一个简单的方法测定一个弱吸收薄膜的光学参数n，k，厚度d

摘要

我们提出了一个新的计算方法，用传统方法来推导一层透明介质薄膜所包围的不吸收介质的传输谱的干涉图样的光学常数。这个方法的特殊处，除了它比较简单外，它使直接的程序计算成为了可能；关于迭代法同序的精确性。

1、介绍

光的传输T的测定，通过一个在透明区域的平行的介电薄膜足够确定复折射率的实数和虚数部分，以及厚度t，Both Hall and Ferguson (1955) and Lyashenko and Miloslavskii(1964)发明了一种使用连续的近似值和插值去计算这三个数量的方法。在同样的适用和精密的范围内我们提出了一个相似的分析方法。我们的方法在两个方向上可以区分开Lyashenko and Miloslavskii（1964）；第一数据处理，计算和估算是更简单的，其次它给了一个关于n，k，t的明确表达式。N和t的精确度将要被强调。

2、原理

图1展示了一个复折射率的薄膜，它的周围是折射率为n0和n1的两个传输介质。

图1 光通过有限厚度透明基底上的吸收薄膜的示意图

考虑到入射波的单位振幅，在正入射的情况下透射波的振幅为

（1）

其中t1，t2，r1，r2分别是薄膜的上下界面的透射系数和反射额系数。该层的透射率为： （2）这个准确的表达式在附录中给出了，在弱吸收（见附录）和的情况下 （3）

式中，且

（4）

K是薄膜的吸收系数。

通常，在基本吸收区域的外面或者自由载流子吸收下，n和k的色散不是很大，T的最大和最小在方程中为

（5）

其中m是顺序号，在通常情况下（n>n1,类似于一个透明的不吸收的半导体薄膜，C2<0），传输极值在方程中给出

（6）

（7）

通过结合方程6和方程7，Lyashenko and Miloslavskii（1964）发明了一个迭代法，允许n和ɑ的测定，并且使用（4）（5）式测定k和t。

我们提出了一个重要的简单化的方法：我们将Tmin和Tmax看作为波长的连续函数通过和。这个函数也就是在透射光谱上最大值和最小值的包络线，图形2显示。方程6和方程7的比率为

（8）

图2 均匀厚度的SnO2薄膜的典型透射光谱

A simple method for the determination of the optical constants n, k and

the thickness of a weakly absorbing thin film

Abstract

We propose a new calculation following traditional methods for deducing optical constants and thickness from the fringe pattern of the transmission spectrum of a thin transparent dielectric film surrounded by non-absorbing media. The particular interest of this method, apart from its easiness, is that it makes a directly programmable calculation possible; the accuracy is of the same order as for the iteration method.

1、Introduction

The measurement of the transmission T of light through a parallel-faced dielectric film in the region of transparency is sufficient to determine the real and imaginary parts of the complex refractive index = n- ik, as well as the thickness t. Both Hall and Ferguson (1955) and Lyashenko and Miloslavskii (1964) developed a method using successive approximations and interpolations to calculate these three quantities. We propose a similar method of analysis in the same range of applicability and precision. Our method can be distinguished from that of Lyashenko and Miloslavski (1964) in two ways: firstly data handling, calculation and computation are easier, and secondly it gives an explicit expression for n, k, t. The accuracy for n and t will be emphasized.

2 Theory

Figure 1 represents a thin film with a complex refractive index = n - ik, bounded by two transparent media with refractive.

Determination of the optical constants of a thin Jilm indices, no and n1. Considering a unit amplitude for the incident light, in the case of normal incidence the amplitude of the transmitted wave is given by

（1）

in which t1, t2, r1, r2 are the transmission and reflection coefficients at the front and rear faces (Heavens 1965). The transmission of the layer is given by

（2）

The exact expression is given in the appendix. In the case of weak absorption (see appendix) with k2<< (n - no)2and k2<<(n - n1)2 ,

(3)

where， and

(4)

K is the absorption coefficient of the thin film.

