Fourier Transforms. Laboratory & Computational Physics 2

Transcription

1 Fourier Transforms Laboratory & Computational Physics 2 Last compiled August 8,

2 Contents 1 Introduction Prelab questions Background theory Physical discussion of Fourier transforms Distance from the object to the image Spatial filtering of diffraction patterns Procedure Investigating optical components Fraunhofer diffraction Inverse Fourier transformations Filtering the diffraction pattern Filtering the centre of the diffraction pattern

3 1 Introduction Over the course of this six hours you will be investigating the wave properties of light, in particular diffraction, interference patterns, and filtering of light waves. You will be introduced to lenses and will investigate Fourier transforms through Fraunhofer diffraction. Unlike the first experiment, you will start by considering the mathematics of the diffraction process before creating and observing different diffraction patterns. 1.1 Prelab questions 1. Given that the length of the Fourier apparatus is around 1.5 m, calculate the time it takes for a photon to traverse the length of the bench. 2. Given this time-of-flight, calculate the average time between photons for an emission rate of photons/second (as may correspond to the count rate of your experiment at the central maximum). 3. What does this say about the number of photons present along the length of the bench at any point in time? How much energy does this equate to? 4. Look at the general, one-dimensional form of a Fourier transform (equation 7) and its inverse (equation 8). What exactly is represented by F(k) and f(x)? What does k represent and what are its dimensions? 5. Consider a rectangular aperture as shown below: This aperture can be described using the function: { 1 if x a/2, y b/2 f(x, y) = 0 otherwise 3

4 b/2 -a/2 a/2 -b/2 Figure 1: A rectangular aperture. a) Substitute this function into equation 9 to obtain an expression describing the Fraunhofer diffraction pattern. b) Compute the diffraction pattern produced by this aperture. Note that the x and y dimensions can be treated separately as the integral is separable. c) Show that the equation describing the intensity, of the Fraunhofer pattern is given by: where sinc(α) = sin(α)/α I(k) = F (k)f (k) (1) I(k x, k y ) = a 2 b 2 sinc 2 (k x a 2 )sinc2 (k y b 2 ) (2) d) Provide a sketch of the expected diffraction pattern. 4

5 2 Background theory Fourier analysis is the method by which any function can be expressed as an infinite sum of sines and cosines. An example of this process of addition of trigonometric functions to approximate another function is shown in figure 2 below. (a) Example square pulse (b) Sum of 2 cosine terms overlaid (c) Sum of 5 cosine terms (d) Sum of 10 cosine terms Figure 2: These graphs illustrate the Fourier series sum for a square wave. A nice gif of this showing the contributions of increasingly small trigonometric functions can be found at For an infinite non-periodic function the Fourier integral is used to calculate the sum of periodic functions. This is written as: f(x) = 1 [ ] A(k) cos(kx)dk + B(k) sin(kx)dk (3) π 0 A discrete version of Fourier integral also exists, known as the discrete Fourier series: f(x) = A A m cos(m k x) + B m sin(m k x) (4) where k = 2π λ. m=1 In both representations, A(k) and B(k) are functions for weighting the sine and cosine components of the Fourier series. They are themselves integrals and depend on our initial function f(x): A(k) = 0 m=1 f(x) cos(kx)dx (5) 5

6 and B(k) = f(x) sin(kx)dx (6) It s important to note here that in equations 3 to 6, we re plugging in our function f(x) to turn it into a Fourier series, not trying to solve for f(x), so don t go re-arranging terms trying to find a solution. These equations can be combined into a single complex integral equation: F (k) = f(x)e ikx dx (7) The corresponding/reciprocal integral is known as the inverse Fourier transform and is given by: f(x) = 1 F (k)e ikx dk (8) 2π Note the (1/2π) term as a consequence of converting from frequency space to real space. In two dimensions the complex and inverse transforms generalise to, respectively: F (k x, k y ) = f(x, y)e i(kxx+kyy) dxdy (9) f(x, y) = 1 F (k (2π) 2 x, k y )e i(kxx+kyy) dk x dk y (10) Although the exact mathematical representation of the Fourier transform varies across the literature, all forms are equivalent (this definition is from Hecht s Optics). Another term commonly used for the function F (k) is the spatial frequency spectrum, or more simply the frequency spectrum. 2.1 Physical discussion of Fourier transforms Okay, so that s the mathematics of Fourier transforms (or at least one form of it). But what does this mean physically? Let s consider the optical case. Consider a photographic slide (such as the one shown in figure 3(a)) illuminated by a monochromatic plane wave. The incoming plane wave is scattered by the image and rays diffract from it depending on the inherent properties of the image (and to an extent, the optical components, but let s assume an ideal case). Each scattered ray scatters in a particular direction, labelled as k i. Each direction k i corresponds to a spatial frequency component of the Fourier transform. The relative strength of each of the scattered rays leaving the slide is given by the Fourier transform of the electric field just after the slide. The light transmitted through the image is the sum of all of the individual scattered rays. 6

