Measurement of surfaces topography

Noël Brunetière

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Noël Brunetière. Measurement of surfaces topography. Doctoral. France. 2022. hal-03710660v2

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Measurement of surfaces topography

Noël Brunetière<fnref n="1" />

Institut Pprime - Poitiers

November 4, 2023

<figure><img src="image_2.png" /></figure>

Labex
Interactifs

<footnote n="1">¹noel.brunetiere@univ-poitiers.fr</footnote>
# Outline

① Introduction

② Description of the surface roughness

③ Roughness parameters
* Definition of the usual parameters
* Influence of the machining process

④ Optical instruments for surface topography
* Some basics of optics
* Interferometry
* Phase shifting interferometry
* Vertical scanning interferometry
* Focus variation

⑤ Other instruments

⑥ Comparison of instruments

⑦ Summary

⑧ Application
# Outline

1 **Introduction**

2 Description of the surface roughness

3 Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4 Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5 Other instruments

6 Comparison of instruments

7 Summary

8 Application
# Why measuring surface topography?

The topography of the surfaces will affect many physical phenomena and surface functions:

* light reflexion

* contact area between solids

* adhesion

* electricity conduction

* heat conduction

* friction

* tactile perception

* liquid spreading on the surface (wettability)

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>2</figcaption></figure>
# Why measuring surface topography?

* **Control and verify the surface:** According to Leach[Lea11], it is essential to measure surface texture: "Surface texture plays a vital role in the functionality of a component. It is estimated that surface effects causes 10% of manufactured parts to fail and can contribute significantly to an advance nation's GDP (Gross Domestic Product)"

* **Check the surface morphology change with time**

<figure><img src="image_7.png" /><figcaption>Figure: New surface</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure: Surface after friction</figcaption></figure>
Content of the course:

* Description for the surface roughness: the different types of surface deviation according to their scales and what will be measured.

* Roughness parameters: the parameters to describe roughness, identify different surfaces topography, link of the parameters with the production method

* Optical instruments for surface topography: different optical methods (mainly interferometry)

* Other instruments: other instruments available at Pprime

* Comparison of instruments: limits of the different instruments and influence on measurements results.
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4. Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
# Topography of surfaces: case of profiles

* Due to the production processes, the conditions of use, the wear, the surfaces exhibit defects which are characterized by deviations in altitude from their theoretical profile;

* Defects are generally identified according to their wavelength *L*:

* *L* ~ surface size: shape defect

* *L* ~ 0.01 to 0.001*m* : waviness

* *L* < 0.001*m* : roughness.

<figure><img src="image_9.png" /></figure>
Example of surfaces measured with different instruments

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

Figure: The surface is digitized by the instrument: a given number of points equally spaced and on given area (or length)

<figure><img src="image_12.png" /></figure>

Figure: Same surface with: a) White Light Interferometer: z scale = 5.8 µm, L = 0.9mm, Δx = 0.9µm, b) Atomic Force Microscope: z scale = 3 µm, L = 14.8µm, Δx = 0.029µm
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters
* Definition of the usual parameters
* Influence of the machining process

4. Optical instruments for surface topography
* Some basics of optics
* Interferometry
* Phase shifting interferometry
* Vertical scanning interferometry
* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
# Topography of surfaces: case of profiles

* Profile: height z as a function of the horizontal length x;

* CDF = Cumulative Distribution Function: ratio of material above a cutting line of height z;

<figure><img src="image_13.png" /></figure>
Topography of surfaces: case of profiles

* PDF = Probability Density Function ∝ derivative with respect to /z of the CDF

* For most rough surfaces, the probability density follows a Gaussian law: :

$$
f(z) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(z-\mu)^2}{2\sigma^2}\right];
$$

* μ is the mean and σ the standard deviation (Rq in ISO metrology);

* The profile points are in the range -3σ to +3σ.

<figure><img src="image_14.png" /></figure>
Height statistical parameters: case of profiles

* The profile is measured by a device which gives a series of *n* points *z*ᵢ laterally spaced of Δx;

* Height (m or µm): Arithmetic average $Ra = \frac{1}{n} \sum_{1}^{n} |z_{i}|$

* Height (m or µm): Quadratic average $Rq = \sqrt{\frac{1}{n} \sum_{1}^{n} z_{i}^{2}}$ is equivalent to $\sigma$;

* Skewness parameter (non-dimensional): $RSk = \frac{1}{nRq^3} \sum_{1}^{n} z_{i}^{3}$;

* Kurtosis parameter (non-dimensional): $RKu = \frac{1}{nRq^4} \sum_{1}^{n} z_{i}^{4}$;

<figure><img src="image_15.png" /></figure>
Height statistical parameters: case of profiles

* Due to the manufacturing process, surfaces can be Non-Gaussian.

