V. Fourier transform 5-1. Definition of Fourier Transform * The Fourier transform of a function f(x) is defined as The inverse Fourier transform,

V. Fourier transform

5-1. Definition of Fourier Transform

* The Fourier transform of a function f(x) is defined as

dxexfuFxfF iux2)()()(

The inverse Fourier transform, 1F

dueuFxf

xfFFxf

iux2

1

)()(

)()(

3-D: the Fourier transform of a function f(x,y,z)

dxdydzezyxfwvuF wzvyuxi )(2),,(),,(

Note that zzyyxxr ˆˆˆ

wwvvuuu ˆˆˆ

ux+vy+wz: can be considered as a scalarproduct of if the following conditions are met!

ur

1ˆˆ ;0ˆˆ ;0ˆˆ

0ˆˆ ;1ˆˆ ;0ˆˆ

0ˆˆ ;0ˆˆ ;1ˆˆ

wzvzuz

wyvyuy

wxvxux

wzvyuxur

Therefore,

rderfuFrfF uri 2)()()(

the vector may be considered as a vector in “Fourier transform space”

u

The inverse Fourier transform in 3-D space:

udeuFrfuFF uri 21 )()()(

5-2. Dirac delta function

ax

axax

for 0

for )(

1)(

dxax

Generalized function: the limit of a sequenceof functions

Start with the normalized Gaussian functions

2

)( nxn e

nxg

2

1

n

: standard Gaussian width parameter

1)(

dxxgn

Gaussian Integration:

dxeG x2

dxdyedyedxeG yxyx )(2 2222

dxdy: integration over a surface change to polar coordinate (r, )

rrd

dr

sin ;cos ryrx 222 ryx

2

0 0

2 2

rdrdeG r

Star from

1/2 GG ;2

?2

dxe nx Let xny dxndy

dyenn

dyedxe yynx 222 1

n

2

)( nxn e

nxg

1)(

dxxgn

Consider sequence of function21

)(1xexg

22

2

2)( xexg

2256

256

256)( xexg

);......( );( 43 xgxg

http://en.wikipedia.org/wiki/Dirac_delta_function

a = 1/n

What happen when n =

(a)(b)(c)(d)

)0(g0)0( xg

the width of the center peak = 01)(

dxxg

The sequence only useful if it appears as partof an integral, e.g.

dxxfe

ndxxfxg nx

nn

n)(lim)()(lim

2

Only f(0) is important

)0(lim)0(2

fdxen

f nx

n

dxxfxfdxxfe

n nx

n)()()0()(lim

2

Dirac Delta Function: limit of Gaussiandistribution function

2

lim)( nx

ne

nx

There are infinitely many sequences that canbe used to define the delta function

dxxdxe

n nx

n)(1lim

2

)'()()'( xfdxxfxx

Dirac delta function is an even function

dueuFxf iux2)()(

')'()( '2 dxexfuF iux

duedxexfxf iuxiux 2'2 ')'()(

')'()( 2'2 dxdueexfxf iuxiux

)'(2 xxiue

Note that

')'()'()( dxxxxfxf =

duexx xxiu )'(2)'(

duey iuy 2)(

y

Similar

dxexfuF iux2)()(

dxedueuFuF iuxxiu 2'2 ')'()(

')'()( )'(2 dudxeuFuF xuui

')'()'()( duuuufuF

dxeuu xuui )'(2)'(

uuy 'Let

dxey iyx 2)(

duey iuy 2)(Compare to )()( yy

5-3. A number of general relationships maybe written for any function f(x)real or complex.

Real Space Fourier Transform Space

f(x) F(u)

f(-x) -F(-u)

f(ax) F(u/a)/a

f(x)+g(x) F(u)+G(u)

f(x-a) e-2iauF(u)

df(x)/dx 2iuF(u)

dnf(x)/dxn (2iu)nF(u)

