Download Chapter 6: Random Signals and Noise

Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
Chapter 6: Random Signals and
Noise
6.1 Basic Concepts in Random Signals
6.1.1 Random Variables
A random variable x is characterized by a probability density function Px (x ) . The probability that x
will have a value in an interval dx centered around x is equal to Px ( x )dx . Normalization requires,

 Px ( x )dx  1

The mean and standard deviation for x are defined as,

Mean of x  x   x Px ( x )dx

Standard deviation  x 2  x
2
The sample space of x is the set of all possible values of x (i.e. for which Px ( x )  0 ).
6.1.2 Random Signals
A random signal x (t ) can be any signal from a set of signals { x1(t ), x 2 (t ), x 3 (t ),} which is the sample
space for the random signal x (t ) . The sample space need not be a discreet set. The probability that x(t )
will equal x n (t ) is Px(t ) x n (t ) . The mean of x(t ) is,
m x (t )  x (t )   x n (t ) Px (t ) x n (t )
n
When the sample space is not discreet and is parameterized by a continuous variable y then,
m x (t )   dy x y (t ) Px (t ) [ x y (t )]
The auto-correlation R xx (t1, t 2 ) of x(t ) is defined as,
R xx (t1, t 2 )  x (t1 )x(t 2 )   x n (t1 )x n (t 2 ) Px X (t ) x n (t )
n
The cross-correlation between two random signals x (t ) and y (t ) is,
R xy (t1, t 2 )  x (t1 )y (t 2 )    x n (t1 )y m (t 2 ) Px (t )y (t ) x n (t1 ), y n t 2 
nm
where Px (t )y (t ) x n (t ) , y m (t ) is the probability that x(t )  x n (t ) and y (t )  y m (t ) .
6.1.3 Stationary and Ergodic Signals
Stationary random signals are those whose characteristics do not depend upon the time origin. This
implies the following,
a) The mean values are independent of time, i.e. m x (t )  x(t )  m x .
b) The auto- and cross-correlation functions are functions of the time difference only, i.e.,
R xx (t1, t 2 )  x(t1 )x (t 2 )  R xx (t1  t 2 )
R xy (t1, t 2 )  x (t1 )y (t 2 )  R xy (t1  t 2 )
Most (but not all) signals studied in this course will be stationary.
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
6.1.4 Ensemble Averages Vs Time Averages
Averages of signals can be done in two ways,
a) Ensemble averages: Ensemble averages are averages with respect to a probability density
function. For example,
x t    x n t  Px t  x n t 
n
b) Time averages: One can time average a signal as follows,
1 T 2
x(t )  limit
 x(t ) dt
T   T T 2
6.1.5 Ergodocity
When all ensemble averages equal the corresponding time averages the signal is called ergodic.
Ergodicity implies stationarity but not the other way around. For ergodic signals,
1 T 2
x(t )  x(t )  limit
 x (t )dt  m x
T   T T 2
And
1 T 2
 x(t  t1  t 2 )y (t ) dt
T   T T 2
R xy (t1  t 2 )  x(t1 )y (t 2 )  x(t1 )y (t 2 )  limit
Ergodicity implies that each signal in the sample set is representative of the whole set.
6.1.5 Fourier Transforms
The following definitions of Fourier transforms will be used,

x( )   e  it x(t ) dt


d
2

Parseval’s Identity: The total energy in a random signal can be written in time and frequency domains
as,


2 d
2
 x t  dt   x  
2


x(t )   e  it x  
6.1.6 Power Spectral Density
The total energy of a real random signal x(t ) , given by,


d
2


can be infinite. So in practice one does not work with total energies of signals but with the signal powers.
Define a truncated signal xT (t ) as,
2
2
 x (t )dt   | x( ) |
T
T

 x (t )   t  
xT (t )  
2
2
 0
otherwise 
The power in the signal x (t ) is,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)

T
2 2
1
1  2
 x (t ) dt 
 xT (t ) dt
T T
T 

2

1
2 d
 | xT ( ) |
2
T 
The ensemble averaged power in the signal is,

 1
1 
2 d
2 d
 
.
xT ( )
 xT ( )
2
2
T 
 T
The integrand in the integral in the limit T   can be interpreted as the power spectral density
S xx ( ) of the signal x (t ) ,
1
2
S xx ( )  limit
xT ( )
T  T
The power spectral density, as the name implies, is the power in the signal per unit frequency bandwidth.

