The collimation adjustment of an unstable resonator requires guiding light. Examination of the oscillation process and propagation characteristics of guiding light assists in the understanding and judgment of the resonator misalignment state. Off-axis unstable resonators are hybrid unstable resonators with high extraction efficiencies and beam quality values. The physical process of guiding light oscillation in an off-axis unstable resonator has rarely been investigated. Therefore, this study investigates the multi-beam interference process and far-field propagation characteristics of guiding light in an off-axis unstable resonator using theoretical calculations and experiments.

A guiding light research device in an off-axis unstable resonator is constructed. The schematic of optical path of guiding light in the off-axis unstable resonator composed of four resonator mirrors is shown in Fig.1. The convex cylindrical mirror in the Y-direction and concave spherical mirror form a stable resonator in the X-direction and a positive-branch confocal unstable resonator in the Y-direction, respectively. Z-shaped folding in the X-direction is achieved by turning plane mirrors. The curvature radii of the concave spherical and convex cylindrical mirrors are 16 m and 14 m, respectively. The magnification in the direction of the unstable resonator is 1.14. The 632.8 nm He-Ne laser (guiding light) is injected through a small hole at the bottom of the concave spherical mirror, oscillates back and forth inside the resonator, and outputs off-axis in the Y-direction of the convex cylindrical mirror. The output guiding light is focused on the charge coupled device (CCD) target through a convex lens with 300 mm focal length to observe its far-field characteristics. In terms of numerical calculations, this study utilizes the diffraction theory of the plane wave angle spectrum to calculate the oscillation process of guiding light in a Z-shaped folded off-axis unstable resonator. In the calculations, the incident beam is a fundamental-mode Gaussian beam with a central wavelength of 632.8 nm. The number of samples in the calculation is 16384×16384. The flowchart of the optical field oscillation calculation is shown in Fig.2. The intracavity output loss, reflection loss and injected guiding light energy during the guiding light oscillation process reach a balanced condition within a limited number of oscillations. Throughout this process, the maximum number of guiding light oscillations is approximately 10.

The calculated and experimental results for the near-field spot and far-field spot of the oscillating guiding light in the concave spherical mirror are shown in Figs.3 and 4, respectively. The concave spherical mirror acts as a convex lens in the X-direction to repeatedly focus and diverge the Gaussian guiding beam. In the Y-direction of a positive-branch confocal unstable resonator, for each round-trip oscillation in the resonator, the waist radius of the Gaussian guiding beam in the Y-direction increases M times (M refers to the confocal unstable resonator magnification), its divergence angle is reduced to 1/M, and its curvature radius increases M² times. Therefore, during multiple oscillations, the Gaussian guiding beam tends to be a plane wave, eventually reaches above the convex cylindrical mirror and outputs. As shown in the calculation results in Fig.3(a), the interference pattern at the injection point of the small hole in the concave spherical mirror presents an elliptical water droplet shape, which differs from the circular interference patterns commonly observed in traditional positive/negative confocal unstable and stable resonators. The guiding light is output from an off-axis unstable resonator and passes through a focusing lens with 300 mm length to obtain a far-field spot pattern, as shown in Fig.3(b). Different order spot patterns are distributed along the Y-direction. As the number of oscillations increases, the divergence angle of the N-order Gaussian beam in the Y-direction in the resonator is reduced to 1/M^N (N refers to the number of oscillations). The higher the order of the Gaussian beam, the closer the beam is to a plane wave. When the beam is a plane wave, the (X, Y) coordinate corresponding to the far-field focal spot is (0,0). The highest-order guiding light far-field spot can be used to indicate the far-field spot of an infrared laser. It should be noted that the number of guiding light oscillations in the resonator is limited because of the output loss and resonator mirror reflectivity loss. The highest-order guiding light far-field spot positions as shown in the calculations and experiments can only approximate infrared laser far-field spot positions.

The calculation and experimental results indicate that the interference pattern of an off-axis unstable resonator (with a flat concave stable resonator in the X-direction) differs from that of traditional confocal unstable or stable resonators. The guiding light in the resonator exhibits the interference pattern with an elliptical water droplet shape at the injection point of the small hole in the resonator mirror, which has strong brightness and can be used to determine the optical resonator collimation state. The far-field spot pattern of the output guiding light exhibits a series of bright spots along the Y-direction, and the higher-order bright spot can be used to indicate the far-field spot position of the infrared laser. Thus, the results of this study provide a reference for understanding the physical process of Gaussian beam oscillation in an optical resonator and for determining the collimation state of an off-axis unstable resonator.