Generally, outside the region of fundamental absorption(hv>EG: thin film gap) or of the free-carrier absorption (for higher wavelengths), the dispersion of n and k is not very large.The maxima and minima of T in equation (3) occur for

(5)

where m is the order number. In the usual case (n>nl, corresponding to a semiconducting film on a transparent nonabsorbing substrate, C2<0), the extreme values of the transmission are given by the formulae

By combining equations (6) and (7), Lyashenko and Miloslavskii (1964) developed an iterative method allowing the determination of n and P. and, using (4) and (5), k and t.

We propose an important simplification of this method: we consider Tmin and Tmax as continuous functions of through n() and ɑ(). These functions which are the envelopes of the maxima Tmax() and the minima Tmin() in the transmission spectrum are shown in figure 2. The ratio of equations (6)and (7) gives

（8）

Then, from equation (6),

where （9）

Equation (9) shows that n is explicitly determined from Tmax, Tmin, nl and no at the same wavelength.

Knowing n, we can determine a from equation (8). The thickness t of the layer can be calculated from two maxima or minima using equation (5):

(10)

where M is the number of oscillations between the two extrema (M=1 between two consecutive maxima or minima);, and,are the corresponding wavelengths and indices of refraction. Knowing t and a we are able to calculate the extinction coefficient k from equation (4).It is worthwhile noting that expressions (8), (9) and (10)can be easily calculated using a programmable pocket calculator.

3 Precision of the method and experimental precautions

The relative error was determined using equation (1 1) obtained

by combining equations (8) and (6):

(11)

giving (12)

We assumed: (i) that we could neglect the comparatively insignificant error in n1, which is usually the case for a glass substrate of known index of refraction, and (ii) that the errors for the two envelopes Tmax() and Tmin() are non-correlated. We then obtained

(13)

withbeing the relative precision of measurements()

The function f(n, no, nl) has been plotted on figure 3 for no= 1 and for two particular values of n1, n1=1.51 and nl= 1.6, corresponding respectively to the two extreme values for a conventional glass substrate, Equation (12) shows that due to the presence of the ratio (Tmax +Tmin)/( Tmax -Tmin) the accuracy is strongly affected when the amplitude of oscilla-tions is weak. A necessary condition for a good fringe pattern is that the difference between n and ni should be as great as possible. Similarly, from equation (l0), we obtain

(14)

This can usually be simplified in the case of a weak dispersion of and leads to (15)

On the other hand, if,which is the case if the number of minima Tmin and maxima Tmax is low, we obtain

(16)

It can then be seen that the error in t can be very important due to the coefficient

Moreover, some experimental care should be taken in the application of the above method: (i) the effective bandwidth of the spectrophotometer should be kept smaller than the halfwidth of the interference maximum when using a 0.2-2 pm spectrophotometer: this leads to an upper limit for the thickness of the film of the order of 10 pm; (ii) the sample must be homogeneous and parallel-faced; (iii) the variation of n and k with the wavelength should be small; this condition fails in the vicinity of the fundamental absorption short-wavelength region.

This method was applied to SnO2 films deposited on a glass substrate by a spray (or vacuum evaporation) technique (Manifacier et al 1975, 1976). It is shown from this work that our method is much more convenient and useful and leads to the same results as the iterative method of Lyashenko and Miloslavskii (1964). Typically, using an Aminco DW-ZUV/VIS and a Beckman DK2A spectrophotometer, with 1%we obtained n/n=2-5 % (nbetween 1.8and 2.2). The accuracy in t is a critical function of the definition of the maxima and minima. In the best cases the precision is of the order of 4%.

In conclusion, we can say that this method provides a very simple way of calculating n, k and t with a precision in the same range as that of the iterative method of Lyashenko and Miloslavskii (1964), in the case of a weakly absorbing film surrounded by non-absorbing media.

一个简单的方法测定一个弱吸收薄膜的光学参数n