Numerical analysis of an end-pumped Yb:YAG thin disk laser with variation of a fractional thermal load Guangzhi Zhu,* Xiao Zhu, Yan Huang, Hailin Wang, and Changhong Zhu School of Optical and Electronic Information, and National Engineering Research Center for Laser Processing, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, Hubei 430074, China *Corresponding author: [email protected] Received 18 April 2014; revised 20 May 2014; accepted 24 May 2014; posted 29 May 2014 (Doc. ID 210257); published 30 June 2014
An analytical model is developed to describe the dynamic behavior of an end-pumped Yb:YAG thin disk laser. Within the model, the rate equations, including the nonradiative relaxation process, are calculated taking into account the dependence of the fractional thermal load on the temperature of the thin disk crystal and intracavity laser intensity. The fractional thermal load is analyzed, or can be evaluated clearly, under lasing or nonlasing conditions. The stable temperature and fractional thermal load in a thin disk crystal for different radiative quantum efficiencies are obtained using the numerical iterative method. Furthermore, the dependence of the laser output intensity on variables such as pumping intensity, coupler reflectivity, radiative quantum efficiency, and the temperature of thin disk crystal is discussed. © 2014 Optical Society of America OCIS codes: (140.3430) Laser theory; (140.3480) Lasers, diode-pumped; (140.6810) Thermal effects; (140.3615) Lasers, ytterbium. http://dx.doi.org/10.1364/AO.53.004349
1. Introduction
With the improvement of high-power laser diodes, quasi-three-level gain media, such as Yb:YAG, has attracted worldwide interest in the last twenty years [1,2]. Because of trivalent ytterbium’s simple twolevel electronic structure, excited-state absorption, upconversion, and concentration quenching can be avoided, and the relative energy difference between a pump and a laser photon is small, typically less than 10%. Comparing with Nd:YAG, Yb:YAG may be more competitive for high average power laser devices operating near 1 μm. So face- and edge-pumped Yb:YAG thin disk configurations have been focused on theoretical and experimental development. A classical multipumped configuration given by Geisen is more suitable for thin disk lasers in the power range from several watts to kilowatts [3]. Nowadays, thin disk lasers with output power of 1559-128X/14/194349-10$15.00/0 © 2014 Optical Society of America
several kilowatts are commercially available for industrial applications [4,5]. Furthermore, the power scaling law of thin disk lasers up to 100 kW in continuous-wave (CW) mode has been already considered [6]. However, the power scaling of such lasers is fundamentally limited by the thermal load in the gain medium. It is important to predict the amount of heat generated in the laser material accurately. In 1993, Fan predicted the heat fraction under lasing or nonlasing conditions in Nd:YAG and Yb:YAG [7]. The prediction was based on the assumption that the radiative quantum efficiency was high enough to approach unity in ytterbium-doped materials. So the fractional thermal load under lasing is higher than the nonlasing condition. However, recent studies have shown that the radiative quantum efficiency was less than unity in Yb:YAG [8–10], which induced more heat generation in this material. Several kinetics models of heat generation in Yb:YAG were built to identify the major heating processing and power conservation [11,12]. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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On the other hand, In order to investigate the performance of quasi-three-level Yb:YAG laser system, various models for an end-pumped CW laser are developed by Contag and co-workers [13–17]. Some analytical solutions about laser output intensity, threshold intensity, optical-to-optical efficiency, and the basic dependence of optimal crystal thickness and optimal pump absorption on various parameters are obtained. However, these analytical models only consider the fractional thermal load in the Yb:YAG crystal is a constant. In actuality, the fractional thermal load is not a constant in the processing of the laser operation, which is more related to radiative quantum efficiency, laser extractive efficiency, and temperature of the Yb:YAG crystal. So it is hard to investigate laser operational properties precisely. In this paper, a novel analytical model of an endpumped Yb:YAG thin disk laser combining the rate equations including the nonradiative relaxation process with the variation of fractional thermal load, which is affected not only by the intracavity laser intensity, but also by the temperature of the thin disk crystal, is presented. The fractional thermal load is known or can be evaluated clearly under lasing or nonlasing conditions, and is obtained on various parameters based on the analysis results, laser output intensity, laser extraction efficiency, and the temperature of the thin disk crystal. 2. Physical Model A.
Theoretical Analysis of the Heat Generation
The energy level diagram of Yb:YAG, which consists of only two manifolds, an upper 2 F 5∕2 and a lower 2 F 7∕2 , is shown in Fig. 1. The common pump and laser wavelengths are 941 and 1030 nm, are described using λP and λL , respectively. Because of the trivalent ytterbium ion’s simple electronic structure, the upconversion, excited-stated absorption, and concentration quenching play no important role. The factors generating the heat can be neglected in the model. The Boltzmann occupation factors for the upper and lower state manifolds depend on the temperature and are given by Eqs. (1a) and (1b), respectively:
f 1i
f 0j
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(1a)
E exp − KT0j ; P E0j 4 j1 exp − KT
(1b)
where K is the Boltzmann constant, T is the absolute temperature, E1i is the energy for each Stark level of excited manifold, and E0j is the energy for each Stark level of ground manifold. The fluorescence branching ratios of Yb:YAG have been calculated at room temperature are shown in Table 1 [18]. Based on the data, the average fluorescence wavelength could be given by 1 ¯λf Δv–
P3
i1
P4
1i –1i − j1 f 1i β 0j v P4 1i j1 f 1i β0j i1
P3
v–0j
;
(2)
where the index pair 1i refers to the upper manifold Stark levels and 0j refers to the lower manifold Stark levels. Δv– is the average wavenumber, and upper manifold wavenumbers are v–1i and lower manifold wavenumbers are v–0j . β1i 0j are the fluorescence branching ratios for the upper level manifold. The average fluorescence wavelength is 1009.1 nm at room temperature. Figure 2 shows the relationship between the average fluorescence wavelength and temperature. It is found that the average fluorescence wavelength changes only slightly with temperature. In order to make the heat generation in the Yb:YAG clear, the various heating generation mechanisms, as well as for fluorescence and laser extraction, must be considered. The processing of heat generation could be divided into two stages. The first stage is the pumping stage. The second stage is the upper manifold population depleted by both fluorescence decay and laser extraction. The heat generation in the pumping stage is shown as [18] Table 1.
Starting level
Fig. 1. Yb:YAG energy level diagram.
E1i exp − KT ; P E1i 3 i1 exp − KT
Yb:YAG Fluorescence Branching Ratios
Ending level
Designation
Value
(1,1)
(0,1)
0.225
(1,1)
(0,2)
(1,1) (1,1)
(0,3) (0,4)
(1,2)
(0,1)
(1,2) (1,2)
(0,2) (0,3)
(1,2)
(0,4)
(1,3) (1,3)
(0,1) (0,2)
(1,3)
(0,3)
(1,3)
(0,4)
β11 01 β11 02 β11 03 β11 04 β12 01 β12 02 β12 03 β12 04 β13 01 β13 02 β13 03 β13 04
0.09 0.342 0.088 0.079 0.084 0.015 0.023 0.009 0.010 0.010 0.025
Fig. 4. Simple explanation of heat generation by fluorescence and simultaneous laser extraction. Fig. 2. Average fluorescence wavelength as a function of temperature.
