v. fourier transform 5-1. definition of fourier transform * the fourier transform of a function f(x)...
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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
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