* Gaussian surface : $Rq \approx \sqrt{\frac{\pi}{2}}Ra \approx 1.25Ra \quad RSk = 0 \quad RKu = 3$

* Non Gaussian surface (process with pics removal): $RSk < 0$ et $RKu > 3$.

<figure><img src="image_16.png" /></figure>
Lateral size of asperities: case of profiles

* ACF = Auto Correlation Function: degree of resemblance of a surface to itself shifted by x;

* $FAC(x = j\Delta x) = \frac{1}{nRq^2} \sum_{i=1}^{n} z_i \times z_{i+j}$

* For many rough surfaces, it is a decreasing function tending to zero.

* The correlation length $\lambda$ is the characteristic size of asperities.

* Is it the horizontal distance to reach ACF = 0.2.

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>
Roughness characterization: PSD

* PSD = Power Spectrum Density is the Fourier transform of the ACF.

* Distribution of the space frequencies;

* Lowest frequency $\frac{1}{L}$ where L is the sample length;

* Highest frequency $f_M = \frac{1}{2\Delta x}$ where $\Delta x$ is the sampling length.

* The slope changes occurs at $\frac{1}{\lambda}$

* The area below the curve corresponds to $\frac{R_q^2}{2}$

<figure><img src="image_21.png" /></figure>
From profiles to surfaces

The roughness parameters are defined in the ISO standard. Many metrological tools measure the surface (and not the profile). Surface parameters have the same definition but different name:

<table><thead><tr><td>Parameter</td><td>Profile</td><td>Surface</td></tr></thead><tbody><tr><td>Arithmetic average</td><td>Ra</td><td>Sa</td></tr><tr><td>Quadratic average</td><td>Rq</td><td>Sq</td></tr><tr><td>Skewness</td><td>RSk</td><td>SSk</td></tr><tr><td>Kurtosis</td><td>RKu</td><td>SKu</td></tr><tr><td>Correlation length (lowest value)</td><td>λ</td><td>Sal</td></tr><tr><td>Correlation length (biggest value)</td><td>λ</td><td>Sal/Str</td></tr></tbody></table>
Usual machining process

<figure><img src="image_22.png" /><figcaption>Milling</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Grinding</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Honing and polishing</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Electroerosion</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Molding</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>3D printing</figcaption></figure>

Figure: Some usual surfaces production methods
Impact of machining on topography

Impact of the machining process on the roughness height Sq (WLI, L = 0.9mm, Δx = 0.9μm):

<figure><img src="image_28.png" /></figure>
Impact of machining on topography

The high surface finish (or low roughness height level) corresponds to processes that require more energy [Has+16]. The amount of energy is inversely proportional to Ra:

<figure><img src="image_29.png" /></figure>
Impact of machining on topography

Impact of the machining process on the Gaussian nature of the surface:

<figure><img src="image_30.png" /></figure>
Examples of surfaces

<figure><img src="image_31.png" /></figure>

<table><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>4.38</td></tr><tr><td>Sa (μm)</td><td>3.55</td></tr><tr><td>SSk</td><td>0.60</td></tr><tr><td>SKu</td><td>2.793</td></tr><tr><td>Sal (μm)</td><td>37</td></tr><tr><td>Str</td><td>0.56</td></tr></tbody></table>

Figure: Surface obtained by electrorosion (z × 5), WLI, L = 0.9mm, Δx = 0.9μm.

<figure><img src="image_32.png" /></figure>

<table><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>3.25</td></tr><tr><td>Sa (μm)</td><td>2.63</td></tr><tr><td>SSk</td><td>-0.04</td></tr><tr><td>SKu</td><td>2.51</td></tr><tr><td>Sal (μm)</td><td>93.8</td></tr><tr><td>Str</td><td>&lt; 0.2</td></tr></tbody></table>

Figure: Surface obtained by rough milling (z × 5), WLI, L = 0.9mm, Δx = 0.9μm.
Examples of surfaces

<figure><img src="image_33.png" /></figure>

<table><caption>Figure: Surface obtained by fine milling (z × 5), WLI, L = 0.9mm, Δx = 0.9μm.</caption><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>1.08</td></tr><tr><td>Sa (μm)</td><td>0.89</td></tr><tr><td>SSk</td><td>0.034</td></tr><tr><td>SKu</td><td>2.49</td></tr><tr><td>Sal (μm)</td><td>24.3</td></tr><tr><td>Str</td><td>&lt; 0.2</td></tr></tbody></table>

