Table 4. Gaussian beam radius( w t = w s) at different positions within the passive Q-switched Nd: YAG laser cavity for 4 cavity configurations.
Setup
Beam radius( lm) at concave mirror( w t ¼ w s)
Beam radius( lm) at input of Nd: YAG crystal( w t ¼ w s)
J. Eur. Opt. Society-Rapid Publ. 22, 7( 2026) 65
Beam radius( lm) at output of Nr: YAG crystal( w t ¼ w s)
Beam radius( lm) at input of Cr: YAG crystal( w t ¼ w s)
Beam radius( lm) at output of Cr: YAG crystal( w t ¼ w s)
Beam radius( lm) at plane mirror( w t ¼ w s)
1 |
393.507 |
356.762 |
323.768 |
299.479 |
298.713 |
290.401 |
1 |
275 |
2 |
453.231 |
393.273 |
352.561 |
308.199 |
307.073 |
288.298 |
1 |
345 |
3 |
518.145 |
428.932 |
379.942 |
314.681 |
313.095 |
276.427 |
1 |
405 |
4 |
617.639 |
479.745 |
418.224 |
320.422 |
318.122 |
250.594 |
1 |
465 |
w t w s
L total
( mm)
Fig. 6. Schematic of the concave – flat Q-switched Nd: YAG laser cavity incorporating a Cr 4 +: YAG saturable absorber and a fusedsilica Brewster plate.
By substituting the values from Table 3 into the round-trip beam transfer matrix and calculating the stability parameter( �1 < AþD < 1), the stability parameters for the four
2
cavities were found to be 0.089, �0.191, �0.431 and �0.671 respectively. As shown in Figure 5, all these values lie within the stable region( �1 < AþD < 1). Even with the
2
cavity length increased to 465 mm, the cavity remains fully stable.
Using equation( 11), the Gaussian beam radius was calculated at six positions inside the Q-switched cavity – namely, at the concave mirror, the input and output faces of the Nd: YAG crystal, the input and output faces of the Cr 4 + YAG saturable absorber, and the flat output mirror – for all four cavity configurations. The results are summarized in Table 4. For every cavity length and at every location, the sagittal and tangential beam radii are identical( w t = w s), demonstrating that the resonator is completely free of astigmatism and that the transverse intensity distribution is fully symmetric in both orthogonal planes. As the total cavity length increases from 275 mm to 465 mm, the beam radius at the concave mirror increases significantly from 393.507 lm to617.639lm, while the beam radius at the flat output mirror decreases from 290.401 lm to 250.594 lm. Within the Cr 4 +: YAG absorber, the beam radius at the output face grows from 298.713 lm to 318.122 lm, and at the input face from 299.479 lm to 320.422 lm. Inside the gain medium, the beam radius at the Nd: YAG output face increases from 323.768 lm to 418.224 lm, and at the Nd: YAG input face from 356.762 lm to479.745lm. These variations are fully consistent with the expected behavior of a stable concave – flat
resonator, where increasing the cavity length shifts the mode toward larger radii near the curved mirror and slightly smaller radii near the flat mirror.
Finally, the beam quality factor M 2 was calculated using the corresponding relation, and for all four configurations with significantly different cavity lengths( 275 – 465 mm), M 2 was found to be equal to 1. This result indicates the generation of a nearly ideal Gaussian beam, even in the presence of the saturable absorber and with large variations in cavity length.
As shown in Figure 6, in the concave-planar cavity configuration of a passive Q-switched Nd: YAG laser, an 8 mm thick Brewster plate made of fused silica with a refractive index of n = 1.458( atk = 1064 nm) is placed near a highly reflective concave mirror( R = 500 mm). The plate is oriented at the Brewster angle( 55.7 °) such that the p-polarized component( parallel to the incident plane) experiences virtually zero reflection loss at both surfaces and is completely transmitted, while the s-polarized component experiences partial reflection at each interface, resulting in a significantly higher round-trip loss. As a result, only the p-polarized mode oscillates and is strongly amplified, forcing the output beam to be linearly polarized with very high polarization purity(> 99.9 %). The use of a Brewster plate in this cavity produces a beam with high linear polarization, which is essential for efficient nonlinear frequency conversion processes such as second harmonic generation( SHG).
The resonator under study is a concave – flat cavity( Fig. 6) whose total physical length is approximately 363 mm. It consists of a high-reflectivity concave mirror