6.1.7 Weiner-Kinchine Theorem
Weiner-Kinchine theorem relates the power spectral density of a stationary signal to the signal autocorrelation,

S xx ( )   e  it R xx (t )dt

In other words, the power spectral density is the Fourier transform of the auto-correlation function.
Proof: We start from,
T 2
| xT ( ) |2 
 dt1 e
it1
T 2
T 2
T 2
x(t1 )  e  it2 x (t 2 ) dt 2
T 2
T 2
 i (t1  t2 )
 dt1  dt 2 x(t1 )x (t 2 ) e

T 2  T 2
T 2

T 2
 i ( t1  t2 )
 dt1  dt 2 R xx (t1  t 2 ) e
T 2  T 2
2
t2
T/2
T/2
T/2
-T/2
t1
-T/2
-T
T
1
-T/2
Let,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
t1  t 2
2
The Figure above shows how the integration region transform when going from the t1-t2 plane to the 1-2
plane. It follows that,
 1  t1  t 2
( xT ( )
2
2 
T 1

2 2
0
 i1
 d 2 R xx ( 1 )e
  d 1
T

T 1

2 2

T
 d 1
0
T 1

2 2
 i1
 d 2 R xx ( 1 )e

T 1

2 2
T
  d 1 R xx ( 1 )T  |  1 | e  i1
T
and therefore,
1
 |  |
| xT ( ) |2  limit TT d 1 R xx ( 1 )1  1  e  i1
T 
T  T
T 

limit

  d 1 R xx ( 1 ) e  i1


 S xx ( )   d 1 R xx ( 1 ) e  i1

Weiner-Kinchine theorem is a very important result and forms the basis of theoretical and experimental
signal analysis.
Consider a stationary random signal x t  . Then,

x( )   dt e  it x(t )

It follows that,
x *  'x( )  2   'S xx  
and,

d' *
x  'x( )
  2
We will often use the above result.
S xx    
6.2 Fluctuations and Noise: Concepts and Models
In quantum optics two types of noise are most commonly encountered:
a) Shot noise
b) Brownian or diffusion noise
In this chapter, we will look at these noise processes from a classical perspective.
6.2.1 Shot Noise
Suppose we are looking at a process that consists of a set of discrete events happening randomly in time.
Suppose the j -th event happens at time t j . At any time t , the rate r (t ) of the events (i.e. the number of
events happening per unit time) is,
r ( t )    (t  t j )
j
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
The times t j constitute the random part. We will assume the following:
i)
The times t j are completely independent of each other
The probability that there is an event in a very short time interval dt is given by  dt where
 is called the average rate of the process.
With these two assumptions we can compute the ensemble average of r (t ) ,
ii)

r ( t )    ( t  t j )    d t   ( t  t )  
j

The above result illustrates why  is the average rate. We want to find the spectral density of r (t ) . We
know that,
d  i
S rr ( )  
e
R rr ( )
2
So we find the auto-correlation of the function r t  ,
R rr ( )  r (t   )r (t )
    (t    t j ) (t  t m )
j m

