Analytical model of thermal effect and optical path difference in end-pumped Yb:YAG thin disk laser Guangzhi Zhu,* Xiao Zhu, Mu Wang, Yufan Feng, 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, China *Corresponding author: [email protected] Received 10 July 2014; revised 1 September 2014; accepted 2 September 2014; posted 4 September 2014 (Doc. ID 216777); published 8 October 2014
An analytical model of the thermal effect and optical path difference (OPD) of a thin disk laser is developed with the combination of the analytical method and commercial finite element analysis software. The distributions of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation are obtained. Based on the analytical model, the production mechanisms, features, and influence of OPD in an end-pumped thin disk laser are discussed, which make the causes of spherical and aspherical parts of the OPD more obvious. Furthermore, the OPD including the spherical and aspherical parts of the thin disk crystal is discussed for various pumping intensities. © 2014 Optical Society of America OCIS codes: (140.6810) Thermal effects; (140.3615) Lasers, ytterbium; (140.3580) Lasers, solid-state; (350.6830) Thermal lensing. http://dx.doi.org/10.1364/AO.53.006756
1. Introduction
The thin disk concept avoids thermal problems compared with conventional high-power rod or slab lasers and enables high power with good beam quality and high efficiency. Typically, the thin-diskshaped active medium is cooled through one of the flat faces of the disk. This ensures a large surfaceto-volume ratio and therefore provides very efficient thermal management. Recently, a record 10 kW of laser power was demonstrated from a single disk with greater than 60% optical-to-optical efficiency. The laser typically operates in a high-order multimode [1]. So thermal lensing and aberration are not negligible in the multikilowatt regime, which induces a decrease in beam quality. In order to further improve the brightness of the thin disk laser, intensive research has been carried out in thermal-optical aberrations, thermal lensing, and compensation techniques [2–4]. 1559-128X/14/296756-09$15.00/0 © 2014 Optical Society of America 6756
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
In general, the main contributions to the optical path difference (OPD) and thermal lensing in thin disk lasers are temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation of the disk crystal. Moreover, the difference between pumped volume and heat sink causes a mainly spherical bending of the disk. The spherical deformation of the OPD could be compensated by the resonator design, while the aspherical part of the thermo-optical aberrations has to reduce to improve the brightness of the output laser. In order to investigate the OPD, thermal lensing, and compensation techniques, various models and experiments have been carried out by Speiser and Geisen [5], Milani et al. [6], Perchermeier and Wittrock [7], Shang et al. [8], Sazegari et al. [9], and others. Many numerical models were built and calculated using finite element analysis (FEA) software. The FEA method is more convenient to calculate the temperature distribution, and thermal stress (strain), especially for mechanical deformation for arbitrary pumping light distributions. But FEA is a complex and time-consuming process. It is also
thin disk crystal mounted on the heat sink and the thermal boundary conditions is shown in Fig. 1. The basic parameters, which are used to calculate temperature, stress, strain displacement, and deformation inside the thin disk crystal, are shown in Table 1.
difficult to compute OPD changes with different system parameters. On the other hand, the analytical method will show the relationship between the OPD and numerous system parameters more clearly. In this paper, the thermal effect and OPD of a thin disk crystal are investigated. Combined with merits of the analytical method and FEA software, the OPD analytical model of the thin disk crystal is built. The distributions of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain (bulging), and thermal strain-induced birefringence are calculated using analytical methods.
A. Temperature Distribution inside the Disk Crystal
We assume that the side surface of the thin disk is heat-insulated and heat generated in the crystal is homogeneous after multipass pumping. The stationary heat conduction equation can be built [12]:
8 < ∂2 T YAG r;z 1 ∂T YAG r;z ∂2 T YAG r;z − r ∂r : ∂2 T Cu-W2 r;z 1 ∂T Cu-W r;z r ∂r ∂r ∂r2
∂z2 ∂2 T Cu-W r;z ∂z2
Qh kYAG
circrP for disk crystal;
0
for CuW heat sink:
1
The boundary conditions are 8 > > ∂T YAG r;z > 0; > > ∂r > > rR > < ∂T YAG r;z 0; ∂z > zLth > > > > > ∂T YAG r;z ∂T Cu-W r;z > k k > YAG Cu-W ∂z ∂z : z0
for disk crystal
(2)
; z0
8 > T YAG r; 0 T Cu-W r; 0; > > > > < ∂T Cu-W r;z 0; ∂r for Cu-W heat sink rR > > > ∂T r;z h Cu-W > kCu-W T Cu-W r; 0 − T f ; > ∂z :
(3)
z−dCuW
Meanwhile, the OPD caused by deformation of the disk crystal is obtained by FEA software. Based on the composite model, the OPD including the spherical and aspherical parts of the thin disk crystal is discussed for various system parameters. 2. Physical Model and Analysis
The numerical model is used to carry out thermal analysis in the gain medium. The Yb:YAG crystal (typical thickness between 0.2 and 0.4 mm) is soldering on the heat sink with an indium layer. The disk is highly reflective coated on its backside for both laser and pumping wavelengths and antireflective coated on the front side for both wavelengths. The heat sink is cooled by impingement cooling using several nozzles inside the heat sink. A schematic diagram of the
