Table 1.
Nd:YAGParameters used in FEAModeling
Parameter  Value 

Fractional thermal load  0.241 
Thermal conductivity 
$K\left(T\right)=\frac{\frac{1.9\times {10}^{6}W}{\left(\mathrm{cm}\bullet K\right)}}{{\left(\mathrm{ln}\left({T\bullet 5.33K}^{1}\right)\right)}^{7.14}}\frac{\frac{331W}{\mathrm{cm}}}{T}\left[8\right]$

Specific heat at 300 K  5.9 × 10^{2} J/(kg K) [9] 
Thin film convection coefficient  1.8 × 10^{4} W/(m^{2} K) 
Table 2.
Coefficients from equation 1, resulting focal length f, and temperature at the rod’s boundary and center for Nd:YAG rod with radius of r_{rod} = 4.5 mm and length L_{rod} = 12 cm corresponding to the refractive index profiles shown in Fig. 3 for 378 W of absorbed pump power.
 C_{4}[m^{4}]  C^{2}[m^{2}]  n_{0}
 f[cm]  T(r_{rod}) [K]  T(0)[K] 

Case 1  1446  4.65  1.82255  92.5  294.5  300.4 
Case 2  24877  5.59  1.82255  83.0  294.5  301.0 
Case 4  52791  6.75  1.82256  73.7  294.5  301.9 
Case 3  119346  9.38  1.82259  58.8  294.5  303.4 
Table 3.
Coefficients from equation 1, resulting focal length f, and temperature at the rod’s boundary and center for Nd:YAG rod with radius of r_{rod} = 2.5 mm and length L_{rod} = 12 cm compared to extrapolated values for a rod with 4.5 mm radius for 378 W of absorbed pump power (compare to Table 2).
 C_{4}[m^{4}]  C^{2}[m^{2}]  n_{0}
 f[cm]  T(r_{rod}) [K]  T(0)[K] 

Case 1  18790  15  1.82256  27.5  295.7  301.6 
Case 2  266151  18  1.82257  24.7  295.7  302.2 
Case 4  560858  22  1.82258  21.9  295.8  303.2 
Case 3  1263272  31  1.82261  17.5  295.7  304.6 

$\ast {\left(\frac{5}{9}\right)}^{4}$

$\ast {\left(\frac{5}{9}\right)}^{2}$
 
$\ast {\left(\frac{9}{5}\right)}^{2}$
  

Case 1  1790  4.62   89.1   
Case 2  25353  5.56   80.0   
Case 4  53427  6.79   71.0   
Case 3  120339  9.56   56.7   
Table 4.
Results for Nd:YAGrod with a radius of r_{rod} = 2.5 mm and a length of 12 cm. The coefficients for the refractive index profiles, the focal length f and the temperatures at the rod’s boundary and center are given at three different cooling water temperatures for 378 W of absorbed pump power.
10°C  C_{4}[m^{4} ]  C_{2}[m^{2} ]  n_{0}
 f [cm]  T(r_{rod})[K]  T(0)[K] 

Case 1  21472  15  1.82240  28.4  285.7  291.5 
Case 2  262019  18  1.82241  25.5  285.7  292.1 
Case 4  548927  21  1.82242  22.5  285.8  293.1 
Case 3  1232954  30  1.82245  18.0  285.7  294.5 
20 °C 

Case 1  18790  15  1.82256  27.5  295.7  301.6 
Case 2  266151  18  1.82257  24.7  295.7  302.2 
Case 4  560858  22  1.82258  21.9  295.8  303.2 
Case 3  1263272  31  1.82261  17.5  295.7  304.6 
30 °C 

Case 1  16072  16  1.82272  26.8  305.7  311.7 
Case 2  269229  19  1.82273  24.1  305.7  312.3 
Case 4  570540  23  1.82275  21.3  305.8  313.3 
Case 3  1288501  31  1.82277  17.0  305.7  314.8 
Table 5.
Coefficients of the refractive index profiles of the core doped rod with r_{doped} = 1.5 mm core radius, r_{rod} = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 6 for 378 W of absorbed pump power.
 C_{4}[m^{4}]  C_{2}[m^{2}]  n_{0}
 f [cm] 

Case 1  148689  43.0  1.819554  9.75 
Case 2  2084599  51.7  1.819563  8.76 
Case 3  9881382  86.5  1.819602  6.22 
Table 6.
Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of r_{rod} = 1.5 mm, for 378 W of absorbed pump power.
 C_{4}[m^{4}]  C_{2}[m^{2}]  n_{0}
 f [cm] 

Case 1  148214  42.6  1.819487  9.82 
Case 2  2082286  51.2  1.819497  8.58 
Case 3  9905294  85.8  1.819535  5.68 
Table 7.
Coefficients of the refractive index profiles of the core doped rod with r_{doped} = 2 mm core radius r_{rod} = 2.5 mm outer radius, and a length of 12 cm as depicted in Fig. 7 for 378 W of absorbed pump power.
 C_{4}[m^{4}]  C_{2}[m^{2}]  n_{0}
 f [cm] 

Case 1  46993  24.0  1.819500  17.5 
Case 2  654599  28.8  1.819509  15.7 
Case 3  3102901  48.2  1.819548  11.2 
Table 8.
Coefficients of the refractive index profiles of a 12 cm long crystalline Nd:YAG rod with a radius of r_{rod} = 2 mm, for 378 W of absorbed pump power.
 C_{4}[m^{4}]  C_{2}[m^{2}]  n_{0}
 f [cm] 

Case 1  46429  23.9  1.819469  17.5 
Case 2  656509  28.7  1.819478  15.3 
Case 3  3124836  48.1  1.819517  10.1 
Table 9.
Calculated beam qualities for a collimated Gaussian beam with different beam radii passing through the core doped Nd:YAGrods with 1.5 mm and 2 mm core radius and 2.5 mm outer radius in comparison to a crystalline rod with 2.5 mm radius. All examples are calculated for the pump distribution case 1. M
^{2}
_{C4
} denotes the beam propagation factor for a propagation with 4^{th}order aberrations, M
^{2} is the beam quality for propagation without 4^{th}order aberration term C_{4} in the refractive index profiles shown in Fig. 6 and Fig. 7.
3 mm core 
M
^{2}
_{C4
}
 M^{2}


W_{0} = 1.0 mm  3.2  3.0 
W_{0} = 1.3 mm  7.5  7.0 
W_{0} = 1.7 mm  13  12 
4 mm core 
M
^{2}
_{C4
}
 M^{2}


W_{0} = 1.0 mm  1.1  1.0 
W_{0} = 1.3 mm  1.6  1.7 
W_{0} = 1.7 mm  3.6  4.1 
5 mm crystalline 
M
^{2}
_{C4
}

M
^{2}


W_{0} = 1.0 mm  1.0  1.0 
W_{0} = 1.3 mm  1.3  1.0 
W_{0} = 1.7 mm  2.2  1.6 