J. Eur. Opt. Society-Rapid Publ. 22, 7( 2026) 63
Fig. 2. Schematic arrangement of Nd: YAG laser cavity in free running mode.
is the sum of all these segments and determines the longitudinal geometry and stability characteristics of the resonator.
To accurately analyze the cavity stability and beam characteristics, the distances L 1( between the concave mirror and the beginning of the crystal rod) and L 2( between the end of the crystal rod and the flat output mirror) were considered as the main variable parameters. According to Table 1, five different configurations with total cavity lengths of 312, 322, 332, 342 and 352 mm were designed and investigated.
The round-trip transfer matrix for the beam passing through each optical element within the cavity( free spaces L 1 and L 2, Nd: YAG rod and two mirrors) is given as follows:
M ¼ 1 0 1 L1 1
d n
1 L2 0 1 0 1 0 1 0 1
1 0
� 2 1 R
1 L2
0 1
1 d n
0 1
1 L1
: ð14Þ
0 1
By substituting the values from Table 1 into the above relationship and using the well-known stability criterion, the stability parameter( �1 < AþD < 1) for cavity lengths of 2
312 to 352 mm was obtained as �0.068, �0.108, �0.148, �0.188 and �0.228 respectively. As shown in Figure 3, all these values lie within the stable region, which confirms that all the designed cavities are fully stable.
Subsequently, using equation( 11), the radius of the Gaussian beam at four points inside the cavity was calculated: on the concave mirror, at the entrance face of the Nd: YAG rod, at the exit face of the Nd: YAG rod, and on the flat exit mirror. As shown in Table 2, the results show that the beam radius in the sagittal and tangential planes is exactly the same at all four points for all five configurations. Therefore, the cavity shows no astigmatism and the intensity distribution in the two perpendicular planes is completely symmetrical and identical. According to Table 2, as the total length of the cavity increases from 312 mm to 352 mm, the beam radius on the concave mirror gradually increases from approximately 425.718 lm to462.089lm, while the beam radius on the flat mirror decreases from approximately 290.647 lm to 287.132 lm. This behavior
Table 1. Longitudinal parameters for five distinct configurations of laser cavity in free running mode.
Setup |
L 1 = L 2( mm) |
L total( mm) |
1 |
106 |
312 |
2 |
111 |
322 |
3 |
116 |
332 |
4 |
121 |
342 |
5 |
126 |
352 |
Fig. 3. Resonator stability parameter as a function of cavity length.
is fully consistent with the expected properties of stable concave-flat resonator cavities.
Finally, the beam quality factor M 2 was calculated and for all five configurations with different cavity lengths, the value M 2 = 1 was obtained. This result shows that the output beam is perfectly Gaussian in all cases and has no significant deviation from the ideal TEM. The fact that M 2 remains constant at 1 despite a ± 20 mm variation over the entire cavity length indicates an optimal resonator design.