# Imaging Microstructure on Optically Rough Surfaces Using Spatially Resolved Acoustic Spectroscopy
## 2 Materials and Methods
### What Is 'Roughness'  
The roughness of a surface depends upon the height variations and the spatial correlation of those variations. As an example, when considering the surface of sample, three distinct roughness domains can be identified: _form_, semi-periodic mesoscopic form or _wavitness_, and (optical-scale) _rougitness_. These are separated using different cut-off lengths, also referred to as the correlation length [23]. For the purposes of this work, this distinction is useful as each regime presents a different challenge to laser ultrasound measurements. Figure 1a shows how the surface influences the probe laser beam. When the surface is smooth, the _form_ is essentially flat with respect to the detection, and so the laser returns to the detector; if the surface is _wavy_, the local gradient means the returning light is reflected at a different angle when the sample moves, which means that it can return outside of the acceptance angle of the collection optics and be lost. For small-scale _roughness_, the light is scattered over a wide range of angles. The changes to the return probe light cause problems for the knife edge detector used in the instrument; Figure 1b shows the ideal case, where the laser reflects from the smooth surface and returns to the detector, where both photodiodes are illuminated equally, allowing balanced detection and cancellation of common mode noise, which produces optimal signals. In this case, the primary challenge is to maintain the focus of the optical system on the specimen surface. Figure 1c shows the influence of a surface with waviness, where the surface normal is changing quickly moving the reflected spot off-centre, away from the optimal balanced position, and in the worst cases reflecting light out of the acceptance angle of the optical system, causing signal dropouts. In Figure 1d, the return light goes everywhere, but speckles are also introduced, which are across both photodiodes, and so when the speckles move due to the presence of the sound, both light and dark areas move across the knife edge and no signal is obtained. The dataset shown in Figure 1e was acquired using Mitutoyo-Surftest SV-600 Pro-filomet [24], whereby a tool is used to measure the variation in height across the surface of the specimen; this can be performed mechanically or optically. From a mechanical perspective, one can use surface profilometry, where a stylus is dragged across the surface of the inspected component [25]. From an optical perspective, a wide variety of microscopy and interferometry methods can be used, including focus variation and confocal and coherence scanning [26]. The data produced by these techniques then allow the extraction of representative roughness parameters. A range of such parameters are defined by ISO 21920, including \(R_{a}\), the arithmetic mean deviation of the measured profile; \(R_{a}\), the arithmetic mean deviation of the measured profile; and \(R_{z}\), the arithmetic mean deviation of the measured profile [27]. Of these, \(R_{a}\) is the most commonly used and is therefore used throughout this work. Where the \(R_{x}\) parameters are concerned with describing roughness over a line profile, the \(S_{x}\) describes the areal roughness; however, these are yet to see widespread industrial use [28]. It is important to convey that the measured roughness is highly dependent on the tool used, and the length scale across which it operates [29]; coordinate measuring machines, for example, are generally used to capture features from the millimetre to metre scale, whilst on the opposite end of the spectrum, atomic force microscopy is concerned with measurements on the micron scale. Therefore, to elucidate a greater understanding of surface roughness, it is often necessary to fuse measurements across length scales. The notion of spatial frequency is essential in surface metrology; the calculation of descriptive roughness parameters requires the filtering of low-frequency components (to remove form and waviness) and is often referred to as multi-scale analysis [30].