IMIT

Swedish original by Leif Ek

English verison by Anna Thaning

Optics exp eriment

Fourier Optics

Royal Institute of Technology

Department of Micro electronics and Information Technology

Optics Section

Intro duction

In this exp eriment we will as implied by the title use optical systems to study

F ourier transforms Fourier analysis describ es how a function may b e divided into

a sum or an integral of sines and cosines This is often done in one dimension

only but the optical system provides us with them means to easily do it in two

functions or images expressed dimensions It is describ ed how twodimensional

in the spatial co ordinates x y are transformed The twodimensional Fourier

transform is dened as

Z Z

1

ff x y g f x y expik x k y dxdy F

x y

1

To day twodimensional Fourier transforms can quite easily b e found numerically

for example using image pro cessing software you can also nd a function for it t

in Matlab But this still requires some time and computational p ower During this

exp eriment we will create an instant Fourier transform of a pattern A diractive

slit in uniform illumination gives rise to a diraction pattern which turns out to

b e the fourier transform of the diractive slit It is placed at innity ie very far

from the slit but a fo cusing lens can b e used to move it to the fo cal plane of the

lens Similarly a lens may b e used to retransform the image Schematically the

i m a g e o f o b j e c t Fourier plane t h e o b j e c t l a s e r

transform ing lens re-transform ing lens

Figure Schematic illustration of the exp eriment

setup is shown in Fig It provides us with the opp ortunity to view the Fourier

transform but it also gives us the p ossibility to meddle with it By blo cking some

parts of the Fourier transform we remove certain frequencies from the image The

eect can then b e studied in the image of the ob ject This pro cess is known as

ltering Unfortunately this metho d for viewing the Fourier transforms has some

limitations The lenses used are not p erfect they have ab errations which will distort

the transform But even with a p erfect lens the true transform will not b e found

since the nite size of the lens will cut o the highest frequences The lens will act

as a lowpass lter The extent of this ltering dep ends on whether the coherent or

the incoherent illumination is used

The aim of this exp eriment is to show

that Fraunhofer diraction gives the Fourier transform of the diracting

ap erture

what twodimensional transforms of some simple images lo ok like

the dierences that app ear for coherent and incoherent illumination

As an extra b onus you also get the Babinets principle

Theory

Fourier transformation by Fraunhofer diraction

Now one would think that the Fourier transform on an ob ject eg a square could b e

obtained just by illuminating the ap erture and then study the diraction pattern In

reality its a little bit more complicated since the diraction pattern of an ap erture

there are changes its shap e dep ending on the distance of observation Generally

two cases Fresnel diraction where the diraction pattern close to the ap erture

is considered and Fraunhofer diraction where the pattern is studied at innity

Fresnel diraction is normally much more complicated so we will stick to Fraunhofer

diraction and show that the Fraunhofer diraction pattern is really the Fourier

transform of the ob ject Assume that the ob ject consists of an ap erture of any

y x r

R

Figure Diraction from an arbitrary ap erture at P X Y Z

shap e in the x y plane We study the diraction pattern in the p oint P of

resnel co ordinates X Y Z as illustrated in Fig Accordning to the HuygensF

principle the ap erture may b e regarded as the sum of many p oint sources that

pro duce spherical waves From the p oint source lo cated in the small surface element

dS we obtain a spherical wave which gives rise to the electric eld dE

A

dE exp ik r tdS

r

where is the strength of the source ie the incident eld strength at this par

A

ticular p oint and r is the distance b etween this p oint on the ap erture and the

observation p oint P as illustrated in Fig This is the normal expression for a

spherical wave on complex form see eg Eq in Hecht To obtain the total

eld at P we must sum all the dierent contributions This is done by integration

ZZ

A

E P exp ik r tdS

r

A

where A is the area of the diracting ap erture This is a fairly complicated ex

pression and we need to do some approximations According to the geometry see

Fig

p

r X x Y y Z

p

R X Y Z

Since the size of the ob ject is small compared to the distance R we can approximate