    (t    t j ) (t  t m )  jm  1   jm
j m
Rrr ( )    (t    t j ) (t  t j )

 diagonal terms
j
    (t    t j ) (t  t m )
j m
 off - diagonal terms
j m

   dt   (t    t )  (t  t )     (t    t j ) (t  t m )
j

m
j m
Noting that the times t j are completely independent, the second term on the right hand side is
approximately,
  (t    t j )
j
2
  (t  t m )  
m
We therefore obtain,
R rr ( )    ( )   2
The spectral density becomes,
Srr ( )     2 2 ( )
The noise in r (t ) is n(t )  r (t )  r (t )  r (t )   . The auto-correlation and the spectral density of noise
is,
R nn ( )  R rr ( )   2     
 S nn ( )  S rr ( )   2 2 ( )
S nn ( )  
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
Noise n(t ) having the above characteristics is called shot noise. Shot noise implies a frequency
independent spectral density with a magnitude equal to the average rate of the process. The noise is not
present because the events are discreet. The noise is there because the events happen randomly in time
(although the average rate is ). If a process consists of discreet events then just this does not mean that it
will have shot noise. Shot noise implies that the timings of different events are random and completely
independent of each other. If the times t j get somehow correlated then the noise is not called shot noise.
For example, consider the process w t  consisting of events happening at times t j where t j  jT . The
events happen after fixed intervals of T . Each event is perfectly correlated in time with the other events.
There is no randomness and therefore,
w (t )    (t  t j )    (t  jT )    (t  jT )  w t 
j
j
The noise in w t  is zero.
j
6.2.2 Poisson Statistics and Shot Noise
For a process with shot noise, one may ask the question, "what is the probability of having n events in
time T ?" Let P (n,T ) be the probability of having n events in time T . Then,
P (n,T  T )  P (n  1,T ) T  P (n,T ) (1  T )
P (n,T  T )  P (n,T )

 P (n  1,T )  P (n,T )
T
d
P (n,T )   P (n,T )   P (n  1,T )

dT
Solution of the above differential equation with the boundary condition P (n,T  0)   n 0 is,
(T )n T
e
n!
When the statistics of events are according to the expression above, we call the process a Poisson process
and the statistics are called Poisson statistics. The number of events in time T is,
P (n,T ) 
T
T
0
0 j
NT   r (t ) dt     (t  t j ) dt
The ensemble average number of events in time T is,
T
T
0
o
NT   r (t ) dt     (t  t j ) dt  T
j
This can also be confirmed from the probability distribution found above,

(T ) n  T
e
n!
n 1

NT   n P (n,T )   n
n 0
(T ) n 1  T
e
(T )  T
n 1 ( n  1)!
The standard deviation in the number of events in time T is,

 
NT2  NT  NT
2
 NT2  NT
2
The first term is,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)

NT2   n 2
n 0
 (T ) n 1
(T ) n  T
e
e  T (T )
 n
n!
n
(
1
)!

n 1

  ( n  1  1)
n 1
(T ) n 1  T
e
(T )  (T ) 2  (T )
(n  1)!
Therefore,
NT2  NT2  NT
2
 (T )  NT
For a process with shot noise, the standard deviation in the number of events during any time interval is
equal to the mean. Of course, we can derive NT
and NT2  NT
2
directly from r (t ) as well. Since
r (t ) is the rate of events, we can write,
NT 
t o T
 r (t )dt
to
where t o is the time from where we start counting. The ensemble average value of NT is,
NT 
t o T
t o T
to
to
 r (t )dt 
  dt  T
and,
NT2 



t o T
t o T
to
to
 r (t1)dt1
 r (t 2 )dt 2
t o T
t o T
to
to
t o T
t o T
to
to
t o T
t o T
to
to
 dt1
 dt1
 dt 2 r (t1)r (t 2 )
 dt1
 dt 2 Rrr (t1  t 2 )

 (t
1  t2 )  
2

 T  (T ) 2
And, therefore, NT2  NT
2
 T  NT
as before.
6.2.3 Partition Noise
Here we will consider noise introduced in particle splitting. Consider a stream of particles arriving at a
splitter as shown below. The splitter decides whether each particle is transmitted in the forward direction
or reflected in the downward direction. The decision is random but on the average the probability of
forward transmission is  and the probability of reflection is 1    .
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
The incident particle rate is i (t ) , the "reflected" particle rate is r (t ) and the rate of particles transmitted in
the forward direction is f (t ) . The average rates satisfy,
i (t )  
f (t )   i (t )  
r (t )  (1   ) i (t )  (1   ) .
Let the noises be,
n i (t )  i (t )  i ( t )  i (t )  
n f (t )  f (t )  
n r (t )  r (t )  (1   )
If the splitter added no noise to the outgoing particles we would have had the following relations,
f (t )   i ( t )
(1)
r (t )  (1   ) i (t )
( 2)
and therefore,
Sff ( )   2S ii ( )

