Joshua Marx and Steve Turley, Department of Physics and Astronomy
Extreme ultraviolet (XUV) light is light with wavelengths between one and 60 nanometers. The shorter wavelengths of light in this range are on the same order of magnitude as atomic dimensions. Currently, XUV optics have many uses and even more potential applications in a variety of fields, such as photolithography, plasma diagnostics, and astrophysics.
To further research the effects of XUV light, our research group has made thin films from various materials that we have tested to see how rough the surfaces are. These materials include thorium, ruthenium, aluminum, uranium, yttrium, and silicon nitride. These optics need to be created with very precise specification of layer thickness and roughness in order to reflect as much XUV light as possible. Many of the thin films are multilayer mirrors which create a composite reflectance in the XUV. For these to work properly, each layer thickness must be controlled to less than one nanometer so that the reflected beams from each layer add constructively.
Another crucial aspect of our thin films is the index of refraction. Not all indices of refraction for materials are well known in XUV, particularly when oxidation, hydration, and phase changes are taken into account. Due to these effects, we do not always know the index of refraction and would like to account for it. To calculate it, we need to understand the layers’ roughness to form an accurate fit. All of these reasons make it crucial that we know how rough the surface of the sample.
To improve upon other methods we are working on finding a new way to map surface roughness using non-specular reflectance. This method analyzes the intensity of light that reflects off at different angles from a set portion of the sample, and calculates the reflectance per unit angle at the specific wavelength. To accomplish this, we went to the Advanced Light Source (ALS) at the Lawrence Berkeley National Labs (LBNL) and took non-specular reflectance scans of the surfaces of our wafers. We then took this data and computed the reflectance per unit angle of the samples in order to map the surfaces of the optics.
The results from the ALS can be compared to different optical methods to determine variations of surface height, however only geometric optics and Huygens’ calculations will be used in the calculation of this data. Huygens’ method samples the surface over a series of equally spaced points. It assumes that current on the surface is constant in a region centered on each point. The phase of the current is equal to the phase of an incident plane wave with an amplitude of one. The integral of the current density is accomplished by summing the currents from each point. For our calculations we only considered a single layer surface and so only took the part of the beam being reflected, and ignored the transmitted portion. A limitation of this method is it considers all of the radiated fields as being of equal amplitude independent of the index of refraction or incidence angle. Also, as we increase the number of surface points to approach the nominal surface height, Huygens’ scattering density increases. This can be approximated as an increasing index of refraction as a function of surface height which is directly related to interfacial roughness, or roughness between the different layers of the thin film discussed earlier.
To compute the reflectance using Huygens’ method, Steve Turley wrote a program that I used during my analysis. The program uses a uniform random number generator to create a surface so that the amplitude at each point has an equal probability of being between -1 and 1. This is done in the Fourier domain so this is the amplitude of the Fourier component. We then multiply by a half-Gaussian of a desired width to attenuate the amplitude of different random frequency components; this biases the surface to favor the lower spatial frequencies. We take the inverse Fourier transform of the amplitude components to put it in the spatial domain. We now compute the RMS roughness of the created surface. To obtain a surface with a specific desired RMS height, we multiply the surface by the desired RMS height over the computed RMS roughness. This allows us to vary the spatial frequency components and the RMS height to create a model surface from which we can calculate the reflectance. Each run creates a new surface that has the same Gaussian envelope but different random amplitudes for the spatial frequency components. This is important because we can average the computed reflectance to find how a class of mirrors work, rather than one point on one particular surface.
The final method used to calculate reflection from a surface is geometrical optics, which is also called ray tracing. This is the simplest method, but also has a limitation that feature size causing the reflection must be much larger than the wavelength. This method applies Snell’s law at the surface to calculate the angle of reflection. We used this method when the surfaces created using Huygens’ method do not match the reflectance from our samples in order to find an approximation of the surface. This method will then be later used to create a better surface model then the one currently used.
We measured the reflectance per unit angle of our samples at the ALS at the LBNL where we took non-specular reflectance measurements. These different runs needed to be joined together and put on the same scale to compensate for the different gains. The combined curve was an equivalent reflectance curve as if all the data were taken with the same detector, filter, and gain. By comparing the non-specular reflectance data to the theoretical calculations found using Huygens’ principle, we were able to discern more information about our surfaces. As we averaged the reflectance from a variety of surfaces with the same parameters, we noticed that we were not able to fit the non-specular curves as well as we hoped. Our model worked for a simple curve that had a specular peak, but could not match the features in the wings as well as we hoped.
To use geometrical optics to calculate the surface of our thin films and the bumps that would create the diffuse reflection, we found what percent of the reflected light was represented by the peak of the non-specular reflectance curve and what percent was represented in the wings. First, we found the modified size of the beam for the sample angle, and found the total area of the beam spot. Using the percent of the reflected light contributing to the plateau and the beam area, we were able to find the area of the bump scattering the light. We were able to create a model surface with this feature and find a histogram of the rays from the surface. Upon computing this, we found that the histogram showed a tall specular peak with a plateau around it, confirming that the majority of the surface was flat with one or more bumps.
Our results were enlightening, but not entirely in the way that we were hoping. We were unable to quantify our surfaces due to the surface variance being higher than we expected. To obtain better quantifications of the surface roughness we need a better surface model than the one we are currently employing. In order to improve the model we will need to find a way to combine different cutoffs to obtain the correct amounts of high and low frequency spatial frequencies to properly describe our thin films. We will also have to use a better model than the geometrical optics and Huygens’ method used for these calculations. This can be done by finding the exact surface current using boundary element methods. We hope to be able to create a model accurate enough to describe the non-specular scattering from surfaces with roughness having spatial frequencies much less than one over the wavelength.