• Introduction
  
• Effects of Surface Roughness
  
• How Does Surface Roughness Arise?
  
• How is Roughness Specified?
  
• How is Surface Roughness Obtained from Light Scatter?
  
• What is a Scatterometer?
  
• How does Scatterometry Compare to Other Methods for Determining Surface Roughness?
  
• Scanning Tunneling & Atomic Force Microscopy
  
• Stylus Profilometry
  
• Optical Profilometry
  
• Advantages of Scatterometry for Quality Control/Process Control Applications
  
• Conclusions

Introduction
All smooth surfaces possess some degree of roughness, even if only at the atomic level. For man-made surfaces, this roughness arises from the manufacturing process which may involve chemical deposition, grinding, polishing, etching or several other commonly used techniques. Correct function of the fabricated component often is critically dependent on its degree of roughness. In this technical note we examine how scatterometers can be used to quantify surface roughness with rapid and repeatable measurements. We also compare scatterometry and its practical implementation with other methodologies.

In optical applications, the primary motivation in measuring surface roughness is to estimate how much the surface will scatter light at the intended wavelength(s) of operation. Excess scatter can result in system nonperformance for sensing optics, imaging optics, laser optics and numerous others. In these instances it is much better to specify and measure scatter directly, using the scatter definitions outlined later in this article. For example, if a polished optic is to be used at 488 nm, then the best specification is the scatter (TIS, BRDF, etc.) measured at 488nm.

In non-optical applications, excess surface roughness can lead to mechanical malfunctions. For example, computer hard disks have a narrow tolerance band for acceptable roughness. If the surface is too smooth, the read/write head may bind to the surface of the disk. If the surface is too rough, the head may be unable to fly over the disk surface on its air cushion in the proper manner. Another example of mechanical malfunction can be found in high performance engine machine parts which are required to move or rotate at high speed without wear. Excess surface roughness can lead to unacceptably high levels of frictional heating, causing damage and even failure.

Rollers used in almost any application from computer printers and plotters to pressing metals, papers and films in factory environments require careful control of surface quality to ensure the quality of the finished product. These applications can benefit from periodic surface testing or in situ surface monitoring to alert manufacturers of the need to refinish the roller surface before the process degrades to an unacceptable level. In addition, by testing the surface quality during the grinding and refinishing process, the roller surface can be brought to the required smoothness with a minimum amount of processing.

Surface roughness can affect a component's chemical and physical stability. Surfaces that have to stand up to hostile environments (temperature, humidity, or hostile chemicals) must be as smooth as possible in order to present the minimum surface area for attack, and to have as few defects or weak spots as possible.

Cosmetic appearance of a surface can be adversely affected by surface roughness. Although this may appear to be the most trivial of the problems caused by surface roughness, it is often important in terms of potential lost revenue. For example, although a rough paint surface may function perfectly well on an automobile, it would reduce the customers' perception of quality and value.

How Does Surface Roughness Arise?
Figure 1 shows examples of the common factors contributing to overall surface roughness. The defects or features which contribute to surface roughness may be random or regular (periodic). These defects can arise in a number of different ways.

For surfaces produced by a grinding or polishing process, the most obvious roughness is the unevenness of the surface itself, i.e. scratches, pits and ridges. These features can range in size from a step defect in the molecular structure measuring a few angstroms, up to a visible scratch whose width could be measured in microns and length measured in millimeters. In the case of nonrandom wear processes such as diamond turning, these surface imperfections may have an inherent symmetrical pattern, rather like a smooth version of a phonograph record.

How is Roughness Specified?
The roughness of a surface can be specified by a number of different parameters, each with its own utility. All these parameters consider roughness in terms of deviations from the mean surface level. One of the most common parameters in RMS roughness. Assuming a surface in the horizontal plane, this is the root mean squared value of all vertical deviations from the mean surface level. Ra roughness, the arithmetic average roughness, is a term used more for machined surfaces than for polished optics. It is the arithmetic average of the absolute deviations from the mean surface level.

In the case of real world measurements, it should be understood that each instrument is limited to a spatial bandwidth of operation. That is, some features are too wide (or far apart) to be detected and some too narrow (or close). It is common to give these limits in terms of spatial frequency (i.e., the inverse of maximum or minimum widths). Therefore, the measured roughness (RMS or Ra) doesn't include features outside this range. When comparisons between measurements (or instruments) are made, they must involve identical spatial bandwidths. There are also limits of operation for measuring the amplitude of surface features. It is therefore important when expressing roughness to not only specify bandwidth, but also to state the specific range of surface amplitudes of the instrument.

In the case of machined surfaces it is very useful to know if the surface has periodic undulations or features. For example, fine sandpaper and a phonograph record may have the same RMS roughness, but have very different overall characteristics. The distribution of roughness versus spatial frequency of features is defined by the Power Spectral Density (PSD) function or Power Spectrum. The sandpaper would have a rather random distribution of power vs. frequency, where as the phonograph record would tend to display at least one dominant frequency or a "spike" in a PSD plot. A similar comparison can be seen in the world of precision optics when comparing a lapped and polished optic to a diamond turned optic. While RMS or Ra roughness parameters have evolved as popular numbers to specify, detailed information about the surface is lost in the "averaged" nature of these numbers.

