Appl. Phys. B 70, 487–490 (2000) / Digital Object Identifier (DOI) 10.1007/s003409900180
Applied Physics B
Lasersand Optics
Springer-Verlag 2000
Determination of the Auger upconversion rate in fiber-coupled diode
end-pumped Nd:YAG and Nd:YVO
4
crystals
Y.F. Chen1,∗, C.C. Liao2, Y.P. Lan2, S.C. Wang2
1Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China 2Institute of Electro-Optics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China
Received: 7 June 1999/Revised version: 3 August 1999/Published online: 30 November 1999
Abstract. An analytical model is developed to study the in-fluence of the Auger upconversion process on the thermal loading under lasing and nonlasing conditions. With the de-veloped model, Auger upconversion rates can be determined by comparing theoretical calculations with experimental re-sults for the ratio of the thermal loading under lasing and nonlasing conditions. The upconversion rates obtained with the present method are compared with the results measured from the fluorescence decay experiment.
PACS: 42.55.Rz
Diode-pumped solid-state lasers have been shown to be ef-ficient, compact, and reliable all-solid-state optical sources. Neodymium-doped laser crystals are widely used in diode-pumped solid-state lasers. The thermal effect is the main fac-tor in scaling diode-end-pumped Nd-doped crystal lasers to high power [1]. Recently, a number of papers [2–6] show that the upconversion process has a significant influence on the population mechanisms of the Nd-doped laser crystals at high excitation densities, leading to much larger thermal load-ing. The high excitation densities are typically found under nonlasing conditions, Q-switched operation, or operation as an amplifier. Therefore, knowledge of the Auger upconver-sion rate is important for designing a laser system with high excitation densities. The fluorescence decay measurement is often used to determine the upconversion rate in laser ma-terials. Guyot et al. [7] performed the fluorescence decay experiment and reported that the values of the upconversion rate in Nd:YLF and Nd:YAG are(1.7± 1) ×10−16cm3/s and
(2.8 ± 1) × 10−16cm3/s, respectively. Ostroumov et al. [8] used the same method and reported that the upconversion rates in Nd:LSB and Nd:GVO4 are(2–6) × 10−16cm3/s and
(1–1.4) × 10−15cm3/s, respectively.
In scaling end-pumped lasers to higher power, the fiber-coupled laser-diodes with circular beam profiles are often ∗Corresponding author.
Present address: Department of Electrophysics, National Chiao Tung
Uni-versity, 1001 TA Hsueh Road, Hsinchu 30050 Taiwan (Fax: +886-35 /729-134, E-mail: [email protected])
used as pump sources because the high-power diode lasers are very asymmetric in their emitting aperture dimensions. In this work, we developed an analytical model to consider the influence of Auger upconversion on the thermal loading in fiber-coupled diode-end-pumped lasers under lasing and nonlasing conditions. With the derived formulae, the values of upconversion rates can be determined from the best fit of theoretical calculations to experimental results for the ther-mal loading without and with laser action. The present model provides a method to estimate the Auger upconversion rates. The practical examples of Nd:YAG and Nd:YVO4 crystals are considered to illustrate the utility of the present model.
1 Theoretical modeling
The Auger upconversion process involves two nearby ions in the metastable 4F3/2. One ion returns to the 4I9/2, 4I11/2 or 4I
13/2level by transferring its energy to the other ion that is, in turn, brought into a higher excited state. To take Auger upcon-version effects into account, the rate equation for inupcon-version population density before laser action is given by [9]:
dn
dt = R(r, z) −
n
τ − γn2, (1)
where n is the population density, R(r, z) is the rate of the pump intensity at any radial location r or axial location z, τ is the emission lifetime, andγ is the upconversion rate. Here we neglect the self-quenching effect, i.e. the cross relaxation, which could be significant at a high dopant concentration [4]. Since the crystals studied here were 1.0% Nd:YAG and 0.5% Nd:YVO4 crystals, the cross relaxation is negligible. Under cw excitation, the population density of the laser level is ob-tained by setting dn/dt = 0; therefore,
n(r, z) =hp1+ 4τ2γR(r, z) − 1i/2τγ . (2) Using a fiber-coupled laser diode in an end-pumping configuration, the pump rate can be expressed as a top-hat
488 distribution: R(r, z) = Pin hνp αe−αz πω2 p 1− e−αlΘ(ω 2 p−r2) , (3)
where Pin is the incident pump power, νp is the pump fre-quency, α is the absorption coefficient at the pump wave-length, ωp is the pump size, l is the length of the active medium, andΘ() is the Heaviside step function.
