Thermal stress effects in pulsed pump solid state lasers with super-Gaussian profile

A. Keshavarz*a, P. Elahia, S. Rezazadehb

aDepartment of Physics, Faculty of Science, Shiraz University of Technology, Shiraz, Iran bDepartment of Physics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Iran

ABSTRACT The thermally induced stress in pulsed pump solid state lasers with super-Gaussian profile has been investigated. An analytical expression for the thermal stress is introduced. We consider the heat deposited in the crystal due to the pump. The temperature distribution in the crystal has been calculated by solving the non-steady state heat conduction equation. A Ti: Sapphire crystal is assumed pumped by a pulse laser. All the stress components have been obtained and discussed in details. The results show that the non-homogenous temperature distribution is induced by the thermal stress in the crystal. Keywords: Thermal effects, Super-Gaussian, Solid State Lasers, Stress, Strain

1. INTRODUCTION The most important problem in designing solid state lasers with high power is the thermal effects in laser active medium. In end-pumped configuration solid state lasers, the pump energy is focused around a small regime of the crystal axis, and this leads to heat production and unsteady distribution of temperature in the laser crystal. Produced temperature gradient causes changes of refractive index in different points of crystal that is called thermal dispersion. End effects, thermal stresses and strains are other thermal effects that are produced in the laser crystal. So, by exact recognition and calculation of the produced heat and the correct management of the heat, we can apply suitable ways to decrease and compensate the thermal effects in solid state laser crystals. These crystals are of high efficiency, with wide tunable region in high powers, which are using in new industrial tools. Ti:sapphire laser crystals is an example of these crystals, because of its physical and optical properties. Beside the excellent thermal conductivity decreases the thermal effects considerably even for high intensities and powers. This crystal is pumped by short pulses and is suitable for producing ultra-short pulses in 660-1100 nm. In this article, a fourth-order multimode Super-Gaussian end-pumped pulse source, with the energy of pulse near the crystal damage threshold has been applied to a cylindrical-shaped Ti:sapphire laser crystal and thermal stress effects have been investigated analytically1-6.

2. TEMPERATURE DISTRIBUTION Theoretical study of the temperature distribution for a cylindrical shaped crystal pumped by a pulse of energy can be seen in Ref.[2,3], and continuous wave in Ref.[4]. In these references an analytical expression introduced, and then must be solved in order to investigate thermal effects. Thermal anisotropy of Ti:sapphire laser crystal, is so low that we can use solving the time- dependant heat transport equation in isotropic materials for it. The time-dependent heat transport equation is given by:

),,(2 trQtTcTK =∂∂

−∇ ρ (1)

where QcK ,,, ρ are thermal conductivity, mass density, specific heat and heat source density respectively. By solving the time-dependant heat transport equation (1) with ,0),( =trQ for mentioned cylindrical crystal, time-dependant radial temperature distribution, while the crystal is pumped by short repetitive pulses is: * keshavarz@sutech.ac.ir; phone +98-711-7261392; P.O. Box 313-71555

16th International School on Quantum Electronics: Laser Physics and Applications, edited by Tanja Dreischuh, Dimitar Slavov, Proc. of SPIE Vol. 7747, 774719 · © 2011

SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.882034

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.)exp(1

)exp(.)(),(1 22

22

00 ∑∞

= −−−

=n n

nnn

akatkarJBTtrT

τααα

(2)

We assumed the temperature changes in z-axis be very slight. The expression coefficient nB can be obtained as:

(3)

where the hypergeometric function );,(11 xcaF introduced as:

,....2,1,0...,!2)1(

)1(!1

1);,(2

11 −−≠+++

++= cxccaax

caxcaF (4)

and

],1[)2(

22

2

22

2/30laser

pump

pp

total

claerf

ETλλ

ηρ

ωωπ

−= (5)

is the temperature difference between the center of the rod and the surface of the crystal at 0=t . Also other parameters introduced as follow: ητα ,,,,,, lEka totaln are the crystal radius, thermal conductivity, zeros of the Bessel function 0J , time between successive pulses, pulse energy, crystal length and quantum efficiency respectively.

3. THERMAL STRESS DISTRIBUTIONS The end-pumped solid state laser crystals with high power has unsteady temperature gradient distributed in the crystal, which is expanded inside the area rather than the cooler outside. These changes lead to mechanical stresses in the crystal. Totally, the stress is the force in a unit of surface that is presented as a second-rank tensor and it can have different directions in crystal, like electric field. Stress components act as two forms: compressive and tensile. If components are negative it will be compressive, and if there are positive it will be tensile. Tensile stress is a kind of stress caused by applied forces against the directions that the crystal tends to be broken. Compressive stress is a kind of stress caused by applied forces against the directions that crystal tends to be compressed. When crystal exposed to stress, its form will change and when the stress stopped, the crystal taken its initial form. Applied stresses should be less than crystal elastic limit. These stresses lead to birefringence and losses of depolarization. Thermally-induced stress components in plan stress approximation are calculated as:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= ∫ ∫

a rT

r rdrrTr

rdrrTa

cr0 0

22 )(1)(11

)(υ

ασ , (7)

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−= ∫ ∫

a rT rTrdrrT

rrdrrT

acr

0 022 )()(1)(1

1)(

υα

σ ϕ, (8)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= ∫

aT

z rTrdrrTa

cr0

2 )()(21

)(υ

ασ . (9)

Where Tαυ, are Poisson ratio and thermal expansion coefficient respectively.