QB –12 − v–11 f 12 Re hcv–12 − v–12 h f 11 Re hcv f 13 Re hcv–12 − v–13 :
(3)
To ensure power conservation, we set the absorbed power density, given by –12 ; QA QUM QB h Re hcv
(4)
where Re is the excitation density, QUM is the upper manifold population power density, and h is Planck’s constant. The power density associated with storage of population density is given by QUM Re hcf 11 v–11 f 12 v–12 f 13 v–13 :
QB h : QA
λ λ ηh 1 − ηp 1 − ηl ηR P ηl P ; λL λf
(7)
(5)
So pump quantum efficiency, which is the fraction of absorption pump photons contributing to inversion, can be described in this stage by ηp 1 −
heat generation by fluorescence alone. QFh is the fluorescence heating power density. On the other hand, the fluorescence and laser extraction occur simultaneously and both depopulate the upper manifold and upper laser level in CW laser oscillators. The contributions of fluorescence and laser extraction to heat are shown in Fig. 4, where QLh is the laser heating density. In order to describe the process of heat generation, Fan presented the equation to describe the fractional thermal load [7]:
(6)
In the second stage, the upper manifold is depleted only by fluorescence with nonlasing generation in the resonator. Figure 3 shows a simple explanation of
where ηp is the pump quantum efficiency and ηR is the radiative quantum efficiency for the upper manifold. Nonunity radiative quantum efficiency can be related to multiphoton relaxation. ηl is the laser extraction efficiency defined as the fraction of excited ions that are extracted by stimulated emission. For the no lasing extraction condition, ηl is 0, in which case the upper manifold is depleted only by fluorescence. If ηl is 1, the equation describes complete laser extraction. B. Thermal Analysis
Fig. 3. Simple explanation of heat generation by fluorescence alone.
The numerical model is used to calculate the temperature distribution in the gain medium. The Yb:YAG crystal (typical thickness between 0.2 and 0.4 mm) is soldered on the heat sink with the indium layer. The back side of the crystal is high reflection (HR) coated for both wavelengths. The heat sink is cooled by impingement water. The geometry of the thin disk crystal mounted on the heat sink and the thermal boundary condition are shown in Fig. 5. Figure 6 shows the normalized pumping light profile from the measurement in our experiments after multipass pumping. So we assume that the side surface of the disk is heat-insulated and the heat generation in the crystal is homogeneous. The stationary heat conduction equation can be built [16]: 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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8 < ∂2 T YAG r;z 1 ∂T YAG r;z ∂2 T YAG r;z − :
∂2 T
∂r2
Cu−W r;z ∂r2
r
∂r
1 ∂T Cu−W r;z r ∂r
∂z2
∂2 T
Cu−W r;z ∂z2
Qh kYAG
circrP
0
for disk crystal;
8
for Cu W heat sin k:
The boundary conditions are: 8 > > > > >
> > > > : kYAG ∂T YAG r;z ∂z
0;
z0
for disk crystal
r;z kCu−W ∂T Cu−W ∂z
z0
;
8 > > T YAG r; 0 T Cu−W r; 0; > > > > < ∂T Cu−W r;z 0; ∂r rR > > > ∂T Cu−W r;z hec > > kCu−W T Cu−W r; 0 − T f ; > ∂z : z−z1
Fig. 5. Schematic diagram and thermal boundary conditions of the thin disk module (A is the pump zone and B is the unpump zone).
(9)
for Cu W heat sink
(10)
where T YAG r; z and T Cu−W r; z are the temperature distribution in the thin disk crystal and CuW heat sink, Lth is the thickness of laser crystal, rP is a radius of pumping spot, kYAG and kCu−W are the thermal conductivity of thin disk crystal and heat sink, respectively, hec is the heat exchange coefficient of the heat sink with the cooling liquid, T f is temperature of the cooling liquid, and Qh is the heat flux in the thin disk crystal. The stationary heat conduction equations of the thin disk crystal and CuW heat sink are a classical Poisson equation and a Laplace equation. Using the methods of mathematical physics and complex derivation, we can obtain analytical expressions of the temperature distribution: 0 ∞ X xn z An exp R n1 0 0 xn xn z · J0 r Bn exp − R R 0 ∞ X 1 xn r ; Gn · J 0 − z2 G 0 R 2 n1
T YAG r; z A0 B0 z
(11)
0 xn z R n1 0 0 xn xn z · J0 r ; (12) B0n exp − R R
T Cu−W r; z T f A00 B00 z
Fig. 6. Normalized pumping light profile after multipass pumping. 4352
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where
∞ X
A0n exp
z0 Qh r2P 0 z0 Qh r2P h z z0 Qh r2P hec z1 z0 Qh r2P 0 A0 T f 1 ec 1 ; B ; A 1 ; B0 ; 0 0 2 2 2 kCu−W hec R kCu−W hec R kYAG R kCu−W R2 hec H 0nb −kCu−W E0nb 2f n kCu−W E0na −hec H 0na 0 An ; 0 0 0 0 kCu−W hec H nb −kCu−W Enb kCu−W kCu−W hec H nb −kCu−W Enb kCu−W 1 1− Ena 1− 1 Enb kYAG kCu−W E0na −hec H 0na kYAG kYAG kCu−W E0na −hec H 0na kYAG 2f n B0n ; h H 0 −k 0 0 0 kCu−W ec nb Cu−W Enb kCu−W kCu−W hec H nb −kCu−W Enb kCu−W Ena 1− Enb 1 1− 1 kYAG kCu−W E0na −hec H 0na kYAG kYAG kCu−W E0na −hec H 0na kYAG hec H 0nb −kCu−W E0nb 1 k k B0 k kCu−W An 1− 1 Cu−W A0n 1− Cu−W B0n n 1 Cu−W ; kYAG kYAG 2 kYAG kCu−W E0na −hec H 0na kYAG 2 hec H 0nb −kCu−W E0nb 1 k k B0 k kCu−W 1 1− Cu−W A0n 1 Cu−W B0n n 1− Cu−W ; Bn kYAG kYAG 2 kYAG kCu−W E0na −hec H 0na kYAG 2 0 Qh r2P 2Qh rP xn J1 r : ; and Gn G0 2 2 0 0 R P kYAG R kYAG J 0 xn xn R
Figure 7 shows the temperature distribution along the radial direction inside the disk module for different layers (the calculated parameters are shown in Table 2). Figure 8 shows the thermal image and temperature curve at 3000 W pumping power using an infrared thermal imager. It can be seen that the maximum temperature would be observed at the center of the top layer and the temperature difference along the radial direction is small. In order to simplify the analytical expression, the average temperature of the thin disk crystal is used in the model, which is defined by 1 hTrYAG i Lth
Z
Lth
0
T YAG r; zdz:
Table 2.