The total population number is found by integrating the population density over the crystal volume
N= l Z 0 dz ωp Z 0 n(r, z)2πrdr . (4)
It can be found that ifγ = 0, the average population num-ber is given by No= τPin/hνp. Therefore, the fractional re-duction of the population inversion due to upconversion can be expressed as
Fuc= (No− N)/No. (5)
Substituting (2)–(4) into (5), and integrating over the medium length and transverse dimensions, the fractional re-duction can be found to be
Fuc(No) = 1 − 2 %(1 − e−αl) 2p1+ %−p1+ %e−αl + ln " e−αl 2+ % − 2√1+ % 2+ %e−αl− 2p1+ %e−αl # − αl , where % =4πωγτα2 p No. (6)
With γ = 0, i.e. no upconversion effects, one can find
Fuc(No) = 0.
Figure 1 shows the dependence of Fuc(No) on the aver-age population number No by using various different Auger upconversion rates and the parameters of the 0.5 at. % Nd:YVO4system in our experiment:ωp= 400 µm, l = 7 mm,
α = 10 cm−1andτ = 100 µs. It can be seen that the reduc-tion factor Fuc(No) is an increasing function of the average population number No. Even though the average population number No below threshold is proportional to the incident pump power Pin, for a cw laser above threshold, the inversion density is clamped to the critical inversion at the thresh-old condition. As a consequence, the reduction factor for a cw laser above threshold should be determined by Fuc(Nth), where Nth= τPth/hνpand Pthis the threshold pump power.
The excited ion involving in the Auger process relaxes down to the 4F
3/2 level mostly via multiphonon emission. Accordingly, the Auger upconversion process reduces the in-version population density and increases the thermal loading in laser materials. Neglecting the contribution from the cross-relaxation process, the total thermal loading in laser materials can be given by [10]
Pload= Pqd+ Puc, (7)
Fig. 1. Dependence of Fuc(No) on the average population number Nowith
ωp= 400 µm, l = 7 mm, α = 10 cm−1andτ = 100 µs
where Pqd is the thermal power deposited by the quantum defect and Puc is the thermal loading caused by the upcon-version. Under lasing conditions, the heat from the quantum defect is determined from the difference between the pump photon energy and the lasing photon energy. On the other hand, under nonlasing conditions, the heat from the quantum defect depends upon the difference between the pump pho-ton energy and the fluorescence phopho-ton energy. Therefore, the thermal loading caused by the quantum defect is given by [10]
Pqd= λl− λp /λl
Pin for lasing condition
λf− λp
/λf
Pin for nonlasing condition , (8) where λl is the laser wavelength, λp is the pump wave-length, and the average fluorescence wavelengthλf is given byRλ f(λ)dλ/R f(λ)dλ, where f(λ) is the relative
fluores-cence spectrum. To calculate the average fluoresfluores-cence wave-length, fluorescence spectra were measured with an optical spectrum analyzer (Advantest Q8347) which enhanced the performance of spectrum analyzer (0.01-nm resolution) em-ploying a Fourier spectrum system with a Michelson interfer-ometer. Figure 2 shows the fluorescence spectrum of the 0.5% Nd:YVO4pumped by a 808-nm fiber-coupled laser diode in the region of 900–1400 nm. With the fluorescence spectra,
λfwas found to be 1032 nm and 1038 nm for Nd:YVO4and Nd:YAG, respectively.