7 (a) Input photographic slide. (b) Amplitude of Fourier transform of the image (log scaled). Incident plane wave Photographic slide Screen (c) Scattered plane waves. Fourier transform in (b) appears on screen. Figure 3: Example of Fourier transforming an image. Figure 4: A diffraction pattern from a crystal - note the points are discrete and bright. Very close to the slide, the image formed on a viewing screen will look very much like the slide because the scattered rays have not yet spread or interfered with each other. As the distance to the viewing screen increases the scattered rays will increasingly overlap and the observed image will no longer resemble the original. The wave in the region beyond the slide is called the diffraction pattern. Note that the rays only scatter to discrete points if the object has periodic structure, for example crystals, as in figure 4. If the object being imaged instead has non-periodic structure, the diffraction pattern may appear featureless or lacking particular identifiers, as in figure 3(b). This then raises a number of questions; What can we immediately determine about an object based on its diffraction pattern? If we slowly increase the periodicity of an object, how does the diffraction pattern respond? What if an object has both periodic and non-symmetric features? Are we able to diffract simple objects and use their diffraction patterns to understand the diffraction patterns of more complex objects? We will investigate some of these questions today. 7

8 2.1.1 Distance from the object to the image Depending on the distance from the object to the viewing screen, two different mathematical approximations can be used to describe the diffraction pattern. The Fresnel approximation describes the diffraction pattern close to the slide but is mathematically complicated, and won t be covered here. The Fraunhofer approximation describes the diffraction pattern when the distance between the slide and the viewing screen is extremely large. In this case any sharp points of light on the viewing screen each correspond to a specific Fourier spatial frequency component. Strictly speaking, a true Fraunhofer diffraction pattern is only produced when the screen is an infinite distance from the object. We can achieve this condition by placing the object at the focal point of a lens and the viewing screen at the back focal point of the lens. A quick ray tracing diagram should convince you that an object placed at the focal point of a lens will have its image reproduced at infinity. So a lens physically performs a Fourier transform (it is a Fourier transformer). We will use two Fourier transforming lenses. One will produce the diffraction patterns from the light exiting our objects, and the other will invert those diffraction patterns to recreate images of the original objects. It is worth noting that although the wave field is in general a complex function, we only observe the intensity (see equation 1), while the phase remains unknown. Obtaining phase information in a reliable fashion remains an ongoing question in research. 8

9 2.1.2 Spatial filtering of diffraction patterns LOW-PASS FILTER F(k) f(x) INPUT IMAGE F(k) f(x) HIGH-PASS FILTER Figure 5: An image Fourier transformed, then filtered with a low- or high-pass filter, and the filtered image inverse transformed. If you look back at figure 2, you can see that the sharp edges of the pulse become more defined as larger numbers of cosines are summed. Adding ten cosines produces much sharper edges than only adding five or two. From this observation we can argue that the high spatial frequencies of a Fourier transform produce the edges in an image. Conversely the lower order frequencies give magnitude to the signal, and smooth out the centre of the square pulse. Low-pass filters block out the edges of the Fourier transform, cutting out the high frequency components. In the top middle image above, the edges of the Fourier transform are removed. The inverse Fourier transform has lost its edges and sharp contours, resulting in a blurring of the image. High-pass filters block the centre of the Fourier transform, as in the bottom row of images above. In this case the inverse Fourier transform contains the edges of the image, but has lost the bright and dark regions. Complementary low- and high-pass filters allow for addition to return the complete inverse Fourier transform. High and low pass filters can be used for a variety of applications. (Side note: a band-pass filter is combination of filters.) Fourier transforms aren t restricted to only electromagnetic waves but also audio, electronic, etc signals. A simple example is playing music from a different room - the low frequencies can diffract more around corners and so are heard more clearly than the high frequencies. 9

