Obtaining local surface profiles

We use a Wyko NT1100 White Light Interferometer (WLI) microscope in Verti-cal Scanning mode (VSI) to obtain loVerti-cal 3-D profiles of the fracture surfaces. The method is based on the short coherence length of white light, and that the inter-ference fringes have peak intensity when the test beam (reflecting off the sample) has the same optical path length as the reference beam (fixed length). The layout of the interferometer is shown in figure 7.3. The objective is moved vertically while tracking its vertical offset with a piezo-electric translation stage. The reflected beam forming an image on the CCD screen is a combination of the reference beam and the test beam (re-combined at the beam splitter), which will have peak intensity if the two beams have travelled the same optical length. As the objective is moved, the CCD image is recorded at each step. An algorithm finds the vertical position with highest intensity for each pixel and makes a 3-D profile of the scanned surface with this information. In a WLI setup with a broadband light source, we will only see interference fringes in the short range of the coherence length, where the wave-lengths in the reflected beam have almost the same phase conditions as the light source. Outside the range of the coherence length, the phases and interferences are randomly distributed, and no interference fringes are observed.

The dependence of the correlogram width on the coherence length and central wavelength is briefly explained by assuming a Gaussian spectrum for the emitted light [86]. The normalized spectral density function is defined according to

S(ν) = 1

√π∆νe^{−ν∆ν−ν0}, (7.1)

where ∆ν is the effective 1/e-bandwidth and ν0 is the mean frequency. The auto-correlation function of the light field, which is measured at the CCD screen for each pixel, is given by the Fourier transform of eq. (7.1) as

k(τ) = Z ^∞

−∞

S(ν)e^−i2πντdν =e^{−(πτ∆ν)2} ·e^−i2πν0τ. (7.2) By rewriting eq. (7.2) with ν₀ =c/λ₀, L_c = c/(π∆ν), and τ = 2(z−z₀)/c, we can describe the intensity measured at the CCD plate as function of z, the distance from the sample to the beam splitter. Here, cis the speed of light,λ0 is the central wavelength, L_c is the correlation length, and 2(z−z₀) is the optical path difference

wherez₀ is the distance of the reference mirror from the beam splitter. By assuming that the reference and test beams have the same intensityI0, the intensity measured at a pixel in the CCD screen is given by

I(z) =I0ℜ{1 +k(z)}=I0

1 +e⁻⁴(^z−_Lc^z0)² ·cos

4π(z−z0) λ0

. (7.3) Equation (7.3) shows that the intensity distribution is formed by a Gaussian envelope, and a periodic modulation of periodλ0/2. While a real correlogram is more distorted than this ideal case, it shows the strong dependence of the correlogram on the central wavelength λ₀ (in the range of 400 - 700 nm for visible light) and coherence lengthLc of the light source (usually on the order of microns). This shows that we will only observe an interference pattern for a small range of (z−z0), and will be focused on different elevations of the sample during scanning.

The output data from the interferometer is an image of width W and height H where the pixels contain elevation values h(x, y) at each pixel position in the measured area (1 pixel size≈2 ➭m). The fracture surfaces are profiled on both pieces of the sample, before and after the flooding experiments. Since the surface profiles are local (areas between 4×5 mm and 8×8 mm), we drill 1.5 mm benchmark holes in the samples to measure roughly similar areas in the succeeding measurements.

Resulting 3-D profiles for similar areas on the opposing fracture surfaces of a sample (side A and side B) are shown in figure 7.4. During data processing, tilt of the profiles is removed by subtracting a linear plane fitted to the profile data. To align surface profiles before and after experiments (same piece of the sample, e.g. side A with side A), the profiles are cross-correlated during translational and rotational displacements, such that they are best aligned at the highest correlation coefficient.

Aligning profiles of opposing fracture surfaces of the same sample (side A with side B) is done in the same fashion, but first the sideB profile is been multiplied with -1 and flipped along the x-axis. This is done to represent the surface elevation hB in the same direction as hA, and to account for the mirroring of the fracture surfaces (it corresponds to rotating the profile 180 degrees about the y-axis). An example of aligned profiles from opposing surfaces is shown in figure 7.5.

Incident beam Reflected beam

Mirror

Beam splitter

Sample Microscope lens (20X)

Tungsten halogen lamp CCD Camera

Test beam (Reflects off sample)

Reference beam

(Reflects off mirror)

}

Sample is at fixed position

Figure 7.3: The Wyko NT1100 optical profiling system. Left: Photo of the equip-ment with the sample under the microscope objective (inside red circle). Right:

Principle of the Mirau interferometer; Incident white light goes through the micro-scope lens and hits a beam splitter plate (semi-transparent mirror), splitting the beam into a reference beam and a test beam. The reference beam reflects off a mir-ror at a fixed position in the objective, while the test beam reflects off the sample.

Then the two beams combine at the beam splitter plate, and goes back through the microscope lens to a CCD camera which records the beam intensity at each pixel.

Due to constructive interference, the beam intensity is highest when the optical path of the test beam is equal to the optical path of the reference beam. By scanning the objective vertically (relative to the sample), and by recording the vertical position of the objective, an algorithm finds the vertical position at each pixel where the beam intensity is highest, and with this information a surface profile is made.

Benchmark holes,

Figure 7.4: Examples of 3-D profiles before reactive flooding. Left: The measured fracture surfaces, where the pieces shown are the opposite sides of the same fractured sample. The brighter areas indicate the profiled regions. Right: The resulting 3-D profiles as found with WLI. We see that details in the profiles match: valleys in the top profile matches peaks in the bottom profile and vice-versa. Note that in this figure, the tilt of the sample is not subtracted, and the profile on the bottom is flipped but not multiplied by -1.

1 mm

Figure 7.5: Examples of aligned profiles. Left: Side A is the profile to be matched by side B. Middle: Side B has been flipped horizontally and aligned with side A.

Right: The original sideB profile before it is flipped along the x-axis and multiplied by -1.