## 1.

## Introduction

One of the central problems in the industry of paper is the roughness of the surface of the paper that is a very important parameter in the manufacture of paper.^{1, 2} The roughness parameter most commonly used is the arithmetic mean [Eq. 1] of the absolute values of the heights measured from the centerline (Fig. 1).

Mathematically, it is very simply defined in terms of the surface height variations *δh*_{i} measured from the mean surface level:

## 1

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} R_a = \frac{1}{N}\sum\limits_{i = 1}^N {\left| {\delta h_i } \right|}. \end{equation}\end{document} $${R}_{a}=\frac{1}{N}\sum _{i=1}^{N}\left|\delta {h}_{i}\right|.$$## 2

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \sigma _R = \left[ {\frac{1}{N}\sum\limits_{i = 1}^N {\delta h_i^2 } } \right]^{1/2}. \end{equation}\end{document} $${\sigma}_{R}={\left[\frac{1}{N}\sum _{i=1}^{N}\delta {h}_{i}^{2}\right]}^{1/2}.$$*R*

_{a}and σ

_{R}. However, this will depend on the particular statistical distribution of surface height present in any given case. In the industry, the relationship usually assumed is

^{3}The multiplying factor will, however, be larger for surfaces with abnormally high numbers of large deviations from the mean surface height.

## 2.

## Measurement of the Surface Roughness

Researches have been investigated and developed techniques of contact and noncontact suitable for measuring the roughness of the paper. Among the most common measurement techniques are: airflow, profilometry (mechanical and optical), interferometry,^{4} atomic force microscopy,^{5} confocal scanning microscopy techniques,^{6} optical correlation,^{7, 8, 9} triangulation, and speckle metrology. In general, many of these techniques can be used only in laboratory conditions, i.e., outside the industrial manufacturing process.

## 2.1.

### Air Leak Methods

Air leak methods currently are the standard test for measuring surface roughness. The air leak rate between the paper surface being measured and a specific flat surface is recorded using specialized pneumatic devices. Methods differ in the pressure on which the measuring surface is pressed to paper, measured quantity (time, volume), softness of the flat surface, or measured area. Bendtsen roughness is achieved by clamping the test piece between a flat glass plate and a circular metal surface and measuring the rate of airflow between the paper and the metal surface.^{10, 11} Bekk smoothness is again measured by the air leak method but, unlike the previous instrument, air is drawn across the surface of the test piece under partial vacuum.^{12}

## 2.2.

### Profilometers

Profilometers measure the actual topography of a surface. This method of measuring surface roughness is to pass a mechanical or optical stylus probe across the surface and measure its movement as it follows the surface profile. This technique has been developed to a very sophisticated level and can achieve surprisingly high precision.^{13} However, there are limits set by the finite radius of the tip of the probe; for example, peaks that are narrower than the probe tip will still be followed by the probe but will be recorded as being broader than they really are, while the probe will be physically incapable of reaching the bottom of the narrow fissures. In other words, some of the surface height information is being filtered out. Results also seem to depend on the actual shape of the stylus and on its loading. In addition, many applications as in paper are unsuitable for contact techniques, which could damage the surface. Finally, profilometers are too slow for online use.

## 2.3.

### Speckle

Surface roughness measurement can also be accomplished by a speckle-based instrument. A surface speckle pattern, which is a grainy structure produced by scattered light from a rough surface when illuminated by coherent light, contains rich information about the surface roughness.

With the advent of lasers in the 1960s, researchers have discussed the relationship between surface roughness and the statistical properties of speckle pattern. We have seen that speckle metrology has the potential to obtain measurements of the surface roughness.^{3, 14, 15}

Researches have been developed and used different methods to study the surface roughness by speckle techniques. These are based on the study of speckle contrast^{16} and others use the speckle correlation,^{17, 18, 19, 20} which have been shown to give good results, for studying metal surfaces, dielectric fluids, and paper.

## Table 1

Results of the measurement of the roughness obtained with confocal microscope, Bendtsen method, sum of the values of μa (r > 20) and the equivalent roughness for 14 papers.

No. | Ra(Confocal) (micres) | Bendtsen (ml/s) | (∑μa(r > 20)10−21 | σ = [ln(1/∑μa)]1/2 |
---|---|---|---|---|

1 | 6,5 | 1150 | 0,0284 | 2,45 |

2 | 6,2 | 1100 | 0,0261 | 2,44 |

3 | 6,1 | 700 | 0,0284 | 2,42 |

4 | 6 | 750 | 0,0304 | 2,41 |

5 | 6 | 750 | 0,0256 | 2,44 |

6 | 5,7 | 850 | 0,0282 | 2,42 |

7 | 5,5 | 800 | 0,0333 | 2,39 |

8 | 4,2 | 75 | 0,0366 | 2,37 |

9 | 4,3 | 80 | 0,0373 | 2,36 |

10 | 3,7 | 190 | 0,0458 | 2,32 |

11 | 3,4 | 120 | 0,0508 | 2,30 |

12 | 3,4 | 60 | 0,0523 | 2,29 |

13 | 3,4 | 55 | 0,0531 | 2,29 |

14 | 3,3 | 125 | 0,0501 | 2,30 |

We multiply by 10−2 to avoid negative numbers in the logarithm.