<figure><img src="image_34.png" /></figure>

<table><caption>Figure: Surface obtained by honing (z × 5), WLI, L = 0.9mm, Δx = 0.9μm.</caption><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>0.912</td></tr><tr><td>Sa (μm)</td><td>0.709</td></tr><tr><td>SSk</td><td>-0.24</td></tr><tr><td>SKu</td><td>3.93</td></tr><tr><td>Sal (μm)</td><td>4.54</td></tr><tr><td>Str</td><td>0.7</td></tr></tbody></table>
Examples of surfaces

<figure><img src="image_35.png" /></figure>

<table><caption>Figure: Surface obtained by polishing (z × 500), WLI, L = 0.9mm, Δx = 0.9μm.</caption><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>0.028</td></tr><tr><td>Sa (μm)</td><td>0.022</td></tr><tr><td>SSk</td><td>-0.92</td></tr><tr><td>SKu</td><td>4.09</td></tr><tr><td>Sal (μm)</td><td>44.8</td></tr><tr><td>Str</td><td>0.35</td></tr></tbody></table>

<figure><img src="image_36.png" /></figure>

<table><caption>Figure: Surface obtained by mirror polishing (z × 500), WLI, L = 0.9mm, Δx = 0.9μm.</caption><thead><tr><td>Paramètre</td><td>Valeur</td></tr></thead><tbody><tr><td>Sq (μm)</td><td>0.0071</td></tr><tr><td>Sa (μm)</td><td>0.0057</td></tr><tr><td>SSk</td><td>-0.09</td></tr><tr><td>SKu</td><td>3.14</td></tr><tr><td>Sal (μm)</td><td>23.22</td></tr><tr><td>Str</td><td>0.6</td></tr></tbody></table>
Can surfaces be fractal ?

* A fractal surface: $PSD \propto f^{2D-5}$ where D is the fractal dimension (between 1 and 2 for a profile)

* According to Whitehouse[Whi01], $\lambda$ or $Sal$ is linked to the process (density of grain during polishing, etc)

* A surface appears to be fractal if $L < \lambda$

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>
Can surfaces be fractal ?

<figure><img src="image_39.png" /><figcaption>Figure: Engineered surface: slope change at 1/λ</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>Figure: Natural material (fractured granite): fractal[PD97]</figcaption></figure>
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4. Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
# Some basics of optics

Monochromatic light wave (in air): $s = a \cos[\frac{2\pi}{\lambda}(ct - z) + \phi]$

* c speed of light

* λ wave length

* t time, z position

* φ phase

<figure><img src="image_41.png" /></figure>
# Some basics of optics

* f focal length

* Magnification $M = \frac{b}{a}$

<figure><img src="image_42.png" /></figure>
# Some basics of optics

* $A_N$ Numerical aperture of a microscope objective

* $n$ refractive index of the media ($n = 1$ in air)

* $A_N = n \sin \alpha$

* This parameter will affect the performance of the objective (working distance $W_d$ and others)

<figure><img src="image_43.png" /></figure>
# Some basics of optics

* Spatial resolution $r$ = minimum distance between two features [Lea11]

* $r = 0.61 \frac{\lambda}{A_N}$ (Rayleigh)

* $r = 0.82 \frac{\lambda}{A_N}$ (Sparrow)

* Depth of field $Z$ = depth along with the object is in focus [Lea11]

* $Z = \frac{n\lambda}{A_N^2}$

The numerical aperture has a strong impact on the optical performance

<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>
# Michelson Interferometer

● $L_2 = L_1 + d$

Beam 1:

$$s_1 = a \cos \left[ \frac{2\pi}{\lambda} (ct - z) + \phi \right]$$

Beam 2:

$$s_2 = a \cos \left[ \frac{2\pi}{\lambda} (ct - z - 2d) + \phi \right]$$

● Resulting beam: $s = s_1 + s_2 = 2a \cos \left[ \frac{2\pi}{\lambda} (ct - z - d) \right] \cos \left( \frac{2\pi}{\lambda} d \right)$

● Light intensity:

● $I = \frac{1}{T} \int_{0}^{T} s^2 dt = 2a^2 \cos^2 \left( \frac{2\pi}{\lambda} d \right)$