Example

(1)

a

uF

aaxfF

1)}({

dxeaxfaxfF iux2)()}({

Set X = ax

a

dXeXfaufF a

Xiu2

)()}({

dXeXf

aaufF

Xa

ui2

)(1

)}({

a

uF

a

1

(2) uFeaxfF iau2)}({

dxeaxfaxfF iux2)()}({

Set X = x - a

)()()}({ a)(2 aXdeXfaxfF Xiu

dXeXfaxfF Xiu a)(2)()}({

dXeXfe iuXiu 2a2 )(

uF

uFeaxfF iau2)}({

(3) uiuFdx

xdfF 2}

)({

dxedx

xdf

dx

xdfF iux2)(

})(

{

dueuFxf

xfFFxf

iux2

1

)()(

)()(

dxedueuF

dx

d

dx

xdfF iuxxiu 2'2 ')'(}

)({

dxedu

dx

deuF

dx

xdfF iux

xiu

2

'2

')'(})(

{

dxedueiuuF

dx

xdfF iuxxiu 2'2 ''2)'(}

)({

')'('2}

)({ )'(2 dudxeuFiu

dx

xdfF xuui

')'()'('2}

)({ duuuuFiu

dx

xdfF

)'( uu

)(2})(

{ uiuFdx

xdfF

5-4. Fourier transform and diffraction

(i) point source or point aperture

A small aperture in 1-D: (x) or (x-a).

Fourier transform the function Fraunhofer diffraction pattern

For (x):

dxxedxexxF iuiux )()()}({ 022

= 1 = 1= 1

The intensity is proportional 1|)(| 2uF

For (x-a):

dxeaxaxF iux 2)()}({

The intensity is proportional 1|)(| 2uF

dXeXXF aXiu )(2)()}({

Set X = x-a

dXeXe iuXiua 22 )(

iuaeaxF 2)}({

The difference between the point source atx = 0 and x = a is the phase difference.

(ii) a slit function

2|| when1

2|| when0)(

bx

bxxf

dxexfuFxfF iux2)()()}({

u

ub

iu

ee

iu

euF

iubiubb

b

iux

)sin(

22)(

2/

2/

2

2/

2/

22/

2/

2 )2(2

1)(

b

b

iuxb

b

iux iuxdeiu

dxeuF

c.f. the kinematic diffraction from a slit

c.f. the kinematic diffraction from a slit

Chapter 4 ppt p.28

ub

ubbuF

)sin(

)(

sin

2

sin2

2

sin bbkbub

sin

u

2sin

2sin

sin~~ )(

'

kb

kb

eR

bE tkRiL

(iii) a periodic array of narrow slits

n

naxxf )()(

dxexfuFxfF iux2)()()}({

dxenaxuF iux

n

2)()(

n

iuxdxenaxuF 2)()(

n

iuna

n

iuna edxnaxeuF 22 )()(

xx

n

n

1

1

1)(0

20

22

n

iuna

n

iuna

n

iuna eeeuF

0

2

n

iunae

1)(0

2

0

2

n

niua

n

niua eeuF

11

1

1

1)(

22

iuaiua ee

uF

11

1

1

1)(

22

iuaiua ee

uF

Discussion12 iuae For

1)1)(1(

11)(

22

22

iuaiua

iuaiua

ee

eeuF

= 1

0)( uF

12 iuae For )(uF

It occurs at the condition

)2sin()2cos(12 uaiuae iua

hua 22 h: integerhua

In other words,

h

huauF )()(

||

)()(

a

xax

Note that

dxxadxax )|(|)( Set xax ||

||

1

||)()|(|

aa

xdxdxxa

1)(

dxx

0for 0

0for )(

x

xx

The Fourier transform of f(x)

hh a

huahuauF )()()(

h a

hu

auF )(

1)( where a > 0

Hence, the Fourier transform of a set ofequally spaced delta functions with a perioda in x space a set of equally spaced delta functionswith a period 1/a in u space

Similarly， a periodic 3-D lattice in real space;(a, b, c)

m n p

pcznbymaxr ),,()(

)()}({ uFrF

rdepcznbymax rui

m n p

2),,(

h k l c

lw

b

kv

a

hu

abc)()()(

1

This is equivalent to a periodic lattice inreciprocal lattice (1/a 1/b 1/c).

(iv) Arbitrary periodic function

h

aihxheFxf /2)(

http://en.wikipedia.org/wiki/Fourier_series

)()}({ uFxfF

dxeeF iux

h

aihxh

2/2

dxeeF iuxaihx

hh

2/2

dxeFxa

hui

hh

)(2

)(a

huF

hh

Hence， the F(u) ； i.e. diffracted amplitude,is represented by a set of delta functions equally spaced with separation 1/a and eachdelta function has “weight”, Fh, that is equalto the Fourier coefficient.