Basic Parameters Used for this Model
Parameter Number of pump beam pass (M) Pump intensity (I P ) Disk thickness (Lth ) Dopant concentration (cYb ) Diameter of pump spot (mm) Thermal conductivity of thin disk crystal (kYAG ) Thermal conductivity (kHS ) Heat transfer coefficient (hec ) Thickness of heat sink (dSink ) Coolant temperature (T f )
Value 16 5 kW∕cm2 200 μm 6 at. % 6 6.5 W∕m · K [1] 385 W∕m · K 105 W∕m2 · K 2 mm 20°C
(13)
Fig. 7. Temperature distribution along the radial direction inside the disk module for different layers. The curve (-o-) shows the average temperature of in the disk crystal.
Fig. 8. Surface temperature of the thin disk crystal at 3000 W pumping power. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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C. Quasi-three-level Kinetics Model and Its Steady State Solution
In order to simplify, the lifetime, τ, is defined by −1 −1 τ τ−1 R τNR . Eq. (14) can be rewritten as
From the analysis of heat generation mechanisms, the upper manifold populations are depleted by stimulated emission, and spontaneous and nonradiative relaxation rates [19]. Therefore, the upper and lower manifold rate equations that govern the quasi-three-level system are given by
∂N 1 t; z ∂N 0 t; z − σ P f P01 N 0 − f P12 N 1 P ∂t ∂t N σ L f L03 N 0 − f L11 N 1 L − 1 : τ
∂N 1 t; z ∂N 0 t; z − σ P f P01 N 0 − f P12 N 1 P ∂t ∂t N N σ L f L03 N 0 − f L11 N 1 L − 1 − 1 ; τR τNR
In the calculation model, the pump and the laser radiation are taken to be uniform. The average temperature of the thin disk crystal is used from Eq. (13). The temperature dependence of the stimulated absorption cross section [20] and emission cross section [21] on wavelength and temperature are shown by
∂I ∓ L z σ L f L03 N 0 z − f L11 N 1 zI ∓ L z; ∂z ∂I ∓ Pi z ∂z
P
(14)
(15)
σ P f P01 N 0 z − f P12 N 1 zI ∓ Pi z;
PN
i1 I P;i
I −P;i
hvP
;
L
− I L IL ; hvL
Qstim Qstim Qspont Qnr I σ × Qstim L L f L03 N 0 − f L11 N 1 ; hvL N N 1; Qnr 1 ; τR τNR
ηR
Qspont τNR : Qspont Qnr τNR τR
APPLIED OPTICS / Vol. 53, No. 19 / 1 July 2014
(21)
(16)
σ L 1030; T 0.95334 33.608 exp − (17)
ηl
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T − 273 σ P 941; T 0.207 0.637 exp − 288 × 10−20 cm2 ;
where I P and I L are pump and laser intensities in the gain medium, and the superscripts and − represent propagation in the forward and backward directions, respectively. i is the pumping number, σ P and σ L are the spectroscopic absorption cross sections at the pump and laser wavelengths, respectively, τR is the spontaneous emission lifetime of the upper manifold, τNR is the nonradiative lifetime of the upper manifold, and f P01 and f P12 denote the Boltzmann occupation factors within the upper and lower manifolds for the upper and lower pump states of the pumping transition, respectively. Also, f L03 and f L11 denote the Boltzmann occupation factors with the upper and lower manifolds for the upper and lower laser states of the laser transition, hvP is the energy of the pump radiation, hvL is the energy of the laser radiation, and N 1 and N 0 denote the population of the upper and lower manifold. On the other hand, based on the rate equations, the laser extraction efficiency and radiative quantum efficiency are given by [19]
Qspont
(20)
(18)
T 92.82465
× 10−20 cm2 :
(22)
Solving Eq. (20) for the steady state condition, the population of the upper and lower manifold for laser operation are obtained: N 0 z
σ P f P12 P σ L f L11 L 1τ Nt; σ P f P01 f P12 P σ L f L03 f L11 L 1τ (23)
N 1 z
σ P f P01
σ P f P01 P σ L f L03 L Nt; f P12 P σ L f L03 f L11 L 1τ (24)
where N t N 0 N 1 is the sum of the two manifold populations. Based on the boundary conditions of the resonator and the approach followed by Rigrod, the absorption pumping intensity, laser output intensity, laser slope efficiency, threshold intensity, and intracavity laser intensity can be obtained by solving Eqs. (15) and (16). The detailed deviations are presented in [16]. I abs I P;0 ×
M X
i−1 fRi−1 r;P · Rf ;P · exp2i − 1δLth g
i1
× expδLth Rf ;P 1 × 1 − expδLth ;
(25)
(19) I out 1 − Rr;L I L L ηslope I P;0 − I th ;
(26)
ηslope
3. Results and Discussion
1 − Rr;L v − L p vP 1 − p 1 R ∕R R R r;L
×
f ;L
r;L
f ;L
M X i−1 fRi−1 r;P · Rf ;P · exp2i − 1δLth g i1
× expδLth Rf ;P 1 × expδLth − 1:
I th PM
i−1 i1 fRr;P
hvP τ
·
Ri−1 f ;P
n
(27)
f L03 N t Lth FL
σ
In the following, the operational characteristics of an end-pumped Yb:YAG thin disk laser are investigated. The basic dependencies of fractional thermal load, and temperature of a thin disk crystal under lasing and nonlasing condition are discussed. Table 2 summarizes the basic parameters used for the model.
1 L L ·F ·Lth
io 1 ln p Rr;L Rf ;L
· exp2i − 1δLth g × expδLth Rf ;P 1 × expδLth − 1
− IL I L L I L L 1 Rr;L I L L 1 Rr;L A C I ; (29) I P;0 1 Rr;L − B A 1 − Rr;L out
where F L f L03 f L11 , Rr;P and Rf ;P are the pump intensity reflectivity of the front surface (including the parabolic mirror and antireflectivity coating on this surface) and rear surface of the thin disk crystal, Rr;L and Rf ;L are the laser intensity reflectivities of the resonator rear mirror and output coupler, and δ is the absorption coefficient of the gain medium. As shown above, using a strictly plane wave analysis for CW quasi-three-level thin disk laser rate equations, the output power is not linearly related to the input pump power above the threshold value. The main reason is the laser slope efficiency is affected by the growth of the temperature of the thin disk crystal and depends on the absorption coefficient of the gain medium. On the other hand, the laser extractive efficiency is affected not only by the intracavity laser intensity, but also by the temperature of the thin disk crystal. So the fractional thermal load is not a constant during the laser generation, which will affect the laser operational properties. The iterative method employed here, is composed of three steps. First, the laser operational parameters, such as absorption intensity, intracavity laser intensity, laser extractive efficiency, and so on are calculated at room temperature. Then, the fractional thermal load could be calculated based on Eq. (7). Second, we can use the absorption intensity and fractional thermal load to obtain the heat flux in the thin disk medium so that the average temperature of the thin disk crystal can be calculated. Third, the rate equations are calculated again with the new temperature of the thin disk crystal. The laser operational parameters are obtained again. We iterate the second and third steps until the stable temperature of the thin disk crystal and laser output intensity are achieved.
