Objective In laser-driven inertial confinement fusion (ICF) drivers used in fusion experiments, shaping and smoothing the target focal spot are key technologies. To improve the irradiation uniformity of the target and reduce various instabilities related to laser-plasma interaction, a distributed and uniform target focal spot is necessary. Moreover, the uniformity of the focal spot must be maintained up to a certain focal depth. Hence, diffractive devices such as continuous phase plates (CPPs) are used to shape the focal spot and beam smoothing techniques are utilized to reduce contrast of the focal spot. Furthermore, polarization smoothing utilizes the birefringence of crystals to separate the beam into two orthogonally polarized beams that are incoherently superimposed, thus reducing the contrast of the focal spot. This study focuses on polarization smoothing in converging beams and presents a mathematical model that fully describes the transmission characteristics of the beam during polarization smoothing. Moreover, this study demonstrates the relationship between the focal spot shape parameters and the smoothing crystal and lens parameters. Furthermore, the longitudinal smoothing effect of the beam far field is quantitatively described for the first time in this study. Herein, the relationship curve between the contrast of the focal spot and crystal parameters is determined and the optimum range of crystal thickness and tilt angle is calculated. The research obtained from this study can provide a reference for the selection of polarization smoothing techniques and the design of smoothing crystals used in laser drivers. Methods A theoretical and numerical model for realizing polarization smoothing in convergent beams is presented in this study. A CPP is used for shaping the focal spot of a thin lens. The front and back surfaces of the polarization smoothing crystal are parallel and the crystal axis is perpendicular to these surfaces. This crystal is placed in the path of the convergent beam behind the lens. The incident light field is a square aperture that is linearly polarized and comprises a monochromatic harmonic wave. The wave vector of the transmitting beam passing through the above elements is calculated with respect to a spherical coordinate system. The vector expression of the crystal output light field is obtained by combining the ellipsoid equation of crystal refractive index with the wave vector. Then, the far-field distribution near the focus of the lens is calculated using the Huygens-Fresnel diffraction formula. Using small-angle approximation, a simplified relationship between the longitudinal separation of far-field focal spots and crystal thickness and angle is derived based on a simplified vector expression. Numerical simulations are conducted to verify the correctness of the simplified expression. Results and Discussions The longitudinal separation of the far-field focal spots is proportional to crystal thickness and is not substantially affected by the crystal tilt angle [Eq. (21), Fig. 9(a)]. By contrast, the transverse separation is proportional to the crystal thickness and tilt angle [Eq. (22), Fig. 9(b)]. The polarization crystal smooths the far-field focal spot distribution in the transverse and longitudinal directions (Fig. 13), without significant change in the overall shape of the focal spot. The maximum relative intensity and contrast of the smoothed focal spot vary with crystal thicknesses and tilt angles (Fig. 14, Fig. 15). Moreover, the polarization smoothing effect obtained using a crystal with small thickness is similar to that obtained using a wedge-based solution. Conclusions This study presents a mathematical model to analyze polarization smoothing in convergent beams. Herein, the formulas for the transverse and longitudinal separations of the focal spot with respect to the thickness and tilt angle of the polarization smoothing crystal are derived. Numerical calculations are performed with respect to the proposed method, and the relationship curves between the focal spot shape parameters and crystal parameters are provided. The numerical-simulation results show that the best smoothing effect with respect to the focal spot can be achieved when the crystal thickness and tilt angle are within the desired range. Further theoretical analysis is required to investigate the relationship between the smoothing effect and other parameters of the system.