10 3 Procedure Remember to always be very careful using any lasers. Minimise stray reflections and NEVER place your head in the path of laser beams. There are a number of optical components in use in this experiment. Do not to touch their surfaces as oil and dust are very hard to remove. If you think any component needs cleaning please ask your demonstrator. Microscope objective O BJ f f LASER Collimating lens E C T Figure 6: Fraunhofer diffraction experimental setup. diffracting lens. 3.1 Investigating optical components Diffracting CAMERA lens f represents the focal length of the Before diving in to examining diffraction patterns, it s important to familiarise yourselves with the optical components you ll be using. Having a conceptual understanding of light paths and ray tracing involving lenses is great, but as always, reality throws in various other considerations Start by removing all of the components except the laser and viewing screen. Make sure the laser beam is parallel along the track right up to the viewing screen, both horizontally and vertically. Describe your method. 2. Next, investigate the behaviour of the laser beam by placing pieces of equipment one at a time on the track. Compare, for example, the laser beam with and without the collimating lens. Does it effectively collimate the light? What is the purpose of the microscope objective? 3. Place an object (a slide) in the path of the laser and note your observations. What does the light immediately leaving the slide look like? What does it look like far away, do you see any evidence of diffraction with no lenses in place? 4. Now you ll need to determine the focal lengths of some of the lenses that are provided. It may seem simple, but there are two important considerations: firstly, that no lens is perfect and won t diffract to a single sharp point, and secondly, the stated focal length of the lenses tends to have a large degree of error Determine the focal lengths of at least four lenses, making sure the focal lengths aren t too long for our setup on the bench. 10

11 Question 1 What qualitative differences do you notice between the long and short focal length lenses? Describe what makes a good lens in your opinion - why would you choose one over another? 6. Now that you are familiar with the apparatus, set up the track as in figure 6 (note that it s not to scale). 7. Make sure the beam is still aligned each time you add a new component. It s possible the beam might be aligned but the microscope or lenses are on an angle - check this. 8. Place a simple object on a mount on the track and make sure again this object is centered in the laser beam. It should be uniformly illuminated. 3.2 Fraunhofer diffraction Okay, if you ve set everything up in the correct order and at the correct locations, you should see a diffraction pattern on the viewing screen. It s very likely you ll need to move the diffracting lens until it s in quite a precise location - how accurately did you measure the focal length earlier? Question 2 Does inserting the second lens after the collimating lens effectively Fourier transform the light leaving the object? Draw your observations. 9. Once you have obtained one diffraction pattern, use a lens with a different focal length and find the diffraction pattern again. Draw a diagram with labelled positions for both of the setups. Question 3 Did the diffraction pattern look different between the two lenses? Could changes be due to aberrations in the lenses themselves, rather than anything to do with the focal lengths? Based on your observations, which focal length is better suited to this experiment? Question 4 Now we can consider more specific details of diffraction patterns. Place each of the items listed below in the object position. Sketch or print out of the diffraction patterns and explain the relationships between each object and its diffraction pattern. A single slit, at three different rotations. In which direction does the diffraction pattern appear relative to the slit? A double slit, both vertically and horizontally. Make sure you can see the fine structure in the bright spots! A two-dimensional rectangle. What features correspond to straight lines and which to the corners of the rectangle? You might want to try highlighting specific corners to investigate further (move the object, not the laser spot). A wavy/jagged edge. Cut some paper and try to create a non-straight edge. An adjustable slit. Observe and comment on the change in diffraction pattern as you open the slit. Record the size of the central peak with aperture opening. 11

