Light’s internal reflectivity near a critical angle is very sensitive to the angle of incidence and the optical properties of the external medium near the interface. Novel applications in biology and medicine of subcritical internal reflection are being pursued. In many practical situations, the refractive index of the external medium may vary with respect to its bulk value due to different physical phenomena at surfaces. Thus, there is a pressing need to understand the effects of a refractive-index gradient at a surface for near-critical-angle reflection. In this work, we investigate theoretically the reflectivity near the critical angle at an interface with glass assuming the external medium has a continuous depth-dependent refractive index. We present graphs of the internal reflectivity as a function of the angle of incidence, which exhibit the effects of a refractive-index gradient at the interface. We analyze the behavior of the reflectivity curves before total internal reflection is achieved. Our results provide insight into how one can recognize the existence of a refractive-index gradient at the interface and shed light on the viability of characterizing it.

中文翻译：

---

通过其对临界角附近的内部反射率的影响，分析界面处波长尺度一维深度相关的折射率梯度

临界角附近的光的内部反射率对入射角和界面附近外部介质的光学特性非常敏感。正在寻求亚临界内反射在生物学和医学中的新应用。在许多实际情况中，由于表面不同的物理现象，外部介质的折射率可能会随着其体积值而变化。因此，迫切需要了解表面折射率梯度对近临界角反射的影响。在这项工作中，我们从理论上研究了玻璃界面处临界角附近的反射率，假设外部介质具有连续的深度相关折射率。我们展示了内部反射率与入射角的关系图，其表现出界面处的折射率梯度的影响。我们在实现全内反射之前分析反射率曲线的行为。我们的结果为人们如何识别界面处折射率梯度的存在提供了见解，并阐明了表征它的可行性。