Fourier transforms of SEM images

We have used a Fourier transform (FT) technique to analyse the SEM pictures of LIFE sites. This mathematical function separates a waveform into sinusoidal components of different frequency from the original waveform. The result of this is to effectively convert a function from a time domain into a frequency domain. The transform can also convert a spatial domain to spatial-frequency domain. The Fourier transform of a time-varying function f(t) to a frequency-varying function

f(v) is:

f(v)=F[f(t)]=

Z ∞

−∞

When using a computer to calculate Fourier transforms, the discrete version of the equation (Discrete Fourier Transform, DFT) is required. This is:

F[f(t)]= N−₁

X

k=0

f_ke−2πNink _(3.6)

The actual algorithm used is the ‘Fast Fourier Transform’ or FFT, which reduces the number of calculations required from 2N2to 2Nlog₂N, whereNis the number of points.

To calculate the Fourier transform of an image, a two-dimensional Fourier transform has to be carried out. This can be done by taking the FT of each column of an image, then taking the FT of the result. The Fourier Transform of an image will hence have the same number of pixels as the original image, and so the number of frequencies present will depend on the original image. The software used for this analysis, Matlab version 6, by The Mathworks Inc, comes with a two-dimensional Fast Fourier Transform function (‘fft2’) built-in.

An example of the usefulness of Fourier transforms in image analysis is shown in Figure 3.29. A series of square lines is decomposed by the Fourier transform into the spatial frequencies that are required to make up the picture. As the lines correspond to a square wave or grating, only the zero and odd terms of the frequency components are present, as would be expected from Fourier theory.

(a) Original image (b) Fourier transform

Figure 3.29: The Fourier transform of a series of lines, with square-wave profile.

To produce useful Fourier transforms of the SEM pictures the following steps need to be taken:

1. The SEM image needs to be cropped to a square, to simplify the resulting Fourier transform (otherwise the axis scales would not be the same, leading to a ‘squashed’ transform. Also the text (scale, magnification etc) is removed. 2. The scale of the transformed image is calculated from the length and value of the scale bar in the SEM image. The size of the pixel (microns) in the SEM image,∆x, is related to the size of the pixel (inverse microns) in the Fourier transform image,∆u, by the relation (Gonzalez and Wintz, 1987):

∆u= _N∆x1 (3.7)

where N is the number of samples along the x axis (ie number of pixels width of the image).

3. The Fourier transform image is produced by taking the absolute values of the resulting transform, and then taking the logarithm of the absolute values. This is a common procedure in Fourier analysis, which although without physical basis, improves the contrast of the FT image which is initially poor due to the large value of the DC-component compared to the harmonics.

4. The axes of the FT image are also rearranged so that the zero-order (DC) term is at the centre of the resulting image.

An example of a typical Fourier transform of a LIFE SEM image is given in Figure 3.30. The star-shaped Fourier transform clearly shows the strong preference of the lines to form along directions at 120° to each other. The length of each lobe of the star pattern shows that there is a range of frequencies present in those particular directions.

Figure 3.31 shows another SEM and Fourier transform image pair. This SEM image was taken closer to the centre of the site, and shows a denser pattern of lines. The widths of the lobes of the star pattern in the Fourier transform are wider than those in Figure 3.30, indicating that while the patterns are still forming along the preferred directions, there is some variation in the exact direction they take. As the density is higher, the lines are trying to fill the available space more efficiently; this requires slight deviations from the preferred direction, and so results in broadening of the Fourier transform lobes.

The Fourier transform of an SEM image taken near the centre of the site on the 40 minute sample (Figure 3.32) shows quite a different pattern however, as the directional dependence has been lost. Due to a greater density of structures, no

(a) Cropped area of original image (b) Fourier transform Figure 3.30: Cropped SEM image taken at the edge of the 40 minute sample and the corresponding Fourier transform.

(a) Cropped area of original image (b) Fourier transform

Figure 3.31:Cropped SEM image taken near the edge of the 40 minute sample and the corresponding Fourier transform.

long lines are able to form, and so the Fourier transform only shows a ‘halo’. There is still a range of periods that have been formed, but these may be in any direction. Figure 3.33 shows the SEM image taken at the centre of the site, where almost total etch frustration has occurred. The only structures that are seen are small dots, and long scratch lines. The Fourier transform gives very little information, as there are no periodic structures in the image.

(a) Cropped area of original image (b) Fourier transform Figure 3.32:Cropped SEM image taken near the centre of the 40 minute sample and the corresponding Fourier transform.

(a) Cropped area of original image (b) Fourier transform

Figure 3.33:Cropped SEM image taken at the centre of the 40 minute sample and corresponding Fourier transform.

In some Fourier transforms of the images, a second series of peaks can be seen at higher spatial frequencies than the first. An example of this is illustrated in Figure 3.34.

The best Fourier transform results were obtained using SEM images of 2000×

magnification, as these had features that were clearly visible yet also presented a large enough area to give a representative result. At this magnification, each pixel of the SEM image represents a distance of 0.137 µm, which corresponds to 0.010 µm−1 for the Fourier transform image pixels. Thus some of the very high

(a) Cropped area of original image (b) Fourier transform Figure 3.34:Cropped SEM image of the 80 minute sample and corresponding Fourier transform.

frequency components will be a consequence of the pixelised nature of the original image, but other high frequency components will arise due to the width of the individual lines. The very low frequency components are a consequence of the rectangular image as a whole.

Fourier transforms were carried out on each 2000×_{magnification SEM image from}

the survey of 20, 40 and 80 minutes LIFE sites and a method was developed to allow automated analysis of these transforms. This was achieved by recording the intensity value of each pixel along a line in the FT image, rotating the line by 1°, and then adding the value of the pixels along the new line to those from the old line. This process was then repeated over 360°, which resulted in a one-dimensional line that represented all of the data from the 2D Fourier transform. Figure 3.35 shows a typical result. A Gaussian curve has been fitted to the broad peak in the data, and the centre of this curve then corresponds to the most common spatial frequency in the SEM picture. While this clearly is not the only frequency that appears in the images, it does give a good indication of the average and so allows for a quick comparison between the transforms of several SEM images.

This process was applied to each Fourier transform of the 2000×_{magnification SEM}

images, and so a value for the average period present in each image was obtained. The results are shown in Figure 3.36, where the average period is plotted against distance from the centre of the site.

The best results were obtained for Fourier transforms of the ‘halo’ type instead of the ‘star’ type, so the results in Figure 3.36 do not represent the full radius of the site.

Figure 3.35:1D Sum of 2D Fourier Transform. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Distance from centre of the spot (mm)

A veragePeriod( m) m 20mins 40mins 80mins

Figure 3.36: The variation of typical spatial period versus distance from the centre of site, for 3 different exposure times.

The results show that the period increases with the length of time the sample has been etched for, but the difference between the 80 and 40 min samples is less than that between the 40 and 20 min samples. Additionally, the periods are generally greater at the edge of the site than at the centre, as would be expected from visual inspection of the SEM images, which show less densely packed structures occur away from the centre of the site.