S rr ( )  (1   ) 2 S ii ( )
S
nf nf

( )   2S
S
nr nr
ni ni
( )
( )  (1   ) 2 S
ni ni
( ).
But (1) and (2) above are not correct since only the average of f t  equals  times the average of i t  .
Let,
i ( t )    (t  t j )
j
and,
f (t )   f j  (t  t j )
j
where f j is a random variable that can take two values, 1 and 0, with probabilities  and ( 1   ),
respectively. There are two sources of randomness in the above expression. The factors f j are random
and the times t j of arrivals of the particles at the splitter are also random. The ensemble average of f j is,
fj 
It should also be noted that f j and t j are independent random variables. Therefore,
f (t )   f j  (t  t j
j
    (t  t j
  i (t )  
j
In evaluating the above expression, we have first performed ensemble averaging with respect to the
random variables f j . The auto-correlation of f (t ) is,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
R ff ( )  f (t   )f (t )
   (t    t j ) (t  t m )f j f m
jm
R ff ( )   f j2 (t    t j ) (t  t j )    f j f m  (t    t j ) (t  t m )
j
j m
j m
We first perform ensemble averaging with respect to the factors f j ,
f j2  
fj 
f j fm
j m
 f j fm   2
It follows that,
R ff ( )     (t    t j ) (t  t j )   2    (t    t j ) (t  t m )
j
j m
j m
  (1   )   (t    t j ) (t  t j )
j



     (t    t j ) (t  t j )     (t    t j ) (t  t m )

j m
 j
j m 

The part in curly brackets is just Rii ( ) . Therefore,
2
Rff ( )   (1   )   (t    t j ) (t  t j )   2Rii ( )
j
The first term on the right hand side is,
  (t    t j ) (t  t j )   dt   (t    t ) (t  t )
j
  ( )
Therefore,
R ff ( )   (1   ) ( )   2 R ii ( )
 Sff ( )   (1   )   2S ii ( )
Since,
n f (t )  f (t )  f (t )  f (t )  
we have,
R
nf nf