R in the denominator However due to the large value of k r very small changes r

to r will cause large changes to the exp onential term so here we need a more accurate

approximation By some manipulation we get

r

x y xX y Y

r R

R R

The term x y R is small and may b e neglected Then expansion in a p ower

series yields

r

xX y Y xX y Y

R r R

R R

Insertion of Eq into Eq yields

ZZ ZZ

xX y Y

A A

E X Y Z exp ik r tdS exp ik r t expik dS

r R R

A A

To simplify the expression we intro duce the ap erture function

A

exp ik r t A Ax y x y expix y

R

which describ es the transmittance of the ob ject It contains an amplitude part

A x y and a phase part x y R is constant with resp ect to the integration

variables x and y Eq may now b e rewritten as

ZZ

1

E X Y Z Ax y expik xX y Y R dxdy

1

The integration can b e extended to innity since the ap erture function is zero

everywhere but in the ap erture In order to identify Eq as a twodimensional

Fourier transform we dene the spatial frequencies

k X k Y

k k

x y

R R

Inserted into Eq they yield

ZZ

1

Ax y expik x k y dxdy E k k

x y x y

1

Except for the names of the variables this is identical to Eq Consequen tly we

have shown that the electric eld at Fraunhofer diraction is the Fourier transform

of the ap erture function ie that

E k k F fAx y g

x y

Since the ap erture function describ es the geometry of the ap erture this means the

diraction pattern far away from the ob ject will b e its Fourier transform

Tw odimensional images and transforms

In order to b etter understand twodimensional images and their transforms well

start with the easier onedimensional case Assume we have a function of one vari

able f t and its transform F The function can always b e written as a sum of

sine and cosine functions that is as a sum of dierent frequency comp onents The

Fourier transform of the function will the b e the sum of the Fourier transforms of

the trigonometric functions that constitute ft In two dimensions the same thing

happ ens An image can b e viewed as a lot of sinusoidal lattices These lattices have

dierent spatial frequencies and dierent orientations as illustrated in Fig The

a ) b )

Figure Two sinusoidal gratings of dierent orientations

Fourier transform of a sinusoidal lattice is three delta functions one in the origin

and one to each side as illustrated in Fig The p ositions of the the delta pulses

are determined by the spatial frequency of the lattice the higher the frequency the

longer the distance to the origin and by its orientation This can easily b e deduced

mathematically but we leave this to the interested student

a ) b )

Figure The Fourier transforms of the gratings in Fig

Task Some simple transforms

Transforms of several identical parts

If the ob ject consists of several identical parts the sup erp osition principle says the

image will b e the sum of the electrical eld from each part Assume there are N

0 0

parts and that each of them gets the lo cal co ordinate system x y where the

origin is at x y j N as illustrated in Fig Intro duce the notation

j j

Figure An ob ject consisting of four identical parts

0 0

Ax y and A x y for the total ap erture function and the ap erture function for

one part resp ectively Then the total eld will b e

ZZ

N

1

0 0

X

X x x Y y y

j j

0 0 0 0

E X Y A x y exp ik dx dy

R

1

j

or

ZZ

N

1

0 0

X

X x Y y X x Y y

j j

0 0 0 0

ik dx dy exp ik E X Y A x y exp

R R

1

j

For simplicity intro duce the spatial frequences k and k as dened in Eq

x y

Then the eld my b e written as

ZZ

0

N

1

X

0 0 0 0 0 0

A x y exp ik x k y dx dy exp ik x expik y E k k

x y x j y j x y

1

j

The sum at the end of the expression is a phase factor only it will not aect the

pattern itself Consequently several identical parts give the same Fourier transform

as one of the parts except for the extra phase factor

This shows that the electrical eld E at the image plane is the Fourier transform

of the ap erture function for one part of the pattern A multiplied by the Fourier

transform of delta pulses at the origin of each part ie that

N

X

0 0

E F fAx y g F fA x y g F x x y y

j j

j

Compare to the convolution theorem which says that

F fAg F fA g F fg g F fA g g

Tee conclusion is that several identical parts of a gure may b e describ ed as the

ap erture function of the part convoluted with delta functions at the centre of each

part as illustrated in Fig

A A 1 g

x= x* x

Figure Several identical parts can mathematically b e describ ed as a convolution

b etween one part and several delta functions

Task The transform of two circles

a Use two delta functions and a circ function to describ e two circles at a distance

a from each other Place one circle at the origin and the other at a A circ

function is dened as

x y D

circx y

x y D

where D is the diameter of the circle

You dont need to calculate the b Find the Fourier transform of the two circles

Fourier transform of a single circle it is an Airy disc as you probably know since

the phase factor is the interesting part Then lo ok at Fig Is this the diraction

pattern of two circles

c Find the distance b etween the two circles expressed in their diameter D In

the diraction pattern of a circle we know that the distance from the centre to the

rst dark ring is

q

k D

p

where k k k

p

x y

Assume that there are n dark lines inside the circle

Figure Is this the diraction pattern of two circles

Babinets principle

and Complementary means the Assume there are two complementary ob jects

rst ob ject is dark where the other is transparent and transparent where the other

