Image Analysis

After capturing the diffraction images, subsequent analysis was required. A program was written in MATLABTM _{to calculate the inverse Fourier transform (IFT) of the images. This}

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Figure 5.7: The effect of missing everynthline within a diffraction grating structure on the diffrac- tion pattern is to introduce (n−1) sub-orders between the main diffraction orders of the complete grating.

displayed the periodicities (and their directions) detected in the input image. Some examples of the captured images and their corresponding IFT plots are shown in Figure 5.8. When large features in the captured image were Fourier transformed they represented small periods in the IFT. The small features in the IFT plot were not visible until the magnification of the plot was increased and centred on the origin (‘zooming in’ on the detail). These results showed some periodicity from within the image, but did not agree with the actual grating periods as designed. The program needed to be adjusted to ensure that small period features could be displayed properly in the IFT plot.

After initial analysis of the trial mask plate, areas for systematic improvements were iden- tified. The software analysis of the images needed to be further refined in order to bring out the best results from the images obtained. What shall be referred to as ‘zero-padding’ is the process of increasing the size of the input image matrix by creating a larger, empty two-dimensional matrix full of zeros, and then adding the image data to the centre of the matrix. This is the equivalent of adding a large black border surrounding the original image. Zero-padding of the image matrix (after having been ‘read-in’ by the program) improved the display of the IFT plot.

5.2 Initial Diffraction Experiments 128 The larger the zero-padding of the input image, the better the resolution of the IFT. There is no more information contained within the input image, but more information becomes pos- sible to display in the Fourier Transform plot. This is because of the reciprocal relationship between feature sizes in the input image to the corresponding feature sizes in the transform (see Figure 5.9). For example, with an image width of 1000 pixels, a feature that is 1000 pixels in size in the image is represented by 1 pixel in the IFT plot. A feature that is 500 pixels is transformed to a feature size of 2 pixels in the IFT plot.

Figure 5.9: A reciprocal relationship exists between the dimensions in the image and the Fourier transform plot.

Most feature separation distances (the diffraction order separation) imaged on the CCD were in the region of 250−1000 pixels, but the distinction is lost in the IFT since this range was only represented by 1−4 pixels. The difference between 400 pixels and 450 pixels separation was clearly visible in the diffraction pattern image, corresponding to different diffraction grating periods, but this detail could not be displayed in the transform, which had a resolution limited by the relative size of the feature in the image to the image size. By increasing the border (zero-padding) around the image, the feature sizes in the image were relatively smaller in relation to the matrix size and were consequently represented by larger features in the IFT. When the image size was zero-padded to larger than 2048×2048 pixels (image sizes were zero-padded to powers of 2 for convenience and faster processing with MATLABTM_{) the file became too large to be processed. Zero padding of images to the}

next higher power of 2 than the original size gave sufficient resolution for transforming most images.

There were two main approaches to capturing the diffraction image, which had an impact upon the best pattern for the grating. One option was to have the CCD detector close to the slide to capture as much of the diffraction pattern as possible in a single image (example images are shown in Figure 5.8). Since even the biggest detector chips were fairly small

(∼8.0mm) and the camera could not be mounted closer than 25mmbecause of the casing and also the angle of the diffracted beam the capture angle was limited (as shown earlier in Figure 5.2). The period of the grating therefore needed to be fairly large (around 20µm) to capture more than one diffraction order on the chip, even at close range (∼ 30−40 mm). In this situation, the maximum angular range of capture was∼10◦.

The alternative image capture method was to mount the detector further away and to scan across the large area of the diverging diffraction pattern (as shown in Figure 5.4), capturing several images and then ‘stitching’ them together. This setup had the advantage of at least a 90◦ capture angle range, and also a high resolution detail of the diffraction pattern obtained by joining together all the images. However, the resulting image had a large file size and created very lengthy processing times to calculate the IFT. The distance of the detector from the slide could be adjusted to work out the distance for the best trade-off between image resolution and file size. Also, it was only necessary to take a 1−D cross-section through the diffraction orders for IFT analysis. This 1−D matrix could easily be zero-padded to give the necessary resolution in the inverse transform plot.

The position of the CCD camera was accurately controlled by mounting it on a mechanical arm attached to a motorised rotation stage. Knowing the constant distance of the CCD from the centre of rotation, where the sample was located, and the size of the CCD chip, enabled calculation of the angular size of the image captured at any given point. From this point the rotation stage was moved to the next image capture point, with 0.001◦ accuracy in the angular position. The images were then stitched together, using a program written with MATLABTM_{, and stretched out to the equivalent image as would be observed on a flat}

screen perpendicular to the zeroth diffraction order (see Figure 5.10) and then subsequently Fourier transformed. The image had to be stretched out to the flat-screen equivalent since the FT simply searches for common periodicities, but in the curved image captured, orders appear progressively closer together at larger angles from the zeroth order.

The best method to confirm the periodicity of the test gratings was to record the angular position of the centre of each diffraction order. The rotation stage was moved until the diffraction order was aligned with its centre over a cross-hair marked on the monitor. The angular position was read from the digital read-out of the motorised stage, an image cap- tured and then the stage was moved on to the next diffraction order. Knowing the angular position of the diffraction orders, together with the angle of incidence, the period was directly calculated using the diffraction equation.

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Figure 5.10: Diffraction images captured in a curved plane were stretched out to the equivalent as would be imaged on a flat screen perpendicular to the zeroth diffraction order.

The diffraction pattern for a 20µm period chrome grating on silica is shown in Figure 5.11. The stitched image shows the relative angular spacing of the diffraction orders. The zeroth order was located at 55.9◦, with the angular range of the image continuing to the m=−13 order located at 23.6◦. This image represents a curved observation surface, with a constant observation distance. The image shown was deliberately saturated to make the detail of all the diffraction orders visible; these were not the images used to calculate the relative intensities of the diffraction orders. To calculate the relative intensities of the diffraction orders the illumination power was adjusted by rotating the variable neutral density filter to ensure that the CCD camera was not saturated and within a linear response range. When the incident power was adjusted two images were collected. The first was taken at the original power (at this stage the intensity was too low to be visible by eye on the monitor). The power was then increased towards the upper limit of the linear response region of the CCD camera, and a second image of the same diffraction order captured. Without adjusting the power from the lowest level (to avoid saturation at the zeroth and first orders), it was not possible to distinguish the higher orders from background light levels in a dark room. Taking two images of the same order when the power was adjusted enabled normalisation of the diffraction order intensities.

Figure 5.11: (a) Diffraction image produced by a 20µm period, 10 µm linewidth chrome grating on silica illuminated with a 632.8 nm wavelength laser. (b) Normalised intensity graph of the diffraction orders.

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