The scattering of high-power probe laser pulses by tightly focused ultra-strong laser pulses has been investigated through a semi-classical approach using a perturbative method based on the Born approximation. Under a $4\pi$-spherically-focused ultra-strong light field, the electric permittivity and magnetic permeability tensors for vacuum are calculated from the Euler-Heisenberg Lagrangian to show the nonlinear birefringent property of vacuum. And, from permittivity and permeability tensors, the scattering potential is derived for the Born approximation. The first-order solution of the Born approximation is taken as an electric field scattered from the $4\pi$-spherically-focused laser pulse when a probe laser pulse propagates through the focused laser field. The differential cross section of the nonlinear birefringent vacuum is derived, and the number of photons scattered from the nonlinear birefringent vacuum is analyzed in the laser power range of 10 - 1000 PW.

中文翻译：

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a4π球形聚焦超强电磁波对光子的散射

紧密聚焦的超强激光脉冲对大功率探测激光脉冲的散射已经通过基于Born近似的微扰方法通过半经典方法进行了研究。下一个$4\pi$利用Euler-Heisenberg Lagrangian计算了球形聚焦超强光场的真空介电常数和磁导率张量，以显示真空的非线性双折射特性。并且，根据介电常数和磁导率张量，可以得出玻恩近似的散射势。Born近似的一阶解被视为从$4\pi$当探测激光脉冲通过聚焦激光场传播时，球聚焦激光脉冲。得出了非线性双折射真空的微分截面，并分析了在10-1000 PW的激光功率范围内从非线性双折射真空散射的光子数。