},3)2;22

5,1(4{)!)(3)(1(

)4()(

)2exp()(

)()2exp(2

4

4

10

14

4

2

2

21

440

21

20

044

0

+++×++

−−=

−= ∑

∫ ∞

=

saSFasssJ

aTJa

rdrarJrTB

ps p

sn

n

p

n

n

a

p

n ωωα

αω

α

αω

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4. APPLYING THE CONCLUSION ON Ti:Sapphire LASER CRYSTAL The stress components are calculated by applying equations (9) on the Ti:sapphire laser crystal with characteristics introduced in table 1.

Table 1: Physical and thermal properties of the Ti:sapphire laser crystal7-9

In figure 1. the radial, azimuthally and z stress components has been plotted as a function of the radius of the crystal ( )(cmr ), which are shown as the solid curves, dot curves and dash curves respectively; at 0=t in figure (a) and

)(st τ= in figure (b), for repetition frequencies 50Hz(up curves) and 100Hz(down curves). As one can see in these figures, by increasing pulse repetition frequency, the stress components in the crystal center area get more negative, and the azimuthally as well as the axial components in the crystal surface will increase. Also by increasing the time, these components at the center and the surface of the crystal will decrease. In order to compare stress component changes with respect to time, we also fix repetition frequencies as 50Hz in figure 2.(a) and 100Hz in figure 2.(b), and evaluate the stress behavior as a function of )(cmr . By comparing these figures, we see that the maximum of all components of the stress will be increased by increasing the pulse repetition frequency rates.

5. CONCLUSION In this paper after finding the temperature distribution in diode pumped Ti:sapphire laser, all the components of thermal induced stress have been investigated. These components plotted as a function of the crystal radius with various repetition rates. Results show that the radial component of the stress at the crystal surface is zero and the axial and azimuthally stress components are equal; and they have tensile actions. At the center of the crystal, the radial and azimuthally stress components are equal and these three components act compressively. We also observe that after finishing final pulse, stress curves are more un-steady than the next time. Finally, by applying pump source on Ti:sapphire laser, less pulse repetition frequency leads to better output quality and efficiency.

(a) (b)

Figure 1. The stress components at 0=t with repetition frequencies 50Hz (up curves of solid, dot and dash of vertical axis) and 100Hz (down curves of solid, dot and dash of vertical axis) in figure (a), and the same results are seen for )(st τ= in figure (b) too .

)(28.0 cmp =ω )(1 cml = )(356.0 cma =

)(335 GPAc = 64.0=η )(1 JE total =

334.0=υ )(105 16 −−×= kTα

)(33.0 11 −−= kwcmk

)(532nmpump=λ

)(800 nmlaser =λ

)(1.3 13 −−== kJcmcC ρ

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(a) (b)

Figure 2. The stress components at )(0 st = (down curves of solid, dot and dash of vertical axis) and )(st τ= (up curves of solid, dot and dash of vertical axis) with repetition frequency 50Hz in figure (a), and the same results are seen for the repetition frequency

100Hz in figure (b) too.

REFERENCES

[1] Koechner, W., [Solid-State Laser Engineering], 2nd ed.Berlin:Springer-Verlag(1988). [2] Nadgaran, H. and Sabaian, M., "Pulsed pump: Thermal effects in solid state lasers under Super-Gaussian pulses", PARAMANA J. of Phys., 67, 119-1126, (2006). [3] Lausten, R., and Balling, P., "Thermal lensing in pulsed laser amplifiers: an analytical model", J. Opt. Soc. Am., B20, 1479-1485 (2003). [4] Nadgaran, H., and Elahi, P., " The analytical investigation of the super-Gaussian pump source on the thermal, stress and thermo-optics properties of double-clad Yb:glass fiber lasers", PRAMANA J. of Phys., 65, 95 (2005). [5] Arfken, G., [Mathematical methods for physics], 3rd edition, Academic Press, (1988). [6] Schmid, M., Graf, Th. and weber, H. P., "Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating", J. Opt. Soc. Am., B17, (2000). [7] Nye, J.F., [Physical properties of Crystals], London: Oxford University, (1985). [8] Schulz, P. A. and Henion, S. R. "Liquid-Nitrogen-cold Ti:Al2O3 laser", IEEE J. Quantum Electron, 27,1093-1047, (1991). [9] Amaranath, G. and Buddhudu, S. "Nonlinearity and elastic properties of Cr(+):Al2O3 and Ti:Al2O3 laser crystals", Science Direct J., 91, 761-763, (1994).

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