In recent years, a variety of coherent beam smoothing technologies, including temporal and spatial smoothing, have been developed to control the spatiotemporal characteristics of focal spots, which have helped reduce the Rayleigh-Taylor instability generated by laser plasma in the process of inertial confinement fusion. Spatial smoothing technologies, such as continuous phase plates (CPP) and lens arrays (LA), along with temporal smoothing technologies, such as induced spatial incoherence (ISI) technology and smoothing by spectral dispersion (SSD) have been widely used in large scientific devices. Considering coherent beam-smoothing technologies, we explored the feasibility of partially coherent beam smoothing. The high fill factor of a multi-transverse mode-flattened Gaussian partially coherent beam (MPCB) is conducive to energy storage extraction from the laser medium and often used in high-power laser systems. Thus, the technical route of MPCB smoothing is of great significance, as discussed in this paper, in analyzing the relationship between the integration time and focal spot of MPCB passing through a CPP designed for coherent beam smoothing. The relationship between the spatial coherence of MPCB and the smoothing effect and profile of the focal spot are also reported.

In this study, a one-dimensional MPCB is constructed through an incoherent superposition of Hermite?Gaussian (HG) modes. This is easily extendable to a two-dimensional MPCB, using an analytical algorithm that recovers the modal weights (Fig. 1). The spatial coherence of the MPCB at any position in space can be evaluated by the modal weights and is determined by the light field order N, representing the number of HG modes involved in the superposition. The larger the N, the lower the spatial coherence of MPCB. Two different spatial coherences of MPCBs are established, with N values of 22 and 46. The CPP designed for coherent beam smoothing is calculated using the Gerchberg-Saxton algorithm. The differences in the focal spots of the coherent beam and MPCBs passing through the CPP are compared (Fig. 2). The parameters η and R₉₀ show the profile of focal spots, whereas FOPAI5 and δ_RMS indicate the smoothing effect of focal spots. Additionally, the power spectral density (PSD) is also calculated to verify that the mid-frequency of focal spots (10?100 μm) is suppressed.

Figure 2 clearly demonstrates that focal spot distributions depend on the relationship between integration time t and coherence time t_c. HG modes interfere coherently to generate a speckle pattern on a timescale of t<t_c, whereas HG modes are mostly uncorrelated with one another and interfere incoherently to generate a relatively more uniform pattern on a timescale of t>t_c. The spatial coherences of MPCBs and CPP are shown in Figs. 3?4, respectively. The focal spots of the coherent beam and MPCBs share a similar profile [Fig. 5(a)]. The parameter η of each focal spot is a little different and over 98%. Similarly, the parameter R₉₀ of a focal spot is hardly distinguishable from that of another and does not exceed 254.4 μm. In detail. the focal spots of MPCBs feature a larger contour profile. When N is 22 and 46, the δ_RMS parameters of the focal spot of MPCB were 5.26% and 4.75%, respectively, with a 40 μm filter, whereas that of the coherent beam was 5.75% with the same filter, which suggests that focal spots of MPCBs are more uniform than that of coherent beam. The higher the order N of MPCB, the lower the spatial coherence of MPCB, and the smaller the δ_RMS, the more uniform the focal spot [Fig. 5(b)]. However, FOPAI5 implies nonuniformity of the focal spots of the MPCBs on a timescale of t<t_c. Correspondingly, the PSD revealed that the midfrequency of the focal spots of the MPCBs is not well suppressed on a timescale of t<t_c [Fig. 5(c)].

In this paper, we propose a technical approach for MPCB smoothing using CPP, explaining the principle based on the characteristics of MPCB. The focal spot of the MPCB and that of a coherent beam passing through the same CPP were compared. The results show that on a timescale of t>t_c, the focal spot of the MPCB can be reshaped by the CPP and basically coincides with that of the coherent beam. The spatial coherence of the MPCB had little impact on the focal spot profile. The smoothing effect of the focal spot of the MPCB was always better than that of the coherent beam with the same filter. The lower the spatial coherence of the MPCB, the more uniform the focal spot. For N over 46, the focal spot of MPCB is expected to achieve δ_RMS<4%. On a timescale of t<t_c, the problems of FOPAI5 and PSD of the MPCB focal spot can be solved by shortening the coherence time and increasing the beam bandwidth. The focal spot of the MPCB would be much more uniform if MPCB smoothing were combined with other beam-smoothing technologies.