Once a suitable method for producing a particular thin film has been determined, the next step is the measurement of the optical properties. Many methods for this exist, and a useful earlier account is given by Heavens. Measurement of the optical constants of thin films is also included in the book by Liddell. A more recent survey is that of Borgogno.

Recently, the measurement of the optical properties of thin films has increased in importance to the extent that special-purpose instruments are now available. These instruments normally include extraction software and are essentially push-button in operation. As always, however, even when automatic tools are available, some understanding of the nature of the process and its limitations is still necessary. Here we shall be concerned with just a few methods that are frequently used.

In all of this, it is important to understand that we never actually measure the optical constants \( n \) and \( k \) directly. Although thickness \( d \) is more susceptible to direct measurement, its value too is frequently the product of an indirect process. The extraction of these properties, and others, involves measurements of thin-film behavior followed by a fitting process in which the parameters of a film model are adjusted so that the calculated behavior of the model matches the measured data. The adjustable parameters of the model are then taken to be the corresponding parameters of the real film. The operation is dependent on a model that corresponds closely to the real film.

The appropriateness of the model would be of less importance were we simply trying to recast the measurements in a more convenient form. Even an inadequate model with parameters appropriately adjusted can be expected to reconstitute the original measurements.

However, the parameters extracted are rarely used in that role. Rather, they are used for predictions of film performance in different situations where film thickness may be quite different, and where the film is part of a much more complex structure.

This leads to the idea of stability of optical constants, a rather different concept from accuracy. Accurate fitting of measured data using an inappropriate model may reproduce the measurements with immense precision yet yield predictions for other film thicknesses that are seriously in error.

Such parameters are lacking in stability. Stable optical constants might reproduce the measured results with only satisfactory precision but would have equal success in a predictive role. A good example might be a case where a film that is really inhomogeneous and free from absorption is modeled by a homogeneous and absorbing film. The extracted film parameters in this case can be completely misleading. It must always be remembered that the film model is of fundamental importance.

Almost as important as the model is the accuracy of the actual measurements. Calibration verification is an indispensable step in the measurement of the performance that will be used for the optical constant extraction.

Remember that only two parameters are required to define a straight line, but to verify linearity requires more. Small errors in measurement can have especially serious consequences in the extinction coefficient and/or assessment of inhomogeneity of the film.

The samples themselves should be suitable for the quality of measurement. For example, a badly chosen substrate may deflect the beam partially out of the system so that the measurement is deficient, or it may introduce scattering losses that are not characteristic of the film.

The calculation of performance given the design of an optical coating is a straightforward matter. Optical constant extraction is quite different. Each film is a separate puzzle. It may be necessary to try different techniques and different models. Repeat films of different thicknesses or on different substrates may be required.

Some films may appear to defy rational explanation. A common film defect is a cyclic inhomogeneity that produces measurements that the usual simpler film models are incapable of fitting with sensible results. It is always worthwhile attempting to recalculate the measurements using the model and extracted parameters to see where deficiencies might lie. Because of all the caveats in this and the previous paragraphs, exact correspondence, however, does not necessarily indicate perfect extraction.

As we saw in previous tutorials, given the optical constants and thicknesses of any series of thin films on a substrate, the calculation of the optical properties is straightforward. The inverse problem, that of calculating the optical constants and thicknesses of even a single thin film, given the measured optical properties, is much more difficult, and there is no general analytical solution to the problem of inverting the equations.

For an ideal thin film, there are three parameters involved: \( n \), \( k \), and \( d \), the real and imaginary parts of refractive index and the geometrical thickness, respectively. Both \( n \) and \( k \) vary with wavelength, which increases the complexity. The traditional methods of measuring optical constants, therefore, rely on special limiting cases that have straightforward solutions.

Perhaps the simplest case of all is represented by a quarter-wave of material on a substrate, both of which are lossless and dispersionless; that is, \( k \) is zero and \( n \) is constant with wavelength. The reflectance is given by:

\[
R = \frac{(n_f – n_\text{sub})^2}{(n_f + n_\text{sub})^2}
\]

where \( n_f \) is the index of the film and \( n_\text{sub} \) that of the substrate, and the incident medium is assumed to have an index of unity. Then \( n_f \) is given by:

\[
n_f = n_\text{sub} \sqrt{\frac{1 + \sqrt{R}}{1 – \sqrt{R}}}
\]

where the refractive index of the substrate, \( n_\text{sub} \), must, of course, be known. The measurement of reflectance must be reasonably accurate. If, for instance, the refractive index is around 2.3 with a substrate of glass, then the reflectance should be measured to around one-third of a percent (absolute \( \Delta R \) of 0.003) for a refractive index measurement accurate in the second decimal place.

It is sometimes claimed that this method gives a more accurate value for refractive index than the original measure of reflectance since the square root of \( R \) is used in the calculation. This may be so, but the value obtained for refractive index will be used in the subsequent calculation of the reflectance of a coating, and therefore the computed figure can be only as good as the original measurement of reflectance.

In the absence of dispersion, the curve of reflectance versus wavelength of the film will be similar to that in Figure 11.20. The extrema correspond to integral numbers of quarter-waves, even numbers being half-wave absentees and giving reflectance equal to that of the uncoated substrate, and odd corresponding to the quarter-wave of Equations 11.1 and 11.2.

Thus, it is easy to pick out those values of reflectance that correspond to the quarter-waves. The technique can be adapted to give results in the presence of slight dispersion. The maxima in Figure 11.20 will now no longer be at the same heights, but, provided the index of the substrate is known throughout the range, the heights of the maxima can be used to calculate values for film index at the corresponding wavelengths. Interpolation can then be used to construct a graph of refractive index against wavelength. Results obtained by Hall and Ferguson for MgF2 are shown in Figure 11.21.

This simple method yields results that are usually sufficiently accurate for design purposes. If, however, the dispersion is somewhat greater, or if rather more accurate results are required, then the slightly more involved formulae given by Hass et al. must be applied. It is still assumed that the absorption is negligible.