● $I = a^2 \cos \left( \frac{4\pi}{\lambda} d \right) + a^2$

<figure><img src="image_46.png" /></figure>

Figure:

https://commons.wikimedia.org/wiki/File:

Michelson Interferometer scheme.png,
Polytec GmbH, CC BY-SA 3.0 via

Wikimedia Commons
# Michelson Interferometer

● $L_2 = L_1 + d$

● Beam 1:

$$s_1 = a_1 \cos \left[ \frac{2\pi}{\lambda} (ct - z) + \phi \right]$$

● Beam 2:

$$s_2 = a_2 \cos \left[ \frac{2\pi}{\lambda} (ct - z - 2d) + \phi \right]$$

● Light intensity $I = \frac{1}{T} \int_{O}^{T} s^2 dt$ with
$s = s_1 + s_2$

$$I = a_1 a_2 \cos \left( \frac{4\pi}{\lambda} d \right) + 0.5 (a_1^2 + a_2^2)$$

● It $a_1 \ll a_2$, there is a problem of signal detection and accuracy

<figure><img src="image_47.png" /></figure>

Figure:

https://commons.wikimedia.org/wiki/File:Michelson Interferometer scheme.png, Polytec GmbH, CC BY-SA 3.0 via Wikimedia Commons
# Michelson Interferometer

* If the mirror is moved, there will be alternate of dark and light signal on the screen

* The wave length is $λ/2$

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

Figure:

https://commons.wikimedia.org/wiki/File:

Michelson Interferometer scheme.png,

Polytec GmbH, CC BY-SA 3.0 via

Wikimedia Commons
# Michelson Interferometer

* If one mirror is spherical, Newton's ring are obtained

* The lines are level lines with a height difference of $λ/2$ (with $λ$ from 380 to 780 nm for visible light.)

* Is it convex or concave shape ?

<figure><img src="image_50.png" /></figure>

Figure: Interferences on ceramic ball (from a ball bearing)
# Michelson Interferometer with white light

* White light has a spectrum of wave lengths (380–780 nm).

* If the mirror is moved, there will be alternate of dark and light signal on the screen but with decreasing amplitude (on spatial range of a few microns)

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

Figure: White light through a prism (https://upload.wikimedia.org/wikipedia公共领域共享/1.0/White-light%20through%20a%20prism.png) dispersion of a mercury vapor lamp with a flint glass prism IPNr 0125.jpg, D-Kuru, CC BY-SA 3.0 via Wikimedia Commons)
Phase shifting interferometry (monochromatic light)

* The real position is find by imposing controlled displacement $u_i$ (using a piezo actuator or an accurate device)

* $I = a^2 \cos(\theta + \phi_i) + a^2$

* $\theta = \frac{4\pi}{d}$ and $\phi_i = \frac{4\pi}{\lambda}u_i$

* It is possible to calculate $\theta$ (with an unknown constant $k\pi$) and then $d$. Unwrapping is necessary

Position is find from relative position to neighbor points

<figure><img src="image_53.png" /><figcaption>Figure: Interferences on ceramic ball at different $\phi$ values</figcaption></figure>
# Phase shifting interferometry (monochromatic light)

## Advantage

* Very fast measurement (limited number of images)

* Vertical accuracy of less than 0.1 *nm*, independent of magnification

## Disadvantages:

* Limited to flat surface (must be in focus everywhere)

* Limited to smooth surface (*Sq* < 30 nm), otherwise interference fringes are not visible.

* Does not work with steps > λ/2

<figure><img src="image_54.png" /></figure>

Figure: Example of surface adapted to phrase shifting interferometry: hard drive disk surface
# Vertical scanning interferometer (white light)

White light interferometer:

* Large vertical displacement (compared to phase shifting)

* Full interferogram for each pixel

* Height calculation independent of the neighbors

<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>
# Vertical scanning interferometer (white light)

## Advantage

* Vertical range not limited by depth of field

* Works with rough and stepped surfaces

## Disadvantages:

* More images to record and analyse

* Slower than phase shifting

Position is independent of relative position to neighbor points

<figure><img src="image_57.png" /></figure>

Figure: Interferences on ceramic ball during vertical scanning measurement
Interferometry lens

