## Abstract

The roughness parameters of diamond-turned metal surfaces are obtained from the Fourier transform of the scattered light intensity. Theoretical consideration of the scattered light intensity and its Fourier transform is made. Under the condition of weak scattering, the Fourier transform of the scattered intensity is equivalent to the autocorrelation of the surface. From this autocorrelation function, we can derive the parameters of the surface roughness, namely, the amplitude and frequency of the periodic tool marks and the rms roughness and correlation length of the random surface component. The surface-roughness parameters of the diamond-turned surfaces of aluminum alloy are determined experimentally from the autocorrelation function. The data obtained by light scattering are compared with those of the conventional stylus method. They show qualitative agreement.

© 1986 Optical Society of America

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### Equations (19)

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(1)
$$A\left(u\right)={\displaystyle \int {A}_{0}\left(x\right)\text{exp}\left(-{x}^{2}/{{w}_{0}}^{2}-ikux/R\right)\mathrm{d}x},$$
(2)
$${A}_{0}\left(x\right)=\text{exp}\left[-i\varphi \left(x\right)-i\theta \left(x\right)\right].$$
(3)
$$\u3008\varphi \left(0\right)\varphi \left(x\right)\u3009={{\sigma}_{\varphi}}^{2}\phantom{\rule{0.2em}{0ex}}\text{exp}\left(-{x}^{2}/{{r}_{0}}^{2}\right).$$
(4)
$$\theta \left(x\right)={\displaystyle \sum _{n=0}^{\infty}{a}_{n}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi nx/p\right)}.$$
(5)
$$\begin{array}{ll}I\left(u\right)=\hfill & {\displaystyle \int \mathrm{d}{x}_{1}\mathrm{d}{x}_{2}\left[1+{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi {x}_{1}/p\right)\text{cos}\left(2\pi {x}_{2}/p\right)\right]}\hfill \\ \hfill & \times \u3008\text{exp}\left[i\varphi \left({x}_{1}\right)-i\varphi \left({x}_{2}\right)\right]\u3009\hfill \\ \hfill & \times \text{exp}\left[-\left({{x}_{1}}^{2}+{{x}_{2}}^{2}\right)/{{w}_{0}}^{2}-iku\left({x}_{1}-{x}_{2}\right)/R\right].\hfill \end{array}$$
(6)
$$\begin{array}{l}\u3008\text{exp}\left[i\varphi \left({x}_{1}\right)-i\varphi \left({x}_{2}\right)\right]\u3009\\ \phantom{\rule{1em}{0ex}}\sim \text{exp}\left({{-\sigma}_{\varphi}}^{2}\right)\{1+\left[\text{exp}\left({{\sigma}_{\varphi}}^{2}\right)-1\right]\\ \phantom{\rule{1em}{0ex}}\times \text{exp}\left[-g\left({{\sigma}_{\varphi}}^{2}\right){\left({x}_{1}-{x}_{2}\right)}^{2}/{{r}_{0}}^{2}\right]\},\end{array}$$
(7)
$$g\left(x\right)=1+0.313x+0.0887{x}^{2}.$$
(8)
$$\begin{array}{ll}I\left(u\right)=\hfill & {{\pi w}_{0}}^{2}\phantom{\rule{0.2em}{0ex}}\text{exp}\left({{-\sigma}_{\varphi}}^{2}\right)\{\text{exp}\left[-{\left(k{w}_{0}u/R\right)}^{2}/2\right]\hfill \\ \hfill & +{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{exp}\left[-{\left(k{w}_{0}u/R\right)}^{2}/2\right]/\hfill \\ \hfill & \times 2*\left[\delta \left(u-2\pi R/kp\right)+\delta \left(u+2\pi R/kp\right)\right]\hfill \\ \hfill & +\left[\text{exp}\left({{\sigma}_{\varphi}}^{2}\right)-1\right]{{r}_{0}}^{\prime}/\sqrt{2}{w}_{0}\phantom{\rule{0.2em}{0ex}}\text{exp}\left[-{\left({{kr}_{0}}^{\prime}u/R\right)}^{2}/4\right]\},\hfill \end{array}$$
(9)
$${{r}_{0}}^{\prime}={r}_{0}/{\left[g\left({{\sigma}_{\varphi}}^{2}\right)\right]}^{1/2}.$$
(10)
$$\begin{array}{ll}C\left(X\right)\hfill & =FT\left[I\left(u\right)\right]\hfill \\ \hfill & =\text{exp}\left(-{Z}^{2}/{{2w}_{0}}^{2}\right)+{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi Z/p\right)\text{exp}\left(-{Z}^{2}/{{2w}_{0}}^{2}\right)\hfill \\ \hfill & \phantom{\rule{0.4em}{0ex}}+\left[\text{exp}\left({{\sigma}_{\varphi}}^{2}\right)-1\right]\text{exp}\left(-{Z}^{2}/{{r}_{0}}^{\prime 2}\right),\hfill \end{array}$$
(11)
$$Z=\mathrm{\lambda}RX$$
(12)
$$C\left(Z\right)=1+{a}^{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\pi Z/p\right)+{{\sigma}_{\varphi}}^{2}\phantom{\rule{0.2em}{0ex}}\text{exp}\left(-{Z}^{2}/{{r}_{0}}^{2}\right).$$
(14)
$${\sigma}_{h}={\sigma}_{\varphi}/2k.$$
(15)
$$\begin{array}{l}R\left({x}_{1},{x}_{2}\right)=\u3008\text{exp}\left[i\varphi \left({x}_{1}\right)-i\varphi \left({x}_{2}\right)\right]\u3009\\ \phantom{\rule{1em}{0ex}}=\text{exp}\left\{-\left[{\u3008\varphi \left(0\right)\u3009}^{2}-{\u3008\varphi \left({x}_{1}\right)\varphi \left({x}_{2}\right)\u3009}^{2}\right]\right\},\end{array}$$
(16)
$$\u3008\varphi \left({x}_{1}\right)\varphi \left({x}_{2}\right)\u3009={{\sigma}_{\varphi}}^{2}\phantom{\rule{0.2em}{0ex}}\text{exp}\left[-{\left({x}_{1}-{x}_{2}\right)}^{2}/{{r}_{0}}^{2}\right],$$
(17)
$$\begin{array}{ll}R\left({x}_{1},{x}_{2}\right)=\hfill & R\left({x}_{1}-{x}_{2}\right)\hfill \\ \hfill & \sim \text{exp}\left({{-\sigma}_{\varphi}}^{2}\right)\{1+\left[\text{exp}\left({{\sigma}_{\varphi}}^{2}\right)-1\right]\hfill \\ \hfill & \times \text{exp}\left[-g\left({{\sigma}_{\varphi}}^{2}\right){\left({x}_{1}-{x}_{2}\right)}^{2}/{{r}_{0}}^{2}\right]\},\hfill \end{array}$$
(18)
$$g\left(x\right)=x/\left[1-\text{exp}\left(-x\right)\right].$$
(19)
$$g\left(x\right)=1+0.313x+0.0887{x}^{2}$$