The present study addresses the question of total internal reflection of a plane wave from an amplifying or attenuating medium. Inspection on the expressions for the modal gain or loss of an asymmetric planar waveguide having an amplifying or attenuating cladding has led us to the coefficients for amplified or attenuated internal reflection. The derived formulas suggest an interpretation that the refracted inhomogeneous plane wave undergoes amplification or attenuation while it travels a distance equal to the Goos-Hanchen shift along the boundary plane before going back into the high-index medium. Furthermore, the evanescent wave in the low-index medium is discussed. Unlike the transparent case, the Poynting vector is shown to have a nonvanishing time-averaged component perpendicular to the boundary plane.

中文翻译：

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对放大或衰减内反射的菲涅耳公式的修改

本研究解决了来自放大或衰减介质的平面波的全内反射问题。对具有放大或衰减包层的非对称平面波导的模态增益或损耗表达式的检查使我们得到了放大或衰减内反射的系数。导出的公式表明，折射的非均匀平面波在返回高折射率介质之前沿边界平面传播的距离等于 Goos-Hanchen 位移时会经历放大或衰减。此外，还讨论了低折射率介质中的倏逝波。与透明情况不同，坡印廷矢量显示为具有垂直于边界平面的非零时间平均分量。