( )  R ff ( )   2 2   1        2 R ii     2
  1        2 R
ni ni

 
The noise spectral density in the transmitted particle stream is,
S
nf nf
( )   (1   )   2S
ni ni
( )
The noise spectral density of the transmitted particles consist of two parts: i) the noise in the input particle
stream multiplied by  2 , ii) the noise added by the splitter which goes as  (1   ) . A similar analysis
shows that,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
R rr ( )   (1   ) ( )  (1   ) 2 R ii ( )
S rr ( )   (1   )  (1   ) 2 S ii ( )
S
nr nr
( )   (1   )  (1   ) 2 S
Let’s look at,
S f f ( )    (1   )   2 S
n n
ni ni
ni ni
( )
( )
Case (i): Suppose  is very small such that  2 S
S
nf nf
( )  
ni ni
( ) can be neglected. Then,
But  is the average rate at which particles go forward. So in the limit   0 , the output stream has
shot noise irrespective of the noise in the input stream.
Case (ii): Suppose the incident stream has shot noise i.e. S i i ( ) is equal to  , then,
nn
S
S
nf nf
nf nf
2
( )   (1   )   
( )  
If the input stream has shot noise then the output stream also has shot noise.
Case (iii): If   1 , then S f f ( )  S i i ( ) which makes sense since if no splitter is present, and no
n n
n n
particle is reflected, then the noise in the incident and forward streams should be the same.
6.3 Introduction to Langevin Equations
6.3.1 Brownian Velocity
The best way to introduce Langevin equations is to look at a particle undergoing Brownian motion.
Consider a particle of mass m in a gas (or a liquid) undergoing random Brownian motion. The particle is
being kicked around randomly. The velocity of the particle is v (t ) which can be considered a random
signal. We limit ourselves to one dimensional Brownian motion. The surrounding medium (other gas or
liquid particles) exert a frictional force on the test particle so that any initial velocity given to the particle
is quickly lost as a result of scattering with other particles. The equation for the particle velocity v (t ) can
be written as,
dv (t )
  v (t )
dt
Here,  describes the damping of the particle velocity due to collisions with other particles. The above
equation cannot be a complete description of the particle motion since it gives,
v (t )  v (t  0) e t
This implies that,
v 2 (t )  v 2 (t  0) e 2t
In the limit t   , the particle velocity and, therefore, energy both become zero. We know that the
average particle energy is given by the equipartition theorem of statistical physics,
K T
1
1
mv 2 (t )  K BT  v 2 (t )  B
2
2
m
One way to fix the equation for the motion of the particle is to add a force that describes the kicks the
particle receives during collisions with other particles,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
dv (t )
  m v (t )  F (t )
dt
The force F t  is a random function. We require that the kicks at different times be completely
uncorrelated and random in direction. In mathematical language, we require the ensemble average values
of F t  and its auto-correlation to obey,
m
F (t )  0
F (t1)F (t 2 )  A (t1  t 2 )
A-priori we don't know the value of A . We will determine A by enforcing the equipartition theorem. We
start from the equation,
dv (t )
F (t )
  v (t ) 
dt
m
The solution is,
1 t
v (t )  v (t  0)e  t   F (t1 ) e   (t  t1) dt1
m0
 v 2 (t )  v 2 (t  0) e  2t 
t
1 t
dt1  dt 2 F (t1 )F (t 2 ) e   ( 2t  t1  t2 )

m2 0 0
e  t t
  ( t  t1)
 dt1 F (t1 ) e
m 0
We perform an ensemble average on both sides,
 2v (t  0)
v 2 (t )  v 2 (t  0) e  2t  2
e t t
  (t  t1)
 dt1 V (t  0)F (t1 ) e
m 0
t
1 t
dt1  dt 2 F (t1 )F (t 2 ) e   ( 2t  t1  t2 )

m2 0
0
In steady state, when t can be assumed to be large, the only term that survives is the last term,
t
1 t
lim v 2 (t )  lim
dt1  dt 2 A (t1  t 2 ) e   ( 2t  t1  t2 )

t 
t  m 2 0
0

A t
 2 ( t  t1)
 dt1 e
2
t  m 0
 lim
A (1  e  2t )
A

.
2
2r
t  m
2 m 2
Since according to the equipartition theorem,
K T
v 2 (t )  B
m
we must have,
K T
A
 B
 A  2 m K BT
2
m
2 m
 lim
We therefore arrive at the following description of the velocity of a particle undergoing Brownian motion,
dv (t )
F (t )
  v (t ) 
dt
m
where the force F t  has a zero mean and the following auto-correlation function,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
F (t1 )F (t 2 )  2 m K BT  (t1  t 2 )
The equation for the particle velocity is an example of a Langevin equation and F (t ) is a Langevin noise
source. In most cases, the auto-correlation function of a Langevin noise source is a delta function in time
and its magnitude is set by fundamental physical considerations irrespective of the microscopic details of
the system. For example, we could have derived the factor 2 mK BT by a detailed microscopic
calculation but it was easier to use a fundamental physical principle (i.e. equipartition theorem). The
Langevin equation for the particle velocity is also an example of the "fluctuation dissipation" principle in
statistical physics according to which every source of dissipation must also introduce fluctuations in a
physical system.
Spectral Density: The spectral density of v (t ) can be derived in two ways:
First find Rvv ( ) and then Fourier transform to obtain Svv ( ) .
i)
ii)
The second method to obtain Svv ( ) is to work directly in the frequency domain.
Both the above methods are illustrated below.
Start from,
1 t
  (t  t1 )
 F (t1) e
m 0
The auto-correlation of the velocity is,
1 t  t 
v (t )v (t )  v 2 (0)e  r (t   t ) 
dt dt F (t1) F (t 2 ) e   (t   t ) e  (t1  t 2 )
2  1 2
m 0
0
v (t )  v (t  0)e  t 