12 (a) Slide 13 (b) Slide 14 Figure 7: Slides 13 and 14. Cover up the sections you don t want to image. Different sized circles. How does this compare to the rectangle? Is the circle simpler or more complex? Try different size circles. Now create a table in your books that tries to assign specific diffraction pattern features to the object features. Then try to predict what a semicircle would look like. We can also look at what might appear to be more complicated objects: Place each of the three wire grids in the object position. Relate the grid spacing to the diffraction pattern features. Make sure to include comments about vertical and horizontal features. Slides 13 and 14 have six sections each, with each one offering a different diffraction pattern. You ll have to cover up the sections of the slide you aren t using (unless you want to try combining various sections). The various other slides available to you. Look at them, consider what the diffraction pattern might look like, and then investigate and comment. 3.3 Inverse Fourier transformations Okay so we ve had a good go at performing Fourier transforms and looking at diffraction patterns. Let s now add a second lens in to look at inverse Fourier transforms. We now want to invert a Fourier transform to obtain an image of the original object. Figure 8 shows the rough setup, remembering that accurate focal lengths and spacings are critical to this experiment. You should use something complicated, like a person s face or other complicated image, and if you are successful you ll see a clear, regular image on the screen. Question 5 Where did you place the imaging lens to produce an inverse Fourier transform on the imaging screen? Draw a diagram with labelled distances. 12

13 Microscope objective O BJ f d f d f i f i LASER E C T Diffracting lens Collimating lens Imaging lens Screen Figure 8: Inverse Fourier transformations experimental setup. f d is the focal length of diffracting lens and f i is the focal length of the imaging lens. Question 6 What differences do you notice between the original object and the image? Do you think differences are due to imperfect optics, or the nature of Fourier transforms, or something else perhaps? Question 7 Block say, the left half of the original object and note how the recovered image changes. What happens if you block more and more of the object? Explain this behaviour. 3.4 Filtering the diffraction pattern As mentioned in the theory, we can introduce filters just after our Fourier transform to alter the recovered image. Make sure you have a number of items on hand that you can use to block various parts of the diffraction pattern, including the adjustable iris and different sized dots and slits and rectangles. You will begin by blocking the outer edges of the diffraction pattern. 1. Place the array of eight slits in the object position, and make sure the image of this object is clear on the imaging screen. 2. Next, place either the adjustable iris or slit at the focal point between the diffracting and imaging lens. Question 8 Is it important that the adjustable iris or slit is centered on the object? How will you centre it accurately? What happens if you intentionally offset the iris from the centre of the object? 3. Once the slit is properly centered, very very slowly make it smaller (don t close it completely) and keep a close eye on the recovered image. 4. If you don t observe any change in the recovered image make even smaller adjustments to the slit width. Have a closer look at the image, particularly the spacings and sizes of the dots. 5. Remove the eight slit object and put in the semitone object, the one marked 1/2 tone. Note your observations. 13

14 6. Now use one of the wire grids (the one you thought most interesting ) with this filter, again noting observations and comparing to other objects. Question 9 Which type of filtering is this? High- or low-pass? Can you suggest anything to improve your filtering methods? Question 10 Did this filtering work as you expected? Explain, without maths, how only including the centre of the diffraction pattern alters the resulting image Filtering the centre of the diffraction pattern Figure 9: Though this is an x-ray diffraction image, you can see the the filter they ve applied in the centre. You could fabricate and apply a similar filter in place of the adjustable iris. Now we want to investing blocking the centre of the diffraction pattern. To do this, there are a number of dots or different sized that can be positioned to block the centre of the pattern, in place of the adjustable iris. You should try using both a semitone image (a face) and again the most interesting wire grid with some of these dots and record your observations. Question 11 Do you expect to see anything different in the re-creation of the wire grid image? How does this filtering of the wire grid compare with the previous filtering your performed? Question 12 Again, which type of filtering is this? How could you improve this particular filtering method? Consider both the equipment and how everything is set up. Question 13 Try moving the dot around the pattern. Does this have any noticeable effect? You may have a bit of trouble filtering the centre of the diffraction patterns. Remember to try to be as precise in measurements and observations as possible! If your filtering wasn t up to your expectations, can you explain why? A good starting comment is whether focal points are actually points, or whether they might be a line, or even a volume... 14