is dark as illustrated in Fig According to Babinets principle the diraction

patterns of those ob jects will b e identical Assume that E is the eld b ehind

Figure Two complementary ob jects

and E the eld b ehind If b oth ob jects are illuminated at the same time no

light at all will get through This means

E E E E I I

This means the intensities b ehind the ob ject are identical

Because of the nite size of the ob jects they are not truly complementary

Because of this Babinets principle do esnt apply to the th diraction order the

centre of the transforms

Transfer theory

oherent or In optical imaging there are two main typ es of illumination namely c

incoherent The coherent system is linear with resp ect to amplitude and the inco

herent is linear with resp ect to intensity ie for coherent light we add amplitudes

but for incoherent light we add intensities The main p oint of this section is to

show that the imaging prop erties of the two systems are dierent a fact that will

also b e conrmed exp erimentally Assume there is a p oint source x y in the ob

y Y x X y x Y X

( 0 , 0 ) h ( X , Y ) ( x , y ) h ( X - x , Y - y )

Figure A p oint source and its distribution in the image plane When the source

is moved the image is moved in the same way

X x Y y in the image plane The ject plane and that it pro duces the eld h

corresp onding intensity is jhX x Y y j as illustrated in Fig If the incident

eld and intensity is E x y and I x y resp ectively the resulting eld or intensity

in the image plane can b e found by adding all those little p oint sources together

integration We nd that

ZZ

0

E X Y hX x Y y E x y dxdy

for coherent light and that

ZZ

0

I X Y jhX x Y y j I x y dxdy

for incoherent light Since b oth expressions are convolutions they can b e Fourier

transformed to yield

0

F fE g H F fE g

0

F fI g H H F fI g

where H F fhg H and H H are the transfer functions of the system For an ideal

lens H is illustrated in Fig This transfer function a attop is valid for coherent

light For incoherent light two attops should b e convoluted to give the triangular

transfer function of Fig Its worth noticing that the limiting frequency the

highest frequency that can b e imaged by the system for the incoherent light is twice

that of the coherent light Incoherent light is b etter for imaging high frequencies

while the lower frequencies are imaged b etter in coherent light H

k x

Figure The transfer function for an ideal lens in coherent illumination

H*H

k x

Figure The transfer function for an ideal lens in incoherent illumination

Summary of the theory

After reading the instructions and b efore p erforming the exp erimental tasks you

should know the following

That the Fourier transform of an ob ject may b e found through Fraunhofer

diraction

That an ob ject containing several identical patterns will give the Fourier

transform of this pattern mo dulated by an interference pattern

Accordning to Babinets principle two complementary ob jects give the

same intensity pattern

That transfer functions can b e used to see what spatial frequencies a

system will image

That coherent and incoherent systems have dierent transfer functions

and consequently dierent imaging prop erties

Exp eriment

t Equipmen

The basic equipment for this exp eriment is displayed in Fig A HeNe laser f i l t e r L 1 o b j e c t L 2 L 3 L 4

l a s e r m - h o l e Fourier plane s c r e e n

Figure Schematic illustration of the exp eriment L and L give a magnied

image of the Fourier plane L retransforms it into an image of the ob ject which

can b e seen on the screen

provides coherent illumination via a microscop e ob jective with a micrometer hole

that creates a spherical wave Then lens L p erforms the Fourier transformation and

