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Semianalytical analysis of thermal effect in LD double-side-pumped rectangular laser crystal
Optics Communications 282 (2009) 3751–3756
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Semianalytical analysis of thermal effect in LD double-side-pumped rectangular laser crystal Wen Chen *, Peng Shi, Zhongwen Hua, Long Li, Ansheng Gan School of Science, Xi’an University of Architecture and Technology, 710055 Xi’an, China
a r t i c l e
i n f o
Article history: Received 16 October 2008 Received in revised form 1 June 2009 Accepted 2 June 2009
Keywords: Nd:YVO4 crystal Double-side-pumped Temperature field Thermal distortion field
a b s t r a c t A semianalytical method to analyze the thermal effect in a LD double-side-pumped rectangular laser crystal is put forward. Through the analysis of working characteristics of the laser crystal, a thermal model that matches actual situations of the laser crystal is established. General expressions of temperature field and thermal distortion field in the laser crystal can be obtained by a novel method to solve the heat conduction equation of orthotropic material. This semianalytical method can be used to calculate the temperature field and thermal distortion field in other LD double-side-pumped laser crystals and is applied to Nd:YVO4 crystal in detail in this paper, and two methods of effectively reducing thermal distortion in the laser crystal are offered. Results show that a maximum temperature rise of 362.2 °C and a maximum thermal distortion of 5.55 lm are obtained in Nd:YVO4 crystal when the output power of the two laser diodes are both 30 W. When the off-center distance is 0.6 mm, the maximum thermal distortion is reduced by 37.7%; when the thickness of the crystal is reduced from 2.0 mm to 1.4 mm, the maximum thermal distortion is reduced by 31.7%. Results in this paper can offer theoretical base for better solving thermal problems in laser system. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Diode-pumped high-power solid-state lasers (DPSSL) are widely used in industry, medicine and military area, etc. [1] due to their high efficiency, compact package and beam quality [2], Though end-pump is the most efficient method in miniature DPSSL system, in the system of high-power DPSSL side-pump is usually adopted [3–6], because high power pump light can be coupled in the quite small area of the laser gain medium when end-pump is used, which can result in strong thermal effect and even thermal fracture damage of laser crystal [7–10]. Yet a key factor that affects the optical conversion efficiency of side-pumped DPSSL is the degree of overlapping between the mode capacity in the cavity and the LD pump capacity. The energy of pump light is absorbed by the laser crystal and results in not only population inversion in pump area but also uneven temperature rise and thermal distortion in laser crystal [11,12]. All these thermal effects can severely affect the optical conversion efficiency of side-pumped DPSSL. In this paper a semianalytical method to analyze the thermal effect in a double-side-pumped rectangular laser crystal is put forward. Since Nd:YVO4 crystal is currently one of the most popular laser crystals for DPSSL, and the thermal effect in Nd:YVO4 is especially severe due to its strong absorption coefficient and poor ther* Corresponding author. Tel.: +86 029 82664971. E-mail address: [email protected] (W. Chen). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.012
mal conductivity [7], this semianalytical method is applied to a Nd:YVO4 crystal and the temperature field and the thermal distortion field in the Nd:YVO4 crystal are calculated in this paper. And two methods of effectively reducing the thermal distortion in the Nd:YVO4 crystal are offered. 2. A semianalytical method to calculate temperature field and thermal distortion field in double-side-pumped laser crystal The semianalytical method to calculate temperature field and thermal distortion field in double-side-pumped laser crystal mainly includes following three steps. Firstly, a physical model that matches actual working state of the laser crystal can be established by analyzing the working characteristics of the laser crystal. Secondly, general expressions of temperature field and thermal distortion field in the laser crystal can be obtained by a novel method to solve the heat conduction equation of orthotropic material based on analytical and semianalytical theory. Lastly, temperature field and thermal distortion field can be obtained by computation software according to their general expressions. 2.1. Thermal model of double-side-pumped Nd:YVO4 crystal Diagrammatic sketch of a double-side-pumped Nd:YVO4 laser is given in Fig. 1 [3]. Nd:YVO4 crystal is cut along a-axis of crystal and evaporated with a layer of reflection reducing film with a
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If the power of this pump light is P1, then
P1 ¼ I01
Z
1
e
2ðxx0 Þ2 w2
dx
rffiffiffiffi
z2
dz ¼ I01 w
z1
1
I01 ¼
Z
pffiffiffi 2P 1 pffiffiffiffi wðz2 z1 Þ p
p 2
ðz2 z1 Þ ð2Þ
According to absorption rule, if the absorption coefficient of laser crystal about pump light is b, on the plane y = y the light intensity of the pump light that propagates along y-direction is:
(
Fig. 1. Diagrammatic sketch of double-side-pumped Nd:YVO4 laser.
I1 ðx; y; zÞ ¼ I1 ðx; 0; zÞe
by
¼
I01 e
2ðxx0 Þ2 by w2
0
ðz1 6 z 6 z2 Þ
ð3Þ
other
Similarly, the light intensity at the center of the pump light on the back side y = b is:
I02 ¼
pffiffiffi 2P 2 pffiffiffiffi wðz4 z3 Þ p
ð4Þ
where P2 is the power of the pump light projected to the back side. The light intensity on the plane y = y is:
( I2 ðx; y; zÞ ¼
I02 e
2ðxx0 Þ2 bðbyÞ w2
0
ðz3 6 z 6 z4 Þ
ð5Þ
other
Therefore, the total light intensity on the plane y = y is:
Fig. 2. Double-side-pumped Nd:YVO4 crystal.