;
(28)
Figure 9 shows the dependence of laser output intensity (left scale) and average temperature of the thin disk crystal (right scale) on pump intensity for different radiative quantum efficiencies. It can be seen that operational properties of the laser have three classical stages. At the early stage, the pump intensity is lower than the threshold intensity, and there is no laser generated in the resonator. The laser extractive efficiency is 0. The fractional thermal load is generated by fluorescence alone. With the increasing of pump intensity, the pump intensity exceeds the threshold intensity and the laser is generated from the resonator. This stage is defined as the operational stage. The fluorescence and laser extraction occur simultaneously. First, with the increasing of pump intensity, the laser output intensity and temperature of the thin disk crystal will increase consistently. But the laser output intensity and the temperature of the thin disk crystal are not linearly related to the pump intensity above the threshold value. The main reason is that the laser slope efficiency is affected by the growth of temperature of the thin disk crystal and depends on the absorption
Fig. 9. Dependence of laser output intensity (left scale) and average temperature of the thin disk crystal (right scale) on the pump intensity for different radiative quantum efficiencies. (a) ηR 100% (b) ηR 97% (c) ηR 90% 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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coefficient of the gain medium and, at the same time, the fractional thermal load is not a constant and is a function of the temperature of the thin disk crystal and the laser extractive efficiency. With the increasing of pump intensity, a further increasing of the temperature of the thin disk crystal will be obtained, so that the pump intensity cannot maintain the necessary population inversion to keep the laser generating at this temperature. The laser output intensity will decrease to zero again. This is the third stage. The fractional thermal load is generated by fluorescence alone at this stage. On the other hand, it can be seen that the radiative quantum efficiency is also one of the important factors that affect the temperature of the thin disk crystal, laser output intensity, and other parameters. As the radiative quantum efficiency increases, the fractional thermal load will decrease. The temperature of the thin disk crystal will decrease constantly at the same pump intensity, which induces the increasing of the laser output intensity. Figure 10 shows the operational temperature and fractional thermal load under lasing or nonlasing conditions as a function of pump intensity for different radiative quantum efficiencies. Although the temperature of the thin disk crystal increases with increasing pump intensity, the fractional thermal load under the nonlasing condition decreases slightly for the different radiative quantum efficiencies. The main reasons are that the average fluorescence wavelength and the Boltzmann occupation factors for the upper and lower state manifolds are both functions of the temperature of the thin disk crystal. On the other hand, because the laser extractive efficiency has a close relationship with the intracavity laser intensity, temperature of the thin disk crystal, and the radiative quantum efficiency, the dependence of the fractional thermal load on the pump intensity under the lasing condition changes more complexly. When the radiative quantum efficiency is less than unity, the fractional thermal load decreases first, and then increases with increment of pump intensity. When the radiative quantum efficiency is unity, the fractional thermal load increases first and then decreases constantly with increment of pump intensity. On the other hand, it can be seen that the temperature of the thin disk crystal and the fractional thermal load have significant differences for the same pump intensity under lasing and nonlasing conditions. The difference has to do with the radiative quantum efficiency. When the radiative quantum efficiency is unity, as shown in Fig. 10(a), the fractional thermal load under the lasing condition is higher than the nonlasing condition. So the temperature of the thin disk crystal under the lasing condition is also higher than the nonlasing condition. This is due to the fact that the average fluorescence wavelength is less than the laser wavelength, which means that the heat generation in fluorescence is lower than the stimulation emission. When the 4356
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Fig. 10. Temperature of a thin disk crystal and fractional thermal load under lasing or nonlasing conditions as a function of pump intensity for different radiative quantum efficiencies.
radiative quantum efficiency is less than unity, as shown in Figs. 10(b) and 10(c), the fractional heat load and the temperature of the thin disk crystal under the lasing condition are both lower than for the nonlasing condition; and the lower the radiative quantum efficiency, the higher the fractional thermal
Fig. 11. Measured temperature of the thin disk crystal versus pumping current for lasing and nonlasing conditions at the radiative quantum efficiency less than unity.
combining the rate equations including the nonradiative relaxation process with variation of fractional thermal load, which is affected not only by intracavity laser intensity, but also by temperature of the thin disk crystal. The fractional thermal load is known, or can be evaluated clearly, under lasing or nonlasing conditions. The analytical results show that the radiative quantum efficiency is one of the important factors of the fractional thermal load. At the same time, the iterative method is demonstrated to calculate the stable temperature and fractional thermal load in the thin disk crystal for different radiative quantum efficiencies. Based on the analytical expressions, the laser output intensity, the slope efficiency, and the optimal coupler reflectivity on various parameters are obtained. The results generated from this model are more accurate than those that do not consider the variation of fractional thermal load. References
Fig. 12. Laser output intensity and temperature of the thin disk crystal as a function of the resonator coupler reflectivity for different radiative quantum efficiencies.
load. The main reason is that, when the laser is on, the stimulated emission short circuits the nonradiative path, causing the thermal load to be lower. Figure 11 shows the measured temperature of the thin disk crystal versus pumping current for lasing and nonlasing conditions at a radiative quantum efficiency less than unity. The results of the measurements are in good agreement with theoretical calculations. Figure 12 shows the laser output intensity and the temperature of the thin disk crystal as a function of resonator coupler reflectivity for different radiative quantum efficiencies. It can be seen that there is an optimal resonator coupler reflectivity for different pump intensity that can acquire the maximum output laser. On the other hand, the optimal coupler reflectivity decreases slightly with an increase of radiative quantum efficiency. 4. Conclusion
In summary, this paper presented a novel analytical model of an end-pumped Yb:YAG thin disk laser,
1. H. W. Bruesselbach, D. S. Sumida, R. A. Reeder, and R. W. Byren, “Low-heat high-power scaling using InGaAs-diodepumped Yb:YAG lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 105–116 (1997). 2. A. K. Sridharan, S. Saraf, and R. L. Byer, “Yb:YAG master oscillator power amplifier for remote wind sensing,” Appl. Opt. 46, 7552–7564 (2007). 3. A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state lasers,” Appl. Phys. B 58, 365–372 (1994). 4. A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13, 598–609 (2007). 5. D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010). 6. J. Speiser, “Scaling of thin-disk lasers—influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26, 26–35 (2009). 7. T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron. 29, 1457–1459 (1993). 8. N. P. Barnes and B. Walsh, “Quantum efficiency measurements of Nd:YAG, Yb:YAG, and Tm:YAG,” in OSA Proceedings of Advanced Solid State Lasers (Optical Society of America, 2002), Vol. 68, pp. 284–287. 9. F. D. Patel, E. C. Honea, J. Speth, S. A. Payne, R. Hutcheson, and R. Equall, “Laser demonstration of Yb3Al5O12(YbAG) and materials properties of highly doped Yb:YAG,” IEEE J. Quantum Electron. 37, 135–144 (2001). 10. S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium lasers—part II: evaluation of quantum efficiencies and thermo-optic coefficients,” IEEE J. Quantum Electron. 40, 1235–1243 (2004). 11. D. C. Brown, T. M. Bruno, and V. Vitali, “Saturated absorption effects in CW-pumped solid-state lasers,” IEEE J. Quantum Electron. 46, 1717–1725 (2010). 12. D. C. Brown, T. M. Bruno, and J. M. Singley, “Heat-fractionlimited CW Yb:YAG cryogenic solid-state laser with 100% photon slope efficiency,” Opt. Express 18, 16573–16579 (2010). 13. K. Contag, S. Erhard, and A. Giesen, “Calculations of optimum design parameters for Yb:YAG thin disk laser,” in OSA Proceedings of Advanced Solid State Lasers (Optical Society of America, 2000), Vol. 34, pp. 124–130. 14. G. L. Bourdet, “Theoretical investigation of quasi-three-level longitudinally pumped continuous wave lasers,” Appl. Opt. 39, 966–971 (2000). 15. C. Lim and Y. Izawa, “Modeling of end-pumped CW quasithree-level lasers,” IEEE J. Quantum Electron. 38, 306–311 (2002). 16. G. Zhu, X. Zhu, C. Zhu, J. Shang, H. Wan, F. Guo, and L. Qi, “Modeling of end-pumped Yb:YAG thin-disk lasers with 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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nonuniform temperature distribution,” Appl. Opt. 51, 2521–2531 (2012). 17. D. A. Copeland, “Optical extraction model and optimal outcoupling for a CW quasi-three level thin disk laser,” Proc. SPIE 7912, 79120D (2011). 18. D. C. Brown and V. A. Vitali, “Yb:YAG kinetics model including saturation and power conservation,” IEEE J. Quantum Electron. 47, 3–12 (2011). 19. S. Chenais, F. Balembois, F. Druon, G. Lucas-Leclin, and P. Georges, “Thermal lensing in diode-pumped ytterbium
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lasers—part I: theoretical analysis and wavefront measurements,” IEEE J. Quantum Electron. 40, 1217–1234 (2004). 20. Q. Liu, X. Fu, M. Gong, and L. Huang, “Effects of the temperature dependence of the absorption coefficients in edge-pumped Yb:YAG slab lasers,” J. Opt. Soc. Am. B 24, 2081–2089 (2007). 21. B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass, “Modeling of high power solid-state slab lasers,” Proc. SPIE 4968, 1–10 (2003).
