JEOS RP ISSN03 | Page 73

66

J. Eur. Opt. Society-Rapid Publ. 22, 7( 2026)

with radius of curvature R = 500 mm, followed by 65 mm of free space( L 4 = 65 mm) to an 8.00 mm thick fused-silica Brewster plate( n = 1.458 at 1064 nm) oriented at Brewster’ s angle 55.6 °. This plate presents an effective optical path of 11.664 mm in the tangential plane and 8.00 mm in the sagittal plane. After the plate there is 65 mm of free space( L 3 = 65 mm) to the front face of a 100 mm long Nd: YAG rod( n 1.82)( d 1 = 100 mm), then 60 mm of free space( L 2 = 60mm) toa5mmthickCr 4 +: YAG saturable absorber( initial transmission T = 40 %, n 1.82))( d 2 = 100 mm) and finally 60 mm of free space( L 1 = 60 mm) to the flat output coupler with 60 % reflectivity( 40 % transmission) at 1064 nm. The cavity consists of three main intracavity optical elements( Brewster plate, Nd: YAG rod, Cr 4 +: YAG crystal) and four free-space propagation regions between the mirrors and these elements. Because the Brewster plate is tilted, its effective optical path and focusing properties differ in the tangential plane( plane of incidence) and the sagittal plane( perpendicular to the plane of incidence). Therefore, the ray-transfer( ABCD) matrix must be evaluated separately for the two planes. The total round-trip matrix( tangential and sagittal plane) is constructed as follows:

M T ¼ 1 0 0 1

1 d2 n

0 1

1 L1 1 d1

n

1 L2 0 1 0 1 0 1

" pffiffiffiffiffiffiffi #

1 L: n 2 þ1

n 4 1 L4

0 1

1 L3

0 1

1 0

� 2 1 L4 1

R

0 1

1

d2 n

0 1

1 L2

0 1 n 2 þ1 n 4

0 1

1

d1 n

0 1

1 L1

0 1

1 L3

0 1

1

d2 n

0 1

1 L2

0 1

1 d1 n

0 1

1 L1

0 1 ð16Þ

ð17Þ

Therefore, using�1 < AþD

2

< 1, the cavity stability parameters in the two principal planes are calculated as follows:

In the tangential plane: �0.243 In the sagittal plane: �0.257

Since both values satisfy the condition�1 < AþD

2

< 1, the resonator lies well within the stable region in the tangential as well as the sagittal plane.

Using the equation( 11) along the previously calculated ABCD matrices for both planes, the evolution of the beam radius inside the cavity is obtained as follows. At the surface of the concave HR mirror the beam is essentially circular: w t = 465.349lm, w s = 465.861lm( w t / w s = 0.998). Immediately before the Brewster plate a pronounced astigmatism appears, with w t = 593.748 lm and w s = 409.267 lm( w t / w s = 1.451), which is directly attributable to the different effective optical paths of the tilted plate in the two principal planes. After passing through the Brewster plate, the spot sizes in the tangential and sagittal planes rapidly converge again owing to the compensating action of the subsequent elements and free-space propagation. At the entrance face of the Nd: YAG rod the beam has already recovered circularity: w t = 355.265lm, w s = 355.065lm( w t / w s = 1.001). At the exit face of the Nd: YAG rod the radii are w t = 321.048 lm andw s = 320.69lm( w t / w s = 1.001). At the entrance face of the Cr 4 +: YAG saturable absorber the values are w t = 295.994 lm and w s = 295.507 lm( w t / w s = 1.002), and at its exit face w t = 295.288 lm andw s = 294. Media 294.798 lm( w t / w s = 1.002). Finally, at the flat output coupler the beam radius is w t = 286.578 lm in the tangential plane and w s = 286.039 lm inthesagittalplane( w t / w s = 1.002). These results confirm that, except for the localized elliptical distortion at the Brewster plate itself, the laser mode remains nearly perfectly circular throughout the entire cavity, with the astigmatism introduced by the Brewster element being very effectively compensated by the rod and the remaining propagation distances.

The beam quality factor M 2, calculated over the entire cavity using the ABCD formalism for a Gaussian mode, remains constant at 1. This confirms that the resonator operates in a pure diffraction-limited TEM mode and that no beam quality degradation is observed despite the presence of the Brewster tilt and the resulting temporary astigmatism.

Intensity-dependent loss of the Cr 4 +: YAG saturable absorber.

The intensity-dependent single-pass absorption coefficient of the Cr 4 +: YAG crystal is described by the standard two-level saturable absorber model [ 1, 2, 7 ]:

aðI

Þ ¼ a 0 1 þ I

I sat

: ð18Þ

Where I is the instantaneous intracavity intensity, a is the small-signal absorption coefficient, and I sat is the saturation intensity. For the 5 mm thick crystal with measured initial transmission T 0 = 40 %, the small-signal absorption coefficient is calculated as:

a 0 ¼� lnðTÞ L

: 40Þ

¼�lnð0 0:005 m ¼ 1: 83 cm�1: ð19Þ

The saturation intensity is determined by the ground-state absorption cross-section r gsa and the excited-state lifetime s a of Cr 4 + ions:

I sat ¼ ht: ð20Þ r gas s a