where T YAG r; z and T Cu-W r; z are the temperature distributions in the thin disk crystal and CuW heat sink. Lth is the thickness of the laser crystal. rP is the radius of the pumping spot. kYAG and kCu-W represent the thermal conductivity of the thin disk crystal and the heat sink, respectively. h is the heat exchange coefficient of the heat sink with cooling liquid. T f is the temperature of cooling liquid. Qh is the heat generation density in the thin disk crystal. The stationary heat conduction equations of the thin disk crystal and CuW heat sink are the classical Poisson equation and the Laplace equation, the solutions of which include general and special solutions. Using the Methods of Mathematical Physics [13] and complex derivation, we can obtain analytical expressions of the temperature distribution: 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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T YAG r; z A0 B0 z
∞ X n1
0 0 0 0 ∞ X xn xn xn 1 2 xn z Bn exp − z · J0 r − z G0 r ; An exp Gn · J 0 R R R R 2 n1 (4)
0 0 0 ∞ X xn xn xn 0 0 0 0 T Cu-W r; z T f A0 B0 z z Bn exp − z · J0 r ; An exp R R R n1
(5)
where the functions J 0 x and J 1 x are the zero-order Bessel function and first-order Bessel function. The x0 n are the roots of J 1 x 0, n 1; 2; …∞, and the coefficients A0 , B0 , A00 , B00 , A0n , B0n , An , Bn , Ena , Enb , E0na , E0nb , H 0na , H 0nb , G0 , and Gn are given by hdCuW Lth Qh r2P ; A0 T f 1 kCu-W hR2
A00
hH 0 −k
A0n
hdCuW Lth Qh r2P 1 ; kCu-W hR2
L Q r2 B0 th h 2P ; kYAG R
1 An 2
Bn
E0
1 2
1 kkCu-W YAG
hH 0nb −kCu-W E0nb kCu-W E0na −hH 0na
2f n
; 0 0 kCu-W kCu-W hH nb −kCu-W Enb kCu-W 1 − kYAG Ena 1 − kYAG kCu-W E0na −hH 0na 1 kYAG Enb
kCu-W 0 kCu-W 0 B0n kCu-W hH 0nb − kCu-W E0nb kCu-W 1 1− 1 An 1 − Bn ; kYAG kYAG 2 kYAG kCu-W E0na − hH 0na kYAG
1−
hH 0nb − kCu-W E0nb kCu-W 0 k B0 k kCu-W 1 An 1 Cu-W B0n n 1 − Cu-W ; kYAG kYAG 2 kYAG kCu-W E0na − hH 0na kYAG
0 0 0 0 xn xn xn xn Lth ; Enb exp − Lth − ; R R R R 0 0 xn xn dCuW − ; exp R R
Ena exp
G0
Qh r2P ; kYAG R2
H 0nb exp
Gn kYAG
0 2Qh rP xn 2 J1 r ; R P 0 0 J 0 xn xn R
0 0 xn xn E0na exp − dCuW ; R R
0 xn H 0na exp − z ; R 1
0 xn z : R 1
Because the thickness of the disk crystal is much smaller than the transverse dimension and the laser propagates along the axis, the axial average temperature of the thin disk crystal is used principally in the model to analyze the temperature distribution, which is defined by 6758
Lth Qh r2P ; kCu-W R2
Cu-W nb nb 2f n kCu-W E0na −hH 0na 0 i 0 i o; h nh 0 hH nb −kCu-W E0nb kCu-W kCu-W hH nb −kCu-W Enb kCu-W 1 kkCu-W E E 1 − 1 − 1 0 0 0 0 na nb kYAG kYAG kCu-W Ena −hH na kYAG kCu-W Ena −hH na YAG
B0n
E0nb
B00
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
1 hTrYAG i Lth
Z
Lth 0
T YAG r; zdz:
(6)
Figure 2 shows the average temperature distribution along the radial direction inside the thin disk crystal for different pumping intensities. It can be
in the cylindrical coordinate system. The strain and displacements are related by [14]
εr ∂u εθ ur ∂r ∂w ∂u ; εz ∂z γ zr ∂w ∂r ∂z
8
where u and w are the radial component of displacement and the axial component of displacement, respectively. And the stresses, strains, and temperatures are related by the generalized Hook’s laws [14]: Fig. 1. Schematic diagram and thermal boundary conditions of thin disk module.
seen that the higher the pumping intensity, the higher the average temperature of the disk crystal. At the same time, the maximum temperature would be observed at the center of the disk crystal depending on the pumping intensity, and the temperature difference along the radial direction is small in the pumping zone. The simulation results from analytical calculation agree with the finite element simulations from relational literature [6,9]. B.
Stress and Strain Distribution inside the Disk Crystal
Based on the temperature distribution, the stress and strain inside the disk crystal will be calculated in the thermo-mechanical model. The equilibrium equations are built in a cylindrical coordinate system [14]: 8 < ∂σr ∂τzr σr −σθ 0 ∂r ∂z r ; ∂τrz τrz z : ∂σ ∂z ∂r r 0
7
where σ r , σ θ , σ z and τzr , τrz represent the normal components of stress and the shear components of stress Table 1.
Basic Parameters of Thermal Analysis Model
Parameter Radius of thin disk crystal (R) Pumping spot (rp ) Disk thickness (Lth ) Dopant concentration (cYb ) Pump intensity (I P ) Dopant concentration (cYb ) Coolant temperature (T f ) Heat fraction [10] Heat exchange coefficient (h) Cu-W plate thickness (dCuW ) Thermal conductivity (kCu-W ) Thermal conductivity of disk crystal (kYAG ) Pumping profile Young’s modulus Poisson ratio ∂n∕∂T Thermal expansion coefficient Elastophoto coefficient p11 ; p12 ; p44 [11]
8 > > εr E1 σ r − νσ θ σ z αΔT > > > < ε 1 σ − νσ σ αΔT θ z r E θ ; > εz E1 σ z − νσ r σ θ αΔT > > > > : γ zr E1 21 ντzr
9
where εr , εθ , and εz are normal components of strain in the cylindrical coordinate system, and γ zr is the shear component of strain. E is the Young modulus. v is the Poisson ratio. As the thickness of the disk crystal is much smaller than the transverse dimension, it may be shown that the axial stress approaches zero (σ z 0). So the plane stress approximation can be used in this calculation to simplify the analytical expressions. So the shear stress connected with axial component τzr also vanishes. Based on the elasticity mechanics theory, these physical quantities can be expressed as 8 R R > 1 rp 1 r > > < σ r αE r2p 0 ΔTrdr − r2 0 ΔTrdr Rr R σ θ αE −T r12 0 p ΔTrdr r12 0r ΔTrdr : > > p > :σ 0 z
10
Figure 3 shows the distributions of radial stress and tangential stress that have been computed using
Value 9 mm 2.75 mm 200 μm 7 at. % 5 kW∕cm2 7 at. % 20°C 14.6% 105 W∕m2 · K 2 mm 385 W∕m · K 6.0 W∕m · K Top-hat 310 GPa 0.3 7.3 × 10−6 K −1 7.8 × 10−6 K −1 −0.029, 0.0091, −0.0615
Fig. 2. Average temperature distribution along radial direction inside the disk crystal for different pumping intensities. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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where ΔT is the temperature shift relative to the cooling liquid or environment temperature: Ar0
1 rp
Z
rp 0
ΔTrdr;
Br0
1 r
Z
r o
ΔTrdr:
All shear strain values are zero in isotropic material with the plane stress approximation. These distributions of nonzero normal strain are plotted in Fig. 4. It can be seen the normal strain inside the disk crystal changes rapidly at the boundary of the pumping spot. The nonzero axial strain is a main cause of refractive index change and the expansion of the crystal. C.