Iðx; y; zÞ ¼ I1 ðx; y; zÞ þ I2 ðx; y; zÞ ¼
wavelength of 1064 nm after polished. One diode laser is projected to the front side of Nd:YVO4 crystal; the other diode laser is projected to the back side of Nd:YVO4 crystal, while the two diode lasers are staggered, shown in Fig. 2. The other two sides of Nd:YVO4 crystal are contacted with heat sink through a layer of soft indium foil to protect the crystal and well dissipate the heat generated in the crystal. Fig. 3 shows the thermal model of Nd:YVO4 crystal, of which three lengths are a, b, and c, respectively. Two pump lights are projected to the range of z1 to z2 of the front side(y = 0) and z3 to z4 of the back side(y = b), respectively, after collimated and focused by a group of cylindrical mirrors. In this paper the two pump lights are supposed to have a Gaussian profile along x-direction and a uniform profile along z-direction [11]. The distribution of light intensity on the front side (y = 0) caused by the pump light projected to this side is:
( I1 ðx; 0; zÞ ¼
I01 e 0
2ðxx0 Þ2 w2
ðz1 6 z 6 z2 Þ
ð1Þ
other
in which w is Gaussian radius, and I01 is the light intensity at the center of the pump light.
8 2ðxx0 Þ2 > by > 2 > < I01 e w > I02 e > > : 0
2ðxx0 Þ2 bðbyÞ w2
ðz1 6 z 6 z2 Þ ðz3 6 z 6 z4 Þ other ð6Þ
Since the energy absorbed by fluorescence quantum effect and internal loss is much more than the energy absorbed due to other reasons in laser crystal, only the heat generated by above two reasons is considered in this paper. The thermal power density on the plane z = z is given by [13]:
qV ðx; y; zÞ ¼ gbIðx; y; zÞ
ð7Þ
in which g is the heat conversion coefficient determined by fluorescence quantum effect and internal loss, and g = 1 kp/kL, where kp is the wavelength (808 nm) of pump light from LD, and kL is the wavelength (1064 nm) of the laser that oscillates in the resonant cavity. The upper side and lower side of the laser crystal are contacted with heat sink and cooled to a relatively constant temperature, which can be supposed to be uw. In mathematical treatment the temperature uw can be set at zero so long as the cooling temperature uw is added to the ultimately calculated temperature field, supposed to be u(x, y, z). Except the upper and lower sides of the crystal, the other two sides and two end faces of the crystal are exposed to air and can be supposed to be adiabatic because the
Fig. 3. Thermal model of Nd:YVO4 crystal.
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W. Chen et al. / Optics Communications 282 (2009) 3751–3756
coefficient of heat transfer between crystal and air is very small. Therefore boundary conditions can be given by:
Z
uð0; y; zÞ ¼ 0; uða; y; zÞ ¼ 0 @uðx; y; zÞ @uðx; y; zÞ ¼ 0; ¼0 @y @y y¼0 y¼b @uðx; y; zÞ @uðx; y; zÞ ¼ 0; ¼0 @z @z z¼0 z¼c
ð8Þ
kx
ð9Þ
(1) a form of the family of the Eigen function of Poisson Equation can be determined according to the boundary conditions, and the family of the Eigen function can constitute the solution of Poisson Equation u(x, y, z), in which there are some constants to be determined. (2) u(x, y, z) is then substituted into the Poisson Equation to obtain the expression of the undetermined constants. (3) u(x, y, z) satisfies both the Poisson Equation and its boundary conditions, and the solution of the Poisson Equation is unique; therefore u(x, y, z), should be the unique solution of the heat conduction equation.
2.2.1. Temperature field u1(x, y, z) inside the crystal generated from the pump light projected to the front side According to the boundary conditions, the Eigen functions of x, y, and z can be derived to be the family of sin nap x; cos mbp y and cos lcp z, respectively. The following equation can be assumed:
Alnml sin
n¼1 m¼0 l¼0
np mp lp x cos y cos z a b c
ð10Þ
In which Alnml is a constant to be determined. Above expression of ul(x, y, z) can be substituted into Eq. (8) and testified to satisfy the boundary conditions. Substituting Eq. (10) into Eq. (9) yields
Bnml sin
n¼1 m¼0 l¼0
np mp lp x cos y cos z ¼ qv ðx; y; zÞ a b c
ð11Þ
h 2 2 i 2 Where Blnml ¼ Alnml kx nap þ ky mbp þ kz lcp According to the orthogonality and normalization of the Eigen function of x,
Z 0
a
m¼0 l¼0
Blnml cos
mp lp 2 ycos z ¼ a b c
Z
a 0
qv l ðx; y; zÞsin
np xdx a
Z
b
Z
0
0
a
qv l ðx; y; zÞsin
np mp xcos ydxdy a b
According to the orthogonality and normalization of the Eigen function of z,
Z
c
cos
lp kp c z cos zdz ¼ dik 2 c c
ð13Þ
ð16Þ
Both sides of Eq. (15) are multiplied by cos kcp z, and integrating to z from 0 to c yields
Z cZ bZ a 8 np mp q ðx; y; zÞsin xcos ycos abc 0 0 0 v l a b lp zdxdydz c
Blnml ¼
ð17Þ
Substituting Eq. (1) into Eq. (17) yields
