Fourier TransformsLaboratory & Computational Physics 2

Last compiled August 8, 2017

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Contents

1 Introduction 31.1 Prelab questions . . . . . . . . . . . . . . . . . . . . 3

2 Background theory 62.1 Physical discussion of Fourier transforms . . . 8

2.1.1 Distance from the object to the image . 102.1.2 Spatial filtering of diffraction patterns . 12

3 Procedure 143.1 Investigating optical components . . . . . . . . . 143.2 Fraunhofer diffraction . . . . . . . . . . . . . . . . 163.3 Inverse Fourier transformations . . . . . . . . . . 183.4 Filtering the diffraction pattern . . . . . . . . . . 19

3.4.1 Filtering the centre of the diffraction pat-tern . . . . . . . . . . . . . . . . . . . . . . . . 20

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1 Introduction

Over the course of this six hours you will be investigat-ing the wave properties of light, in particular diffraction,interference patterns, and filtering of light waves.You will be introduced to lenses and will investigate Fouriertransforms through Fraunhofer diffraction. Unlike the firstexperiment, you will start by considering the mathematicsof the diffraction process before creating and observingdifferent diffraction patterns.

1.1 Prelab questions

1. Given that the length of the Fourier apparatus is around1.5 m, calculate the time it takes for a photon to tra-verse the length of the bench.

2. Given this time-of-flight, calculate the average timebetween photons for an emission rate of 2 × 106 pho-

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tons/second (as may correspond to the count rate ofyour experiment at the central maximum).

3. What does this say about the number of photonspresent along the length of the bench at any pointin time? How much energy does this equate to?

4. Look at the general, one-dimensional form of a Fouriertransform (equation 7) and its inverse (equation 8).What exactly is represented by F(k) and f(x)? Whatdoes k represent and what are its dimensions?

5. Consider a rectangular aperture as shown below:

b/2

-b/2

-a/2 a/2

Figure 1: A rectangular aperture.

This aperture can be described using the function:

f(x, y) =

{1 if |x| ≤ a/2, |y| ≤ b/20 otherwise

a) Substitute this function into equation 9 to obtainan expression describing the Fraunhofer diffrac-tion pattern.

b) Compute the diffraction pattern produced by thisaperture. Note that the x and y dimensions canbe treated separately as the integral is separable.

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c) Show that the equation describing the intensity,

I(k) = F (k)F ∗(k) (1)

of the Fraunhofer pattern is given by:

I(kx, ky) = a2b2sinc2(kxa2)sinc2(ky b

2) (2)

where sinc(α) = sin(α)/αd) Provide a sketch of the expected diffraction pat-

tern.

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2 Background theory

Fourier analysis is the method by which any function canbe expressed as an infinite sum of sines and cosines. Anexample of this process of addition of trigonometric func-tions to approximate another function is shown in figure 2below.

(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 sumfor a square wave. A nice gif of this showing thecontributions of increasingly small trigonometricfunctions can be found at https://goo.gl/ZxWVvT

For an infinite non-periodic function the Fourier integral isused to calculate the sum of periodic functions. This iswritten as:

f(x) =1

π

[∫ ∞

0

A(k)cos(kx)dk +

∫ ∞

0

B(k)sin(kx)dk]

(3)

A discrete version of Fourier integral also exists, known

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as the discrete Fourier series:

f(x) =A0

2+

∞∑m=1

Am cos(m|k|x) +∞∑

m=1

Bm sin(m|k|x) (4)

where |k| = 2π

λ.

In both representations, A(k) and B(k) are functions forweighting the sine and cosine components of the Fourierseries. They are themselves integrals and depend on ourinitial function f(x):

A(k) =

∫ ∞

−∞f(x)cos(kx)dx (5)

andB(k) =

∫ ∞

−∞f(x)sin(kx)dx (6)

It’s important to note here that in equations 3 to 6, we’replugging 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 termstrying to find a solution.These equations can be combined into a single complexintegral equation:

F (k) =

∫ ∞

−∞f(x)eikxdx (7)

The corresponding/reciprocal integral is known as the in-verse Fourier transform and is given by:

f(x) =1

2π

∫ ∞

−∞F (k)e−ikxdk (8)

Note the (1/2π) term as a consequence of converting fromfrequency space to real space.In two dimensions the complex and inverse transformsgeneralise to, respectively:

F (kx, ky) =

∫ ∞

−∞

∫ ∞

−∞f(x, y)ei(kxx+kyy)dxdy (9)

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f(x, y) =1

(2π)2

∫ ∞

−∞

∫ ∞

−∞F (kx, ky)e

−i(kxx+kyy)dkxdky (10)

Although the exact mathematical representation of theFourier transform varies across the literature, all formsare equivalent (this definition is from Hecht’s Optics). An-other term commonly used for the function F (k) is the‘spatial frequency spectrum’, or more simply the ‘frequencyspectrum’.