From the definition of the factor Fuc, the number of ions involving in the Auger process for the nonlasing condition can be expressed as Fuc(No)(Pin/hνp). On the other hand, the number of ions involving in the Auger process for the lasing condition is given by Fuc(Nth)(Pth/hνp) because the inver-sion density is clamped to the critical inverinver-sion at the thresh-old condition. From the known Stark-level splitting [11], the possible upconversion processes in Nd:YVO4crystal consist of the transitions 4F
3/2→ 4D3/2, 4F3/2→ 4G9/2+4G11/2+ 2K
15/2, and 4F3/2 → 4G7/2 which result from the down-conversion transitions 4F3/2→ 4I9/2, 4F3/2→ 4I11/2, and 4F
3/2→ 4I13/2, respectively. In Nd:YAG the possible upcon-version processes include 4F3/2→ 2K15/2+2D3/2, 4F3/2→ 4G
9/2+4G11/2, and 4F3/2→ 4G7/2+2K13/2+2G7/2, which arise from the downconversion transitions 4F3/2→ 4I9/2,
489
Fig. 2. Fluorescence spectrum of the 0.5 at. % Nd:YVO4in the region of
900–1400 nm
4F
3/2→ 4I11/2, and 4F3/2→ 4I13/2, respectively. The heat
generated from the upconversion processes is due to the mul-tiphonon relaxation from the excited level back to the upper laser level. Therefore, the energy of the multiphonon relax-ation is equal to the energy of the correlated downconversion transitions. Since it is rather difficult to estimate the rela-tive contribution of each upconversion channel for nonlasing conditions, we assume that the probability of the upconver-sion processes is proportional to the fluorescence spectrum. In terms of the average fluorescence wavelength, the contri-bution of the downconverted ion of an upconversion process to the thermal load can be given by
Puc= Fuc(No) Pin/hνp hνf= λp/λf Fuc(No) Pin for nonlasing conditions
Fuc(Nth) Pth/hνp
hνf= λp/λf
Fuc(Nth) Pth for lasing conditions
.
(9) With (7)–(9), the total thermal loading can be analytically expressed as Pload= 1− λp/λf Pin+ λp/λf Fuc(No) Pin for nonlasing conditions
1− λp/λl Pin+ λp/λf Fuc(Nth) Pth for lasing conditions
.
(10) Then the ratio of the thermal loading without and with laser action can be given by
P= 1 below threshold [1−(λp/λf)]Pin+(λp/λf)Fuc(No)Pin [1−(λp/λl)]Pin+(λp/λf)Fuc(Nth)Pth above threshold . (11)
2 Determination of upconversion rate
With the present model, we have considered the upconversion-induced heat generation in Nd:YVO4 and Nd:YAG crystals
end-pumped by a fiber-coupled laser diode under both las-ing and nonlaslas-ing conditions. The laser wavelength for laslas-ing conditions was 1064 nm for both Nd:YVO4 and Nd:YAG crystals. The laser crystals were wrapped with indium foil and were press-fitted into a copper housing with cylindri-cal symmetry. A thermocouple was embedded in the copper housing and near to the edge of the laser crystal. The pe-riphery of the copper housing was held at T0= 293 K. Under edge-cooled conditions [12], the end faces of the laser crystal and the copper housing are virtually insulating in compari-son to the heat transfer across the radial surface. Based on this approximation, the steady-state temperature distributions in the laser crystal and copper housing are given by the so-lution of the linear heat diffusion equation. Therefore, the heat flow through the laser crystal is equal to the heat flow through the copper housing and is proportional to K(T1− T0), where T1 is the temperature of the embedded thermocouple and K is the thermal conductivity of the copper housing. In other words, the ratios of the thermal loading in laser crys-tals without and with laser action can be determined from the ratios of the (T1− T0) under nonlasing and lasing con-ditions. Strictly speaking, the thermal conductivity is not a constant but depends upon temperature. For pure copper, the thermal conductivity varies from 387 W m−1K−1at 273 K to 379 W m−1K−1at 373 K. To avoid a significant deviation from the temperature dependence of the thermal conductivity, the thermal resistance of the copper housing was designed to lead to the maximum(T1− T0) in