Z Z b 8gbI01 a 2ðxx20 Þ2 np mp e w sin eby cos xdx ydy abc a b 0 0 Z z2 lp cos zdz c z1 8I01 gb sin lpcz2 sin lpcz1 Z a 2ðxx0 Þ2 np e w2 sin ¼ xdx abpl a 0 Z b mp eby cos ydy b 0
B1nml ¼
Rb
2
Because of 0 eby cos mbp ydy ¼ bb be obtained as follows:
B1nml ¼
A1nml
ð1ebb cos mpÞ ; Blnml b2 b2 þm2 p2
ð18Þ
and Alnml can
8I01 gb2 b sin lpcz2 sin lpcz1 ð1 ebb cos mpÞ 2
aplðb2 b þ m2 p2 Þ np e sin xdx a 0 8I01 gb2 b sin lpcz2 sin lpcz1 ð1 ebb cos mpÞ ¼ 2 2 2 k m2 ap3 lðb2 b þ m2 p2 Þ kxan2 þ yb2 þ kcz2l Z a 2ðxx Þ2 0 np e w2 sin xdx a 0 Z
a
2ðxx0 Þ2 w2
ð19Þ
ð20Þ
2.2.2. Temperature field inside the crystal generated from the pump light projected to the back side The pump light projected to the back side of the crystal propagates along –y-direction, and the temperature field in the crystal generated from this pump light is 1 X 1 X 1 X n¼1 m¼0 l¼0
ð12Þ
Both sides of Eq. (11) are multiplied by sin kap z, and integrating to x from 0 to a yields 1 X 1 X
lp 4 z¼ ab c
u2ðx; y; zÞ ¼
np kp a sin x sin xdx ¼ dnk 2 a a
ð14Þ
ð15Þ
In which kx, ky, kz are the coefficients of heat conductivity along x, y, z directions, respectively. Due to the complexity of qV(x, y, z) and boundary conditions, this equation can not be solved by general method that is used to solve Poisson Equation. A novel method to solve the Eq. (9) with boundary conditions as Eq. (8) is put forward with detailed steps as follows:
1 X 1 X 1 X
Blnml cos
l¼0
2
1 X 1 X 1 X
mp kp b y cos ydy ¼ dmk 2 b b
Both sides of Eq. (13) are multiplied by cos kbp y, and integrating to y from 0 to b yields
0
@ u @ u @ u þ ky 2 þ kz 2 ¼ qv ðx; y; zÞ @x2 @y @z
ulðx; y; zÞ ¼
cos
1 X
Since Nd:YVO4 crystal is orthotropic with an internal heat source, heat conduction inside the crystal should observe the heat conduction equation of an orthotropic solid (Poisson Equation) [12]: 2
b
0
2.2. Calculation of temperature field in Nd:YVO4 crystal
2
According to the orthogonality and normalization of the Eigen function of y,
A2nml sin
np mp lp x cos ðb yÞ cos z a b c
ð21Þ
Similarly following equation can be obtained
Z cZ bZ a 8 np mp q ðx; y; zÞsin xcos ðb yÞ abc 0 0 0 v 2 a b lp cos zdxdydz c
B2nml ¼
ð22Þ
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Substituting Eq. (5) into Eq. (22) yields
Z Z b 8gbI02 a 2ðxx20 Þ2 np mp e w sin ebðbyÞ cos xdx ðb yÞdy abc a b 0 0 Z z4 lp cos zdz c z3 8I02 gb sin lpcz4 sin lpcz3 Z a 2ðxx0 Þ2 np e w2 sin ¼ xdx abpl a 0 Z b mp ebðbyÞ cos ðb yÞdy ð23Þ b 0
B2nml ¼
Because of
Z
2
b
ebðbyÞ cos
0
mp bb ð1 ebb cos mpÞ ðb yÞdy ¼ 2 b b2 b þ m2 p2
ð24Þ
Following equations can be obtained
B2nml ¼
8gb2 bI02 sin lpcz4 sin lpcz3 ð1 ebb cos mpÞ
A2nml
Fig. 4. Three-dimensional temperature field on the middle plane (x = 1 mm) of double-side-pumped Nd:YVO4 crystal.
2
aplðb2 b þ m2 p2 Þ a 2ðxx0 Þ2 np e w2 sin xdx a 0 8gb2 bI02 sin lpcz4 sin lpcz3 ð1 ebb cos mpÞ ¼ 2 2 2 k m2 ap3 lðb2 b þ m2 p2 Þ kxan2 þ yb2 þ kcz2l Z a 2ðxx Þ2 0 np xdx e w2 sin a 0 Z
ð25Þ
ð26Þ
Therefore total temperature field u(x, y, z) inside the crystal is composed of the two temperature fields generated by two pump lights:
uðx; y; zÞ ¼ u1ðx; y; zÞ þ u2ðx; y; zÞ
ð27Þ
which can be achieved by computation software. 2.3. Calculation of thermal distortion in Nd:YVO4 crystal The upper side and lower side of the Nd:YVO4 crystal are appressed to soft indium foil, so heat expansion of the laser crystal can be approximately considered to be free. Consider a small volume of a rectangular solid in the crystal. Let dx, dy, and dz be the lengths of the three sides of the rectangular solid. If the initial temperature of the rectangular solid is zero (relatively) and the temperature in steady state is u(x, y, z), then the difference in temperature is u(x, y, z), and differences of thermal distortion are as follows: Fig. 5. Distribution of isothermal line on the middle plane of Nd:YVO4 crystal.
dlx ¼ ax uðx; y; zÞdx dly ¼ ay uðx; y; zÞdy dlz ¼ az uðx; y; zÞdz
ð28Þ
In which ax, ay, az are thermal expansion coefficients along the x, y, and z directions, respectively, and the thermal expansion coefficient along each direction is constant, therefore the total differences of the thermal distortion along different directions become
lx ¼
Z
a
dlx ¼ ax
0
ly ¼ lz ¼
Z
a
uðx; y; zÞdx
0
b
Z0c 0
Z
dly ¼ ay dlz ¼ az
Z
b
Z 0c
uðx; y; zÞdy
ð29Þ
uðx; y; zÞdz
0
According to Eq. (29), total thermal distortion field, thermal strain field and total thermal distortion in the crystal can be obtained by computation software.
Fig. 6. Distribution of three-dimensional temperature field on the pump side.
W. Chen et al. / Optics Communications 282 (2009) 3751–3756
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Fig. 8. Distribution of thermal distortion on the end face z = 0.
Fig. 7. Distribution of temperature along the central line on the front pump side.
