The Hermite-Gauss eigensolution of confocal or spherical cavities significantly affects lasers, and the transmission laws of these cavities have been extensively investigated. When multiple transverse-mode oscillations are present, lasers are partially spatially coherent. In 1969, Arnuaud proposed a range of degenerate cavities that can yield a low-coherence laser output. In 2013, Nixon et al. experimentally demonstrated that a 4F degenerate cavity can generate low-coherence lasers, which can effectively reduce the scattering contrast while simultaneously controlling the diaphragm size in the intracavity spectrum to achieve a spatially coherent optical-field output of the 4F degenerate cavity. In 2015, Chriki et al. demonstrated that altering the geometry of the diaphragm in the spectral plane of the cavity can alter the shape of the spatial-coherence function of the optical field output. These studies showed that a 4F degenerate cavity can generate low-spatial-coherence lasers with adjustable spatial coherence to achieve scatter suppression during imaging. Most studies analyzed the 4F degenerate cavity’s output field using a multitransverse mode laser. This approximation explains the spatial properties of the 4F degenerate cavity’s output field. However, some conflicts exist, such as the shape of the beam output from the 4F degenerate cavity changing during transmission, whereas the shape of the Hermite-Gauss beam remains constant during linear transmission. Therefore, the transverse light-field distribution in a 4F degenerate cavity must be re-analyzed.

A 4F degenerate cavity laser is constructed with circular diaphragms of varying sizes on its spectral plane. The near and far fields are measured in different cases using a charge-coupled device (CCD) and spatial coherence distributions are measured at various locations within the optical field. This study analyze the transverse and longitudinal modes in a 4F degenerate cavity using the Fresnel-Kirchhoff diffraction equation. Finally, the light intensity and coherence distribution of the 4F degenerate cavity’s light field are compared with those of the Schell model.

The near- and far-field light intensity distributions of the output light field of the 4F degenerate laser cavity in the three cases are shown in Figs. 2(a)?(f). Near-field distributions were compared when diaphragms of different sizes were inserted into the spectral plane. The edges of the near field become clearer as the small hole in the spectral plane enlarges, thus resulting in a more homogeneous overall light intensity. However, a strong spot appears at the center of the light spot owing to residual reflection. When a diaphragm with a small diameter is present in the spectrum, the shape of the far-field light matches that of the diaphragm. When no diaphragm is present in the spectral plane of the cavity, the near-field light intensity of the 4F degenerate cavity’s light field shows an approximately flat-top distribution [Fig. 2(g)], whereas the far-field light intensity shows a conical distribution, as shown in Fig. 2(h). The shape of the optical field changes significantly, thus demonstrating that the Hermite-Gauss mode is not an inherent mode of the 4F degenerate cavity. In the ideal case, when no diaphragm is present in the cavity, the resonance of the 4F degenerate cavity (Fig. 3) is formed between point A on M₁ and point A' on M₂. No correlation is indicated among the different points. When a diaphragm is inserted into the spectral plane of the cavity, the electric field at point A' is altered. Meanwhile, the shape and width of the electric field change with the diaphragm shape. This results in a certain shape and width of the electric field at point A', which is noncoherently superimposed on the light field of the surrounding points. Consequently, spatial coherence is created in the light field. The spatial-coherence function of the partially coherent light in the Schell model is shown in Fig. 5. This is consistent with the experimental results, where the spatial coherence initially decreases to zero, followed by a discernible difference between the two.

In this study, the transverse and longitudinal modes of a 4F degenerate cavity were investigated using the Fresnel-Kirchhoff diffraction formula and the formation of laser coherence. The results indicate that the intrinsic mode of the 4F degenerate cavity is not the Hermite-Gauss mode. Additionally, the intracavity spectral plane diaphragm affects not only the spatial coherence of the 4F degenerate cavity’s output light field but also the width of its longitudinal mode. The near- and far-field shapes of the output optical field differ significantly in the absence of an intracavity spectral-plane diaphragm. This inconsistency with the transmission law of the Hermite-Gauss mode confirms the theoretical analysis of the transverse mode of the 4F degenerate cavity. The spatial-coherence distribution was measured after circular diaphragms of varying sizes were inserted into the spectral plane of the cavity. The results show that the spatial coherence of the output optical field decreases as the size of the spectral-plane diaphragm increases. The spatial coherence of the output light field of the 4F degenerate cavity was compared with the partially coherent light of the Schell model. The results show that the output light field of the 4F degenerate cavity with a circular diaphragm inserted in the spectral plane differs slightly from that of the Schell model but can be approximated using the Schell model.