν) are only ν_if, Δν_if, f_i and b_if. Dependences of f_i on T is given by Boltzmann distribution as following:

(B10)$$f_i\left(T\right)=\exp \left[\frac{h{\nu }_{if}\left(T\right)}{kT}\right]/{\displaystyle \sum _{i\in j}\exp \left[\frac{h{\nu }_{if}\left(T\right)}{kT}\right]},$$

where h and k are Plank's constant, and Boltzmann's constant, respectively.

If the emission peak in the fluorescence consists of one transition between Stark level-i and -f, and not overlapped with other transitions, σ_em(T) are expressed by [13]

(B11)$${\sigma }_{em}\left(T\right)={\sigma }^{jk}\left({\nu }_{if}\left(T\right)\right)=\frac{{\lambda }^2}{4{\pi }^2{n}^2}\frac{a_i{b}_{if}{f}_i}{\Delta {\nu }_{if}}{\displaystyle \sum _{i\in j}{A}_i}\propto \frac{a_i{b}_{if}{f}_i}{\Delta {\nu }_{if}}.$$

In this work, we assumed that A_i is independent on temperature as mentioned in introduction. The temperature dependence of b_if for the transition from ⁴F_3/2 to ⁴I_11/2 in Nd:YAG, Nd:YVO₄, and Nd:GdVO₄ was not able to be experimentally confirmed. Even if this temperature variation existed, it is too small to detect under experiments with accuracy of our spectroscopic measurements.

Acknowledgments

This work was partially supported by Genesis Research Institute and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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