How is Surface Roughness Obtained from Light Scatter?
In the case of reflective surfaces, surface roughness causes incident light to be scattered in directions other than the specular reflection direction (angle of reflection = angle of incidence), as shown in Figure 2. Clearly, the rougher the surface, the higher the proportion of incident light that is scattered. For example, a highly polished aluminum mirror may show little scatter to the naked eye and is able to form a high quality image in reflection, whereas a sheet of white paper scatters so much light that any specular reflection becomes indistinguishable from scatter.

 
Figure 2

Scattered light is defined by its magnitude and its angular distribution, both of which may be used to derive important surface roughness data. There are two commonly accepted measurements used to quantify these functions: TIS (Total Integrated Scatter) and BRDF (Bi-directional Reflectance Distribution Function).

For a reflective surface, TIS is defined as the ratio of the total scattered power to the total reflected power. The total scattered power is the sum of all light scattered outside the specular direction. TIS is related to RMS roughness.

Figure 3

BRDF is determined from the ration of the scatter per unit solid angle to the incident power, i.e. normalized scatter density. BRDF is commonly presented as a function of angle. This BRDF function contains valuable information about the amplitude and width of the surface features (see figure 3).

The amount of scattered light is a result of the amplitude of the scattering features, whereas the scatter angular distribution is a result of the surface spatial frequency. TIS can be a very good yardstick by which to measure surface roughness, but it is the angular distribution, as defined by BRDF, which carries additional information about the distribution, shape and size of surface imperfections. Diffraction theory has been used to accurately relate the surface PSD to the BRDF. These two functions are nearly proportional, which makes determination of surface parameters from the BRDF straightforward. Additionally, simple integration of the PSD results in RMS roughness values of the surface over a selected range of surface wavelengths.

There are a number of alternative methods for deriving surface roughness; the matrix in Figure 4 shows a condensed comparison of these. The final choice of methods depends on the type of data that is required, the speed with which the data must be acquired, the skill level or training time of the operator and the environment in which the measurements must be made. All of these techniques are indirect methods of obtaining a surface profile or roughness measurement.

That is to say they "calculate" the surface roughness based on optical or physical effects measured by the instrument. Also, each method has its own unique and finite range of heights (RMS roughness) and spatial frequencies. Detailed knowledge of these techniques and their limitations is critical to properly measuring, understanding, and interpreting the results from any system.

d=2(pi)(square root of)ar Where d is spatial wavelength, a is the height (amplitude) of the feature, and r is the radius of the stylus tip. Although capable of high resolution, the sharper tips are more likely to scratch the test surface.

Profilometry and scatterometry also have major differences. In the case of profilometers, profiles need to be sampled in order to calculate surface roughness values such as PSD and RMS roughness. The sampling process and subsequent calculations introduce spatial bandwidths (and small errors) that must be taken into consideration to create the PSD. Scatterometry, on the other hand, starts with the PSD (which is directly derived from the BRDF) thus eliminating this source of error. From the PSD, RMS roughness over required spatial bandwidths is calculated with either technique. In general, scatterometers can measure to slightly higher spatial frequencies, while some optical profilometers can measure to slightly lower spatial frequencies.

Operator skill is another important factor. Although optical profilometers, STM's and AFM's can be powerful tools in the hands of a skilled operator, it requires experience and skill to correctly focus and align these instruments on the sample. Consequently, operators must be trained to a high proficiency level to avoid making erroneous measurements. Scatterometers require less skill to operate, especially for the repetitive measurements generally associated with QC operations.

Vibration sensitivity is another important difference between these techniques. Optical profilometry depends on precision interferometry and the instrument must be mounted on some type of vibration isolation table. STM's, AFM's and stylus profilometers can be ultra sensitive to vibrations, erroneously interpreting them as roughness on the surface. Scatterometers are relatively insensitive to vibration. This has lead to the development of portable and hand held devices which can be operated on the factory floor. In a machining process, parts can actually be tested in situ, i.e. without removing from the machine.

Finally, when testing large surfaces (e.g. magnetic disks or large mirrors) the best way to reduce testing time and increase statistical validity is to test the roughness of several small areas at a variety of locations on the surface. Since the alignment is critical for optical profilometers, STM's and AFM's, this can be a very slow process. Stylus profilometers measure only a very fine line, typically 1mm in length, which is an exceptionally small area relative to the entire surface. In the case of scatterometry, the inspection spot can be quickly and automatically rastered over the entire surface.

The differences between scatterometers and other optical techniques (i.e. STM's, AFM's and optical profilometers) can be summarized as follows. STM's, AFM's and optical profilometers require the intervention of a skilled operator and perform measurements relatively slowly, whereas the scatterometer is much simpler and, consequently, can make measurements more rapidly while still maintaining excellent repeatability and accuracy.

Stylus profilometers, like scatterometers, are simpler devices which require less skilled operators. However, there are a number limitations that effect a stylus type system. For example, measurements must be conducted under vibration isolation conditions, and large areas require numerous scans. While these instruments are well suited to single measurements of hard materials, such as machined surfaces and parts, they are generally destructive to soft materials such as most coated optics. Furthermore, as they are very sensitive to vibration, they are not well suited for QC or process control applications.