2.1 Physical discussion of Fourier transforms

Okay, so that’s the mathematics of Fourier transforms (orat least one form of it). But what does this mean physi-cally? Let’s consider the optical case.Consider a photographic slide (such as the one shown infigure 3(a)) illuminated by a monochromatic plane wave.The incoming plane wave is scattered by the image andrays diffract from it depending on the inherent propertiesof the image (and to an extent, the optical components,but let’s assume an ideal case).Each scattered ray scatters in a particular direction, la-belled as k⃗i. Each direction k⃗i corresponds to a spatialfrequency component of the Fourier transform.The relative strength of each of the scattered rays leavingthe slide is given by the Fourier transform of the electricfield just after the slide. The light transmitted through theimage is the sum of all of the individual scattered rays.Very close to the slide, the image formed on a viewingscreen will look very much like the slide because the scat-tered rays have not yet spread or interfered with eachother. As the distance to the viewing screen increasesthe scattered rays will increasingly overlap and the ob-served image will no longer resemble the original. The

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(a) Input photographic slide. (b) Amplitude of Fourier trans-form of the image (log scaled).

Incidentplane wave

Photographicslide

Screen

(c) Scattered plane waves. Fourier trans-form in (b) appears on screen.

Figure 3: Example of Fourier transforming an image.

wave in the region beyond the slide is called the diffrac-tion pattern.Note that the rays only scatter to discrete points if the ob-ject has periodic structure, for example crystals, as in fig-ure 4. If the object being imaged instead has non-periodicstructure, the diffraction pattern may appear featurelessor lacking particular identifiers, as in figure 3(b).This then raises a number of questions;

• What can we ‘immediately’ determine about an objectbased on its diffraction pattern?

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Figure 4: A diffraction pattern from a crystal - note thepoints are discrete and bright.

• If we slowly increase the periodicity of an object, howdoes the diffraction pattern respond?

• What if an object has both periodic and non-symmetricfeatures?

• Are we able to diffract simple objects and use theirdiffraction patterns to understand the diffraction pat-terns of more complex objects?

We will investigate some of these questions today.

2.1.1 Distance from the object to the image

Depending on the distance from the object to the viewingscreen, two different mathematical approximations can beused to describe the diffraction pattern.

• The Fresnel approximation describes the diffractionpattern close to the slide but is mathematically com-plicated, and won’t be covered here.

• The Fraunhofer approximation describes the diffrac-tion pattern when the distance between the slide andthe viewing screen is extremely large. In this caseany sharp points of light on the viewing screen eachcorrespond to a specific Fourier spatial frequency com-ponent.

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Strictly speaking, a true Fraunhofer diffraction pattern isonly produced when the screen is an ‘infinite’ distancefrom the object. We can achieve this condition by plac-ing the object at the focal point of a lens and the viewingscreen at the ‘back focal point’ of the lens. A quick raytracing diagram should convince you that an object placedat the focal point of a lens will have its image reproducedat infinity. So a lens physically performs a Fourier trans-form (it is a Fourier transformer).We will use two Fourier transforming lenses. One will pro-duce the diffraction patterns from the light exiting ourobjects, and the other will invert those diffraction pat-terns to recreate images of the original objects.It is worth noting that although the wave field is in generala complex function, we only observe the intensity (seeequation 1), while the phase remains unknown. Obtainingphase information in a reliable fashion remains an ongoingquestion in research.

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2.1.2 Spatial filtering of diffraction patterns

INPUT IMAGE

LOW-PASS FILTER

HIGH-PASS FILTER

F(k)f(x)

F(k)f(x)

Figure 5: An image Fourier transformed, then filtered witha low- or high-pass filter, and the filtered imageinverse transformed.

If you look back at figure 2, you can see that the sharpedges of the pulse become more defined as larger num-bers of cosines are summed. Adding ten cosines producesmuch sharper edges than only adding five or two. Fromthis observation we can argue that the high spatial fre-quencies of a Fourier transform produce the edges in animage. Conversely the lower order frequencies give mag-nitude to the signal, and smooth out the centre of thesquare pulse.