<table><caption>Table: Lens of the Talysurf CCI with δz < 0.1 nm</caption><thead><tr><th>Lens</th><th>A<sub>N</sub></th><th>W<sub>d</sub> (mm)</th><th>L (mm)</th><th>Δx (μm)</th><th>r (μm)</th><th>Z (μm)</th><th>Max slope</th></tr></thead><tbody><tr><td>5x</td><td>0.13</td><td>9.3</td><td>3.6</td><td>3.6</td><td>2.12</td><td>20</td><td>3.5°</td></tr><tr><td>10x</td><td>0.30</td><td>7.4</td><td>1.8</td><td>1.8</td><td>0.92</td><td>3.73</td><td>8°</td></tr><tr><td>20x</td><td>0.40</td><td>4.7</td><td>0.9</td><td>0.9</td><td>0.69</td><td>2.10</td><td>14.5°</td></tr><tr><td>50x</td><td>0.55</td><td>3.6</td><td>0.36</td><td>0.36</td><td>0.50</td><td>1.11</td><td>22°</td></tr></tbody></table>
Focus variation

* A focus variation microscope uses confocal light and actuator for the lens

* The light reflected will vary depending on the focus level

<figure><img src="image_58.png" /><figcaption>Figure: Focus variation microscope</figcaption></figure>
Focus variation

<figure><img src="image_59.png" /><figcaption>Figure: Effect of displacement of the objective on the image</figcaption></figure>
# Focus variation

* Identification of the maximum by curve fitting

* Vertical resolution dependent on the depth of field Z

* Necessity to use neighbor points information

<figure><img src="image_60.png" /></figure>
Focus variation lens

<table><caption>Table: Lens of the Alicona Infinte focus (Low vertical resolution but high surface slope limit)</caption><thead><tr><th>Lens</th><th>A<sub>N</sub></th><th>Δx (μm)</th><th>r (μm)</th><th>Z (μm)</th><th>Δz (nm)</th><th>Max slope</th></tr></thead><tbody><tr><td>10x</td><td>0.30</td><td>1</td><td>1.09</td><td>8.2</td><td>100</td><td>87°</td></tr><tr><td>20x</td><td>0.40</td><td>0.5</td><td>0.82</td><td>4.6</td><td>50</td><td>87°</td></tr><tr><td>50x</td><td>0.6</td><td>0.2</td><td>0.54</td><td>2.05</td><td>20</td><td>87°</td></tr></tbody></table>
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4. Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
Stylus profiler:

* Vertical resolution $\Delta z = 1nm$;

* Lateral resolution $\Delta x = 2\mu m$ = curvature radius of the tip;

* Sample length $L_x = qqcm$;

* Requires profile or surface scan (can create scratches in soft materials)

<figure><img src="image_61.png" /></figure>
Atomic force microscope

* Vertical resolution $\Delta z < 0.1nm$;

* Lateral resolution $\Delta x \approx 1nm$ = curvature radius of the tip;

* Sample length $L_x < 0.1mm$;

* Requires profile or surface scan.

<figure><img src="image_62.png" /><figcaption>Figure: Left: Principle of an AFM - Right: AFM tip<fnref n="3" /></figcaption></figure>

<footnote n="3">³: https://commons.wikimedia.org/wiki/File:AFM_(used)_cantilever_in_Scanning_Electron_Microscope,_magnification_3000x.JPG: SecretDisc, derivative work: Materialscientist, CC BY-SA 3.0, via Wikimedia Commons</footnote>
# Outline

① Introduction

② Description of the surface roughness

③ Roughness parameters
* Definition of the usual parameters
* Influence of the machining process

④ Optical instruments for surface topography
* Some basics of optics
* Interferometry
* Phase shifting interferometry
* Vertical scanning interferometry
* Focus variation

⑤ Other instruments

⑥ Comparison of instruments

⑦ Summary

⑧ Application
Comparison of instrument

<figure><img src="image_63.png" /><figcaption>Figure: Usual limitations of optical and contacting measurement systems</figcaption></figure>
Comparison of instrument

<figure><img src="image_64.png" /><figcaption>Figure: Comparison of the ranges of measurement of instruments (including slope and radius limits) [Ste87]</figcaption></figure>
Comparison of instruments

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /><figcaption>Figure: WLI (White Light Interferometer),
$Sq = 0.93\mu m$, $L = 0.9mm$, $\Delta x = 0.9\mu m$</figcaption></figure>

<figure><img src="image_67.png" /><figcaption>Figure: AFM (Atomic Force Microscope),
$Sq = 0.37\mu m$, $L = 14.8\mu m$, $\Delta x = 0.029\mu m$</figcaption></figure>

Necessity to specify the instrument
(sample length and lateral sampling)
when giving $Sq$ values
Comparison of instrument

