An analytical formalism is developed for the nonlinear frequency shift of intense laser pulse, due to relativistic mass nonlinearity (in the sub-relativistic regime), on reflection from the critical layer in an inhomogeneous plasma. As a higher and higher intensity front of the pulse approaches the critical layer, the reflection layer moves forward to higher densities, due to the relativistic increase in the electron mass, causing a red shift in the reflected wave frequency. The frequency shift increases with shortening laser pulse duration and laser intensity. For the rear portion of the pulse, the critical layer moves backward, causing a blue shift. The effect is more pronounced for shorter pulses and longer density scale lengths, consistent with recent experiments.

中文翻译：

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非均匀等离子体中临界层反射激光脉冲的相对非线性频移

由于相对论质量非线性（在亚相对论范围内），从非均匀等离子体中的临界层反射，强激光脉冲的非线性频移被开发了一种分析形式。随着越来越高强度的脉冲前沿接近临界层，由于电子质量的相对论增加，反射层向前移动到更高的密度，导致反射波频率发生红移。频移随着激光脉冲持续时间和激光强度的缩短而增加。对于脉冲的后部，关键层向后移动，导致蓝移。与最近的实验一致，对于较短的脉冲和较长的密度尺度长度，这种影响更为明显。