## 3.

## Theory and System Set-up Configuration. Computation of the Normalized Autocorrelation Function *μ*_{a}

When we project a monochromatic plane wave over a surface that is assumed to be rough and its correlation function depends only on the differences of measurement coordinates, then the relationship between the height variations of the surface and the amplitude variations of the scattered wave is, in general, an extremely complex one, influenced by variations of surface slope on reflection, multiple scattering, and shadowing. We adopt an oversimplified model that implies that the scattered complex amplitude just above the surface is related to the surface height by a geometrical approximation that assigns a phase φ to the scattered complex amplitude that is the phase delay associated with propagating to the surface and scattering from the surface.

Then

## 4

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \varphi (\alpha,\beta) = \frac{{2\pi }}{\lambda }( - \vec i.\vec n + \vec o.\vec n)h(\alpha,\beta) \end{equation}\end{document} $$\varphi (\alpha ,\beta )=\frac{2\pi}{\lambda}(-\stackrel{\u20d7}{i}.\stackrel{\u20d7}{n}+\stackrel{\u20d7}{o}.\stackrel{\u20d7}{n})h(\alpha ,\beta )$$*h*(α, β) is the surface height, λ is the wavelength and [TeX:] $\vec i$ $\stackrel{\u20d7}{i}$ , [TeX:] $\vec o$ $\stackrel{\u20d7}{o}$ , and [TeX:] $\vec n$ $\stackrel{\u20d7}{n}$ are the unitary vectors of incidence, reflection, and normal to the surface, see Fig. 2. With this approximation the variance of the phase shifts σ

_{φ}is related to the variance of the surface height fluctuations

*σ*

_{h}through

^{17}

## 5

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \sigma _\varphi ^2 = \left[ {\frac{{2\pi }}{\lambda }( - \vec i.\vec n + \vec o.\vec n)} \right]^2 \sigma _h^2. \end{equation}\end{document} $${\sigma}_{\varphi}^{2}={\left[\frac{2\pi}{\lambda}(-\stackrel{\u20d7}{i}.\stackrel{\u20d7}{n}+\stackrel{\u20d7}{o}.\stackrel{\u20d7}{n})\right]}^{2}{\sigma}_{h}^{2}.$$## 6

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \mu _a (\Delta \alpha,\Delta \beta) = \frac{{\Gamma _a (\alpha _1,\beta _1,\alpha _2,\beta _2)}}{{\left| r \right|^2 I_{inc} }}, \end{equation}\end{document} $${\mu}_{a}(\Delta \alpha ,\Delta \beta )=\frac{{\Gamma}_{a}({\alpha}_{1},{\beta}_{1},{\alpha}_{2},{\beta}_{2})}{{\left|r\right|}^{2}{I}_{inc}},$$_{a}is the autocorrelation function of the field,

*r*is the average amplitude reflectivity of the surface, and

*I*

_{inc}is the intensity incident on the scattering area.

Making the assumptions that φ(α, β) is a Gaussian random variable and that the surface height correlation function is also Gaussian, then it can be shown^{17} that:

## 7

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \mu _a (r) = \exp \left[ { - \sigma _\varphi ^2 (1 - e^{ - (r/r_c)^2 })} \right], \end{equation}\end{document} $${\mu}_{a}\left(r\right)=\mathrm{exp}\left[-{\sigma}_{\varphi}^{2}(1-{e}^{-{(r/{r}_{c})}^{2}})\right],$$*r*

_{c}is a constant. The greater σ

_{φ,}more rapidly

*μ*

_{a}will tend to zero as

*r*grows. So, we can relate the sum of the values of

*μ*

_{a}far from the center (

*r*> 20) to σ

_{φ}through Eq. 7 and we may define σ as:

## 8

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{equation} \sigma = \sqrt {\ln \left( {\frac{1}{{\sum\nolimits_{r > 20} {} \mu _a (r)}}} \right)}. \end{equation}\end{document} $$\sigma =\sqrt{\mathrm{ln}\left(\frac{1}{{\sum}_{r>20}{\mu}_{a}\left(r\right)}\right)}.$$## 3.1.

### Calculation of μ_{a}

The Van Cittert–Zernike theorem is a result of the coherence theory that states that under certain conditions the Fourier transform of the mutual coherence function of a distant, incoherent source is equal to its intensity.