3. Results and discussion 3.1. Temperature field in double-side-pumped Nd:YVO4 crystal For a Nd:YVO4 crystal doped with 1.1at.% Nd3+, the coefficient of heat conductivity is 5.23 Wm1 K1(kc), 5.10 Wm1 K1(\c) and the coefficient of the thermal expansion is 11.37 106 K1(k c), 4.43 106 K1(\c); the absorption coefficient of the crystal about the pump light is 31.4 cm1 [14]. c is the direction of optical axis of Nd:YVO4 crystal, which is in the x-direction in this paper. The powers of the two diode lasers are both 30 W. The Gaussian radius of the pump light is 0.20 mm, and the size of the crystal is 2 mm 2 mm 10 mm, viz. a = b = 2 mm and c = 10 mm. The center of light intensity x0 of pump light along x-direction is at x0 = 1 mm. The front pump light is projected to front side of the crystal between z1 = 0 mm and z2 = 5 mm. The back pump light is projected to the back side of the crystal between z3 = 5 mm and z4 = 10 mm. Calculated three-dimensional distribution of the temperature field on the middle plane of the crystal x = 1 mm is shown in Fig. 4, and the distribution of isothermal line on the middle plane is shown in Fig. 5. Three-dimensional distribution of the temperature field on the front pump side y = 0 is shown in Fig. 6, and the distribution of the temperature along the central line x = 1 mm of the front pump side y = 0 is shown in Fig. 7. From Fig. 7 it can be obtained that the maximum temperature rise is 362.2 °C. The influence of pump power and the radius of pump light on the temperature field can be further investigated using the obtained expression of temperature field.
Fig. 9. Contrast of maximum thermal distortions on the pump side with the change of x0.
3.2. Thermal distortion on the end faces of Nd:YVO4 crystal Since thermal distortion on the two end faces of laser crystal is the main cause of thermal lens effect, distribution of thermal distortion on the end face z = 0, same as that on the end face z = c due to symmetry is calculated and shown in Fig. 8. From Fig. 8 it can be obtained that a maximum thermal distortion of 5.55 lm occurs at the point y = 0, x = x0 = 1 mm. Fig. 10. Change of maximum thermal distortion with thickness.
3.3. Two methods of reducing thermal distortion Thermal distortion is the main factor that affects the stability of the resonant cavity and output power and mode of laser. Following two methods can effectively reduce the thermal distortion on the end faces of laser crystal. First method is to let the center of the pump light intensity, supposed to be x0, deviate from the central line of the pump side along x-direction, which means to change x0. Fig. 9 shows the distributions of thermal distortions on the side y = 0 when x0 is 0.4 mm, 0.6 mm, 0.8 mm and 1.0 mm, respectively.
From Fig. 9 it can be found that the maximum thermal distortion is reduced when x0 is reduced. And this is because the center of the maximum pump light intensity is nearer to the cooling end of the crystal x = 0 when x0 is reduced. When x0 is 0.4 mm, 0.6 mm and 0.8 mm, respectively, the maximum thermal distortion is 3.46 lm, 4.62 lm and 5.32 lm, respectively. Compared with the thermal distortion of center-pumped (x0 = 1.0 mm) crystal, which is 5.55 lm, the thermal distortion is reduced by 37.7%, 16.8%, 4.1%, respectively.
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The second method is to reduce the thickness of the crystal along x-direction, namely to change a, one of the three lengths of crystal, while the crystal is still center-pumped (x0 = a/2). Change of the maximum thermal distortion with the thickness of crystal is shown in Fig. 10. From Fig. 10 it can be obtained that compared with the thermal distortion of a = 2.0 mm, the maximum thermal distortion is reduced by 10.5%, 20.7%, 31.7%, respectively, when a is1.8 mm, 1.6 mm, 1.4 mm, respectively. And this is resulted from the shortening of the distance between the maximum light intensity and the cooling side when a is reduced.
4. Conclusions In this paper a semianalytical method to analyze the thermal effect in LD double-side-pumped rectangular laser crystal is put forward and applied to Nd:YVO4 crystal. Temperature field inside the crystal and thermal distortion field on the pump sides are achieved and analyzed quantitatively, and two methods of effectively reducing thermal distortion are offered. The semianalytical method in this paper can be used to calculate the temperature field and thermal distortion field in other LD double-side-pumped rectangular laser crystals and offer theoretical base for the stability design of LD side-pumped high-power solid-state laser.
Acknowledgement This research was supported by the Special Scientific Research Project of Shaan Xi Education Office through Grant 06JK251. References [1] T.Y. Fan, R.L. Byer, IEEE J. Quant. Electron. 24 (1988) 895. [2] W. Koechner, Solid-state Laser Engineering, fifth ed., Springer-Verlag, Berlin, 1999. [3] J. Hong, C. Yelland, M.H. Dunn, W. Sibbett, B.D. Sinclair, J. Mod. Opt. 41 (6) (1994) 1227. [4] M.J. Damzen, M. Trew, E. Rosas, G.J. Crofts, Opt. Commun. 196 (1) (2001) 237. [5] H. Zimer, K. Albers, U. Wittrock, Opt. Lett. 29 (23) (2004) 2761. [6] R.J.St. Pierre, D.W. Mordaunt, H. Injeyanetc, IEEE J. Sel. Top. Quant. Electron. (1997) 53. [7] Z. Ma, D.J. Li, J.C. Gao, N.L. Wu, K.M. Du, Opt. Commun. 275 (2007) 179. [8] W.J. Xie, S.-C. Tam, Y.-L. Lam, K.S. Lai, R.F. Wu, Y.L. Lim, E. Lau, IEEE J. Quant. Electron. 37 (10) (2001) 1368. [9] M. Eichhorn, Appl. Phys. B: Lasers Opt. 94 (3) (2009) 451. [10] F. Li, T. Fan, J.T. Bai, X. Hou, Acta Photonica Sinica 37 (3) (2008) 425 (in Chinese). [11] J.C. Bermudez, G.V.J. Pinto-robledo, A.V. Kir’yanov, M.J. Damzen, Opt. Commun. 210 (1) (2002) 75. [12] A. Agnesi, F. Pirzio, A. Tomaselli, F. Bonfiqli, T. Marolo, Proc. SPIE 577 (2005) 753. [13] P. Shi, W. Chen, L. Li, A.S. Gan, Appl. Opt. 46 (26) (2007) 6655. [14] P. Shi, L. Li, A.S. Gan, Chin. J. Lasers 32 (2) (2007) 197 (in Chinese).