a holder is placed at the Fourier plane Two lenses L and L provide a magnication

of the Fourier transform on the screen The lens L will b e inserted and removed

many times during the exp eriment since it pro jects the image on the screen For

incoherent illumination the laser is replaced by a lamp

er lo ok into the laser b eam Nev

The intensity is high enough to cause eye damage

Dont touch the lenses or the ob jects other than at the edges

Tasks

Setting up the exp eriment

part Turn the laser on

Move the hole in the transverse direction until the transmit

ted intensity is maximized

Move the hole towards the microscop e ob jective Be careful

that you dont move them close enough to touch each other

Move the hole in the transverse direction again Rep eat the

pro cedure until the hole is in the fo cal p oint of the micro

scop e ob jective ie when there is no interference pattern in

the transmitted intensity

Insert lens L

Place a diusely reecting surface a piece of aluminum in the

xyholder and nd the Fourier plane by maximizing the sp eckle

size

Insert lenses L and L so that the Fourier plane is magnied

on the screen Use for example ob ject

Insert lens L so that the ob ject is imaged on the screen

Some simple transforms

part Place ob ject in the ob ject holder It consists of four geomet

rical patterns Lo ok at their images on the screen

Blo ck all patterns except one at the ob ject Then view the

Fourier transform on the screen ie remove L The trans

form is dicult to see so you need almost complete darkness

There might b e a b etter version of the Fourier pattern at the

xyholder

ask Observe and draw the transforms of all four patterns T

Sev eral identical images

part Insert ob ject two circular holes Observe the Fourier trans

form

Task Draw the Fourier transform of two circles Do es it agree

with your predictions in task of the theory section use the

Fourier transform to determine the distance b etween the circles

expressed in their diameter Check your results by measuring

the distance in the image of the ob ject Think of p ossible causes

for errors

part Replace ob ject by ob ject three circles Try to predict

the Fourier transform then lo ok at it Was your prediction

accurate

Task Draw the Fourier transform

part Insert ob ject random circles

Task Draw the Fourier transform Explain what you see

Insert ob ject random circles part

Task Draw the Fourier transform Explain what you see Are

the circles really random

principle Babinets

Insert ob ject two gratings part

Insert an aluminum piece that allows you to blo ck the left or

the right side of the ob ject

Task Regard the two Fourier transforms from the left and

the right part Are they dierent from each other

Task Remove the aluminum piece and instead blo ck the th

order in the Fourier plane make a black dot on a piece of glass

Lo ok at the image of the ob ject What has happ ened Explain

Task Now blo ck everything except the th order in the Fourier

plane make a small hole in a piece of pap er Lo ok at the im

age and explain what has happ ened

Task Make a slightly bigger hole which lets orders and

through Explain what you see

Frequency ltering

part Insert ob ject a sector star

Image the star on the screen Then watch the Fourier trans

form Dierent parts of the Fourier transform corresp ond to

dierent parts of the image Try to understand how they are

connected

Task Around order in the Fourier tranform there is a dark

area Why

Task Perform lowpass and highpass ltering using holes

and dots Regard the image of the ob ject and explain what

happ ens

Task Use a piece of pap er to blo ck the lower half of the Fourier

plane but place it low so that the central part of the transform

can pass through The use the xyholder to move the pap er

slowly until the central part is blo cked to o Watch the image

of the ob ject while doing this Explain what happ ens

Transfer theory

Use the same ob ject but p erform lowpass ltering in the part

Fourier plane so that the limiting frequency is at half the radius

of the star

Insert lters in front of the ob ject to reduce the intensity of

the light

ONLY CONTINUE IF THE ASSISTANT IS PRESENT

The assistant will check that the intensity level is low enough

Lo ok into the system through the last lens L You will see an

image of the sector star You can replace L by a telescop e if

you want to

NOBODY SHOULD TOUCH THE SYSTEM WHILE SOME

BODY IS LOOKING INTO IT Changes might lead to sudden

changes in intensity level

Insert the diusing glass b ehind the ob ject This is allowed

since it will only reduce the intensity

Increase the intensity slowly by moving the lter until the

p erson watching can see a sp eckle pattern

Move the diusing glass back and forth in the holder careful

not to unblo ck the op ening Now you have made the light

incoherent

Task What has happ ened to the image and the limiting fre

quency Which was b etter for imaging the high frequences the

coherent or the incoherent light

part Replace ob ject by ob ject remove all lters and the diusing

glass and insert L Ob ject consists of two gratings whose

amplitude transmittance is shown in Fig

t t

x x

Figure The amplitude transmittance of the two gratings in ob ject

Task What is the intensity transmittance of the two gratings

Observe each grating and their Fourier transforms b oth sepa

rately blo ck part of the image and together

Task Explain the dierences b etween the two Fourier trans

forms Identify the two gratings This is a dicult task ask

your assistant if you cant solve it

Remove the aluminum piece that blo cks half the image and

insert a lter in the Fourier plane It should just barely let

three orders pass

Task Then move the lter in the Fourier plane along the

Fourier transform and see what happ ens to the image Explain

Filters and pattern recognition

part Insert ob ject dierent gratings

Watch the image and the Fourier transform Try to explain how

dierent parts of the Fourier transform corresp ond to dierent

parts of the image

Insert a lter that lets only order through

Task Move the hole using the xyholder so that dierent

parts of the Fourier transform are transmitted Watch the im

age Can you explain what happ ens to it

Task Make a lter in the Fourier plane which removes all

horizontal gratings but not the others

part Insert ob ject three circles

Observe the image of the ob ject What do you think the Fourier

transform will lo ok like Lo ok at the Fourier transform to see

if you were right

Task Explain the Fourier transform

Task Make a lter on the Fourier plane which removes two

of the circles but lets the third one through

part Keep ob ject but remove the lter

Turn the laser o and intro duce white light Dont change

the system just intro duce the white light through a mirror at



This way you can remove the mirror and turn the laser

on and the system will still b e aligned

Regard the image using L You might need to insert some

lters

Task Make the three circles app ear in dierent colours for

example one red one yellow and one blue

part If there is still time there are extra tasks for those interested

Ask the assistant