Complex-valued index of refraction functions

4.6.1. Theoretic background and model

The following paragraphs cover some fundamentals of physical optics relevant to the implementation. The author chose the form and order of the relations presented so that their application in the later sections seems logical. There are, however, many approaches and ways to derive and formulate the presented relations. For further information, please refer to the standard works of optical theory, e.g. Hecht (2015), Bergmann-Schaefer (2019) or Saleh and Teich (1991).
Reformulating Maxwell’s equations and applying them to an uncharged medium, the following differential equation is found:

The last term of the equation represents a damping force that leads to an exponential attenuation in the medium if the conductivity is not equal to zero. Following Clark Maxwell’s approach leading to the pivotal discovery (see section 4.1), it is obvious that rest of the equation represents the classical wave equation:

By comparison of the two equations, the phase speed of light 𝑣 in the medium can be found as:

Introducing the (measured) constant 𝑐0 for the speed of light,

a definition for the index of refraction 𝑛 is established:

The index of refraction represents, therefore, simply the ratio of the vacuum speed of light to the speed of light in the medium.
In a very efficient – but surprisingly quite rarely applied – approach, this scalar definition of the index of refraction 𝑛 can be extended to the more general concept of a complex index of refraction 𝑛̃, which also includes an imaginary part, often referred to as the extinction coefficient. Since these coefficients show, in general, a dependence on the wavelength, it is stated as:

In order to show this definition’s usefulness but also to link the extinction coefficient to an empirically accessible quantity, it is substituted into the plane wave solution. It can be shown that a plane wave Light, sun and optics – applied principles, models and methods RadiCal, D. Rüdisser 59 is a solution to equation (30). In exponential form, a plane wave travelling in the x-direction can be written as:

A complex wave-number 𝑘̃, complex phase speed 𝑣̃ is introduced, and equations (34) and (35) are used to bring the complex index of refraction into the wave equation:

The real part of this equation equals:

The solution now reflects a sinusoidal plane wave with a phase speed of 𝑐0 𝑛 ⁄ (cosine term), which is exponentially attenuated in the medium (exponential term).
At this point, the concept of attenuation and Beer-Lambert’s law (see section 4.3.6) is recalled:

Considering that irradiance is proportional to the square of the field amplitude 𝐸, and by comparing the exponential decay term of (39) with (25), it becomes obvious how the imaginary part of the complex index of refraction (or extinction coefficient) can be determined empirically by attenuation measurements:

For further practical application in this work, equations (35), (25) and (40) are most relevant. In the RadiCal approach, all materials are described by their spectrally resolved, complex-valued index of refraction functions. The two components of this function, the real part, representing the index of refraction 𝑛 and the imaginary part reflecting the extinction coefficient, are not independent of each other as both are results of the electromagnetic interaction of light with the medium, affecting its “speed 𝑛” and its “loss”. The fundamental link between them is provided in form of the Kramers-Kronig relation (de L. Kronig, 1926; Kramers, 1927; Bohren, 2010). The connection between them also leads to dispersion relations that are essential for measuring and describing these properties. Dispersion relations for many types of materials can e.g. be found in the Handbook of optical materials (Weber M. J. et al., 2003).

4.6.2. Implementation

The implementation in the library rc_refractivityIndex features a base class TRIbaseclass that provides the 𝑛 and values for any given wavelengths through the function getnk(const wl : Double;var n,k : Double). By applying polymorphism, an abstract base class is inherited to child classes, which carry the specific definitions. Table 5 provides an overview of the currently implemented classes and the definitions used to model the 𝑛 and values as function of wavelength.

The class TRIconst is used to create simplified generic materials; the two spline classes are used to approximate refractive index functions of transmittance and reflectance spectra by inversion (see section 4.12). The rest of the classes are used to import data provided by data sources in various formats. In the flexible implementation, only terms with defined values for the constants Ci are evaluated. Therefore, the omission of constants allows an even wider range of potential dispersion relations. The primary source for refractive index data is the excellent website and database RefractiveIndex.INFO (Polyanskiy, 2022). The author M.N. Polyanskiy collects and provides data for a wide range of materials and spectral ranges. The information is provided for free access using YAML (Ben-Kiki, Evans and döt Net, 2021) format. The implemented TRIclasses allow a direct import of the provided data in the YAML files for use with the RadiCal method.

4.6.3. Testing and validation

Each of the implemented classes has been tested by the import of sample data. The spectrally resolved 𝑛 and values were then calculated based on the functions implemented in the specific classes. The results were plotted and compared with the charts and data provided on RefractiveIndex.INFO.
Spectra in the global radiation wavelength range of relevant materials, imported to RadiCal in various formats, are presented in Figure 32.