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Semianalytical analysis of thermal effect in LD double-side-pumped rectangular laser crystal Wen Chen *, Peng Shi, Zhongwen Hua, Long Li, Ansheng Gan School of Science, Xi’an University of Architecture and Technology, 710055 Xi’an, China
a r t i c l e
i n f o
Article history: Received 16 October 2008 Received in revised form 1 June 2009 Accepted 2 June 2009
Keywords: Nd:YVO4 crystal Double-side-pumped Temperature field Thermal distortion field
a b s t r a c t A semianalytical method to analyze the thermal effect in a LD double-side-pumped rectangular laser crystal is put forward. Through the analysis of working characteristics of the laser crystal, a thermal model that matches actual situations of the laser crystal is established. General expressions of temperature field and thermal distortion field in the laser crystal can be obtained by a novel method to solve the heat conduction equation of orthotropic material. This semianalytical method can be used to calculate the temperature field and thermal distortion field in other LD double-side-pumped laser crystals and is applied to Nd:YVO4 crystal in detail in this paper, and two methods of effectively reducing thermal distortion in the laser crystal are offered. Results show that a maximum temperature rise of 362.2 °C and a maximum thermal distortion of 5.55 lm are obtained in Nd:YVO4 crystal when the output power of the two laser diodes are both 30 W. When the off-center distance is 0.6 mm, the maximum thermal distortion is reduced by 37.7%; when the thickness of the crystal is reduced from 2.0 mm to 1.4 mm, the maximum thermal distortion is reduced by 31.7%. Results in this paper can offer theoretical base for better solving thermal problems in laser system. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Diode-pumped high-power solid-state lasers (DPSSL) are widely used in industry, medicine and military area, etc. [1] due to their high efficiency, compact package and beam quality [2], Though end-pump is the most efficient method in miniature DPSSL system, in the system of high-power DPSSL side-pump is usually adopted [3–6], because high power pump light can be coupled in the quite small area of the laser gain medium when end-pump is used, which can result in strong thermal effect and even thermal fracture damage of laser crystal [7–10]. Yet a key factor that affects the optical conversion efficiency of side-pumped DPSSL is the degree of overlapping between the mode capacity in the cavity and the LD pump capacity. The energy of pump light is absorbed by the laser crystal and results in not only population inversion in pump area but also uneven temperature rise and thermal distortion in laser crystal [11,12]. All these thermal effects can severely affect the optical conversion efficiency of side-pumped DPSSL. In this paper a semianalytical method to analyze the thermal effect in a double-side-pumped rectangular laser crystal is put forward. Since Nd:YVO4 crystal is currently one of the most popular laser crystals for DPSSL, and the thermal effect in Nd:YVO4 is especially severe due to its strong absorption coefficient and poor ther* Corresponding author. Tel.: +86 029 82664971. E-mail address: [email protected] (W. Chen). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.06.012
mal conductivity [7], this semianalytical method is applied to a Nd:YVO4 crystal and the temperature field and the thermal distortion field in the Nd:YVO4 crystal are calculated in this paper. And two methods of effectively reducing the thermal distortion in the Nd:YVO4 crystal are offered. 2. A semianalytical method to calculate temperature field and thermal distortion field in double-side-pumped laser crystal The semianalytical method to calculate temperature field and thermal distortion field in double-side-pumped laser crystal mainly includes following three steps. Firstly, a physical model that matches actual working state of the laser crystal can be established by analyzing the working characteristics of the laser crystal. Secondly, general expressions of temperature field and thermal distortion field in the laser crystal can be obtained by a novel method to solve the heat conduction equation of orthotropic material based on analytical and semianalytical theory. Lastly, temperature field and thermal distortion field can be obtained by computation software according to their general expressions. 2.1. Thermal model of double-side-pumped Nd:YVO4 crystal Diagrammatic sketch of a double-side-pumped Nd:YVO4 laser is given in Fig. 1 [3]. Nd:YVO4 crystal is cut along a-axis of crystal and evaporated with a layer of reflection reducing film with a
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If the power of this pump light is P1, then
P1 ¼ I01
Z
1
e
2ðxx0 Þ2 w2
dx
rffiffiffiffi
z2
dz ¼ I01 w
z1
1
I01 ¼
Z
pffiffiffi 2P 1 pffiffiffiffi wðz2 z1 Þ p
p 2
ðz2 z1 Þ ð2Þ
According to absorption rule, if the absorption coefficient of laser crystal about pump light is b, on the plane y = y the light intensity of the pump light that propagates along y-direction is:
(
Fig. 1. Diagrammatic sketch of double-side-pumped Nd:YVO4 laser.
I1 ðx; y; zÞ ¼ I1 ðx; 0; zÞe
by
¼
I01 e
2ðxx0 Þ2 by w2
0
ðz1 6 z 6 z2 Þ
ð3Þ
other
Similarly, the light intensity at the center of the pump light on the back side y = b is:
I02 ¼
pffiffiffi 2P 2 pffiffiffiffi wðz4 z3 Þ p
ð4Þ
where P2 is the power of the pump light projected to the back side. The light intensity on the plane y = y is:
( I2 ðx; y; zÞ ¼
I02 e
2ðxx0 Þ2 bðbyÞ w2
0
ðz3 6 z 6 z4 Þ
ð5Þ
other
Therefore, the total light intensity on the plane y = y is:
Fig. 2. Double-side-pumped Nd:YVO4 crystal.
