lecture 10 fourier transforms remember homework 1 for submission 31/10/08 remember phils problems...
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Lecture 10Lecture 10Fourier TransformsFourier Transforms
Remember homework 1 for submission 31/10/08
http://www.hep.shef.ac.uk/Phil/PHY226.htmRemember Phils Problems and your notes = everything
Today• More Fourier fun with examples
Homework 1 hints Homework 1 hints
For part (c) look in your notes or remember from lecture
For part (d) The answer should be the first line of the integral that you would perform to find bn. For example ….
L
L
L
n
L
n dxL
xnLx
Ldx
L
xnx
Lbsodx
L
xnxf
Lb
2
2
00sin)(
2sin
2sin)(
2
Since an integral is the limit of a sum, and a Fourier series is made up of an infinite sum of discrete terms (harmonics), the sum can be manipulated to form the Fourier transform which describes the continuum of frequencies present in the original function.
Fourier Transforms Fourier Transforms
deFtf ti)(
2
1)(
dtetfF ti
)(
2
1)(
dkekFxf ikx)(
2
1)(
dxexfkF ikx)(2
1)(
where
where
time
amplitude
frequency
amplitudeThey are used extensively in all of physics.
Fourier Transforms Fourier Transforms
pxandxp
pxpxf
0
1)(
Example 1: A rectangular (‘top hat’) function
kp
kppee
ikkF ikpikp sin
2
21
2
1)(
dkekFxf ikx)(
2
1)(
X domain
K domain
If we take the width of the pulse in the X domain as 2p
then the width ΔK of the pulse in K domain is p
2
kpneikpwhenkF ..0sin0)(
pknF(k)
at 1for 0
pk
We find the following important result: 4
2)2(
ppkx
The product of the widths of any function and its Fourier transform is a constant, its exact value determined by how the width is defined.
Therefore …
Heisenberg’s Uncertainty Principle Heisenberg’s Uncertainty Principle The broader the function in real space (x space), the narrower the transform in k space. Or similarly, working with time and frequency, .
constant t
2
px
In quantum physics, the Heisenberg uncertainty principle states that the position and momentum of a particle cannot both be known simultaneously. The more precisely known the value of one, the less precise is the other.
Remember that momentum is related to wave number by
Thus and so
p k
kp 2 pxkx
Eso E tEt
Can you plot exponential functions? Can you plot exponential functions? The ‘one-sided exponential’ function
0
00)(
xe
xxf
x
What does this function look like?
Can you plot exponential functions? Can you plot exponential functions? The ‘one-sided exponential’ function
0
00)(
xe
xxf
x
What does this function look like?
Can you plot exponential functions? Can you plot exponential functions?
For any real number a the absolute value or modulus of a is denoted by | a | and is defined as
The ‘one-sided exponential’ function
0
00)(
xe
xxf
x
What does this function look like?
The ‘****’ function
What does this function look like?
xBexf x)(
Can you plot exponential functions? Can you plot exponential functions?
For any real number a the absolute value or modulus of a is denoted by | a | and is defined as
The ‘one-sided exponential’ function
0
00)(
xe
xxf
x
What does this function look like?
The ‘****’ function
What does this function look like?
xBexf x)(
Can you plot exponential functions? Can you plot exponential functions?
For any real number a the absolute value or modulus of a is denoted by | a | and is defined as
The ‘one-sided exponential’ function
0
00)(
xe
xxf
x
What does this function look like?
The ‘tent’ function
What does this function look like?
xBexf x)(
Fourier Transforms Fourier Transforms Example 4: The ‘one-sided exponential’ function
Show that the function has Fourier transform
0
00)(
xe
xxf
x ikkF
1
2
1)(
0
)(
0 2
1
2
1)(
2
1)( dxedxeedxexfkF ikxikxxikx
ik
eeik
dxekF ikx
1
2
11
2
1
2
1)( 0
0
)(
222
11
2
1)(
k
ik
ik
ik
ikkF
dkekFxf ikx)(
2
1)(
Fourier Transforms Fourier Transforms Example 3: A pulse of radiation
otherwise
ttAtf
0
cos)( 0 Consider a pulse of light given by
The frequency spectrum F(ω) is given by :-
dteee
Adtet
AdtetfF tititititi )(
2
1
2cos
2)(
2
1)( 00
0
)()(2222 0
)(
0
)()()(
00
00
i
e
i
eAdtee
A titititi
)()()()(22)()(22)(
0
)(
0
)(
0
)(
0
)(
0
)(
0
)( 000000
iiiititi eeee
i
Aee
i
AF
)()()()(22)(
0
)(
0
)(
0
)(
0
)( 0000
iiii eeee
i
AFGrouping terms
dtetfF ti
)(
2
1)(
Fourier Transforms Fourier Transforms Example 3: A pulse of radiation
i
ee ii
2sin
But remember that so we write …
)(
)sin(
)(
)sin(
2)(
)sin(2
)(
)sin(2
22)(
0
0
0
0
0
0
0
0
Aii
i
AF
Now let’s multiply both top and bottom by .