all experiments below 50 K. The experimental results for 0.5% Nd:YVO4 are com-pared in Fig. 3 to the theoretical results (solid lines) calculated from (1) and (11) by using the parameters used in Fig. 1. Experimental results show that the thermal loading at 10-W pump power increases by a factor of≈ 2.0 times owing to the upconversion process. The value of the upconversion rate de-duced from Fig. 3 is about(1.5 ± 0.5) × 10−15cm3/s. Since Nd:YVO4 crystal is similar to Nd:GVO4 crystal, we made a comparison between these two crystals. It can be found that the obtained upconversion rate in Nd:YVO4 crystal is close to the value in Nd:GVO4crystal,(1–1.4) × 10−15cm3/s, ob-tained by Ostroumov et al. from the fluorescence decay meas-urement [8]. To our knowledge, this is the first time the up-conversion rate in Nd:YVO4 has been estimated. Figure 3 also shows that the upconversion macroparameterγ is not a universal characteristic of the given laser crystal at differ-ent pump powers. Generally, the energy transfer mechanism consists of static and migration-assisted processes. As shown in [8], the description of upconversion in terms of γn2 is not applicable in the case of static upconversion and is also limited in the case of migration-assisted upconversion in the diffusion regime. If the upconversion parameterγ is deduced from the experiments that cannot be precisely fitted in terms of γn2, different values of γ can be obtained, depending on many experimental factors and other material parameters. This is one of the most probable reasons why there is a sys-tematic deviation between the experimental data points and the calculated curve. Even though the present experimental results cannot be completely explained with the model by using a simple termγn2 in the rate equation, we can use it to estimate the effective value ofγ .
Figure 4 shows the experimental data and theoretical calculations for the ratios of the thermal loading in 1.0% Nd:YAG crystal without and with laser action. The
theor-490
Fig. 3. Ratios of the thermal loading in Nd:YVO4 crystal without and
with laser action: experimental results (symbol) and theoretical calculations (solid lines)
etical results are calculated by using the following parame-ters:ωp= 400 µm, l = 10 mm, α = 8 cm−1andτ = 230 µs. The value of the upconversion rate deduced from Fig. 4 is about(1.8 ± 0.2) × 10−16cm3/s. This result is between the value (2.8 ± 1) × 10−16cm3/s reported by Guyot et al. [7] and the value 0.5 × 10−16cm3/s deduced from the data of Guy et al. [4]. The reason why there is a deviation between the present result and other works may be due to the defi-ciency of the model with a simple termγn2, as discussed in the Nd:YVO4result.
Finally, it is worthwhile to mention that the present for-mulae are similar to those in the recent article by Hardman et al. [13] and that article appeared after the present work was performed. Even so, Hardman et al. investigate Nd:YLF crystal whereas the present work focuses on the estimation of upconversion parameters for Nd:YAG and Nd:YVO4 crys-tal. In addition, we use the temperature rise to determine the ratios of thermal loading instead of using the thermal lens pa-rameter.
3 Conclusions
A theoretical model has been developed to consider the in-fluence of Auger upconversion on the thermal loading under lasing and nonlasing conditions. With this model, upconver-sion rates can be determined from the best fit of theoretical calculations to experimental results. The practical examples of Nd:YAG and Nd:YVO4crystals are performed to illustrate
Fig. 4. Ratios of the thermal loading in Nd:YAG crystal without and
with laser action: experimental results (symbol) and theoretical calculations (solid lines)
the utility of the present model. We believe that the present model can be applied to other Nd-doped laser materials and used to minimize the negative influence of upconversion in laser design.
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