The most important influence that the temperature gradient inside the laser rod has on the mechanical properties of the material, is the stresses that are induced. When the induced stresses in the laser rod exceed the tensile strength of the material the rod will fracture and the laser will no longer operate. For this reason it is important to operate the laser so that the thermally induced stresses are well below the tensile strength of the material.
By making use of the analytical equations 2.30 Sz 2.31, the stresses on the pump face of the Nd:YAG rod can be determined (see Figure 3.6). As expected, the centre of the rod is under compression while the outer regions are in tension.
-1 -0.5 0 0.5 1 r [mm]
Figure 3.6: The radial (yellow line) and tangential (blue line) stresses on the pump face of the Nd:YAG rod with an 8.5 W incident top-hat pump beam. The green and red regions indicate compression and tension respectively.
With an 8.5 W incident top-hat pump beam, the maximum tensile stress of 16 MPa is found on the edge of the crystal. This value is well below the tensile strength of YAG which is 280 MPa (VLOC YAG Brochure, 2008). The maximum stress in the
laser rod has also been calculated numerically with ABAQUS and found to be within 5% of the analytical value.
The stress intensity distribution in the laser rod while it is pumped by a Gaussian beam has also been investigated numerically and the maximum tensile stress was found to be within 1% of the maximum tensile stress induced by a top-hat pump-beam of the same power.
Now that the stresses have been calculated analytically and verified numerically for an 8.5 W incident pump-beam, it is very important to determine how hard the YAG rod can be pumped before fracture will occur. This power limit can be calculated by making use of equation 2.34 and is referred to as the critical incident pump power.
Once the critical incident pump power is known, the laser can be operated below this limit to ensure that the crystal will not fracture.
The critical incident pump power has been calculated analytically and numerically for the Nd:YAG crystal under lasing as well as non-lasing conditions. While the laser is not lasing, the fractional heat load (rj) is taken as 0.4 (Fan, 1993) as opposed to the 0.27 (Didierjean et ai, 2003) under lasing conditions. Figure 3.7 shows the maximum stress intensity in the laser rod as a function of incident pump power. There is a very good agreement between analytical and numerical values for the maximum stress in the laser rod. When there is no laser beam to extract the pump energy from the rod, the critical incident pump power is ~100 W. When the laser is functional, the critical incident pump power increases to ~150 W since there is a laser beam that can extract energy from the laser rod so that less heat is generated. To prevent thermal fracture, the rod cannot be pumped with more than 150 W under lasing conditions.
A possible method that can be applied to increase the total power that can be deposited into the rod before fracture occurs, is to pump the rod from both sides.
Since the maximum tensile stress occurs on the pump face, there will be a very small contribution to the stresses on a particular rod face due to a pump beam that is pumping the opposite rod face. This will make it possible to deposit almost twice as much power inside the rod before fracture will occur.
It is clear from equation 2.34 that the the ratio of the rod radius to the pump beam radius is very important in determining the critical incident pump power. By changing the pump beam radius, many other important aspects of the laser's operation will be influenced, such as the laser threshold. For this discussion, only the effect that the pump beam radius has on the critical incident pump power will be considered. Figure 3.8 shows the critical incident pump power as a function of pump beam radius. As an example, when the pump beam radius is increased from 650 //m to 800 fim, the critical incident pump power will increase from 150 W to about 170 W. The numerical and analytical solutions agree very well for wp/R larger than 0.5. For wp/R smaller than 0.5, there is a deviation between the numerical and analytical solutions which increases as the the pump beam radius decreases. For wp << R, the analytical solution suggests a constant critical pump power while the numerical solution shows a decrease in the
600
0 50 100 150
Incident Pump Power [W]
200
Figure 3.7: The analytical maximum stress intensity (blue lines) in the Nd:YAG laser rod under lasing and non-lasing conditions. In each case the analytical solution is compared to numerical results (red squares: non-lasing and red dots: lasing). The horizontal red line indicates the tensile strength of Nd:YAG.
critical pump power. If the critical pump power was independent of pump beam size for small pump beam radii the pump beam intensity would go to infinity as the pump beam radius goes to zero.
Figure 3.9 shows the pump beam radius dependence of the maximum stress in the laser rod for an incident pump power of 100 W. As before, there is good correspondence between the analytical and numerical values for the maximum stress only for wp/R greater than 0.5. For wp << R, the analytical solution suggests that the maximum stress is independent of the pump beam size while the numerical solution shows an increase in the maximum stress as the pump beam size decreases (increase in pump intensity).
The analytical solutions (equations 2.33 k, 2.34) for the maximum stress and critical pump power of the laser rod are a very good approximation for wp/R greater than 0.5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w /R
P
Figure 3.8: The critical incident pump power as a function of pump beam radius. (Blue line: analytical and red dots: numerical)
350 300 a. 250 as
</)
g.
CO
I 150
£
| 100 50
0, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w /R
p
Figure 3.9: The maximum stress as a function of the pump beam radius for a 100 W incident pump beam. (Blue line: analytical and red dots: numerical)
3.4 S u m m a r y
The theory surrounding the thermal effects in solid-state lasers that have been discussed in the previous chapter were applied to a CW pumped Nd:YAG laser. The temperature distribution in the laser rod as well as the thermal lens and stresses that are induced due to the temperature gradient have been discussed for this particular laser. The considerations and suggestions is this chapter are purely based on the thermal effects.
Pump beams with Gaussian and top-hat transverse intensity profiles were considered.
As a verification of the results, analytical and numerical solutions were compared where possible.
The temperature gradient in the laser rod compared very well with experimental data. For a CW pumped laser, the transverse intensity profile of the pump beam seems to make no significant difference in the temperature gradient inside the laser rod.
The thermal lens in the laser rod and its dependence on the incident pump power and pump beam radius was investigated.
The analytical and numerical solutions for the maximum tensile stress in the laser rod are in very good agreement for wp/R greater than 0.5. For this reason the use of the analytical solutions to determine the maximum stress and critical pump power should be restricted to larger pump beam radii where wp/R is greater than 0.5. It is not clear why the analytical expression (equation 2.33) cannot accurately determine the maximum stress in the rod for wp/R < 0.5 but it might be due to the maximum stress that occurs elsewhere in the rod and not on the pump face.
An incident pump power of 8.5 W induced a maximum tensile stress of 16 MPa which is well below the tensile strength of YAG. For the pump beam radius of 650 /im, the critical incident pump power is estimated to be 150 W. This power limit can somewhat be increased by using a larger pump beam.
The analysis in this chapter shows how a good thermal model can act as a design tool to address thermally induced effects in end-pumped solid-state lasers.