V. Fourier transform5-1. Definition of Fourier Transform
* The Fourier transform of a function f(x) is defined as
dxexfuFxfF iux2)()()(
The inverse Fourier transform, 1F
dueuFxf
xfFFxfiux2
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 ;0ˆˆ0ˆˆ ;1ˆˆ ;0ˆˆ0ˆˆ ;0ˆˆ ;1ˆˆ
wzvzuzwyvyuywxvxux
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
axax
axfor 0for
)(
1)(
dxax
Generalized function: the limit of a sequenceof functions
Start with the normalized Gaussian functions2
)( nxn enxg
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 1n
2
)( nxn enxg
1)(
dxxgn
Consider sequence of function21)(1
xexg
222
2)( xexg
2256256
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.
dxxfendxxfxg nx
nnn)(lim)()(lim
2
Only f(0) is important
)0(lim)0(2
fdxenf nx
n
dxxfxfdxxfen nx
n)()()0()(lim
2
Dirac Delta Function: limit of Gaussiandistribution function
2
lim)( nx
nenx
There are infinitely many sequences that canbe used to define the delta function
dxxdxen 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 Spacef(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)
auF
aaxfF 1)}({
dxeaxfaxfF iux2)()}({
Set X = ax
adXeXfaufF a
Xiu2)()}({
dXeXf
aaufF
Xaui2
)(1)}({
auF
a1
(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
xdfdx
xdfF iux2)(})({
dueuFxf
xfFFxfiux2
1
)()(
)()(
dxedueuF
dxd
dxxdfF iuxxiu 2'2 ')'(})({
dxedu
dxdeuF
dxxdfF iux
xiu
2
'2
')'(})({
dxedueiuuF
dxxdfF iuxxiu 2'2 ''2)'(})({
')'('2})({ )'(2 dudxeuFiu
dxxdfF xuui
')'()'('2})({ duuuuFiu
dxxdfF
)'( 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|| when12|| when0
)(bxbx
xf
dxexfuFxfF iux2)()()}({
uub
iuee
iueuF
iubiubb
b
iux
)sin(22
)(2/
2/
2
2/
2/
22/
2/
2 )2(21)(
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
ububbuF
)sin()(
sin
2sin2
2sin bbkbub
sin
u
2sin
2sinsin~
~ )('
kb
kb
eRbE 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
11
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
11
1)( 22
iuaiua eeuF
11
11
1)( 22
iuaiua eeuF
Discussion12 iuae For
1)1)(1(
11)( 22
22
iuaiua
iuaiua
eeeeuF
= 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 )()(
||)()(
axax Note that
dxxadxax )|(|)( Set xax ||
||1
||)()|(|
aaxdxdxxa
1)(
dxx
0for 00for
)(xx
x
The Fourier transform of f(x)
hh ahuahuauF )()()(
h a
hua
uF )(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
lwbkv
ahu
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
dxeF xahui
hh
)(2
)(ahuF
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. ConvolutionThe 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 theoremIf )()}({ );()}({ 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
uauNaxfF
1
0
2)1(2/)1(
2/)1(
2Nn
n
iunaaNiuNn
Nn
iuna eee
iua
iuNaaNiu
eee
2
2)1(
11
)()()1(
iuaiuaiua
iuNaiuNaiuNaaNiu
eeeeeee
)sin()sin(
uauNa
Supplement # 1Fourier transform of a Gaussian function isalso a Gaussian function.Suppose that f(x) is a Gaussian function
22
)( xaexf
dxeeuFxfF iuxxa 222
)()}({
dxee aiu
aiuax
22
22
2
2)(
)(
yiuxax
yax
aiuy
iuxaxy
22Define aiuax
adxd
a
deeuF aiu
2
2
)(
deea
uF au
2
2
1)(
12 i
Chapter 5 pptpage 5
2
)(
au
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
21
ax
xax
21
2
)(
au
ea
uF
212
au
21
au
uau
2
21
221
a
aux
c.f hpx ~hkx ~
hkhx ~2
2~kx
22
),( xaexaf
),2( xf),8( xf
2
22
),( au
ea
uaf
),2( uf
),8( uf
Supplement #2Consider 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
dxexfuFiux
iux
2
2
)()(
)()(:
System 2
dueuFxf
dxexfuF
iux
iux
)(21)(
)()(:
System 3
dueuFxf
dxexfuF
iux
iux
)(21)(
)(21)(
:
System 4
System 5
System 6
relationship among Fourier transform, reciprocallattice, and diffraction condition
System 1, 4Reciprocal lattice
)(;
)(;
)(***
bacbac
acbacb
cbacba
**** clbkahGhkl
*
*
2 hkl
hkl
Gkk
GSS
Diffraction condition
System 2, 3, 5, 6Reciprocal lattice
)(2;
)(2;
)(2 ***
bacbac
acbacb
cbacba
**** clbkahGhkl
*
*2
hkl
hkl
Gkk
GSS
Diffraction condition
