Inertial confinement fusion (ICF) requires uniform incident light distribution, and the continuous phase plate (CPP) modulates the incident wavefront distribution by employing the phase fluctuation on the surface, which can realize the shaping and smoothing of focal spots and improve target light uniformity. However, the actual processing ability and detection accuracy of CPP are limited by the surface gradient, which is related to the shaping and smoothing requirements of the far-field focal spot. The researchers studied the relationship between the surface gradient and the far-field focal spot size by geometric optics, but it is necessary to control the size and energy distribution of the focal spot with the requirement improvement in laser focal spot uniformity and energy distribution in high energy density physical experiments. The different focal spot distributions further increase the complexity of the surface gradient distribution and reduce the processing and detection accuracy of the components. Therefore, it is necessary to study and analyze the influence of focal spot size and energy distribution on gradient distribution, and to obtain the surface shape distribution characteristics to predict the processing performance of CPP.

Based on the theory of the near-field and far-field transmission relationship of the beams, we derive the relationship between surface gradient distribution and far-field focal spot energy distribution, and statistically analyze the relationship between the focal spot energy distribution and gradient distribution. Firstly, the focal spots are divided into three types according to the distribution of different target types and the relationship between the light field distribution of the back focal plane and the target plane, including circular focal spots, elliptical focal spots, and eccentric focal spots, with the G-S (Gauss-Seidel) iterative algorithm adopted to design CPP. Secondly, the characteristic parameters of the focal spots, such as elliptic eccentricity and asymmetry, are proposed to characterize the characteristics of the focal spots, and the influence of the gradient characteristics is evaluated by the characteristic parameters. Then we analyze the distribution relationship between the energy distribution of the three types of focal spots and the corresponding surface gradients by histogram statistical and quantitative analysis. Finally, the relationship between random wavefront distribution and far-field energy distribution is employed to verify the universality of the relationship, and the distribution characteristics of surface periods and amplitude are analyzed.

The results show that the histogram of surface depth gradient distribution is consistent with that of light intensity proportion under different radii of focal spots, and the radius of focal spots determines the gradient distribution range. Meanwhile, the energy distribution of focal spots determines the proportion of gradient distribution, and the theoretical analysis is in accordance with the numerical simulation results. The one-dimensional gradient of circular or elliptical focal spots with uniform energy distribution is normally distributed, the mean gradient is close to zero (Fig. 3), and the gradient variance positively correlates with the focal spot radius (Fig. 4). The gradient value corresponding to the peak CPP two-dimensional gradient of the elliptical focal spot decreases with the increasing eccentricity (Fig. 6), and the gradient corresponding to the peak gradient proportion of CPP with eccentric focal spots rises (Fig. 7). Additionally, the relationship between the random wavefront gradient distribution and its far-field focal spot distribution (Fig. 9) is verified, and the minimum spatial period of CPP positively correlates with the surface depth when the target focal spot is determined (Table 2).

The relationship between the surface gradient distribution and far-field focal spot energy distribution is obtained based on the light field transmission theory, and the histogram of the surface depth gradient distribution is consistent with that of the light intensity under different radii of the focal spot. Meanwhile, the radius of the focal spot determines the gradient distribution range, and the energy distribution of the focal spot determines the proportion of the gradient distribution, with the theoretical analysis consistent with the numerical simulation results. The mean gradient is close to zero and the gradient variance positively correlates with the focal spot radius when the energy is uniformly distributed. The smaller elliptical eccentricity of the focal spot leads to a greater proportion of the large gradient value. Eccentric focal spots also increase the proportion of large gradient values. Additionally, the random wavefront is adopted to verify the generality. In conclusion, the relationship between the surface gradient distribution and far-field focal spot energy distribution is determined, and the research results can help estimate the surface distribution and machining error of CPP, thus providing a new idea for wavefront detection.