Objective Structured light has been exploited in several fields in the past two decades, such as optical tweezers, spiral interferometers, and phase-contrast microscopes. The generation of structured light is generally achieved by utilizing optical elements such as spatial light modulators or spiral phase plates. Because of the limit on output power when utilizing these optical elements to generate structured light, a direct generation of structured light is proposed in the laser cavity. Intracavity generation of structured light could be achieved based on the theory of transverse mode degeneracy. The degenerate range (or lock range), which means a special short range of cavity length, is an important condition for exploiting transverse mode degeneracy to generate structured light. When the cavity length is adjusted to the degenerate range, the degeneracy condition is met. When the cavity meets the degeneracy condition, resonant frequencies of specific transverse modes become equal. In this situation, these specific modes will coherently superpose each other and generate a light field that may exhibit spatial structure. Therefore, the accurate degenerate range plays an important role in utilizing this method to generate structured light. Thus, to the approach of measuring the accurate degenerate range becomes a significant issue, which is what we mainly consider in this research.

Methods The degenerate range is determined by the variation of output power or threshold of pump power in some articles. However, the widths of the degenerate range determined using these methods are somewhat approximate; their precision could be improved using other measurement methods. To more accurately measure the degenerate range, we try to exploit the frequency spectra of laser beams to characterize the degenerate range in this paper. As the cavity length is tuned slowly in the experiment, the frequency spectra of laser beams are detected in detail. Two peaks in the spectra that particularly considered in this study to show an obvious process of merging as the confocal resonator approaches the degenerate position. In this paper, the range of cavity length in which the two peaks become superposed is defined as the degenerate range. This type of measurement method is more accurate than the methods that utilize the variation of output power or threshold of pump power to determine the degenerate range because it directly monitors the variation of frequency spectra. We use the frequency spectra measured in experiment to analyze the change of dynamic behavior of transverse modes with different orders.

Results and Discussions The frequency data measured in the experiment are shown in Fig. 3. As the cavity length changes, each of these frequencies (Δf_L,2Δf_T,Δf_L-Δf_T,Δf_T, and Δf_L-2Δf_T) show a variation tendency consistent with the theoretical variation tendency. Variations of these detected frequencies indicate the change of dynamic behavior of transverse modes with different orders. By analyzing these frequency spectra, we find that even though competition between modes exists, transverse modes that belong to different degenerate families are not suppressed when degeneracy occurs. The frequency difference of modes with the same order is clearly detected in the vicinity of the degeneration point, and it is not observed away from the degeneration point. From the beam patterns (Fig. 4), we can observe that higher-order transverse modes are generated when the confocal resonator is tuned in the vicinity of the degeneration point. This phenomenon implies that as the resonant frequency of the high-order mode becomes equal to that of the low-order mode, the high-order mode becomes easier to generate in the cavity. The detailed frequency spectra are presented in Fig. 5, using which we determine the degenerate range. As the confocal resonator is tuned to the degeneration point, the two peaks (Δf_L-Δf_T and Δf_T) become closer to each other and finally superpose. The state in which these two peaks are superposed sustains for a short range of cavity lengths. Accordingly, we could obtain the width of the degenerate range, which is approximately 0.20 mm in this experiment. The degenerate range defined in this way ought to be more accurate because it is directly determined from frequency spectra.

Conclusions In this paper, the phenomenon of transverse mode degeneracy is researched based on the variation of frequency spectra, and a more accurate method for measuring the degenerate range (or lock range) is proposed. As the cavity length varies in the experiment, frequency spectra of laser beams are recorded in detail. The variation of dynamic behavior of transverse modes before and after the degeneracy is analyzed based on the frequency spectra. When the cavity satisfies the degeneracy condition, the two degenerate families of transverse modes (in 1/2 degeneracy) coexist in the cavity, neither of them is suppressed because of the competition between them. Meanwhile, the beam patterns measured in the experiment show that higher-order transverse modes are generated when degeneracy occurs. As the resonant frequency of the high-order transverse mode becomes degenerate with that of the low-order mode, the high-order mode becomes easier to generate, thus resonating with the low-order mode in the cavity. We achieve a more accurate measurement of the degenerate range with the frequency spectra measured in the experiment. The degenerate range is approximately 0.20 mm in this experiment. It exhibits a higher precision and is more accurate than the degenerate range determined using other methods. The results of this paper may provide some reference value for the application of transverse mode degeneracy.