ABC's of the Specular Reflection Method and Kramer's-Koenig Analysis

(1) Kramer's-Koenig Analysis

The explanation below applies to the irradiation of substance at an almost vertical angle of incidence. A more detailed explanation is given in the References ¹⁾, ²⁾ below. The complex index of refraction of the substance n^* is set as follows:

n^*= n + ik (where n and k are the refractive index and absorbance coefficient of the substance, respectively)

Then, the amplitude reflectance (r) under vertical irradiation and energy reflectance (R) determined directly from the specular reflection spectrum are expressed by the following equations:

Where, φ is the phase change. This results because both reflection and absorption occur at the sample surface. r^* is the complex conjugate of r.
Dividing Equation (1) into real and imaginary parts and solving for n and k yields the following equations:

Taking the logarithm of Equation (1), gives

If the energy reflectance is measured across the entire wavenumber range, the phase change φ(ν_g) can be calculated at any required wavenumber ν_g using this relational expression, and then the optical constants n and k can be determined from Equations (3) and (4). Consequently, by calculating k at constant wavenumber intervals across the normal infrared range between 4600 and 400 cm^-1 for example, an absorbance coefficient spectrum equivalent to the transmission spectrum can be calculated from the specular reflection spectrum.
In the integration of Equation (6), a pole exists at ν = ν_g but several methods have been proposed to handle this integration. Two typical methods are the Maclaurin method and the double Fourier transform method. In the Maclaurin method, the phase change φ(ν_g) is given by Equation (7)

j = 2, 4, 6, ........., g-1, g+1, ......... (if g is an odd number)
j = 1, 3, 5, ........., g-1, g+1, ......... (if g is an even number)

The ν_j start point is set so that ν = ν_g does not occur and alternate data points are used. With the double Fourier transform method, φ(ν_g) is determined by Equation (8), a double Fourier transformation which is an approximation of Equation (6).

The Maclaurin method provides better calculation accuracy, but because it is so time-consuming, the quicker double Fourier transformation method is normally used.

(2) Specular Reflectance Accessory

Fig. 1 shows the appearance of the SRM-8000 specular reflection accessory. Fig. 2 shows the optical system. Light reflected from plane mirrors M₁ and M₂ reflects from the spherical mirror M₃ and forms an image on the sample surface. The light beam has a mean 10-degree angle of incidence with respect to the sample. Light reflected from the sample reflects from spherical mirror M₄ and plane mirrors M₅ and M₆ onto the detector in the FTIR instrument.
The normal references are aluminum and gold vapor-deposited mirror.

(3) Examples of Measurements

Fig. 3 shows the specular reflection spectrum for acrylic sheet. It exhibits absorption up to 2000 cm^-1 but the peaks are deformed toward the first-order differential form due to the anomalous dispersion of the refractive index. Fig. 4 shows the spectrum in Fig. 3 after Kramer's-Koenig transformation. This spectrum indicates that the sample is polymethyl methacrylate (PMMA).