Iðx; y; zÞ ¼ I1 ðx; y; zÞ þ I2 ðx; y; zÞ ¼
wavelength of 1064 nm after polished. One diode laser is projected to the front side of Nd:YVO4 crystal; the other diode laser is projected to the back side of Nd:YVO4 crystal, while the two diode lasers are staggered, shown in Fig. 2. The other two sides of Nd:YVO4 crystal are contacted with heat sink through a layer of soft indium foil to protect the crystal and well dissipate the heat generated in the crystal. Fig. 3 shows the thermal model of Nd:YVO4 crystal, of which three lengths are a, b, and c, respectively. Two pump lights are projected to the range of z1 to z2 of the front side(y = 0) and z3 to z4 of the back side(y = b), respectively, after collimated and focused by a group of cylindrical mirrors. In this paper the two pump lights are supposed to have a Gaussian profile along x-direction and a uniform profile along z-direction [11]. The distribution of light intensity on the front side (y = 0) caused by the pump light projected to this side is:
( I1 ðx; 0; zÞ ¼
I01 e 0
2ðxx0 Þ2 w2
ðz1 6 z 6 z2 Þ
ð1Þ
other
in which w is Gaussian radius, and I01 is the light intensity at the center of the pump light.
8 2ðxx0 Þ2 > by > 2 > < I01 e w > I02 e > > : 0
2ðxx0 Þ2 bðbyÞ w2
ðz1 6 z 6 z2 Þ ðz3 6 z 6 z4 Þ other ð6Þ
Since the energy absorbed by fluorescence quantum effect and internal loss is much more than the energy absorbed due to other reasons in laser crystal, only the heat generated by above two reasons is considered in this paper. The thermal power density on the plane z = z is given by [13]:
qV ðx; y; zÞ ¼ gbIðx; y; zÞ
ð7Þ
in which g is the heat conversion coefficient determined by fluorescence quantum effect and internal loss, and g = 1 kp/kL, where kp is the wavelength (808 nm) of pump light from LD, and kL is the wavelength (1064 nm) of the laser that oscillates in the resonant cavity. The upper side and lower side of the laser crystal are contacted with heat sink and cooled to a relatively constant temperature, which can be supposed to be uw. In mathematical treatment the temperature uw can be set at zero so long as the cooling temperature uw is added to the ultimately calculated temperature field, supposed to be u(x, y, z). Except the upper and lower sides of the crystal, the other two sides and two end faces of the crystal are exposed to air and can be supposed to be adiabatic because the
Fig. 3. Thermal model of Nd:YVO4 crystal.
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coefficient of heat transfer between crystal and air is very small. Therefore boundary conditions can be given by:
Z
uð0; y; zÞ ¼ 0; uða; y; zÞ ¼ 0 @uðx; y; zÞ @uðx; y; zÞ ¼ 0; ¼0 @y @y y¼0 y¼b @uðx; y; zÞ @uðx; y; zÞ ¼ 0; ¼0 @z @z z¼0 z¼c
ð8Þ
kx
ð9Þ
(1) a form of the family of the Eigen function of Poisson Equation can be determined according to the boundary conditions, and the family of the Eigen function can constitute the solution of Poisson Equation u(x, y, z), in which there are some constants to be determined. (2) u(x, y, z) is then substituted into the Poisson Equation to obtain the expression of the undetermined constants. (3) u(x, y, z) satisfies both the Poisson Equation and its boundary conditions, and the solution of the Poisson Equation is unique; therefore u(x, y, z), should be the unique solution of the heat conduction equation.
2.2.1. Temperature field u1(x, y, z) inside the crystal generated from the pump light projected to the front side According to the boundary conditions, the Eigen functions of x, y, and z can be derived to be the family of sin nap x; cos mbp y and cos lcp z, respectively. The following equation can be assumed:
Alnml sin
n¼1 m¼0 l¼0
np mp lp x cos y cos z a b c
ð10Þ
In which Alnml is a constant to be determined. Above expression of ul(x, y, z) can be substituted into Eq. (8) and testified to satisfy the boundary conditions. Substituting Eq. (10) into Eq. (9) yields
Bnml sin
n¼1 m¼0 l¼0
np mp lp x cos y cos z ¼ qv ðx; y; zÞ a b c
ð11Þ
h 2 2 i 2 Where Blnml ¼ Alnml kx nap þ ky mbp þ kz lcp According to the orthogonality and normalization of the Eigen function of x,
Z 0
a
m¼0 l¼0
Blnml cos
mp lp 2 ycos z ¼ a b c
Z
a 0
qv l ðx; y; zÞsin
np xdx a
Z
b
Z
0
0
a
qv l ðx; y; zÞsin
np mp xcos ydxdy a b
According to the orthogonality and normalization of the Eigen function of z,
Z
c
cos
lp kp c z cos zdz ¼ dik 2 c c
ð13Þ
ð16Þ
Both sides of Eq. (15) are multiplied by cos kcp z, and integrating to z from 0 to c yields
Z cZ bZ a 8 np mp q ðx; y; zÞsin xcos ycos abc 0 0 0 v l a b lp zdxdydz c
Blnml ¼
ð17Þ
Substituting Eq. (1) into Eq. (17) yields