• Low-pass filters block out the edges of the Fouriertransform, cutting out the high frequency components.In the top middle image above, the edges of theFourier transform are removed. The inverse Fouriertransform has lost its edges and sharp contours, re-sulting in a blurring of the image.

• High-pass filters block the centre of the Fourier trans-form, as in the bottom row of images above. In thiscase the inverse Fourier transform contains the edgesof the image, but has lost the bright and dark regions.

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Complementary low- and high-pass filters allow for ad-dition to return the complete inverse Fourier transform.High and low pass filters can be used for a variety of ap-plications. (Side note: a band-pass filter is combination offilters.) Fourier transforms aren’t restricted to only elec-tromagnetic 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 andso are heard more clearly than the high frequencies.

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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 thisexperiment. Do not to touch their surfaces as oil and

dust are very hard to remove. If you think anycomponent needs cleaning please ask your demonstrator.

Microscopeobjective

Collimating lens

f f

LASER

OBJECT

CAMERA

lensDiffracting

Figure 6: Fraunhofer diffraction experimental setup. f rep-resents the focal length of the diffracting lens.

3.1 Investigating optical components

Before diving in to examining diffraction patterns, it’s im-portant to familiarise yourselves with the optical compo-nents you’ll be using. Having a conceptual understandingof light paths and ray tracing involving lenses is great, butas always, reality throws in various other considerations...

1. Start by removing all of the components except thelaser and viewing screen. Make sure the laser beamis parallel along the track right up to the viewingscreen, both horizontally and vertically. Describeyour method.

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2. Next, investigate the behaviour of the laser beamby placing pieces of equipment one at a time on thetrack. Compare, for example, the laser beam withand without the collimating lens. Does it effectivelycollimate the light? What is the purpose of the mi-croscope objective?

3. Place an object (a slide) in the path of the laser andnote your observations. What does the light immedi-ately leaving the slide look like? What does it looklike far away, do you see any evidence of diffractionwith no lenses in place?

4. Now you’ll need to determine the focal lengths ofsome of the lenses that are provided. It may seemsimple, but there are two important considerations:firstly, that no lens is perfect and won’t diffract toa single sharp point, and secondly, the stated focallength of the lenses tends to have a large degree oferror...

5. Determine the focal lengths of at least four lenses,making sure the focal lengths aren’t too long for oursetup on the bench.

Question 1 What qualitative differences do you notice be-tween the long and short focal length lenses? Describewhat makes a ‘good lens’ in your opinion - why would youchoose 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 bealigned but the microscope or lenses are on an angle- check this.

8. Place a simple object on a mount on the track andmake sure again this object is centered in the laserbeam. It should be uniformly illuminated.

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3.2 Fraunhofer diffraction

Okay, if you’ve set everything up in the correct order andat the correct locations, you should see a diffraction pat-tern on the viewing screen. It’s very likely you’ll need tomove the diffracting lens until it’s in quite a precise lo-cation - how accurately did you measure the focal lengthearlier?

Question 2 Does inserting the second lens after the colli-mating lens effectively Fourier transform the light leavingthe object? Draw your observations.

9. Once you have obtained one diffraction pattern, use alens with a different focal length and find the diffrac-tion pattern again. Draw a diagram with labelled po-sitions for both of the setups.

Question 3 Did the diffraction pattern look different be-tween the two lenses? Could changes be due to aberra-tions in the lenses themselves, rather than anything to dowith the focal lengths? Based on your observations, whichfocal length is better suited to this experiment?

Question 4 Now we can consider more specific details ofdiffraction patterns.Place each of the items listed below in the object position.Sketch or print out of the diffraction patterns and explainthe relationships between each object and its diffractionpattern.

• A single slit, at three different rotations. In whichdirection does the diffraction pattern appear relativeto the slit?

• A double slit, both vertically and horizontally. Makesure you can see the fine structure in the bright spots!

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• A two-dimensional rectangle. What features corre-spond to straight lines and which to the corners ofthe rectangle? You might want to try highlightingspecific corners to investigate further (move the ob-ject, not the laser spot).

• A wavy/jagged edge. Cut some paper and try to cre-ate a non-straight edge.

• An adjustable slit. Observe and comment on thechange in diffraction pattern as you open the slit.Record the size of the central peak with apertureopening.

