For a multi-transverse mode-stabilized cavity, the transverse modes can be written as an incoherent superposition of Hermite?Gaussian (HG) beams. The modal weights, λmn, can completely characterize the light field of partially coherent beams, and the CSD (cross spectral density) of the beam at any position in space can be evaluated using the modal weights as well as any global beam parameter such as beam quantity factor M2. The problem involves determining the weights of the underlying modes based on experimentally determined quantities. Various methods such as coherence measurements and best-fitting procedures have been proposed to address this problem. In many applications, such as material processing and speckle reduction, uniformity of the laser power distribution on the beam cross and low spatial coherence are desired qualities, and the model of circular flat-topped intensity distributions for partially coherent beams (FTIPCBs) is useful. Determining the modal weights of a circular FTIPCB in a simple manner and exploring the characteristics of the circular FTIPCB are important.

In this study, an analytical algorithm for recovering modal weights was extended from one-dimensional flat-topped intensity distribution beams to a circular FTIPCB. The beam quality factor M2 of the circular FTIPCB was evaluated in two-dimensions using model weights to investigate the relationship between beam quality factor M2 and light field order N. The coherence function of uniformly correlated partially coherent beams with the same near- and far-field intensity distributions as those of the circular FTIPCB was calculated based on the transmission function of the partially coherent beams. The coherence function of the circular FTIPCB was then compared with that of the uniformly correlated partially coherent beams, leading to the following conclusion.

We provided an equation for circular flat-topped intensity distributions and cross-sectional one-dimensional light intensity curves (Fig. 1). A numerical deduction process was presented for the modal weights of the circular FTIPCB. Considering N=16 and w0=4 mm as an example, normalized modal weights λmn were provided (Fig. 2). The near-field light intensity distribution, INF, and far-field light intensity distribution, IFF, were calculated using the mode weights shown in Fig. 2 and the Collins formula (Fig. 3). Beam quality factor in x direction, Mx2, was calculated for various values of light field order N, and the data were fitted to determine the relationship between Mx2 and N in the circular FTIPCB. The coherence functions of the circular FTIPCB were calculated using three starting points at different positions (Fig. 4). We constructed a uniformly correlated partially coherent beam with the same near- and far-field intensity distributions as those of the circular FTIPCB. The coherence functions of the uniformly correlated partially coherent beams were calculated based on the transmission functions of the partially coherent beams (Fig. 4). Compared with the coherence function of the circular FTIPCB, the results indicate that the spatial coherence length of the uniformly correlated partially coherent beams is longer than that of the round FTIPCB.

In this study, we determine the modal weights of a circular FTIPCB and evaluated various properties of the circular FTIPCB based on the modal weights. For the circular FTIPCB, the intermediate modes have a larger weight, and the lower- and higher-order modes have smaller weights; this is considerably different from the mode weight distribution of the one-dimensional FTIPCB. The relationship between Mx2 and N in the circular FTIPCB differs significantly from that in a one-dimensional FTIPCB. The shape of the light intensity distribution and the order, N, determine the mode weights that, in turn, determine beam quality factor M2; therefore, the relationship between beam quality factor M2 and N changes as the overall shape of the light intensity distribution changes. The circular FTIPCB maintains a flat-topped distribution after transmission to the far field, maintaining the same shape as that of the near field. The coherence function of the circular FTIPCB changes when the position changes, revealing that the circular FTIPCB is nonuniformly correlated. Based on equation (13), notably, coherence function μ is determined by mutual intensity Jr1,r2 and the light intensity. In the top region where the light intensity is constant, μ=Jr1,r2. Jr1,r2 is determined by both ?mnr and λmn. As ?mnr is a function of the position, Jr1,r2 is also a function of the position. However, in special cases such as the Gaussian Schell model, special mode weights λmn cause the coherence function to become uncorrelated with position. Because of the transmission properties of Hermite?Gaussian beams, when the circular FTIPCB is linearly transmitted to any position, only the beam width changes, leading to changes in the spatial coherence length, and the shape and mode weights of the beam do not change; therefore, the spatial coherence of the beam must be characterized by beam quality factor M2. A clear relationship should exist between the spatial coherence length of the circular FTIPCB and the beam half-width and the beam quality factor M2. Notably, the simulation results indicate that the round FTIPCB has a shorter spatial coherence length than the uniformly correlated partially coherent beams with the same near- and far-field intensities.