Transcript Solid State III, Lecture 23

Lecture 8
 Aims:
Fourier Analysis.
 Fourier Theory:
Description of waveforms in terms of a
superposition of harmonic waves.
Fourier series (periodic functions);
Fourier transforms (aperiodic
functions).
Wavepackets
 Convolution
convolution theorem.
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Fourier Theory
 It is possible to represent (almost) any function
as a superposition of harmonic functions.
 Periodic functions:
 Fourier series
 Non-periodic functions:
 Fourier transforms
 Mathematical formalism
 Function f(x), which is periodic in x, can be
written:

f x 
1
Ao 
2
where,
2
An 
l
Bn 
2
l

n 1
l/2

 2nx 
 2nx 
A
cos

B
sin




n
 n
 l 
 l 

 2nx 
f  x  cos
dx n  0,1,2...
 l 

 2nx 
f  x  sin 
dx n  1,2...
 l 
l / 2
l/2
l / 2
 Expressions for An and Bn follow from the
“orthogonality” of the “basis functions”, sin and
cos.
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Complex notation
 Example: simple case of 3 terms
y  cos2x l   sin 2x l   cos4x l 
cos2x l 
 sin 2x l 
cos4x l 
y
 Exponential
representation:

f x 
Cn 

Cn ei 2nx / l
n  
1 l/2

f  x e i 2nx / l d x
l l / 2
f x 
 with k=2n/l.
C k  
3


C k eikx
n  
1 l/2

f  x e ikx d x
l l / 2
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Example
 Periodic top-hat:
f x  A  l / 8  x  l / 8
 0  l / 2  x  l / 8, l / 8  x  l / 2
f(x)
Fourier transform
1 l /8
l /8

ikx

A e



l l / 8
l   ik 
l / 8
A
2A
 eikl / 8  e ikl / 8  
sin kl / 8
ikl
kl
 C k  

Aeikxdx 
A
A
sinc kl / 8  sinc n / 4 
4
4
sin  x 
x
 N.B.
  1
x 0
x
x
4
Zero when n
is a multiple of 4
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Fourier transform variables
 x and k are conjugate variables.
t
 Analysis applies to a periodic function in any
variable.
and w are conjugate.
F t  

Cn 


Cn ei 2nt / T
n  
1 T /2

F t e i 2nt / T
[6.1]
w n  2n / T
T T / 2
 Example: Forced oscillator
 Response to an arbitrary, periodic, forcing
function F(t). We can represent F(t) using [6.1].
 If the response at frequency nwf is R(nwf), then
the total response is

inw t
R nw f C n e f
n  



Linear in both response and driving amplitude
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Fourier Transforms
 Non-periodic functions:
 limiting case of periodic function as period .
The component wavenumbers get closer and
merge to form a continuum. (Sum becomes an
integral)
1 
ikx
f ( x) 
g (k ) 
g (k )e dk

2  
1 
f ( x)e ikxdk
2  
 This is called Fourier Analysis.
f(x) and g(k) are Fourier Transforms of each
other.
 Example:
f ( x)  A
 x / 2  x  x / 2
0
x  x / 2
x / 2

ikx

x
/
2

1
A e
 ikx
g (k ) 
Ae
dk 


2  x / 2
2   ik 
Top hat

 x / 2
2A
Ax

sin( kx / 2) 
sinc( kx / 2)
2 k
2
 Similar to Fourier series but now a continuous
function of k.
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Fourier transform of a Gaussian
 Gaussain with r.m.s. deviation x=s.
A
 x 2 / 2s 2
f ( x) 
e
s 2


f ( x)dx  A
 Note
 Fourier transform
1 

A
 x 2 / 2s 2 ikx
g (k ) 
e
e
dx
2   s 2
 Integration can be performed by completing the
square of the exponent -(x2/2s2+ikx).
2
2
2 2
ik 2s  k s
 x
 ikx  

 
2
2 
2
2s
 2s
x
 k 2s 2 / 2
u2

 where,
x
ik 2s
u

;
du  dx / 2s
2s
2

A  u 2  k 2s 2 / 2
g (k ) 
e e
2sdu
=
2s  


A
2s
2se
 k 2s 2 / 2
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

e
u 2
A  k 2s 2 / 2
du 
e
2
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Transforms
 The Fourier transform of a Gaussian is a
Gaussian.
 Note: k=1/s. i.e. xk=1
 Important general result:
“Width” in Fourier space is inversely related
to “width” in real space. (same for top hat)
d-function
 Common functions (Physicists crib-sheet)

d-function




cosine
sine
infinite lattice
of d-functions
top-hat 
Gaussian 
 In pictures………...
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constant
2 d-functions
2 d-functions
infinite lattice
of d-functions
sinc function
Gaussian
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Pictorial transforms
 Common transforms
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Wave packets
 Localised waves
 A wave localised in space can be created by
superposing harmonic waves with a narrow
range of k values.


1
f ( x) 
2
g (k )eikxdk

 The component harmonic waves have
amplitude

1
ikx
g (k ) 
2

2

f ( x )e
dx

 At time t later, the phase of component k will be
kx-wt, so

1
i ( kx wt )
f ( x, t ) 
g ( k )e
dk

 Provided w/k=constant (independent of k) then
the disturbance is unchanged i.e. f(x-vt).
 We have a non-dispersive wave.
 When w/k=f(k) the wave packet changes shape
as it propagates.
 We have a dispersive wave.
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Convolution
 Convolution: a central concept in Physics.
Convolution symbol
h( x)  f1 ( x) * f 2 ( x) 

Convolution integral
 f1(u) f2 ( x  u)du

h is the convolution of f1 and f2
 It is the “smearing” or “blurring” of one function
by the other.
 Examples occur in all experimental situations
where the limited resolution of the apparatus
results in a measurement “broader” than the
original.
 In this case, f1 (say) represents the true signal
and f2 is the effect of the measurement. f2 is the
point spread function.
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Convolution theorem
 Convolution and Fourier transforms
 Convolution theorem:
 The Fourier transform of a PRODUCT of two
functions is the CONVOLUTION of their Fourier
transforms.
 Conversely:
The Fourier transform of the CONVOLUTION of
two functions is a PRODUCT of their Fourier
transforms.
 Proof: 
1
h( k ) 
f1 ( x) f 2 ( x)e ikxdx
2
F.T.
of
f1.f2
1

2

1
2
1

2


 


 1
iux 
 ikx
g
(
u
)
e
du
f
(
x
)
e
dx
1
 2
 2



 





 1

g1 (u )du 
f 2 ( x)e i ( k u ) x dx 
 2











g1 (u ) g 2 (k  u )du

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Convolution
of g1 and g2
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Convolution………….
 Summary:
FTf1 ( x)  g1 (k ), FTf 2 ( x)  g 2 (k )
1
FT

f
(
x
)
f
(
x
)


g1 (k ) * g 2 (k )
then
1
2
2
and FT f1 ( x) * f 2 ( x)   2 g1 ( k ) g 2 ( k )
 If,
 Examples:
 Optical instruments and resolution
1-D idealised spectrum of “lines” broadened
to give measured spectrum
2-D: Response of camera, telescope. Each
point in the object is broadened in the image.
 Crystallography. Far field diffraction pattern is a
Fourier transform. A perfect crystal is a
convolution of “the lattice” and “the basis”.
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Convolution Summary
 Must know….
 Convolution theorem
 How to convolute the following functions.
d-function and any other function.
Two top-hats
Two Gaussians.
s 2  s 12  s 22
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