62
J. Eur. Opt. Society-Rapid Publ. 22, 7( 2026)
The oscillation of the ray in the cavity is stable when the Sylvester angle, defined by the relation:
cosh ¼ A þ D: ð3Þ 2
– remains real( i. e., lies between �1 and + 1). Consequently, the classical stability condition for the optical resonator takes the following form:
�1 < A þ D < 1: ð4Þ 2
Although ray tracing provides an excellent geometric description of light propagation in optical systems, the transverse field distribution in laser cavities can only be accurately modeled using Gaussian modes. The intensity profile of the fundamental Gaussian mode( TEM) is expressed in radial coordinates as [ 7 ]:
I ðÞ¼I r 0 exp � 2r2 w 2 ðzÞ
1 2
: ð5Þ
Where w( z) is the beam radius( spot size) at axial position z, and I 0 is the peak intensity at the beam axis. A complete description of Gaussian beam evolution along the cavity is provided by the complex beam parameter q( z), defined as
qz ðÞ¼zþizR: ð6Þ
Or equivalently in its reciprocal form: 1 qðzÞ ¼ 1
RðzÞ � i k pw 2 ðzÞ:
Where R( z) is the radius of curvature of the wavefront at position z, z R ¼ pw2 0 is the Rayleigh range, w
k 0 is the beam waist radius, and k is the wavelength of the light. When a Gaussian beam passes through an optical element or system characterized by an ABCD ray-transfer matrix, its complex beam parameter transforms according to relation:
q 2 ¼ Aq 1 þ B Cq 1 þ D; ð8Þ
here q 1 and q 2 are the complex parameters before and after the element, respectively. This transformation rule – identical in form to the ray-transfer equation – is the cornerstone of Gaussian beam propagation through arbitrary paraxial optical systems. In a stable resonator, the Gaussian beam must exactly reproduce itself after each complete round trip. This self-consistency( or self-reproduction) condition leads to the following quadratic equation for the complex parameter q:
Cq 2 þ ðD � AÞ�B ¼ 0: ð9Þ ð7Þ
Solving this equation yields two fundamental parameters of the resonator mode:
the wavefront radius of curvature:
R ¼
2B D � A: ð10Þ
the spot size: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk jBj w ¼ t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p �: ð11Þ 1 � AþD 2
2
These relations directly connect the wavefront curvature radius and the laser mode spot size to the geometric design parameters of the resonator, thereby playing a fundamental role in determining the spatial characteristics of the intracavity field. The far-field divergence angle of the beam is given by h ¼ k; ð12Þ pw 0
where w 0 is the beam waist( minimum spot) radius. Furthermore, the deviation of a real beam from an ideal diffraction-limited Gaussian beam is quantified by the beam quality factor( or beam propagation factor) M 2, defined as [ 7 ]:
M 2 ¼ pw 0h k: ð13Þ
For a more accurate analysis of laser beams, the mode behavior is examined separately in the tangential( x – z) and sagittal( y – z) planes. These two planes – one parallel to the optical table and the other perpendicular to it – generally support different modes due to birefringence and astigmatic effects introduced by non-rotationally symmetric elements, such as Brewster-cut crystals. The presence of such a crystal inside the cavity induces astigmatism, meaning that the mode size, wavefront curvature, and even the stability condition differ between the two planes. The ABCD matrix formalism enables independent analysis of each plane, allowing the laser modes in the tangential and sagittal directions to be determined separately. In many solid-state lasers, the tangential plane experiences greater beam divergence because of the larger effective refractive index seen by the p-polarized light at Brewster incidence. As a result, the spot size in the tangential plane is typically larger than in the sagittal plane. Employing the foregoing ABCD-matrix formalism and Gaussian-mode self-consistency relations, we now investigate the resonator of a concave – flat Nd: YAG laser under two distinct operating regimes: freerunning operation and Q-switched operation [ 7 ].
A) Cavity analysis in free-running operation
In this section, the stability of the Nd: YAG laser cavity in free-running mode is investigated. In this mode, the laser continues to emit radiation as long as the optical pump provides sufficient energy to maintain population inversion. Figure 2 shows the schematic diagram of the resonator configuration. The cavity consists of a concave high-reflectivity mirror( radius of curvature 500 mm, R 99.9 % at 1064 nm), a plane output coupler with 40 % transmission at 1064 nm and a Nd: YAG rod of 100 mm length( refractive index n = 1.82). In the schematic, L 1 and L 2 represent the free-space optical path lengths between the intracavity components, while d denotes the physical length of the Nd: YAG rod( d = 100 mm). The total cavity length L total