Optical Path Difference in the Thin Disk Crystal
Based on the calculation of temperature, stress, and strain inside the disk crystal, the OPD of the thin disk will be analyzed—specifically, the OPD caused by temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation of the HR-coated surface of the disk. In this section, all parts of the OPD will be analyzed respectively. First, the OPD caused by nonuniform temperature distribution in the disk crystal and defined in two passes in the crystal, which can be calculated as Z Δt r 2 ×
Fig. 3. Distributions of normal stress inside the disk crystal in the plane stress approximation with different pumping intensities.
Eqs. (10) and the given parameters shown in Table 1 for different pumping intensities. It can be seen that the radial stress σ r is compressive stress in the whole crystal, and the maximum absolute value is at the center of the disk. On the other hand, tangential stress σ θ properties are different in the pumping region and the nonpumping region. At the boundary of the pumping spot, the stress changes sharply. The main reason lies in the stress inside the crystal being caused by the radial temperature difference, and the most dangerous tangential stress corresponds with the sharp change of temperature at the edge of the pumping region. At the same time, as the pumping intensity increases, the radial stress and tangential stress increase constantly. The strain field will be calculated based on the distributions of temperature and stress that have been computed by the generalized Hooke law [14]:
Lth 0
Tr; z − T c
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
(12)
Figure 5 shows the OPD of the temperature gradient for different pumping intensities. It can be seen that the OPD caused by a nonuniform temperature distribution is linearly related to the axial average temperature. The higher the temperature of thin disk crystal, the more serious the OPD caused by the nonuniform temperature distribution. Second, the OPD caused by thermal expansion and defined in two passes in the crystal can be written as Z Δexp 2 · n − 1 · Δz 2 · n − 1 ·
Lth 0
εz rdz: (13)
Figure 6 shows the distribution of this part of the OPD. According to Eqs. (11) and (13), it is obvious that the OPD is linearly related to the distribution of the temperature shift. Third, the OPD caused by the photoelastic effect and strain-induced birefringence. The strain-induced changes of the refractive index in the thin disk crystal are investigated with consideration of the photoelastic effect in the isotropic cubic crystal. The strain-induced photoelastic effect is described by the following expression:
8 > < εr E1 σ r − vσ θ αΔT α1 − vAr0 − 1 vBr 1 vΔT ; εθ E1 σ θ − vσ r αΔT α1 − vAr0 1 vBr > : εz −v σ r σ θ αΔT α−2vAr0 1 vΔT E 6760
∂n dz: ∂T
11
Fig. 5. OPD of temperature gradient for different pumping intensities.
According to the Neumann–Curie theorem, the principal axes of all the involved tensors are radial and tangential. The expression of strain-induced index change could be written as [8]
hΔnr;θ i
∂nr;θ ∂n ∂n hε i r;θ hεθ i r;θ hεz i; ∂εr r ∂εθ ∂εz
(15)
where the symbol hi denotes the average along the disk thickness. For a cubic crystal, only elasto-optical coefficients p11 , p12 , and p44 are nonzero. The radial and tangential reflection index changes affect the light with the polarization direction x and y in Fig. 7, and the their values are different. Based on the correct change of coordinates, hΔnr;θ i could be calculated as n30 fp 3ε εθ 2εz p12 3εr 5εθ 4εz 12 11 r (16) p44 6εr − 2εθ − 4εz g;
Δnr −
Fig. 4. Distributions of normal strain inside the disk crystal with different pumping intensities.
ΔBij pijkl εkl ;
(14)
where the relative dielectric impermeability Bij is a second-rank tensor defined in the disk cylindrical coordinate system [111]. The elastophoto coefficient p is a fourth-rank tensor. In the crystal axis coordinate system [001], the new tensors can be written with transformation of rotation [8,15]. Using Nye’s convention, coordination transformation is carried out. The derivation process is described in Refs. [8,15]. The relations among these coordinate systems and the refractive index ellipse are plotted in Fig. 7.
Fig. 6. OPD of thermal expansion for different pumping intensities. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 7. Orientation of the indicatrix caused by the photoelastic effect in plane perpendicular to the z axis.
n30 fp ε 3εθ 2εz p12 5εr 3εθ 4εz 12 11 r (17) − p44 2εr − 6εθ 4εz g;
Δnθ −
where pij is the elastophoto coefficient. Figure 8 shows the distribution of the OPD caused by the photoelectric effect for different pumping intensities. The birefringence effect can be observed, but the influence is slight in the pumping zone. The value of this part of the OPD is still much smaller compared with the other parts. Finally, the OPD caused by deformation of the thin disk crystal is analyzed. Because it is difficult to calculate the OPD caused by deformation using analytical methods, FEA is used to obtain the OPD based on the same calculated parameters. Figure 9 shows the OPD for different pumping intensities. It can be seen that the deformation of the disk crystal is one of the major factors in the OPD. The maximum bending value Δδ is at the center of the thin disk crystal. With increasing pumping intensity, the deformation of the OPD will increase constantly.