v (0)  t  t 
v (0)  rt  t 
  (t   t 2 )
  (t   t1 )
e
dt 2 
e
dt1
 F (t 2 ) e
 F (t1) e
m
m
0
0
As both t  and t  become large, the correlation function becomes,
2 K BT t ' t "
  (t ' t " )  ( t1  t2 )
v (t ' )v (t " )  lim
e
 dt1  dt 2  (t1  t 2 ) e
m 0 0
t'
t " 
Some care needs to be exercised in performing the double integral above:
t"
a) Suppose t "  t ' then we can do the  dt 2 integral first and the result is,
0
2rK BT t '
  (t ' t " ) 2t1
e
 lim
 dt1 e
m 0
t ', t "  

K BT   ( t "  t ' )
e
 e   (t ' t " )
t ', t "   m
K T
 B e   ( t " t ' )
m

lim

t'
b) Suppose t '  t " then we can do the  dt1 integral first,
0
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
2K BT t "
 r ( t ' t " ) 2rt2
e
 dt 2 e
m 0
t
v (t )v (t )  lim
t 

K BT  r (t 't " )
e
 e  r (t ' t " )

m
t 
 lim

t 
K BT  r (t 't " )
e
m
Combining (a) and (b), one gets,
K T
v (t )v (t )  B e   |t 't "|
m
In the limit of large t ' , t " (when all initial correlations have died out) the random signal v (t ) is stationary

(since v (t )  0 and v (t ' )v (t " ) depends only on the time difference (t 't " ) ),
K BT   | |
e
m
The spectral density of v (t ) is,
Rvv ( ) 
 Svv ( ) 
K BT 
 i   | |
e
 d e
m 
2 K BT
m
Svv ( ) 
2
(   2 )
The forces acting on the particle are delta-correlated (meaning their correlation function is proportional to
a delta function in the time difference) but the particle velocity "remembers" the kicks it gets on a time
scale of the order of 1  . Therefore, the velocity-velocity correlation function is not delta-correlated but
looks like,
Rvv()

and decays on a time scale 1  .
The second method to obtain Svv ( ) in steady state is to work directly in the frequency domain. Define,

F ( )   dt e  it F (t )

Then,

F ( )   dt e  it F (t )  0

and,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)


F * (1)F (2 )   dt1 e  i1t1  dt 2 e  i 2t 2 F (t1)F (t 2 )







 2 m K BT  dt1  dt 2 e  i1t1 e  i 2t 2  (t1  t 2 )
 2 m K BT  dt1 e  i (1  2 )t1

F * (1)F (2 )  2 m K BT 2  (1  2 )
Therefore, functions delta-correlated in time are also delta-correlated in frequency. Start from,
dv (t )
F (t )
  v (t ) 
dt
m
and take the Fourier transform of both sides,
 i v ( )   v ( )  F ( )
F ( ) / m
v ( ) 
 i [  i ]
The spectral density Svv ( ) is,

Svv ( )   v * ( ' ) v ( )


1
m
2
d'
2

F  ' F  

'i    i 

d '
2
2 K BT
d '
m
2 ( ' )
 
2
  [ ' i ][  i ]