An analytical model is developed to describe the dynamic behavior of an end-pumped Yb:YAG thin disk laser. Within the model, the rate equations, including the nonradiative relaxation process, are calculated taking into account the dependence of the fractional thermal load on the temperature of the thin disk crystal and intracavity laser intensity. The fractional thermal load is analyzed, or can be evaluated clearly, under lasing or nonlasing conditions. The stable temperature and fractional thermal load in a thin disk crystal for different radiative quantum efficiencies are obtained using the numerical iterative method. Furthermore, the dependence of the laser output intensity on variables such as pumping intensity, coupler reflectivity, radiative quantum efficiency, and the temperature of thin disk crystal is discussed. © 2014 Optical Society of America OCIS codes: (140.3430) Laser theory; (140.3480) Lasers, diode-pumped; (140.6810) Thermal effects; (140.3615) Lasers, ytterbium. http://dx.doi.org/10.1364/AO.53.004349
1. Introduction
With the improvement of high-power laser diodes, quasi-three-level gain media, such as Yb:YAG, has attracted worldwide interest in the last twenty years [1,2]. Because of trivalent ytterbium’s simple twolevel electronic structure, excited-state absorption, upconversion, and concentration quenching can be avoided, and the relative energy difference between a pump and a laser photon is small, typically less than 10%. Comparing with Nd:YAG, Yb:YAG may be more competitive for high average power laser devices operating near 1 μm. So face- and edge-pumped Yb:YAG thin disk configurations have been focused on theoretical and experimental development. A classical multipumped configuration given by Geisen is more suitable for thin disk lasers in the power range from several watts to kilowatts [3]. Nowadays, thin disk lasers with output power of 1559-128X/14/194349-10$15.00/0 © 2014 Optical Society of America
several kilowatts are commercially available for industrial applications [4,5]. Furthermore, the power scaling law of thin disk lasers up to 100 kW in continuous-wave (CW) mode has been already considered [6]. However, the power scaling of such lasers is fundamentally limited by the thermal load in the gain medium. It is important to predict the amount of heat generated in the laser material accurately. In 1993, Fan predicted the heat fraction under lasing or nonlasing conditions in Nd:YAG and Yb:YAG [7]. The prediction was based on the assumption that the radiative quantum efficiency was high enough to approach unity in ytterbium-doped materials. So the fractional thermal load under lasing is higher than the nonlasing condition. However, recent studies have shown that the radiative quantum efficiency was less than unity in Yb:YAG [8–10], which induced more heat generation in this material. Several kinetics models of heat generation in Yb:YAG were built to identify the major heating processing and power conservation [11,12]. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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On the other hand, In order to investigate the performance of quasi-three-level Yb:YAG laser system, various models for an end-pumped CW laser are developed by Contag and co-workers [13–17]. Some analytical solutions about laser output intensity, threshold intensity, optical-to-optical efficiency, and the basic dependence of optimal crystal thickness and optimal pump absorption on various parameters are obtained. However, these analytical models only consider the fractional thermal load in the Yb:YAG crystal is a constant. In actuality, the fractional thermal load is not a constant in the processing of the laser operation, which is more related to radiative quantum efficiency, laser extractive efficiency, and temperature of the Yb:YAG crystal. So it is hard to investigate laser operational properties precisely. In this paper, a novel analytical model of an endpumped Yb:YAG thin disk laser combining the rate equations including the nonradiative relaxation process with the variation of fractional thermal load, which is affected not only by the intracavity laser intensity, but also by the temperature of the thin disk crystal, is presented. The fractional thermal load is known or can be evaluated clearly under lasing or nonlasing conditions, and is obtained on various parameters based on the analysis results, laser output intensity, laser extraction efficiency, and the temperature of the thin disk crystal. 2. Physical Model A.
Theoretical Analysis of the Heat Generation
The energy level diagram of Yb:YAG, which consists of only two manifolds, an upper 2 F 5∕2 and a lower 2 F 7∕2 , is shown in Fig. 1. The common pump and laser wavelengths are 941 and 1030 nm, are described using λP and λL , respectively. Because of the trivalent ytterbium ion’s simple electronic structure, the upconversion, excited-stated absorption, and concentration quenching play no important role. The factors generating the heat can be neglected in the model. The Boltzmann occupation factors for the upper and lower state manifolds depend on the temperature and are given by Eqs. (1a) and (1b), respectively:
f 1i
f 0j
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(1a)
E exp − KT0j ; P E0j 4 j1 exp − KT
(1b)
where K is the Boltzmann constant, T is the absolute temperature, E1i is the energy for each Stark level of excited manifold, and E0j is the energy for each Stark level of ground manifold. The fluorescence branching ratios of Yb:YAG have been calculated at room temperature are shown in Table 1 [18]. Based on the data, the average fluorescence wavelength could be given by 1 ¯λf Δv–
P3
i1
P4
1i –1i − j1 f 1i β 0j v P4 1i j1 f 1i β0j i1
P3
v–0j
;
(2)
where the index pair 1i refers to the upper manifold Stark levels and 0j refers to the lower manifold Stark levels. Δv– is the average wavenumber, and upper manifold wavenumbers are v–1i and lower manifold wavenumbers are v–0j . β1i 0j are the fluorescence branching ratios for the upper level manifold. The average fluorescence wavelength is 1009.1 nm at room temperature. Figure 2 shows the relationship between the average fluorescence wavelength and temperature. It is found that the average fluorescence wavelength changes only slightly with temperature. In order to make the heat generation in the Yb:YAG clear, the various heating generation mechanisms, as well as for fluorescence and laser extraction, must be considered. The processing of heat generation could be divided into two stages. The first stage is the pumping stage. The second stage is the upper manifold population depleted by both fluorescence decay and laser extraction. The heat generation in the pumping stage is shown as [18] Table 1.