<figure><img src="image_68.png" /></figure>

Figure: Comparison of different instruments on a surface: black = stylus profiler, blue = AFM, red = Transmission Electron Microscope, figure from [Guj+18] (open access on https://d-scholarship.pitt.edu/37274/)
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4. Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
# Summary

* Solid surface = Interface between a solid and another media;

* The topography is affected by the machining process, wear, etc;

* The topography influences surface functions and physical phenomena;

* The surface topography is characterized by several parameters including $Rq$ or $Sq$ (roughness height) and $Sal$ (lateral scale of asperities);

* These parameters depend on the measuring instrument;

* The instruments are defined by their bandwidth of spatial frequencies;

* This information must be given with the surface roughness parameters;

* Optical instruments offer areal measurements but some artifacts due to light reflection are possible

* Contact instruments requires a sampling of the surface (long process)
# Outline

1. Introduction

2. Description of the surface roughness

3. Roughness parameters

* Definition of the usual parameters

* Influence of the machining process

4. Optical instruments for surface topography

* Some basics of optics

* Interferometry

* Phase shifting interferometry

* Vertical scanning interferometry

* Focus variation

5. Other instruments

6. Comparison of instruments

7. Summary

8. Application
Application

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /><figcaption>Figure: WLI (White Light Interferometer),
$Sq = 0.93\mu m$, $L = 0.9mm$, $\Delta x = 0.9\mu m$</figcaption></figure>

<figure><img src="image_71.png" /><figcaption>Figure: AFM (Atomic Force Microscope),
$Sq = 0.37\mu m$, $L = 14.8\mu m$, $\Delta x = 0.029\mu m$</figcaption></figure>

If the ACF is exponential, the PSD can be expressed: $PSD = \frac{2Rq_0^2\lambda}{1+4\pi^2\lambda^2f^2}$. Here $Rq_0 = 0.95\mu m$ and $\lambda = 6\mu m$
# Application

<figure><img src="image_72.png" /><figcaption>Figure: WLI (White Light Interferometer),
$Sq = 0.93\mu m$, $L = 0.9mm$, $\Delta x = 0.9\mu m$</figcaption></figure>

① We assume that $PSD = \frac{2Rq_0^2\lambda}{1+4\pi^2\lambda^2f^2}$

② Express the $Rq$ value for frequencies in the range $f_1$ and $f_2$. We use:

$$ \frac{Rq^2}{2} = \int_{f_1}^{f_2} PSDdf $$

③ Calculate the theoretical values of $Rq$ for the AFM and WLI and compare to the measured values.

<figure><img src="image_73.png" /><figcaption>Figure: AFM (Atomic Force Microscope),
$Sq = 0.37\mu m$, $L = 14.8\mu m$, $\Delta x = 0.029\mu m$</figcaption></figure>
[Guj+18] Abhijeet Gujrati et al. "Combining TEM, AFM, and Profilometry for Quantitative Topography Characterization Across All Scales". In: *ACS Applied Materials & Interfaces* 10.34 (2018). PMID: 30052425, pp. 29169-29178. DOI: 10.1021/acsami.8b09899.

[Has+16] Fukuo Hashimoto et al. "Abrasive fine-finishing technology". In: *CIRP Annals* 65.2 (2016), pp. 597-620. ISSN: 0007-8506. DOI: https://doi.org/10.1016/j.cirp.2016.06.003. URL: https://www.sciencedirect.com/science/article/pii/S0007850616301950.

[Lea11] R Leach, ed. *Optical Measurement of Surface Topography*. Springer, 2011.

[PD97] W.L. Power and W.B. Durham. "Topography of natural and artificial fractures in granitic rocks: Implications for studies of rock friction and fluid migration". In: *International Journal of Rock Mechanics and Mining Sciences* 34.6 (1997), pp. 979-989. ISSN: 1365-1609. DOI: https://doi.org/10.1016/S1365-1609(97)80007-X. URL: https://www.sciencedirect.com/science/article/pii/S136516099780007X.
[Ste87] M. Stedman. "Basis for comparing the performance of surface-measuring machines". In: *Precision Engineering* 9.3 (1987), pp. 149-152. ISSN: 0141-6359. DOI: https://doi.org/10.1016/0141-6359(87)90032-8.

[Whi01] DJ Whitehouse. "Fractal or fiction". In: *Wear* 249.5-6 (2001), pp. 345-353.

[Whi03] D.J. Whitehouse. *Handbook of Surface and Nanometrology*. IOP Publishing, 2003.