2 K BT
m
Svv ( ) 
2
 2
After inverse Fourier transforming, we can find the correlation function as well,

d K BT   | |
Rvv ( )   Svv ( ) e  it
e

m
2

Working in frequency domain is often easier, and we don't have to deal with process of taking limits of
large times in order to throw away the initial correlations.
6.3.2 Diffusive Motion (Random Walk)
The diffusion of a particle making random jumps is also described by a Langevin equation,
dx (t )
 W (t )
dt
where,
W (t )  0
and,
W (t1 )W (t 2 )  2D  (t1  t 2 )
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
where D is the diffusion coefficient. The particle motion can also be described by a time-dependent
probability density function P ( x, t ) that gives the probability of finding the particle at position x at time
t subject to some initial condition. P ( x, t ) obeys the standard diffusion equation,
P ( x, t )
 2 P ( x, t )
D
t
x 2
The Langevin equation and the diffusion equation are equivalent descriptions of the same process.
Consider,
dx (t )
 W (t )
dt
Since there is no dissipation, initial conditions cannot be ignored. The solution can be written as,
t
x(t )  x(0)   W (t )dt
0
 x ( t )  x (0 )
The particle has an equal chance of diffusing in any direction, so the ensemble average of x(t ) equals
x(0) . We first find the equal-time auto-correlation of the particle position as follows,
t
t
t
0
0
0
x 2 (t )  x 2 (0)  2 x(0)  W (t )dt   dt1  dt 2 W (t1 )W (t 2 )
t
t
x 2 (t )  x 2 (0)   dt1  dt 2 2D (t1  t 2 )
0
0
t
 x 2 (0)  2D  dt1
0
2
 x (0)  2Dt
This gives,
x 2 (t )  x 2 (0)  2Dt
 x (t )  x(0)2  2Dt
The ensemble average of the square of the displacement of the particle from its initial location increases
linearly with time. This property is a signature of diffusion. Next, we find the more general autocorrelation function. We have,
t1
x (t1)  x (0)   w (t )dt
0
t2
x(t 2 )  x(0)   w (t )dt
0
Therefore,
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
t
t
1
2
x(t1) x(t 2 )  x 2 (0)   dt1  dt 2 w (t1)w (t 2 )
0
0
t1
t
0
0
2
 x 2 (0)   dt1  dt 2 2D (t1  t 2 )
2
x(t1) x(t 2 )  x (0)  2D min (t1, t 2 )

smaller
of t1 and t 2
 x(t1)  x(t 2 ) 2  2D t1  t 2
The process x (t ) is obviously not stationary.
We can get the same results as above from the diffusion equation,

 2 P ( x, t )
P ( x, t )  D
t
x 2
The solution, subject to the initial condition, P ( x, t o )   ( x  x(t o )) , written as P ( x, t | x(t o ), t o ) for
t  t o is,
P ( x, t | x(t o ), t o ) 
 ( x  x(t o )) 2 
exp  

2 2D(t  t o )
 2[2D(t  t o )] 
1
This means,
x (t )  x (t 0 )  0
and,
x(t )  x(t 0 )2
 2 D (t  t 0 )
for t  t 0 
Now consider the diffusion equation in the frequency domain,
 i x( )  W ( )
W ( )
 i
 d '
S xx ( )  
x * ( ' ) x( )
  2
x( ) 
d ' W * ( ' )W ( )
' 
2
d ' 2D 2  (' )

2
' 
2D
S xx ( ) 

2
Spectral density that goes as 1  2 is a signature of a diffusion process. If you see the spectral density of
a signal y t  going as 1  2 then it means that
y (t1 )  y (t 2 )2
| t1  t 2 | . Since x (t ) is not
stationary, and x(t1 )x (t 2 ) is not a function of t1  t 2 , it makes little sense to take the inverse Fourier
transform of S xx ( ) for a diffusion process.
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
6.4 Phasor Representation of Signals and Signal Quadratures
6.4.1 Signal Quadratures and Phasors
Consider a narrowband real signal with spectrum centered near  o , as shown below.
y()





Any real signal y (t ) with bandwidth centered at frequency o can be written as,
d
y (t )  
y ( ) e  it
2
 o   / 2
d
d
y ( ) e  it 
y ( ) e  it