Fig. 8. Distribution of the OPD caused by photoelectric effect for different pumping intensities. 6762
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
Fig. 9. OPD caused by deformation of thin disk crystal for different pumping intensities.
3. Analysis of Total OPD and Experiments
The total OPD includes four parts, which have been computed in previous sections. The summation of each part can be calculated as OPDTotal r Δt Δexp Δelas Δdeform :
(18)
So the optical phase distortion in the thin disk crystal could be separated into a spherical and an
Fig. 10. Total OPD of thin disk crystal, including spherical part and aspherical part, for the different pumping intensities.
Fig. 11. Experimental setup to measure OPD of thin disk crystal.
aspherical part, which is obtained from the following formula:
Φr
2π 2πr2 · OPDr ΔΦr; λ λRL
(19)
where RL is the radius of the spherical part. The spherical part of the optical phase distortion can be easily compensated by appropriate resonator design. The aspherical part is the main reason for decreasing beam quality. Figure 10 shows the total OPD of the thin disk crystal, including the spherical part and the aspherical part, for the different pumping intensities. It can be seen that the total OPD increases with the increment of pumping intensity. The main reason is that higher pumping intensity results in a higher temperature difference inside the disk crystal under the same cooling conditions. Each part of the OPD is also changing with the increasing temperature difference. Increasing the heat exchange coefficient or using a higher thermal conductivity heat sink will be one of the important methods to obtain smoother temperature and strain profiles. On the other hand, the thermal lens of the thin disk crystal is obtained from the spherical part for different pumping intensities. It is important to design a resonator to achieve better performance. It can be seen that, as the pump intensity increases, the curvature radius of the thin disk crystal also decreases from 8.24 m at 2 kW∕cm2 to 3.15 m at 5 kW∕cm2 . At the same time, the aspherical OPD will become serious. This is one of the main reasons to induce a decrease of the beam quality. The simulation results from analytical calculation agree with the finite element simulations from relational literature [6,9].
Figure 11 shows the measuring setup consisting of a Twyman–Green interferometer and a thin disk laser. The reflectivity of the coupler mirror is 95%. The measuring laser beam passes a neutral attenuator, a beam expander, and a pinhole, which is split by a 50% beam splitter. One part of the split beam is guided to the thin disk crystal and the other part to the reference mirror. The interferogram is recorded
Fig. 12. Measurement of OPD of the thin disk with laser oscillation at 256, 896, and 1451 W pumping power. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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by a CCD camera. Several experiments are carried out using the measuring setup. Figure 12(a) shows the measurements of the OPD of the thin disk with laser oscillation at 256, 896, and 1451 W pumping power. Figure 12(b) shows the spherical part of the measurements of the total OPD. It can be seen that the total OPD can be separated into the spherical and aspherical parts. At the same time, the total OPD and the spherical part of the OPD of the thin disk crystal increase with the increment of pumping intensity. The experimental results agree with the analytical results and the finite element results [6,9]. 4. Conclusion
In summary, the thermal effect and OPD of a thin disk crystal are investigated using the analytical model and FEA software. Based on the analytical model, the distribution of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain, thermal strain-induced birefringence, and deformation of the thin disk crystal can be calculated. The analytical results show the total OPD of the thin disk crystal including the spherical part and the aspherical part; the aspherical OPD is the main reason to induce a decrease of the beam quality. The model could be used to predict the maximum pump power limit for different mounting designs of thin disk lasers. References 1. T. Gottwald, C. Stolzenburg, D. Bauer, J. Kleinbauer, V. Kuhn, T. Metzger, S.-S. Schad, D. Sutter, and A. Killi, “Recent disk laser development at Trumpf,” Proc. SPIE 8547, 85470C (2012). 2. J. Mende, G. Sphindle, E. Schmid, J. Speiser, and A. Giesen, “Thin-disk laser: power scaling to the kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
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3. D. B. Sanchez, B. Weichelt, A. Austerschulte, A. Voss, T. Graf, A. Killi, H.-C. Eckstein, M. Strumpf, A. L. Matthes, and U. D. Xeitner, “Improving the brightness of a multi-kilowatt single thin-disk laser by an aspherical phase front correction,” Opt. Lett. 36, 799–801 (2011). 4. A. Aleknavicius, M. Gabalis, A. Michailovas, and V. Girdauskas, “Aberrations induced by anti-ASE cap on thin disk active element,” Opt. Express 21, 14530–14538 (2013). 5. J. Speiser and A. Geisen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WB9. 6. M. R. J. Milani, V. Sazegari, and A. K. Jafari, “Calculating optical path difference in end-pumped Yb:YAG thin disk lasers,” Proc. SPIE 7747, 774718 (2009). 7. J. Perchermeier and U. Wittrock, “Precise measurements of the thermo-optical aberrations of an Yb:YAG thin disk laser,” Opt. Lett. 38, 2422–2424 (2013). 8. J. Shang, X. Zhu, and G. Zhu, “Analytical approach to thermal lensing in end-pumped Yb:YAG thin disk laser,” Appl. Opt. 50, 6103–6120 (2011). 9. V. Sazegari, M. R. J. Milani, and A. K. Jafari, “Structural and optical behavior due to thermal effects in end-pumped Yb: YAG disk laser,” Appl. Opt. 49, 6910–6916 (2010). 10. M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009). 11. C. Sebastien, D. Frederic, F. Sebastien, B. Francois, and G. Patrick, “On thermal effects in solid state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 3, 89–126 (2006). 12. 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 nonuniform temperature distribution,” Appl. Opt. 51, 2521–2531 (2012). 13. M. N. Ozisik, Heat Conduction, 2nd ed. (Wiley, 1993). 14. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (Intext Educational, 1973). 15. W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
