Note: Refractive index sensing of turbid media by differentiation of the reflectance profile: Does error-correction work?

Abstract

A widely used method for determining refractive index postulates that the derivative of the angular profile for light reflected from the sample is maximum at the critical angle for total internal reflection (TIR). It is well-known that in turbid media this "differentiation method" yields errors in refractive index. Unexplained anomalies in previous error-calculations are eliminated if one uses a recent model of TIR which departs from traditional Fresnel theory. However we find that, in practical situations, the refractive index obtained by differentiation even after error-correction is significantly different from the best estimate for the refractive index obtained by curve-fitting the reflectance data. Thus the differentiation method lacks scientific validity in turbid media.

Critical Angle Refractometry for Lossy Media with a Priori Known Extinction Coefficient

“…The most interesting feature in Figure 2 is the turning point of the isoangular curve, which vividly demonstrates that the systematic error, introduced by use of Equation ( 1) with lossy media, does not increase monotonically with attenuation but decreases above the turning point, and even becomes zero when the isoangular curve crosses the transparency limit marked by the dashed vertical line. This behaviour, which now emerges as a natural consequence of the universal a-TIR condition, had been previously observed [13,20] and labeled as inexplicable by other authors [14].…”

“…The underlying assumption is that Equation (1) is still valid (at least approximately) and that it can be used to obtain an estimate of the real index of the lossy sample [11,12]. This assumption introduces systematic errors that have been the subject of extensive discussion; see, e.g., [13][14][15] Let us note that fitting the reflectance profile, R(θ), to Fresnel equations is another way to compute the complex optical constants [16][17][18]. However, bearing its own strengths and weaknesses, data regression is not a point method and, thus, cannot be taken as a "critical angle" refractometry approach in itself.…”

The derivative method of critical-angle refractometry for attenuating media

“…However, determining the complex refractive index of attenuating media from reflectance profiles reliably and accurately still presents some practical problems. The derivative method does not yield the imaginary part and becomes increasingly inaccurate even for the real part when attenuation grows [5,7,8]. Regression of reflectance data to the relevant Fresnel equation or empirical [9][10][11] and phenomenological [12] generalizations are often sensitive to the range of incidence angles, to the point where using the same model [9], better accuracy was reported both for a wider [10] and a narrower [13] range of incidence angles.…”

Extended derivative method of critical-angle refractometry for attenuating media: error analysis

“…The effective critical angle, thereafter referred to plainly as "critical angle", is assumed to coincide with the critical angle of transparent media so as to directly apply the TIR condition. The systematic errors introduced by SDA and the potential to correct for them have been a topic of interest [20][21][22].…”