Starting level
Fig. 1. Yb:YAG energy level diagram.
E1i exp − KT ; P E1i 3 i1 exp − KT
Yb:YAG Fluorescence Branching Ratios
Ending level
Designation
Value
(1,1)
(0,1)
0.225
(1,1)
(0,2)
(1,1) (1,1)
(0,3) (0,4)
(1,2)
(0,1)
(1,2) (1,2)
(0,2) (0,3)
(1,2)
(0,4)
(1,3) (1,3)
(0,1) (0,2)
(1,3)
(0,3)
(1,3)
(0,4)
β11 01 β11 02 β11 03 β11 04 β12 01 β12 02 β12 03 β12 04 β13 01 β13 02 β13 03 β13 04
0.09 0.342 0.088 0.079 0.084 0.015 0.023 0.009 0.010 0.010 0.025
Fig. 4. Simple explanation of heat generation by fluorescence and simultaneous laser extraction. Fig. 2. Average fluorescence wavelength as a function of temperature.
QB –12 − v–11 f 12 Re hcv–12 − v–12 h f 11 Re hcv f 13 Re hcv–12 − v–13 :
(3)
To ensure power conservation, we set the absorbed power density, given by –12 ; QA QUM QB h Re hcv
(4)
where Re is the excitation density, QUM is the upper manifold population power density, and h is Planck’s constant. The power density associated with storage of population density is given by QUM Re hcf 11 v–11 f 12 v–12 f 13 v–13 :
QB h : QA
λ λ ηh 1 − ηp 1 − ηl ηR P ηl P ; λL λf
(7)
(5)
So pump quantum efficiency, which is the fraction of absorption pump photons contributing to inversion, can be described in this stage by ηp 1 −
heat generation by fluorescence alone. QFh is the fluorescence heating power density. On the other hand, the fluorescence and laser extraction occur simultaneously and both depopulate the upper manifold and upper laser level in CW laser oscillators. The contributions of fluorescence and laser extraction to heat are shown in Fig. 4, where QLh is the laser heating density. In order to describe the process of heat generation, Fan presented the equation to describe the fractional thermal load [7]:
(6)
In the second stage, the upper manifold is depleted only by fluorescence with nonlasing generation in the resonator. Figure 3 shows a simple explanation of
where ηp is the pump quantum efficiency and ηR is the radiative quantum efficiency for the upper manifold. Nonunity radiative quantum efficiency can be related to multiphoton relaxation. ηl is the laser extraction efficiency defined as the fraction of excited ions that are extracted by stimulated emission. For the no lasing extraction condition, ηl is 0, in which case the upper manifold is depleted only by fluorescence. If ηl is 1, the equation describes complete laser extraction. B. Thermal Analysis
Fig. 3. Simple explanation of heat generation by fluorescence alone.
The numerical model is used to calculate the temperature distribution in the gain medium. The Yb:YAG crystal (typical thickness between 0.2 and 0.4 mm) is soldered on the heat sink with the indium layer. The back side of the crystal is high reflection (HR) coated for both wavelengths. The heat sink is cooled by impingement water. The geometry of the thin disk crystal mounted on the heat sink and the thermal boundary condition are shown in Fig. 5. Figure 6 shows the normalized pumping light profile from the measurement in our experiments after multipass pumping. So we assume that the side surface of the disk is heat-insulated and the heat generation in the crystal is homogeneous. The stationary heat conduction equation can be built [16]: 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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8 < ∂2 T YAG r;z 1 ∂T YAG r;z ∂2 T YAG r;z − :
∂2 T
∂r2
Cu−W r;z ∂r2
r
∂r
1 ∂T Cu−W r;z r ∂r
∂z2
∂2 T
Cu−W r;z ∂z2
Qh kYAG
circrP
0
for disk crystal;
8
for Cu W heat sin k:
The boundary conditions are: 8 > > > > >
> > > > : kYAG ∂T YAG r;z ∂z
0;
z0
for disk crystal
r;z kCu−W ∂T Cu−W ∂z
z0
;
8 > > T YAG r; 0 T Cu−W r; 0; > > > > < ∂T Cu−W r;z 0; ∂r rR > > > ∂T Cu−W r;z hec > > kCu−W T Cu−W r; 0 − T f ; > ∂z : z−z1
Fig. 5. Schematic diagram and thermal boundary conditions of the thin disk module (A is the pump zone and B is the unpump zone).
(9)
for Cu W heat sink
(10)
where T YAG r; z and T Cu−W r; z are the temperature distribution in the thin disk crystal and CuW heat sink, Lth is the thickness of laser crystal, rP is a radius of pumping spot, kYAG and kCu−W are the thermal conductivity of thin disk crystal and heat sink, respectively, hec is the heat exchange coefficient of the heat sink with the cooling liquid, T f is temperature of the cooling liquid, and Qh is the heat flux in the thin disk crystal. The stationary heat conduction equations of the thin disk crystal and CuW heat sink are a classical Poisson equation and a Laplace equation. Using the methods of mathematical physics and complex derivation, we can obtain analytical expressions of the temperature distribution: 0 ∞ X xn z An exp R n1 0 0 xn xn z · J0 r Bn exp − R R 0 ∞ X 1 xn r ; Gn · J 0 − z2 G 0 R 2 n1
T YAG r; z A0 B0 z
(11)
0 xn z R n1 0 0 xn xn z · J0 r ; (12) B0n exp − R R
T Cu−W r; z T f A00 B00 z
Fig. 6. Normalized pumping light profile after multipass pumping. 4352
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where
∞ X
A0n exp
z0 Qh r2P 0 z0 Qh r2P h z z0 Qh r2P hec z1 z0 Qh r2P 0 A0 T f 1 ec 1 ; B ; A 1 ; B0 ; 0 0 2 2 2 kCu−W hec R kCu−W hec R kYAG R kCu−W R2 hec H 0nb −kCu−W E0nb 2f n kCu−W E0na −hec H 0na 0 An ; 0 0 0 0 kCu−W hec H nb −kCu−W Enb kCu−W kCu−W hec H nb −kCu−W Enb kCu−W 1 1− Ena 1− 1 Enb kYAG kCu−W E0na −hec H 0na kYAG kYAG kCu−W E0na −hec H 0na kYAG 2f n B0n ; h H 0 −k 0 0 0 kCu−W ec nb Cu−W Enb kCu−W kCu−W hec H nb −kCu−W Enb kCu−W Ena 1− Enb 1 1− 1 kYAG kCu−W E0na −hec H 0na kYAG kYAG kCu−W E0na −hec H 0na kYAG hec H 0nb −kCu−W E0nb 1 k k B0 k kCu−W An 1− 1 Cu−W A0n 1− Cu−W B0n n 1 Cu−W ; kYAG kYAG 2 kYAG kCu−W E0na −hec H 0na kYAG 2 hec H 0nb −kCu−W E0nb 1 k k B0 k kCu−W 1 1− Cu−W A0n 1 Cu−W B0n n 1− Cu−W ; Bn kYAG kYAG 2 kYAG kCu−W E0na −hec H 0na kYAG 2 0 Qh r2P 2Qh rP xn J1 r : ; and Gn G0 2 2 0 0 R P kYAG R kYAG J 0 xn xn R
Figure 7 shows the temperature distribution along the radial direction inside the disk module for different layers (the calculated parameters are shown in Table 2). Figure 8 shows the thermal image and temperature curve at 3000 W pumping power using an infrared thermal imager. It can be seen that the maximum temperature would be observed at the center of the top layer and the temperature difference along the radial direction is small. In order to simplify the analytical expression, the average temperature of the thin disk crystal is used in the model, which is defined by 1 hTrYAG i Lth
Z
Lth
0
T YAG r; zdz:
Table 2.