5-5. Convolution

The convolution integral of f(x) and g(x) isdefined as

dXXxgXfxgxfxc )()()()()(

examples：(1) prove that f(x) g(x) = g(x) f(x)

dXXxgXfxgxfxc )()()()()(

Set Y = x - X

)()()()()( YxdYgYxfxgxf

dYYgYxfxgxf )()()()(

)()( xfxg

(2) Prove that )()()( xfxxf

dXXxXfxxf )()()()(

)()()()()( xfdXXxxfxxf

(3) Multiplication theorem

If )()}({ );()}({ uGxgFuFxfF

then )()()}()({ uGuFxgxfF

(4) Convolution theorem (proof next page)If )()}({ );()}({ uGxgFuFxfF

then )()()}()({ uGuFxgxfF

Proof: Convolution theorem

dxedXXxgXfxgxfF iux2)()()}()({

dxedXXxgeXf XxiuiuX )(22 )()(

)()()( )(22 XxdedXXxgeXf XxiuiuX

)()()( )(22 XxdeXxgdXeXf XxiuiuX

= F(u) = G(u))()( uGuF

Example： diffraction grating

2/)1(

2/)1(

)()()(Nn

Nn

xgnaxxf a single slit orruling function

a set of N delta function

2/)1(

2/)1(

)()()}({Nn

Nn

xgnaxFxfF

dxenaxnaxF iunaNn

Nn

Nn

Nn

22/)1(

2/)1(

2/)1(

2/)1(

)()(

dxnaxeNn

Nn

iuna )(2/)1(

2/)1(

2

2/)1(

2/)1(

2Nn

Nn

iunae

)()( uGxgF

)()sin(

)sin()}({ uG

ua

uNaxfF

1

0

2)1(2/)1(

2/)1(

2Nn

n

iunaaNiuNn

Nn

iuna eee

iua

iuNaaNiu

e

ee

2

2)1(

1

1

)(

)()1(iuaiuaiua

iuNaiuNaiuNaaNiu

eee

eeee

)sin(

)sin(

ua

uNa

Supplement # 1Fourier transform of a Gaussian function isalso a Gaussian function.

Suppose that f(x) is a Gaussian function22

)( xaexf

dxeeuFxfF iuxxa 222

)()}({

dxee a

iu

a

iuax

22

22

2

2)(

)(

yiuxax

yax

a

iuy

iuxaxy

22Define a

iuax

adxd

a

deeuF a

iu

2

2

)(

deea

uF a

u2

2

1)(

12 i

Chapter 5 pptpage 5

2

)(

a

u

ea

uF Gaussian Function

in u space

Standard deviation is defined as the range of thevariable (x or u) over which the function dropsby a factor of of its maximum value.2/1e

22

)( xaexf 2/122 ee xa

2

1 ax

xa

x 2

1

2

)(

a

u

ea

uF

2

12

a

u2

1

a

u

u

au

2

2

1

22

1

a

aux

c.f hpx ~hkx ~

hkh

x ~2

2~kx

22

),( xaexaf

),2( xf

),8( xf

2

22

),( a

u

ea

uaf

),2( uf

),8( uf

Supplement #2

Consider the diffraction from a single slit

2/

2/

sin2)(

'~~ b

b

iztkRiL dzeeR

E

The result from a single slit

dxexfuF iux2)()(

The expression is the same as Fourier transform.

Supplement #3Definitions in diffractionFourier transform and inverse Fourier transform

System 1

dueuFxf

dxexfuF

iux

iux

2

2

)()(

)()(:

System 2

dueuFxf

dxexfuF

iux

iux

)(2

1)(

)()(:

System 3

dueuFxf

dxexfuF

iux

iux

)(2

1)(

)(2

1)(

:

System 4

System 5

System 6

relationship among Fourier transform, reciprocallattice, and diffraction condition

System 1, 4

Reciprocal lattice

)(;

)(;

)(***

bac

bac

acb

acb

cba

cba

**** clbkahGhkl

*

*

2 hkl

hkl

Gkk

GSS

Diffraction condition

System 2, 3, 5, 6

Reciprocal lattice

)(

2;

)(

2;

)(

2 ***

bac

bac

acb

acb

cba

cba

**** clbkahGhkl

*

*2

hkl

hkl

Gkk

GSS

Diffraction condition