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Biophysics 2S03

Home Experiment #2

Part 1: Calculating Fourier transforms with ImageJ

The diffraction pattern of an object can be calculated by performing a rather complex mathematical operation called a Fourier transform. Reciprocally, the object can be reconstructed by performing an inverse Fourier transform on the image of the diffraction pattern.

ImageJ can calculate the Fourier transform of images using an algorithm called Fast Fourier Transform, or FFT. In other words, ImageJ allows you to perform a virtual diffraction experiment for two-dimensional objects you have an image of. It also allows you to perform an inverse Fourier transform on the image of a diffraction pattern, to reconstruct the object.

Part 1a: Fourier transform of a single slit

Follow the instructions below to calculate the diffraction pattern created by a single slit:

๏ Open the image called One-Segment.tif in ImageJ. This image is a square (2048 x 2048 pixels) grayscale 8-bit image (meaning that each pixel has an intensity value between 0 and 28-1 = 255), showing the image of a single vertical slit (or segment).

๏ Check that the pixel width and height (which should be 2.5 μm) are entered properly in the image property windows, which you can open by going to Image > Properties.

Question 1: Use the line tool of ImageJ to measure the width ( ) and height ( ) of the a b slit shown on the image. Give the values of and in μm, including an error (you can a b estimate this error to be equal to the width of two pixels, because the precision with which you decide on where the edges of the slits are cannot be better than 1 pixel).

๏ Adjust the settings of the Fast Fourier Transform algorithm by going to: Process > FFT > FFT Options… . Check "Raw power spectrum” (the result of the Fourier transform operation) and uncheck all the other boxes, as shown below.

๏ Calculate the Fourier transform of the image by going to: Process > FFT > FFT. After a few seconds, a new window should pop up showing you the Fourier transform (below, left).

๏ Zoom in on the centre of this new image and adjust the contrast by playing with the “Maximum” value of the contrast in Image > Adjust > Brightness/Contrast… , until you obtain a clear image of the diffraction pattern (as shown above, right).

๏ Draw a horizontal line through the diffraction pattern, then use the command Analyze > Plot Profile to display the classic profile of the diffraction pattern of a single slit (as shown below).

Question 2: Use the line tool again (then Analyze > Measure) to measure the value of the central peak of the diffraction pattern, , and give this value in pixels (including an error of ± 2 pixels).

๏ In a diffraction pattern, the size of features is inversely related to the size of the corresponding features in the original object. For example, in the diffraction pattern of a slit, the half-width of the central peak is given by: , where is the wavelength of the light used for the experiment, and is the distance between the object and the screen on which the diffraction pattern is projected. In order to be able to relate the size of features in the diffraction pattern to that of features in the original image, we need to know two things:

• ImageJ calculate the Fourier transform assuming that , where is the width of the original image (here = 2048 x 2.5 = 5120 μm). For example, in the case of the single slit diffraction pattern, it means that: .

• ImageJ generates the image of the Fourier transform using a pixel size of . Therefore when measuring the width of the central peak in the profile above in pixel we should have:

Question 3: Using the formula above, calculate the value of from the value of nW you measured on the diffraction pattern. Include an error analysis (treat as a constant, meaning that its value is known with infinite precision). Is this value consistent with the one you obtained directly from the image in question 1?

Question 4: Can you obtain the value of the height of the slit, b, from the diffraction pattern generated by ImageJ?

Part 1b: Fourier transform of a double slit

Repeat the procedure outlined in the previous section to calculate the Fourier transform of the image Two-Segments.tif, which shows the image of a double slit. We now expect to see interferences between the light ray coming from both slits in the diffraction pattern.

Question 5: Measure the width of the slits (a) and the distance between them (c) directly from the image, and give their value in μm (including an error).

Question 6: Include a screenshot of the horizontal diffraction pattern profile of this image. Indicate on it with an arrow the characteristic distance, nW, which is related to the width of the slits and can be used to calculate it. Indicate with another arrow the characteristic distance, nD, which is inversely related to the distance between the slits and can be used to calculate it.

Question 7: Measure nW and nD, then calculate a and c (including error) and compare these values to the ones obtained in question 5.

Part 1c: Fourier transform of a grating

Repeat the procedure outlined in the previous section to calculate the Fourier transform of the image Many-Segments.tif, which shows the image of a grating composed of many slits which have the same dimensions and distance between them as those in the double slit image analyzed in the previous section.

Question 8: Include a screenshot of the horizontal diffraction pattern profile of this image. How does this profile compares to the one you obtained in Part 1b for the double slit (discuss both similarities and differences)?

Question 9: The Fourier transform of the diffraction pattern should give us back the original image. Try this with the Fourier transform of the grating you have obtained in this part, and include a screenshot of the image you obtain. Is it in any way similar to the original image Many-Segments.tif?

Part 2: Fourier transform of a helix

In this section, we apply what we have learnt in the previous section to compute the diffraction pattern of a helix. This is a case which is relevant to molecular biology, because many biomolecules (starting with DNA) fold into helical structures.

Question 10: The image called Zig.tif shows a line of short segments (width and spacing ) inclined from the vertical by an angle . Use ImageJ Line Tool to verify that and have the same values as in question 5. Use ImageJ Angle tool to measure and give the value you measured in degrees.

Question 11: Calculate the Fourier transform of the image Zig.tif, and include a screenshot of this Fourier transform in your report (after magnifying it and adjusting the contrast so that the diffraction pattern is clearly visible). Show on this image:

- the distance inversely related to the width of the slits

- the distance inversely related to the spacing between the slits

Check that you obtain the correct values of and from the measured values of and on the diffraction pattern.

Question 12: Calculate the Fourier transform of the image called Zag.tif, which shows a similar line of short segments, but this time inclined from the vertical by an angle . Include a screenshot of this Fourier transform, and indicate on this image how you could directly measure the angle . Measure this angle. Is it consistent with the value you reported in question 10?

Question 13: Now open the image called ZigZag.tif which shows a helical shape object (obtained by a superposition of the images Zig.tif and Zag.tif). Measure the pitch ( ) of this “helix” and report its value in microns.

Question 14: Calculate the Fourier transform of the image ZigZag.tif and include a screenshot of this Fourier transform in your report. Indicate on it which distance ( ) is related to the helical pitch of the helical object. Calculate again from the value of - do you get the same value as in question 13?

Question 15: Indicate on the Fourier transform of ZigZag.tif how you can easily measure the angle .