• Different sized circles. How does this compare to therectangle? Is the circle simpler or more complex? Trydifferent size circles.

Now create a table in your books that tries to assign spe-cific 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 com-plicated’ objects:

• Place each of the three wire grids in the object po-sition. Relate the grid spacing to the diffraction pat-tern features. Make sure to include comments aboutvertical and horizontal features.

• Slides 13 and 14 have six sections each, with each oneoffering a different diffraction pattern. You’ll haveto 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 atthem, consider what the diffraction pattern mightlook like, and then investigate and comment.

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(a) Slide 13 (b) Slide 14

Figure 7: Slides 13 and 14. Cover up the sections you don’twant to image.

3.3 Inverse Fourier transformations

Okay so we’ve had a good go at performing Fourier trans-forms and looking at diffraction patterns. Let’s now adda second lens in to look at inverse Fourier transforms.We now want to invert a Fourier transform to obtain an im-age of the original object. Figure 8 shows the rough setup,remembering that accurate focal lengths and spacings arecritical to this experiment. You should use something com-plicated, like a person’s face or other complicated image,and if you are successful you’ll see a clear, regular imageon the screen.Question 5 Where did you place the imaging lens to pro-duce an inverse Fourier transform on the imaging screen?Draw a diagram with labelled distances.Question 6 What differences do you notice between theoriginal object and the image? Do you think differencesare due to imperfect optics, or the nature of Fourier trans-forms, or something else perhaps?Question 7 Block say, the left half of the original objectand note how the recovered image changes. What hap-pens if you block more and more of the object? Explainthis behaviour.

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LASER

Microscopeobjective O

BJECT

Collimating lens

Diffractinglens

fd fd fi fi

Imaginglens

Screen

Figure 8: Inverse Fourier transformations experimentalsetup. fd is the focal length of diffracting lensand fi is the focal length of the imaging lens.

3.4 Filtering the diffraction pattern

As mentioned in the theory, we can introduce filters justafter our Fourier transform to alter the recovered image.Make sure you have a number of items on hand that youcan use to block various parts of the diffraction pattern,including the adjustable iris and different sized dots andslits and rectangles.You will begin by blocking the outer edges of the diffrac-tion pattern.1. Place the array of eight slits in the object position,

and make sure the image of this object is clear onthe imaging screen.

2. Next, place either the adjustable iris or slit at thefocal point between the diffracting and imaging lens.

Question 8 Is it important that the adjustable iris or slit iscentered on the object? How will you centre it accurately?What happens if you intentionally offset the iris from thecentre of the object?3. Once the slit is properly centered, very very slowly

make it smaller (don’t close it completely) and keepa close eye on the recovered image.

4. If you don’t observe any change in the recovered im-age make even smaller adjustments to the slit width.

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Have a closer look at the image, particularly the spac-ings and sizes of the dots.

5. Remove the eight slit object and put in the semitoneobject, the one marked ‘1/2 tone’. Note your obser-vations.

6. Now use one of the wire grids (the one you thoughtmost ‘interesting’) with this filter, again noting obser-vations and comparing to other objects.

Question 9 Which type of filtering is this? High- or low-pass? Can you suggest anything to improve your filteringmethods?

Question 10 Did this filtering work as you expected? Ex-plain, without maths, how only including the centre of thediffraction pattern alters the resulting image.

3.4.1 Filtering the centre of the diffraction pattern

Now we want to investing blocking the centre of the diffrac-tion pattern. To do this, there are a number of dots ordifferent sized that can be positioned to block the centreof the pattern, in place of the adjustable iris.You should try using both a semitone image (a face) andagain the most interesting wire grid with some of thesedots and record your observations.

Question 11 Do you expect to see anything different inthe re-creation of the wire grid image? How does this fil-tering of the wire grid compare with the previous filteringyour performed?

Question 12 Again, which type of filtering is this? Howcould you improve this particular filtering method? Con-sider both the equipment and how everything is set up.

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Figure 9: Though this is an x-ray diffraction image, you cansee the the filter they’ve applied in the centre.You could fabricate and apply a similar filter inplace of the adjustable iris.

Question 13 Try moving the dot around the pattern. Doesthis have any noticeable effect?

You may have a bit of trouble filtering the centre of thediffraction patterns. Remember to try to be as precise inmeasurements and observations as possible!If your filtering wasn’t up to your expectations, can youexplain why? A good starting comment is whether focalpoints are actually ‘points’, or whether they might be aline, or even a volume...

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