We report the results of a rigorous calculation on the total internal reflection when the second medium is absorbing. A complex dielectric constant given by, \(\hat{\varepsilon }^{\text{Re}} + i\hat{\varepsilon }^{\text{Im}}\) is introduced to represent the effect of the absorption. Starting from Maxwell’s equations, we show that the angle of refraction becomes progressively smaller than \(\frac{\pi }{2}\) with increasing \(\hat{\varepsilon }^{\text{Im}}\). The reflection and transmission coefficients for both of s- and p-polarized incident beams are calculated on the same footing. The absolute value of the reflection coefficient becomes smaller than unity and begins to depend on the incident angle φ with increasing \(\hat{\varepsilon }^{\text{Im}}\). When \(\hat{\varepsilon }^{\text{Im}} = 0\), the phase of the reflection coefficient changes from 0° at the critical angle to −180° at φ = 90°. With increasing \(\hat{\varepsilon }^{\text{Im}}\), the phase becomes less dependent on φ. From a calculation of the time average of the Poynting vector, we confirm that the law of conservation of energy holds at the boundary between the two media.

中文翻译：

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吸收介质中的全内反射

我们报告了第二种介质吸收时对内部全反射进行严格计算的结果。由下式给出的复数介电常数：\（\ hat {\ varepsilon} ^ {\ text {Re}} + i \ hat {\ varepsilon} ^ {\ text {Im}} \）引入代表吸收的效果。从麦克斯韦方程开始，我们证明了折射角逐渐小于\（\ frac {\ pi} {2} \） 随着增加 \（\ hat {\ varepsilon} ^ {\ text {Im}} \）。s-和p偏振入射光束的反射和透射系数都是在相同的基础上计算的。反射系数的绝对值小于1，并且随着入射角φ的增加而开始变化。\（\ hat {\ varepsilon} ^ {\ text {Im}} \）。什么时候\（\ hat {\ varepsilon} ^ {\ text {Im}} = 0 \），反射系数的相位从临界角的0°变为φ = 90°的-180 °。随着增加\（\ hat {\ varepsilon} ^ {\ text {Im}} \），则相位对φ的依赖性减小。通过计算Poynting向量的平均时间，我们可以确定能量守恒定律在两种介质之间的边界上。