An analytical model of the thermal effect and optical path difference (OPD) of a thin disk laser is developed with the combination of the analytical method and commercial finite element analysis software. The distributions of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation are obtained. Based on the analytical model, the production mechanisms, features, and influence of OPD in an end-pumped thin disk laser are discussed, which make the causes of spherical and aspherical parts of the OPD more obvious. Furthermore, the OPD including the spherical and aspherical parts of the thin disk crystal is discussed for various pumping intensities. © 2014 Optical Society of America OCIS codes: (140.6810) Thermal effects; (140.3615) Lasers, ytterbium; (140.3580) Lasers, solid-state; (350.6830) Thermal lensing. http://dx.doi.org/10.1364/AO.53.006756
1. Introduction
The thin disk concept avoids thermal problems compared with conventional high-power rod or slab lasers and enables high power with good beam quality and high efficiency. Typically, the thin-diskshaped active medium is cooled through one of the flat faces of the disk. This ensures a large surfaceto-volume ratio and therefore provides very efficient thermal management. Recently, a record 10 kW of laser power was demonstrated from a single disk with greater than 60% optical-to-optical efficiency. The laser typically operates in a high-order multimode [1]. So thermal lensing and aberration are not negligible in the multikilowatt regime, which induces a decrease in beam quality. In order to further improve the brightness of the thin disk laser, intensive research has been carried out in thermal-optical aberrations, thermal lensing, and compensation techniques [2–4]. 1559-128X/14/296756-09$15.00/0 © 2014 Optical Society of America 6756
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
In general, the main contributions to the optical path difference (OPD) and thermal lensing in thin disk lasers are temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation of the disk crystal. Moreover, the difference between pumped volume and heat sink causes a mainly spherical bending of the disk. The spherical deformation of the OPD could be compensated by the resonator design, while the aspherical part of the thermo-optical aberrations has to reduce to improve the brightness of the output laser. In order to investigate the OPD, thermal lensing, and compensation techniques, various models and experiments have been carried out by Speiser and Geisen [5], Milani et al. [6], Perchermeier and Wittrock [7], Shang et al. [8], Sazegari et al. [9], and others. Many numerical models were built and calculated using finite element analysis (FEA) software. The FEA method is more convenient to calculate the temperature distribution, and thermal stress (strain), especially for mechanical deformation for arbitrary pumping light distributions. But FEA is a complex and time-consuming process. It is also
thin disk crystal mounted on the heat sink and the thermal boundary conditions is shown in Fig. 1. The basic parameters, which are used to calculate temperature, stress, strain displacement, and deformation inside the thin disk crystal, are shown in Table 1.
difficult to compute OPD changes with different system parameters. On the other hand, the analytical method will show the relationship between the OPD and numerous system parameters more clearly. In this paper, the thermal effect and OPD of a thin disk crystal are investigated. Combined with merits of the analytical method and FEA software, the OPD analytical model of the thin disk crystal is built. The distributions of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain (bulging), and thermal strain-induced birefringence are calculated using analytical methods.
A. Temperature Distribution inside the Disk Crystal
We assume that the side surface of the thin disk is heat-insulated and heat generated in the crystal is homogeneous after multipass pumping. The stationary heat conduction equation can be built [12]:
8 < ∂2 T YAG r;z 1 ∂T YAG r;z ∂2 T YAG r;z − r ∂r : ∂2 T Cu-W2 r;z 1 ∂T Cu-W r;z r ∂r ∂r ∂r2
∂z2 ∂2 T Cu-W r;z ∂z2
Qh kYAG
circrP for disk crystal;
0
for CuW heat sink:
1
The boundary conditions are 8 > > ∂T YAG r;z > 0; > > ∂r > > rR > < ∂T YAG r;z 0; ∂z > zLth > > > > > ∂T YAG r;z ∂T Cu-W r;z > k k > YAG Cu-W ∂z ∂z : z0
for disk crystal
(2)
; z0
8 > T YAG r; 0 T Cu-W r; 0; > > > > < ∂T Cu-W r;z 0; ∂r for Cu-W heat sink rR > > > ∂T r;z h Cu-W > kCu-W T Cu-W r; 0 − T f ; > ∂z :
(3)
z−dCuW
Meanwhile, the OPD caused by deformation of the disk crystal is obtained by FEA software. Based on the composite model, the OPD including the spherical and aspherical parts of the thin disk crystal is discussed for various system parameters. 2. Physical Model and Analysis
The numerical model is used to carry out thermal analysis in the gain medium. The Yb:YAG crystal (typical thickness between 0.2 and 0.4 mm) is soldering on the heat sink with an indium layer. The disk is highly reflective coated on its backside for both laser and pumping wavelengths and antireflective coated on the front side for both wavelengths. The heat sink is cooled by impingement cooling using several nozzles inside the heat sink. A schematic diagram of the
where T YAG r; z and T Cu-W r; z are the temperature distributions in the thin disk crystal and CuW heat sink. Lth is the thickness of the laser crystal. rP is the radius of the pumping spot. kYAG and kCu-W represent the thermal conductivity of the thin disk crystal and the heat sink, respectively. h is the heat exchange coefficient of the heat sink with cooling liquid. T f is the temperature of cooling liquid. Qh is the heat generation density in the thin disk crystal. The stationary heat conduction equations of the thin disk crystal and CuW heat sink are the classical Poisson equation and the Laplace equation, the solutions of which include general and special solutions. Using the Methods of Mathematical Physics [13] and complex derivation, we can obtain analytical expressions of the temperature distribution: 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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T YAG r; z A0 B0 z
