连续变化介质中光路的解析研究

The Analytical Analysis of the Light Pathin Medium with Continuously Various Index

本文给出了光折射定律的微分式;利用折射率及其梯度与光路的切向量和曲率向量的联系,得到了"折射率场梯度向量对光路曲率的影响是线性的"的重要结论;再由光折射定律微分式和上述结论,利用微分几何的方法,对刻画光路的向量微分进行正交分解,使其转化为数量微分,从而导出描述折射率场与光路关系的解析式;进而对该解析式进行了合理的近似处理。这一结果可为空气动力学测量、声学测量领域中的实际应用提供新的工具。

In this paper,the author,in terms of Geometrical Optics,gives an analytical description of the effect of a continuously changing optical medium on the path the light passes through.With the medium represented by n as afield of scalar quantity,rewriting the well-known refraction formula in a differential way as well as in a vector form,and relating the index n and its gradient vector ▽ n to the tangent vector and the curvature vector of the light path,the author arrives at an important conclusion that the effect of ▽ n to the light path is linear.The differential formula of refraction makes it possible to analyze the light path in terms of Differential Geometry and the linearity as has been proved provides an orthogonal way to de- compound the differential equation about the light path.The result of the two is a simplification of the integral of vectors to that of scalars.With these the author obtains the group of analytical equations wanted.The author foresees that the possible approximation of the equations as shown above is highly applicable in both Aerodynamic Measurement and Acoustical Measurement.