Basic Parameters Used for this Model
Parameter Number of pump beam pass (M) Pump intensity (I P ) Disk thickness (Lth ) Dopant concentration (cYb ) Diameter of pump spot (mm) Thermal conductivity of thin disk crystal (kYAG ) Thermal conductivity (kHS ) Heat transfer coefficient (hec ) Thickness of heat sink (dSink ) Coolant temperature (T f )
Value 16 5 kW∕cm2 200 μm 6 at. % 6 6.5 W∕m · K [1] 385 W∕m · K 105 W∕m2 · K 2 mm 20°C
(13)
Fig. 7. Temperature distribution along the radial direction inside the disk module for different layers. The curve (-o-) shows the average temperature of in the disk crystal.
Fig. 8. Surface temperature of the thin disk crystal at 3000 W pumping power. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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C. Quasi-three-level Kinetics Model and Its Steady State Solution
In order to simplify, the lifetime, τ, is defined by −1 −1 τ τ−1 R τNR . Eq. (14) can be rewritten as
From the analysis of heat generation mechanisms, the upper manifold populations are depleted by stimulated emission, and spontaneous and nonradiative relaxation rates [19]. Therefore, the upper and lower manifold rate equations that govern the quasi-three-level system are given by
∂N 1 t; z ∂N 0 t; z − σ P f P01 N 0 − f P12 N 1 P ∂t ∂t N σ L f L03 N 0 − f L11 N 1 L − 1 : τ
∂N 1 t; z ∂N 0 t; z − σ P f P01 N 0 − f P12 N 1 P ∂t ∂t N N σ L f L03 N 0 − f L11 N 1 L − 1 − 1 ; τR τNR
In the calculation model, the pump and the laser radiation are taken to be uniform. The average temperature of the thin disk crystal is used from Eq. (13). The temperature dependence of the stimulated absorption cross section [20] and emission cross section [21] on wavelength and temperature are shown by
∂I ∓ L z σ L f L03 N 0 z − f L11 N 1 zI ∓ L z; ∂z ∂I ∓ Pi z ∂z
P
(14)
(15)
σ P f P01 N 0 z − f P12 N 1 zI ∓ Pi z;
PN
i1 I P;i
I −P;i
hvP
;
L
− I L IL ; hvL
Qstim Qstim Qspont Qnr I σ × Qstim L L f L03 N 0 − f L11 N 1 ; hvL N N 1; Qnr 1 ; τR τNR
ηR
Qspont τNR : Qspont Qnr τNR τR
APPLIED OPTICS / Vol. 53, No. 19 / 1 July 2014
(21)
(16)
σ L 1030; T 0.95334 33.608 exp − (17)
ηl
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T − 273 σ P 941; T 0.207 0.637 exp − 288 × 10−20 cm2 ;
where I P and I L are pump and laser intensities in the gain medium, and the superscripts and − represent propagation in the forward and backward directions, respectively. i is the pumping number, σ P and σ L are the spectroscopic absorption cross sections at the pump and laser wavelengths, respectively, τR is the spontaneous emission lifetime of the upper manifold, τNR is the nonradiative lifetime of the upper manifold, and f P01 and f P12 denote the Boltzmann occupation factors within the upper and lower manifolds for the upper and lower pump states of the pumping transition, respectively. Also, f L03 and f L11 denote the Boltzmann occupation factors with the upper and lower manifolds for the upper and lower laser states of the laser transition, hvP is the energy of the pump radiation, hvL is the energy of the laser radiation, and N 1 and N 0 denote the population of the upper and lower manifold. On the other hand, based on the rate equations, the laser extraction efficiency and radiative quantum efficiency are given by [19]
Qspont
(20)
(18)
T 92.82465
× 10−20 cm2 :
(22)
Solving Eq. (20) for the steady state condition, the population of the upper and lower manifold for laser operation are obtained: N 0 z
σ P f P12 P σ L f L11 L 1τ Nt; σ P f P01 f P12 P σ L f L03 f L11 L 1τ (23)
N 1 z
σ P f P01
σ P f P01 P σ L f L03 L Nt; f P12 P σ L f L03 f L11 L 1τ (24)
where N t N 0 N 1 is the sum of the two manifold populations. Based on the boundary conditions of the resonator and the approach followed by Rigrod, the absorption pumping intensity, laser output intensity, laser slope efficiency, threshold intensity, and intracavity laser intensity can be obtained by solving Eqs. (15) and (16). The detailed deviations are presented in [16]. I abs I P;0 ×
M X
i−1 fRi−1 r;P · Rf ;P · exp2i − 1δLth g
i1
× expδLth Rf ;P 1 × 1 − expδLth ;
(25)
(19) I out 1 − Rr;L I L L ηslope I P;0 − I th ;
(26)
ηslope
3. Results and Discussion
1 − Rr;L v − L p vP 1 − p 1 R ∕R R R r;L
×
f ;L
r;L
f ;L
M X i−1 fRi−1 r;P · Rf ;P · exp2i − 1δLth g i1
× expδLth Rf ;P 1 × expδLth − 1:
I th PM
i−1 i1 fRr;P
hvP τ
·
Ri−1 f ;P
n
(27)
f L03 N t Lth FL
σ
In the following, the operational characteristics of an end-pumped Yb:YAG thin disk laser are investigated. The basic dependencies of fractional thermal load, and temperature of a thin disk crystal under lasing and nonlasing condition are discussed. Table 2 summarizes the basic parameters used for the model.