∞ X n1
0 0 0 0 ∞ X xn xn xn 1 2 xn z Bn exp − z · J0 r − z G0 r ; An exp Gn · J 0 R R R R 2 n1 (4)
0 0 0 ∞ X xn xn xn 0 0 0 0 T Cu-W r; z T f A0 B0 z z Bn exp − z · J0 r ; An exp R R R n1
(5)
where the functions J 0 x and J 1 x are the zero-order Bessel function and first-order Bessel function. The x0 n are the roots of J 1 x 0, n 1; 2; …∞, and the coefficients A0 , B0 , A00 , B00 , A0n , B0n , An , Bn , Ena , Enb , E0na , E0nb , H 0na , H 0nb , G0 , and Gn are given by hdCuW Lth Qh r2P ; A0 T f 1 kCu-W hR2
A00
hH 0 −k
A0n
hdCuW Lth Qh r2P 1 ; kCu-W hR2
L Q r2 B0 th h 2P ; kYAG R
1 An 2
Bn
E0
1 2
1 kkCu-W YAG
hH 0nb −kCu-W E0nb kCu-W E0na −hH 0na
2f n
; 0 0 kCu-W kCu-W hH nb −kCu-W Enb kCu-W 1 − kYAG Ena 1 − kYAG kCu-W E0na −hH 0na 1 kYAG Enb
kCu-W 0 kCu-W 0 B0n kCu-W hH 0nb − kCu-W E0nb kCu-W 1 1− 1 An 1 − Bn ; kYAG kYAG 2 kYAG kCu-W E0na − hH 0na kYAG
1−
hH 0nb − kCu-W E0nb kCu-W 0 k B0 k kCu-W 1 An 1 Cu-W B0n n 1 − Cu-W ; kYAG kYAG 2 kYAG kCu-W E0na − hH 0na kYAG
0 0 0 0 xn xn xn xn Lth ; Enb exp − Lth − ; R R R R 0 0 xn xn dCuW − ; exp R R
Ena exp
G0
Qh r2P ; kYAG R2
H 0nb exp
Gn kYAG
0 2Qh rP xn 2 J1 r ; R P 0 0 J 0 xn xn R
0 0 xn xn E0na exp − dCuW ; R R
0 xn H 0na exp − z ; R 1
0 xn z : R 1
Because the thickness of the disk crystal is much smaller than the transverse dimension and the laser propagates along the axis, the axial average temperature of the thin disk crystal is used principally in the model to analyze the temperature distribution, which is defined by 6758
Lth Qh r2P ; kCu-W R2
Cu-W nb nb 2f n kCu-W E0na −hH 0na 0 i 0 i o; h nh 0 hH nb −kCu-W E0nb kCu-W kCu-W hH nb −kCu-W Enb kCu-W 1 kkCu-W E E 1 − 1 − 1 0 0 0 0 na nb kYAG kYAG kCu-W Ena −hH na kYAG kCu-W Ena −hH na YAG
B0n
E0nb
B00
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
1 hTrYAG i Lth
Z
Lth 0
T YAG r; zdz:
(6)
Figure 2 shows the average temperature distribution along the radial direction inside the thin disk crystal for different pumping intensities. It can be
in the cylindrical coordinate system. The strain and displacements are related by [14]
εr ∂u εθ ur ∂r ∂w ∂u ; εz ∂z γ zr ∂w ∂r ∂z
8
where u and w are the radial component of displacement and the axial component of displacement, respectively. And the stresses, strains, and temperatures are related by the generalized Hook’s laws [14]: Fig. 1. Schematic diagram and thermal boundary conditions of thin disk module.
seen that the higher the pumping intensity, the higher the average temperature of the disk crystal. At the same time, the maximum temperature would be observed at the center of the disk crystal depending on the pumping intensity, and the temperature difference along the radial direction is small in the pumping zone. The simulation results from analytical calculation agree with the finite element simulations from relational literature [6,9]. B.
Stress and Strain Distribution inside the Disk Crystal
Based on the temperature distribution, the stress and strain inside the disk crystal will be calculated in the thermo-mechanical model. The equilibrium equations are built in a cylindrical coordinate system [14]: 8 < ∂σr ∂τzr σr −σθ 0 ∂r ∂z r ; ∂τrz τrz z : ∂σ ∂z ∂r r 0
7
where σ r , σ θ , σ z and τzr , τrz represent the normal components of stress and the shear components of stress Table 1.
Basic Parameters of Thermal Analysis Model
Parameter Radius of thin disk crystal (R) Pumping spot (rp ) Disk thickness (Lth ) Dopant concentration (cYb ) Pump intensity (I P ) Dopant concentration (cYb ) Coolant temperature (T f ) Heat fraction [10] Heat exchange coefficient (h) Cu-W plate thickness (dCuW ) Thermal conductivity (kCu-W ) Thermal conductivity of disk crystal (kYAG ) Pumping profile Young’s modulus Poisson ratio ∂n∕∂T Thermal expansion coefficient Elastophoto coefficient p11 ; p12 ; p44 [11]
8 > > εr E1 σ r − νσ θ σ z αΔT > > > < ε 1 σ − νσ σ αΔT θ z r E θ ; > εz E1 σ z − νσ r σ θ αΔT > > > > : γ zr E1 21 ντzr
9
where εr , εθ , and εz are normal components of strain in the cylindrical coordinate system, and γ zr is the shear component of strain. E is the Young modulus. v is the Poisson ratio. As the thickness of the disk crystal is much smaller than the transverse dimension, it may be shown that the axial stress approaches zero (σ z 0). So the plane stress approximation can be used in this calculation to simplify the analytical expressions. So the shear stress connected with axial component τzr also vanishes. Based on the elasticity mechanics theory, these physical quantities can be expressed as 8 R R > 1 rp 1 r > > < σ r αE r2p 0 ΔTrdr − r2 0 ΔTrdr Rr R σ θ αE −T r12 0 p ΔTrdr r12 0r ΔTrdr : > > p > :σ 0 z
10
Figure 3 shows the distributions of radial stress and tangential stress that have been computed using
Value 9 mm 2.75 mm 200 μm 7 at. % 5 kW∕cm2 7 at. % 20°C 14.6% 105 W∕m2 · K 2 mm 385 W∕m · K 6.0 W∕m · K Top-hat 310 GPa 0.3 7.3 × 10−6 K −1 7.8 × 10−6 K −1 −0.029, 0.0091, −0.0615
Fig. 2. Average temperature distribution along radial direction inside the disk crystal for different pumping intensities. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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where ΔT is the temperature shift relative to the cooling liquid or environment temperature: Ar0
1 rp
Z
rp 0
ΔTrdr;
Br0
1 r
Z
r o
ΔTrdr:
All shear strain values are zero in isotropic material with the plane stress approximation. These distributions of nonzero normal strain are plotted in Fig. 4. It can be seen the normal strain inside the disk crystal changes rapidly at the boundary of the pumping spot. The nonzero axial strain is a main cause of refractive index change and the expansion of the crystal. C.