When applied to the fields leaving the scattering surface of paper, the Van Cittert–Zernike theorem^{17} provides us with a connection between the normalized autocorrelation function of the fields leaving the scattering surface of paper and the average intensity distribution observed in the focal plane of a positive lens

## 9

[TeX:] \documentclass[12pt]{minimal}\begin{document}\begin{eqnarray} \overline {I(x,y)} \propto \int\nolimits_{ - \infty }^\infty \!{\int {\mu _a } (\Delta \alpha,\Delta \beta)e^{ - j(2\pi (x\Delta \alpha + y\Delta \beta)/\lambda f)} d\Delta \alpha\, d\Delta \beta } \!\!\!\!\nonumber\\ \end{eqnarray}\end{document} $$\begin{array}{c}\hfill \overline{I(x,y)}\propto {\int}_{-\infty}^{\infty}\int {\mu}_{a}(\Delta \alpha ,\Delta \beta ){e}^{-j\left(2\pi \right(x\Delta \alpha +y\Delta \beta )/\lambda f)}d\Delta \alpha \phantom{\rule{0.16em}{0ex}}d\Delta \beta \end{array}$$*μ*

_{a}is the normalized autocorrelation function,

*f*is the focal length of the positive lens used, and λ is the wavelength of the laser used. Thus, if [TeX:] $\overline {I(x,y)}$ $\overline{I(x,y)}$ is carefully measured, using a detector that is large enough to average over many individual speckles, a normalized inverse Fourier transform of the intensity distribution would provide

*μ*

_{a}at the scattering surface.

## 4.

## System Set-up and Configuration

The system used for measurement consisted of the following elements (Fig. 3): A He–Ne laser of 5 mW with a wavelength of 632.8 nm, a charge-coupled device (CCD) camera UNIQ UM-301, 752 × 582 pixels and a beam expander. The camera is located in the direction normal to the sample. To form the speckle pattern, subjective speckle is used.^{6, 13, 15} The field of view is 10 × 10 mm. The format of the images was 200×200 pixels, and 256 gray levels.

The laser beam is expanded and projected on paper, with an angle Θ = 15 deg from normal, obtaining a speckle pattern of about 1 cm in diameter. Figure 4 shows one of the images obtained. Computer programs used for capturing and analyzing images of the speckle pattern are: Matrox Imaging (Intellicam module) and MATLAB (Digital Image Processing Toolbox).

## 5.

## Results

We characterized the roughness of 14 different types of paper with *R*_{a} between 3.3 and 6.5 *μ*m with similar optical proprieties using an optical method [a confocal microscope optical three-dimensional profilometer operating in confocal mode, an air leak method (Bendtsen), and the sum of the values of μ_{a} (*r* > 20)]. In Fig. 5 we show the values of *μ*_{a} (*r* > 20) for three different papers. In Fig. 6 we show the relationship between *R*_{a} (Confocal) and Bendtsen, and in Fig. 7 we show the comparison between the measurements using confocal microscope and our method

## 6.

## Conclusions

In this work, we have developed a method to find the roughness of paper using speckle. Using the proprieties of the normalized covariance function that we obtain from the image of the speckle through the inverse Fourier transform, we relate the values of the normalized covariance function far from the center with the roughness of the paper. In order to test the method we have characterized the roughness of 14 papers in the range between 3.3 and 6.5 μm using a confocal microscope. The results present a good correlation between them (*R*^{2} = 0.9545). The results are better than the comparison between a standard method (Bendtsen) and the confocal microscope. The method is easy to implement and can be used during the manufacturing process.

## References

## Biography

**Abdiel Pino** obtained a BSc degree in teaching physics from the University of Panama in 2004. He received a scholarship in the Scholarships Program of Professional Excellence SENACYT-IFARHU by the Panama's government, to be enrolled in the PhD program in optical engineering at Technical University of Catalonia. He has participated in national and international congresses and also in publications in journals. Currently, he is developing a PhD thesis in the area of optical metrology and image processing.

**Josep Pladellorens** received his MS in physics and his PhD degree in physics from Barcelona University and Autonomous University of Barcelona in 1983 and 1992, respectively. He is currently head of the Department of Optical and Opthometry, of Technical University of Catalonia in Terrassa, Spain. His research interests include optical metrology, imaging processing technology, and interferometry.

**Oriol Cusola** received his BS and MS degrees in paper and graphic engineering from the Technical University of Catalonia in 2008 and 2010, respectively. He is currently a PhD candidate and associate professor in the Department of Textile and Paper Engineering, Technical University of Catalonia. His research interests include papermaking technology, paper surface biomodification, printing technology, and surface physical chemistry.

**Jesus Caum** received his MS degree in electronic engineering and a PhD degree in optical engineering from Technical University of Catalonia in 1998 and 2010, respectively. He is currently a professor in the Department of Optical and Opthometry, of Technical University of Catalonia in Terrassa, Spain. His research interests include optical metrology, imaging processing technology, and interferometry.