Z Z b 8gbI01 a 2ðxx20 Þ2 np mp e w sin eby cos xdx ydy abc a b 0 0 Z z2 lp cos zdz c z1 8I01 gb sin lpcz2 sin lpcz1 Z a 2ðxx0 Þ2 np e w2 sin ¼ xdx abpl a 0 Z b mp eby cos ydy b 0
B1nml ¼
Rb
2
Because of 0 eby cos mbp ydy ¼ bb be obtained as follows:
B1nml ¼
A1nml
ð1ebb cos mpÞ ; Blnml b2 b2 þm2 p2
ð18Þ
and Alnml can
8I01 gb2 b sin lpcz2 sin lpcz1 ð1 ebb cos mpÞ 2
aplðb2 b þ m2 p2 Þ np e sin xdx a 0 8I01 gb2 b sin lpcz2 sin lpcz1 ð1 ebb cos mpÞ ¼ 2 2 2 k m2 ap3 lðb2 b þ m2 p2 Þ kxan2 þ yb2 þ kcz2l Z a 2ðxx Þ2 0 np e w2 sin xdx a 0 Z
a
2ðxx0 Þ2 w2
ð19Þ
ð20Þ
2.2.2. Temperature field inside the crystal generated from the pump light projected to the back side The pump light projected to the back side of the crystal propagates along –y-direction, and the temperature field in the crystal generated from this pump light is 1 X 1 X 1 X n¼1 m¼0 l¼0
ð12Þ
Both sides of Eq. (11) are multiplied by sin kap z, and integrating to x from 0 to a yields 1 X 1 X
lp 4 z¼ ab c
u2ðx; y; zÞ ¼
np kp a sin x sin xdx ¼ dnk 2 a a
ð14Þ
ð15Þ
In which kx, ky, kz are the coefficients of heat conductivity along x, y, z directions, respectively. Due to the complexity of qV(x, y, z) and boundary conditions, this equation can not be solved by general method that is used to solve Poisson Equation. A novel method to solve the Eq. (9) with boundary conditions as Eq. (8) is put forward with detailed steps as follows:
1 X 1 X 1 X
Blnml cos
l¼0
2
1 X 1 X 1 X
mp kp b y cos ydy ¼ dmk 2 b b
Both sides of Eq. (13) are multiplied by cos kbp y, and integrating to y from 0 to b yields
0
@ u @ u @ u þ ky 2 þ kz 2 ¼ qv ðx; y; zÞ @x2 @y @z
ulðx; y; zÞ ¼
cos
1 X
Since Nd:YVO4 crystal is orthotropic with an internal heat source, heat conduction inside the crystal should observe the heat conduction equation of an orthotropic solid (Poisson Equation) [12]: 2
b
0
2.2. Calculation of temperature field in Nd:YVO4 crystal
2
According to the orthogonality and normalization of the Eigen function of y,
A2nml sin
np mp lp x cos ðb yÞ cos z a b c
ð21Þ
Similarly following equation can be obtained
Z cZ bZ a 8 np mp q ðx; y; zÞsin xcos ðb yÞ abc 0 0 0 v 2 a b lp cos zdxdydz c
B2nml ¼
ð22Þ
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Substituting Eq. (5) into Eq. (22) yields
Z Z b 8gbI02 a 2ðxx20 Þ2 np mp e w sin ebðbyÞ cos xdx ðb yÞdy abc a b 0 0 Z z4 lp cos zdz c z3 8I02 gb sin lpcz4 sin lpcz3 Z a 2ðxx0 Þ2 np e w2 sin ¼ xdx abpl a 0 Z b mp ebðbyÞ cos ðb yÞdy ð23Þ b 0
B2nml ¼
Because of
Z
2
b
ebðbyÞ cos
0
mp bb ð1 ebb cos mpÞ ðb yÞdy ¼ 2 b b2 b þ m2 p2
ð24Þ
Following equations can be obtained
B2nml ¼
8gb2 bI02 sin lpcz4 sin lpcz3 ð1 ebb cos mpÞ
A2nml
Fig. 4. Three-dimensional temperature field on the middle plane (x = 1 mm) of double-side-pumped Nd:YVO4 crystal.
2
aplðb2 b þ m2 p2 Þ a 2ðxx0 Þ2 np e w2 sin xdx a 0 8gb2 bI02 sin lpcz4 sin lpcz3 ð1 ebb cos mpÞ ¼ 2 2 2 k m2 ap3 lðb2 b þ m2 p2 Þ kxan2 þ yb2 þ kcz2l Z a 2ðxx Þ2 0 np xdx e w2 sin a 0 Z
ð25Þ
ð26Þ
Therefore total temperature field u(x, y, z) inside the crystal is composed of the two temperature fields generated by two pump lights:
uðx; y; zÞ ¼ u1ðx; y; zÞ þ u2ðx; y; zÞ
ð27Þ
which can be achieved by computation software. 2.3. Calculation of thermal distortion in Nd:YVO4 crystal The upper side and lower side of the Nd:YVO4 crystal are appressed to soft indium foil, so heat expansion of the laser crystal can be approximately considered to be free. Consider a small volume of a rectangular solid in the crystal. Let dx, dy, and dz be the lengths of the three sides of the rectangular solid. If the initial temperature of the rectangular solid is zero (relatively) and the temperature in steady state is u(x, y, z), then the difference in temperature is u(x, y, z), and differences of thermal distortion are as follows: Fig. 5. Distribution of isothermal line on the middle plane of Nd:YVO4 crystal.
dlx ¼ ax uðx; y; zÞdx dly ¼ ay uðx; y; zÞdy dlz ¼ az uðx; y; zÞdz
ð28Þ
In which ax, ay, az are thermal expansion coefficients along the x, y, and z directions, respectively, and the thermal expansion coefficient along each direction is constant, therefore the total differences of the thermal distortion along different directions become
lx ¼
Z
a
dlx ¼ ax
0
ly ¼ lz ¼
Z
a
uðx; y; zÞdx
0
b
Z0c 0
Z
dly ¼ ay dlz ¼ az
Z
b
Z 0c
uðx; y; zÞdy
ð29Þ
uðx; y; zÞdz
0
According to Eq. (29), total thermal distortion field, thermal strain field and total thermal distortion in the crystal can be obtained by computation software.
Fig. 6. Distribution of three-dimensional temperature field on the pump side.
W. Chen et al. / Optics Communications 282 (2009) 3751–3756
3755
Fig. 8. Distribution of thermal distortion on the end face z = 0.
Fig. 7. Distribution of temperature along the central line on the front pump side.