Optical Path Difference in the Thin Disk Crystal
Based on the calculation of temperature, stress, and strain inside the disk crystal, the OPD of the thin disk will be analyzed—specifically, the OPD caused by temperature gradient, axial thermal strain (bulging), thermal strain-induced birefringence, and deformation of the HR-coated surface of the disk. In this section, all parts of the OPD will be analyzed respectively. First, the OPD caused by nonuniform temperature distribution in the disk crystal and defined in two passes in the crystal, which can be calculated as Z Δt r 2 ×
Fig. 3. Distributions of normal stress inside the disk crystal in the plane stress approximation with different pumping intensities.
Eqs. (10) and the given parameters shown in Table 1 for different pumping intensities. It can be seen that the radial stress σ r is compressive stress in the whole crystal, and the maximum absolute value is at the center of the disk. On the other hand, tangential stress σ θ properties are different in the pumping region and the nonpumping region. At the boundary of the pumping spot, the stress changes sharply. The main reason lies in the stress inside the crystal being caused by the radial temperature difference, and the most dangerous tangential stress corresponds with the sharp change of temperature at the edge of the pumping region. At the same time, as the pumping intensity increases, the radial stress and tangential stress increase constantly. The strain field will be calculated based on the distributions of temperature and stress that have been computed by the generalized Hooke law [14]:
Lth 0
Tr; z − T c
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
(12)
Figure 5 shows the OPD of the temperature gradient for different pumping intensities. It can be seen that the OPD caused by a nonuniform temperature distribution is linearly related to the axial average temperature. The higher the temperature of thin disk crystal, the more serious the OPD caused by the nonuniform temperature distribution. Second, the OPD caused by thermal expansion and defined in two passes in the crystal can be written as Z Δexp 2 · n − 1 · Δz 2 · n − 1 ·
Lth 0
εz rdz: (13)
Figure 6 shows the distribution of this part of the OPD. According to Eqs. (11) and (13), it is obvious that the OPD is linearly related to the distribution of the temperature shift. Third, the OPD caused by the photoelastic effect and strain-induced birefringence. The strain-induced changes of the refractive index in the thin disk crystal are investigated with consideration of the photoelastic effect in the isotropic cubic crystal. The strain-induced photoelastic effect is described by the following expression:
8 > < εr E1 σ r − vσ θ αΔT α1 − vAr0 − 1 vBr 1 vΔT ; εθ E1 σ θ − vσ r αΔT α1 − vAr0 1 vBr > : εz −v σ r σ θ αΔT α−2vAr0 1 vΔT E 6760
∂n dz: ∂T
11
Fig. 5. OPD of temperature gradient for different pumping intensities.
According to the Neumann–Curie theorem, the principal axes of all the involved tensors are radial and tangential. The expression of strain-induced index change could be written as [8]
hΔnr;θ i
∂nr;θ ∂n ∂n hε i r;θ hεθ i r;θ hεz i; ∂εr r ∂εθ ∂εz
(15)
where the symbol hi denotes the average along the disk thickness. For a cubic crystal, only elasto-optical coefficients p11 , p12 , and p44 are nonzero. The radial and tangential reflection index changes affect the light with the polarization direction x and y in Fig. 7, and the their values are different. Based on the correct change of coordinates, hΔnr;θ i could be calculated as n30 fp 3ε εθ 2εz p12 3εr 5εθ 4εz 12 11 r (16) p44 6εr − 2εθ − 4εz g;
Δnr −
Fig. 4. Distributions of normal strain inside the disk crystal with different pumping intensities.
ΔBij pijkl εkl ;
(14)
where the relative dielectric impermeability Bij is a second-rank tensor defined in the disk cylindrical coordinate system [111]. The elastophoto coefficient p is a fourth-rank tensor. In the crystal axis coordinate system [001], the new tensors can be written with transformation of rotation [8,15]. Using Nye’s convention, coordination transformation is carried out. The derivation process is described in Refs. [8,15]. The relations among these coordinate systems and the refractive index ellipse are plotted in Fig. 7.
Fig. 6. OPD of thermal expansion for different pumping intensities. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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Fig. 7. Orientation of the indicatrix caused by the photoelastic effect in plane perpendicular to the z axis.
n30 fp ε 3εθ 2εz p12 5εr 3εθ 4εz 12 11 r (17) − p44 2εr − 6εθ 4εz g;
Δnθ −
where pij is the elastophoto coefficient. Figure 8 shows the distribution of the OPD caused by the photoelectric effect for different pumping intensities. The birefringence effect can be observed, but the influence is slight in the pumping zone. The value of this part of the OPD is still much smaller compared with the other parts. Finally, the OPD caused by deformation of the thin disk crystal is analyzed. Because it is difficult to calculate the OPD caused by deformation using analytical methods, FEA is used to obtain the OPD based on the same calculated parameters. Figure 9 shows the OPD for different pumping intensities. It can be seen that the deformation of the disk crystal is one of the major factors in the OPD. The maximum bending value Δδ is at the center of the thin disk crystal. With increasing pumping intensity, the deformation of the OPD will increase constantly.
Fig. 8. Distribution of the OPD caused by photoelectric effect for different pumping intensities. 6762
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
Fig. 9. OPD caused by deformation of thin disk crystal for different pumping intensities.
3. Analysis of Total OPD and Experiments
The total OPD includes four parts, which have been computed in previous sections. The summation of each part can be calculated as OPDTotal r Δt Δexp Δelas Δdeform :
(18)
So the optical phase distortion in the thin disk crystal could be separated into a spherical and an
Fig. 10. Total OPD of thin disk crystal, including spherical part and aspherical part, for the different pumping intensities.