)(
)sin(
)(
)sin(
2)(
0
0
0
0AF
Often so the second term is very small and we need only consider
the first term:
10
)(
)sin(
2)(
0
0AF
otherwise
ttAtf
0
cos)( 0
Fourier Transforms Fourier Transforms Example 3: A pulse of radiation
The time duration of the pulse is usually longer than the period T so 10
Hzfsofc 146
8
10103
103
1400 1022 sof
otherwise
ttAtf
0
cos)( 0
12
0 T
Fourier Transforms Fourier Transforms Example 3: A pulse of radiation
i
ee ii
2sin
But remember that so we write …
)(
)sin(
)(
)sin(
2)(
)sin(2
)(
)sin(2
22)(
0
0
0
0
0
0
0
0
Aii
i
AF
Now let’s multiply both top and bottom by .
)(
)sin(
)(
)sin(
2)(
0
0
0
0AF
Often so the second term is very small and we need only consider
the first term:
10
)(
)sin(
2)(
0
0AF
The frequencies present are essentially those in the range
i.e. where so full width of frequencies is
)( 0
0 / /22
If pulse is long, the frequency spread is very small and only frequency observed will be 0. This is as expected. But for a short pulse there is significant broadening and we can no longer say it is made up of a single frequency.
otherwise
ttAtf
0
cos)( 0
This result is relevant to pulsed lasers. Ti-sapphire lasers have been developed which give very short pulses of light – pulses lasting just a few femtoseconds to probe relaxation phenomena in solids.
However the frequency of the light itself is only a little greater than 1015Hz so this means that we really have a short cosine wave pulse in time, and the frequency is therefore spread.
While CW (continuous wave) lasers can emit light with an extremely narrow line-width, pulsed laser light must, by its very nature, have a broader line-width. And the shorter the pulse, the broader the line-width.
Fourier Transforms Fourier Transforms Example 3: A pulse of radiation
otherwise
ttAtf
0
cos)( 0
)(
)sin(
2)(
0
0AF
0 /ttt 0 t t
short pulse
long pulsebig
smallt
narrower
smalllesst
Complexity, Symmetry and the Cosine TransformComplexity, Symmetry and the Cosine Transform
kxdxxf
ikxdxxfkF sin)(
2cos)(
2
1)(
dxexfkF ikx)(2
1)(
We know that the Fourier transform from x space into k space can be written as:-
We also know that we can write sincos iei
So we can say:-
What is the symmetry of an odd function x an even function ?
If f(x) is real and even what can we say about the second integral above ?Will F(k) be real or complex ?
If f(x) is real and odd what can we say about the first integral above ?Will F(k) be real or complex ?
Odd
2nd integral is odd (disappears) and F(k) is real
1st integral is odd (disappears), F(k) is complex
What will happen when f(x) is neither odd nor even ?
Neither 1st nor 2nd integral disappears, and F(k) is usually complex
Complexity, Symmetry and the Cosine TransformComplexity, Symmetry and the Cosine Transform
kxdxxf
ikxdxxfkF sin)(
2cos)(
2
1)(
f(x)f(x) is real and even is real and even
Since we say
As before the 2nd integral is odd, disappears, and F(k) is real
let’s see if we can shorten the amount of maths required for a specific case …
kxdxxfkF e cos)(
2
1)(
so
X
e
X
X e dxxfdxxf0
)(2)(But remember that
So
0
cos)(2
)( kxdxxfkF e
0
cos)(2
)( kxdxkFxf e
Now since F(k) is real and even it must be true that were we to then find the Fourier transform of F(K) , this can be written:-
LET’S GO BACKWARDS
Complexity, Symmetry and the Cosine TransformComplexity, Symmetry and the Cosine Transform
0
cos)(2
)( kxdxxfkF e
0
cos)(2
)( kxdxkFxf e
Fourier cosine transform
Here is the pair of Fourier transforms which may be used when f(x) is real and even only
Example 5: Repeat Example 1 using Fourier cosine transform formula above.
kpk
kxk
dxkxkxdxxfkFp
p
ee sin12
sin12
cos2
cos)(2
)(0
00
pxandxp
pxpxf
0
1)(Find F(k) for this function