新的数值积分法及其在光学常数计算中的应用

Novel Numerical Integration Method and Its Application in Calculating Optical Constants

针对利用K-K转换关系计算光学常数时,需要对实验测得光反射谱的离散数据进行数值积分,而利用通常的数值积分方法存在较大的偏差,提出了一种新的数值积分法,即双抛物线比例内插法(DPPIM).该方法在分段二次插值的基础上,将两条二次插值曲线在其重叠区间按比例组合,得到一条光滑的三次插值曲线.将该光滑曲线在整个区间上积分,就可有效解决无边界条件、复杂离散数据的数值积分问题.利用实验测量的单晶硅光反射谱,计算了单晶硅的光学常数、能带特性.计算结果表明,DPPIM方法在处理复杂离散实验数据的数值积分方面具有优越性.

The optical constants of semiconductors are important both in describing the interaction process between light and condensed matters and in designing semiconductor photoelectric devices. In the calculation of the optical constants by the Kramers-Kronig transform relations, it is essential to carry out a numerical integration on the discrete experimental data of the reflection spectrum. The popularly used integral methods might bring a result which is seriously deviated from the practical data. In an effort of solving this problem, a novel numerical integration method, the double parabola proportioned interpolation method （DPPIM） ,is put forward based on the piecewise quadratic interpolation method, in which two interpolation functions are combined proportionally in their overlapping interval and a smooth curve is obtained. Therefore, the numerical integration without deterministic boundary conditions of the complex discrete data is solved. The optical constants and band properties of single crystal silicon are calculated utilizing its experimentally measured reflection data. It is shown that the DPPIM method possesses obvious pre- dominance in dealing with the numerical integration of complex discrete data.