3. Results and discussion 3.1. Temperature field in double-side-pumped Nd:YVO4 crystal For a Nd:YVO4 crystal doped with 1.1at.% Nd3+, the coefficient of heat conductivity is 5.23 Wm1 K1(kc), 5.10 Wm1 K1(\c) and the coefficient of the thermal expansion is 11.37 106 K1(k c), 4.43 106 K1(\c); the absorption coefficient of the crystal about the pump light is 31.4 cm1 [14]. c is the direction of optical axis of Nd:YVO4 crystal, which is in the x-direction in this paper. The powers of the two diode lasers are both 30 W. The Gaussian radius of the pump light is 0.20 mm, and the size of the crystal is 2 mm 2 mm 10 mm, viz. a = b = 2 mm and c = 10 mm. The center of light intensity x0 of pump light along x-direction is at x0 = 1 mm. The front pump light is projected to front side of the crystal between z1 = 0 mm and z2 = 5 mm. The back pump light is projected to the back side of the crystal between z3 = 5 mm and z4 = 10 mm. Calculated three-dimensional distribution of the temperature field on the middle plane of the crystal x = 1 mm is shown in Fig. 4, and the distribution of isothermal line on the middle plane is shown in Fig. 5. Three-dimensional distribution of the temperature field on the front pump side y = 0 is shown in Fig. 6, and the distribution of the temperature along the central line x = 1 mm of the front pump side y = 0 is shown in Fig. 7. From Fig. 7 it can be obtained that the maximum temperature rise is 362.2 °C. The influence of pump power and the radius of pump light on the temperature field can be further investigated using the obtained expression of temperature field.
Fig. 9. Contrast of maximum thermal distortions on the pump side with the change of x0.
3.2. Thermal distortion on the end faces of Nd:YVO4 crystal Since thermal distortion on the two end faces of laser crystal is the main cause of thermal lens effect, distribution of thermal distortion on the end face z = 0, same as that on the end face z = c due to symmetry is calculated and shown in Fig. 8. From Fig. 8 it can be obtained that a maximum thermal distortion of 5.55 lm occurs at the point y = 0, x = x0 = 1 mm. Fig. 10. Change of maximum thermal distortion with thickness.
3.3. Two methods of reducing thermal distortion Thermal distortion is the main factor that affects the stability of the resonant cavity and output power and mode of laser. Following two methods can effectively reduce the thermal distortion on the end faces of laser crystal. First method is to let the center of the pump light intensity, supposed to be x0, deviate from the central line of the pump side along x-direction, which means to change x0. Fig. 9 shows the distributions of thermal distortions on the side y = 0 when x0 is 0.4 mm, 0.6 mm, 0.8 mm and 1.0 mm, respectively.
From Fig. 9 it can be found that the maximum thermal distortion is reduced when x0 is reduced. And this is because the center of the maximum pump light intensity is nearer to the cooling end of the crystal x = 0 when x0 is reduced. When x0 is 0.4 mm, 0.6 mm and 0.8 mm, respectively, the maximum thermal distortion is 3.46 lm, 4.62 lm and 5.32 lm, respectively. Compared with the thermal distortion of center-pumped (x0 = 1.0 mm) crystal, which is 5.55 lm, the thermal distortion is reduced by 37.7%, 16.8%, 4.1%, respectively.
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The second method is to reduce the thickness of the crystal along x-direction, namely to change a, one of the three lengths of crystal, while the crystal is still center-pumped (x0 = a/2). Change of the maximum thermal distortion with the thickness of crystal is shown in Fig. 10. From Fig. 10 it can be obtained that compared with the thermal distortion of a = 2.0 mm, the maximum thermal distortion is reduced by 10.5%, 20.7%, 31.7%, respectively, when a is1.8 mm, 1.6 mm, 1.4 mm, respectively. And this is resulted from the shortening of the distance between the maximum light intensity and the cooling side when a is reduced.
4. Conclusions In this paper a semianalytical method to analyze the thermal effect in LD double-side-pumped rectangular laser crystal is put forward and applied to Nd:YVO4 crystal. Temperature field inside the crystal and thermal distortion field on the pump sides are achieved and analyzed quantitatively, and two methods of effectively reducing thermal distortion are offered. The semianalytical method in this paper can be used to calculate the temperature field and thermal distortion field in other LD double-side-pumped rectangular laser crystals and offer theoretical base for the stability design of LD side-pumped high-power solid-state laser.
Acknowledgement This research was supported by the Special Scientific Research Project of Shaan Xi Education Office through Grant 06JK251. References [1] T.Y. Fan, R.L. Byer, IEEE J. Quant. Electron. 24 (1988) 895. [2] W. Koechner, Solid-state Laser Engineering, fifth ed., Springer-Verlag, Berlin, 1999. [3] J. Hong, C. Yelland, M.H. Dunn, W. Sibbett, B.D. Sinclair, J. Mod. Opt. 41 (6) (1994) 1227. [4] M.J. Damzen, M. Trew, E. Rosas, G.J. Crofts, Opt. Commun. 196 (1) (2001) 237. [5] H. Zimer, K. Albers, U. Wittrock, Opt. Lett. 29 (23) (2004) 2761. [6] R.J.St. Pierre, D.W. Mordaunt, H. Injeyanetc, IEEE J. Sel. Top. Quant. Electron. (1997) 53. [7] Z. Ma, D.J. Li, J.C. Gao, N.L. Wu, K.M. Du, Opt. Commun. 275 (2007) 179. [8] W.J. Xie, S.-C. Tam, Y.-L. Lam, K.S. Lai, R.F. Wu, Y.L. Lim, E. Lau, IEEE J. Quant. Electron. 37 (10) (2001) 1368. [9] M. Eichhorn, Appl. Phys. B: Lasers Opt. 94 (3) (2009) 451. [10] F. Li, T. Fan, J.T. Bai, X. Hou, Acta Photonica Sinica 37 (3) (2008) 425 (in Chinese). [11] J.C. Bermudez, G.V.J. Pinto-robledo, A.V. Kir’yanov, M.J. Damzen, Opt. Commun. 210 (1) (2002) 75. [12] A. Agnesi, F. Pirzio, A. Tomaselli, F. Bonfiqli, T. Marolo, Proc. SPIE 577 (2005) 753. [13] P. Shi, W. Chen, L. Li, A.S. Gan, Appl. Opt. 46 (26) (2007) 6655. [14] P. Shi, L. Li, A.S. Gan, Chin. J. Lasers 32 (2) (2007) 197 (in Chinese).