1 L L ·F ·Lth
io 1 ln p Rr;L Rf ;L
· exp2i − 1δLth g × expδLth Rf ;P 1 × expδLth − 1
− IL I L L I L L 1 Rr;L I L L 1 Rr;L A C I ; (29) I P;0 1 Rr;L − B A 1 − Rr;L out
where F L f L03 f L11 , Rr;P and Rf ;P are the pump intensity reflectivity of the front surface (including the parabolic mirror and antireflectivity coating on this surface) and rear surface of the thin disk crystal, Rr;L and Rf ;L are the laser intensity reflectivities of the resonator rear mirror and output coupler, and δ is the absorption coefficient of the gain medium. As shown above, using a strictly plane wave analysis for CW quasi-three-level thin disk laser rate equations, the output power is not linearly related to the input pump power above the threshold value. The main reason is the laser slope efficiency is affected by the growth of the temperature of the thin disk crystal and depends on the absorption coefficient of the gain medium. On the other hand, the laser extractive efficiency is affected not only by the intracavity laser intensity, but also by the temperature of the thin disk crystal. So the fractional thermal load is not a constant during the laser generation, which will affect the laser operational properties. The iterative method employed here, is composed of three steps. First, the laser operational parameters, such as absorption intensity, intracavity laser intensity, laser extractive efficiency, and so on are calculated at room temperature. Then, the fractional thermal load could be calculated based on Eq. (7). Second, we can use the absorption intensity and fractional thermal load to obtain the heat flux in the thin disk medium so that the average temperature of the thin disk crystal can be calculated. Third, the rate equations are calculated again with the new temperature of the thin disk crystal. The laser operational parameters are obtained again. We iterate the second and third steps until the stable temperature of the thin disk crystal and laser output intensity are achieved.
;
(28)
Figure 9 shows the dependence of laser output intensity (left scale) and average temperature of the thin disk crystal (right scale) on pump intensity for different radiative quantum efficiencies. It can be seen that operational properties of the laser have three classical stages. At the early stage, the pump intensity is lower than the threshold intensity, and there is no laser generated in the resonator. The laser extractive efficiency is 0. The fractional thermal load is generated by fluorescence alone. With the increasing of pump intensity, the pump intensity exceeds the threshold intensity and the laser is generated from the resonator. This stage is defined as the operational stage. The fluorescence and laser extraction occur simultaneously. First, with the increasing of pump intensity, the laser output intensity and temperature of the thin disk crystal will increase consistently. But the laser output intensity and the temperature of the thin disk crystal are not linearly related to the pump intensity above the threshold value. The main reason is that the laser slope efficiency is affected by the growth of temperature of the thin disk crystal and depends on the absorption
Fig. 9. Dependence of laser output intensity (left scale) and average temperature of the thin disk crystal (right scale) on the pump intensity for different radiative quantum efficiencies. (a) ηR 100% (b) ηR 97% (c) ηR 90% 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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coefficient of the gain medium and, at the same time, the fractional thermal load is not a constant and is a function of the temperature of the thin disk crystal and the laser extractive efficiency. With the increasing of pump intensity, a further increasing of the temperature of the thin disk crystal will be obtained, so that the pump intensity cannot maintain the necessary population inversion to keep the laser generating at this temperature. The laser output intensity will decrease to zero again. This is the third stage. The fractional thermal load is generated by fluorescence alone at this stage. On the other hand, it can be seen that the radiative quantum efficiency is also one of the important factors that affect the temperature of the thin disk crystal, laser output intensity, and other parameters. As the radiative quantum efficiency increases, the fractional thermal load will decrease. The temperature of the thin disk crystal will decrease constantly at the same pump intensity, which induces the increasing of the laser output intensity. Figure 10 shows the operational temperature and fractional thermal load under lasing or nonlasing conditions as a function of pump intensity for different radiative quantum efficiencies. Although the temperature of the thin disk crystal increases with increasing pump intensity, the fractional thermal load under the nonlasing condition decreases slightly for the different radiative quantum efficiencies. The main reasons are that the average fluorescence wavelength and the Boltzmann occupation factors for the upper and lower state manifolds are both functions of the temperature of the thin disk crystal. On the other hand, because the laser extractive efficiency has a close relationship with the intracavity laser intensity, temperature of the thin disk crystal, and the radiative quantum efficiency, the dependence of the fractional thermal load on the pump intensity under the lasing condition changes more complexly. When the radiative quantum efficiency is less than unity, the fractional thermal load decreases first, and then increases with increment of pump intensity. When the radiative quantum efficiency is unity, the fractional thermal load increases first and then decreases constantly with increment of pump intensity. On the other hand, it can be seen that the temperature of the thin disk crystal and the fractional thermal load have significant differences for the same pump intensity under lasing and nonlasing conditions. The difference has to do with the radiative quantum efficiency. When the radiative quantum efficiency is unity, as shown in Fig. 10(a), the fractional thermal load under the lasing condition is higher than the nonlasing condition. So the temperature of the thin disk crystal under the lasing condition is also higher than the nonlasing condition. This is due to the fact that the average fluorescence wavelength is less than the laser wavelength, which means that the heat generation in fluorescence is lower than the stimulation emission. When the 4356
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Fig. 10. Temperature of a thin disk crystal and fractional thermal load under lasing or nonlasing conditions as a function of pump intensity for different radiative quantum efficiencies.
radiative quantum efficiency is less than unity, as shown in Figs. 10(b) and 10(c), the fractional heat load and the temperature of the thin disk crystal under the lasing condition are both lower than for the nonlasing condition; and the lower the radiative quantum efficiency, the higher the fractional thermal
Fig. 11. Measured temperature of the thin disk crystal versus pumping current for lasing and nonlasing conditions at the radiative quantum efficiency less than unity.
combining the rate equations including the nonradiative relaxation process with variation of fractional thermal load, which is affected not only by intracavity laser intensity, but also by temperature of the thin disk crystal. The fractional thermal load is known, or can be evaluated clearly, under lasing or nonlasing conditions. The analytical results show that the radiative quantum efficiency is one of the important factors of the fractional thermal load. At the same time, the iterative method is demonstrated to calculate the stable temperature and fractional thermal load in the thin disk crystal for different radiative quantum efficiencies. Based on the analytical expressions, the laser output intensity, the slope efficiency, and the optimal coupler reflectivity on various parameters are obtained. The results generated from this model are more accurate than those that do not consider the variation of fractional thermal load. References
Fig. 12. Laser output intensity and temperature of the thin disk crystal as a function of the resonator coupler reflectivity for different radiative quantum efficiencies.
load. The main reason is that, when the laser is on, the stimulated emission short circuits the nonradiative path, causing the thermal load to be lower. Figure 11 shows the measured temperature of the thin disk crystal versus pumping current for lasing and nonlasing conditions at a radiative quantum efficiency less than unity. The results of the measurements are in good agreement with theoretical calculations. Figure 12 shows the laser output intensity and the temperature of the thin disk crystal as a function of resonator coupler reflectivity for different radiative quantum efficiencies. It can be seen that there is an optimal resonator coupler reflectivity for different pump intensity that can acquire the maximum output laser. On the other hand, the optimal coupler reflectivity decreases slightly with an increase of radiative quantum efficiency. 4. Conclusion
In summary, this paper presented a novel analytical model of an end-pumped Yb:YAG thin disk laser,
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