Fig. 11. Experimental setup to measure OPD of thin disk crystal.
aspherical part, which is obtained from the following formula:
Φr
2π 2πr2 · OPDr ΔΦr; λ λRL
(19)
where RL is the radius of the spherical part. The spherical part of the optical phase distortion can be easily compensated by appropriate resonator design. The aspherical part is the main reason for decreasing beam quality. Figure 10 shows the total OPD of the thin disk crystal, including the spherical part and the aspherical part, for the different pumping intensities. It can be seen that the total OPD increases with the increment of pumping intensity. The main reason is that higher pumping intensity results in a higher temperature difference inside the disk crystal under the same cooling conditions. Each part of the OPD is also changing with the increasing temperature difference. Increasing the heat exchange coefficient or using a higher thermal conductivity heat sink will be one of the important methods to obtain smoother temperature and strain profiles. On the other hand, the thermal lens of the thin disk crystal is obtained from the spherical part for different pumping intensities. It is important to design a resonator to achieve better performance. It can be seen that, as the pump intensity increases, the curvature radius of the thin disk crystal also decreases from 8.24 m at 2 kW∕cm2 to 3.15 m at 5 kW∕cm2 . At the same time, the aspherical OPD will become serious. This is one of the main reasons to induce a decrease of the beam quality. The simulation results from analytical calculation agree with the finite element simulations from relational literature [6,9].
Figure 11 shows the measuring setup consisting of a Twyman–Green interferometer and a thin disk laser. The reflectivity of the coupler mirror is 95%. The measuring laser beam passes a neutral attenuator, a beam expander, and a pinhole, which is split by a 50% beam splitter. One part of the split beam is guided to the thin disk crystal and the other part to the reference mirror. The interferogram is recorded
Fig. 12. Measurement of OPD of the thin disk with laser oscillation at 256, 896, and 1451 W pumping power. 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
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by a CCD camera. Several experiments are carried out using the measuring setup. Figure 12(a) shows the measurements of the OPD of the thin disk with laser oscillation at 256, 896, and 1451 W pumping power. Figure 12(b) shows the spherical part of the measurements of the total OPD. It can be seen that the total OPD can be separated into the spherical and aspherical parts. At the same time, the total OPD and the spherical part of the OPD of the thin disk crystal increase with the increment of pumping intensity. The experimental results agree with the analytical results and the finite element results [6,9]. 4. Conclusion
In summary, the thermal effect and OPD of a thin disk crystal are investigated using the analytical model and FEA software. Based on the analytical model, the distribution of temperature, stress, strain, and OPD caused by temperature gradient, axial thermal strain, thermal strain-induced birefringence, and deformation of the thin disk crystal can be calculated. The analytical results show the total OPD of the thin disk crystal including the spherical part and the aspherical part; the aspherical OPD is the main reason to induce a decrease of the beam quality. The model could be used to predict the maximum pump power limit for different mounting designs of thin disk lasers. References 1. T. Gottwald, C. Stolzenburg, D. Bauer, J. Kleinbauer, V. Kuhn, T. Metzger, S.-S. Schad, D. Sutter, and A. Killi, “Recent disk laser development at Trumpf,” Proc. SPIE 8547, 85470C (2012). 2. J. Mende, G. Sphindle, E. Schmid, J. Speiser, and A. Giesen, “Thin-disk laser: power scaling to the kW regime in fundamental mode operation,” Proc. SPIE 7193, 71931V (2009).
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3. D. B. Sanchez, B. Weichelt, A. Austerschulte, A. Voss, T. Graf, A. Killi, H.-C. Eckstein, M. Strumpf, A. L. Matthes, and U. D. Xeitner, “Improving the brightness of a multi-kilowatt single thin-disk laser by an aspherical phase front correction,” Opt. Lett. 36, 799–801 (2011). 4. A. Aleknavicius, M. Gabalis, A. Michailovas, and V. Girdauskas, “Aberrations induced by anti-ASE cap on thin disk active element,” Opt. Express 21, 14530–14538 (2013). 5. J. Speiser and A. Geisen, “Numerical modeling of high power continuous-wave Yb:YAG thin disk lasers, scaling to 14 kW,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WB9. 6. M. R. J. Milani, V. Sazegari, and A. K. Jafari, “Calculating optical path difference in end-pumped Yb:YAG thin disk lasers,” Proc. SPIE 7747, 774718 (2009). 7. J. Perchermeier and U. Wittrock, “Precise measurements of the thermo-optical aberrations of an Yb:YAG thin disk laser,” Opt. Lett. 38, 2422–2424 (2013). 8. J. Shang, X. Zhu, and G. Zhu, “Analytical approach to thermal lensing in end-pumped Yb:YAG thin disk laser,” Appl. Opt. 50, 6103–6120 (2011). 9. V. Sazegari, M. R. J. Milani, and A. K. Jafari, “Structural and optical behavior due to thermal effects in end-pumped Yb: YAG disk laser,” Appl. Opt. 49, 6910–6916 (2010). 10. M. Najafi, A. Sepehr, A. H. Golpaygani, and J. Sabbaghzadeh, “Simulation of thin disk laser pumping process for temperature dependent Yb:YAG property,” Opt. Commun. 282, 4103–4108 (2009). 11. C. Sebastien, D. Frederic, F. Sebastien, B. Francois, and G. Patrick, “On thermal effects in solid state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. 3, 89–126 (2006). 12. 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 nonuniform temperature distribution,” Appl. Opt. 51, 2521–2531 (2012). 13. M. N. Ozisik, Heat Conduction, 2nd ed. (Wiley, 1993). 14. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed. (Intext Educational, 1973). 15. W. Koechner and D. K. Rice, “Effect of birefringence on the performance of linearly polarized YAG:Nd lasers,” IEEE J. Quantum Electron. 6, 557–566 (1970).
