White Light Reflectance Spectroscopy (WLRS) methodology resembles SWI but involves, instead of a laser (single wavelength), a broad-band (either in UV, VIS or NIR) light source and a PC-driven miniaturized spectrometer, operating in the corresponding spectral range, instead of a photodetector (fig. 1). The white light emitted from the light source is guided to a reflection probe through the fibers a1-a6 (fig. 1a) that incident vertically onto the sample under investigation. The typical sample consists of a stack of transparent and semi-transparent films over an appropriate reflective or transmitting substrate (e.g. Si wafer, glass slide). At the same time the reflection probe collects the reflected light (through the fiber b, fig. 1a), directing it to the spectrometer. The beam from the light source interacts with the sample (fig. 1b) and produces a reflectance signal that is continuously recorded, by the spectrometer.




Figure 1: Typical WLRS set up. (a) detail of the configuration of Reflection Probe, (b) detail of the optical path through a two layer sample over a reflecting substrate

The total reflection coefficient for a sample consisted of k-layers, can be calculated by various models. In our case the Abeles approach was followed where the effective Fresnel coefficient for the k-1 layer is:

where ρ is the amplitude and Δ the phase. From this set of equations the reflectance from any k-th layer can be calculated. In the particular case of the present study i.e. two transparent layers between the substrate and the environment the total energy can be written as:

where ni is the refractive index of the ith layer (i=0 for air, 1 for the first film, 2 for the second film and 3 for the substrate, in the case of suspending film stack the n3 correspond to air), and di of the ith layer and λ the corresponding wavelength. The light propagation at the various interfaces is illustrated in fig. 2. A typical reflectance spectrum for a two transparent layer stack on Si substrate is illustrated in fig. 3. In the same figure, the incident (reference) light spectrum in the VIS-NIR range is also illustrated. The number and the shape of interference fringes depend on the thickness and the refractive index of the film(s). The fitting of the experimental spectrum with the above equation is performed by using the Levenberg-Marquardt algorithm. The refractive indices ni of all layers vs. wavelength have been calculated with a spectroscopic ellipsometer and fitted to a three parameter (A, B, C) Cauchy model and stored in the database. By applying eq. 2.2 for all wavelengths in the desirable spectrum range, the film thickness is calculated. Fitting can be applied for more than one film thickness, or reversely for known thickness can be applied to calculate the refractive index, or a combination of film thickness and refractive index. The increment of the calculated parameters must be done with care, because it is also increased the results uncertainty.


Figure 2: Light propagation schematic in a stack consisting of two layers (1, 2) on the substrate (3), and a media (0) above the films stack (e.g. air or water).
Figure 3: Reflectance (interference) spectrum of a two layer    stack (PHEMA/SiO2) over Si and the incident (reference) white light spectrum

Figure 4 shows a screen shot of the FR-Monitor software with the Reference and Reflectance spectra as acquired from a Si3N4/SiO2 stack over Si substrate. In Figure 5, the normalized Reflectance spectrum (black line) for the stack of Fig. 4 appear together with the fitted spectrum(red line). The Si3N4 and SiO2 thicknesses as calculated by the fitting are 138.4nm and 577.3nm respectively.
Recorded Spectrum  
Fitting Spectrum
Figure 4. Screenshot of FR-Monitor with the acquired spectrum of the Si3N4/SiO2/Si stack and the incident (reference) spectrum.

Figure 5. Normalized (0-100%) reflectance spectrum of Figure 4, after the fitting of the thickness of Si3N4 and SiO2 layers (red line). The normalized value is greater than 100% at 740nm, due to the positive interference.


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