2

2

o   / 2
 o   / 2
Where  is large enough so that y ( ) is zero outside the bandwidth  centered at  o . One can remove
the fast time dependence from inside the integral,
y (t ) 
o   / 2

y (t )  e  io t
o   / 2
 o   / 2
d
 i (  o )t
 io t
y
(

)
e
e
y ( ) e  i (  o )t



2

 o   / 2
o   / 2
and write,
y (t )  e  iot
x (t )
x * (t )
 e iot
2
2
where,
o   / 2 d
x (t )  2

o   / 2 2
y ( ) e  i (  o )t
The above relation expresses y (t ) as a product of a slowly time varying function x(t ) and a rapidly time
varying complex exponential that has the center frequency of y (t ) ,


y (t )  Re x(t ) e  io t
Since x(t ) is complex, we can write,
x ( t )  x1( t )  i x 2 ( t )
Then,


y (t )  Re x1(t )  i x 2 (t ) e iot
y (t )  x1(t ) cos  o t  x 2 (t ) sin  o t
x1(t ) and x 2 (t ) are called the quadratures of the signal y (t ) . Usually, it is easier to deal with the slowly
varying complex signal x(t ) then with the real signal y (t ) . One can represent x(t ) as a vector (called
phasor) in the complex plane, as shown below, and the quadratures are the orthogonal components of the
phasor along the x-axis and the y-axis.
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Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
x(t)
x2(t)
x1(t)
6.4.2 Generalized Quadratures
More often one is not interested in the components of the phasor along the x-axis or the y-axis but along
axes that are rotated with respect to the x-y axes. Suppose we are interested in the components of the
phasor that are along axes that are rotated at an angle  with respect to the x-y axes, as shown below.
x(t)
x+/ 2(t)

x(t)
One can write x(t ) as,


x(t )  x (t ) e i  x   2 (t )e i    2  x (t )  i x   2 (t ) e i
x (t ) and x   2 (t ) are the two orthogonal components of the phasor x (t ) along the rotated axes.
6.4.3 Examples
Case (i): Suppose,
x(t )  xo e i
Here, a is a complex constant. We can write the signal as,
x t   xo cos   i xo sin 
It follows that,


 Re x e  e  
y (t )  Re x(t )e io t
o
i
i ot
y (t )  xo coso t     xo sino t   
Case (ii): Suppose,
x ( t )  x o   1( t )
where xo and 1(t ) are real and 1(t )  x o . Then,
18
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
y (t )  x(t ) cos  o t
 ( x o  1(t )) cos  o t
1(t ) thus represents amplitude modulation of the signal y (t ) , as depicted below.
Case (ii): Amplitude modulation
1(t)
x(t)=xo
Case (iii): Suppose,
x(t )  x o  i 2 (t )
where xo and  2 (t ) are real and  2 (t )  x o . Then,
y (t )  x o cos  0 t   2 (t ) sin  o t
The amplitude of the phasor is,
| x(t ) |2  x o2   22 (t )  x o2  x o
So a small imaginary part does not affect the amplitude of the phasor. We can also write,

 (t ) 
x(t )  xo  i 2 (t )  xo 1  i 2 
xo 

 xo
  (t ) 
i 2 
x
e o 

 t 
 y t   xo coso t  2 
xo 

Therefore,  2 (t ) represents phase modulation of the signal y (t ) , as depicted below.
Case (iii): Phase modulation
x(t)=xo 2(t)
Case (iv): If we have two zero-mean real random signals 1(t ) and  2 (t ) , and,
x(t )  x o  1(t )  i 2 (t ) , where x o is real and 1(t ), 2 (t )  x o , then 1(t ) describes the in-phase
fluctuations or the amplitude fluctuations in the signal y (t ) and  2 (t ) describes the out-of-phase
fluctuations or, simply, the phase fluctuations in the signal y (t ) . The tip of the phasor will be randomly
19
Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)
moving in time (for example, the tip is shown to be moving in time in the shaded region in the Figure
below). The shaded region is called the “error region.”
1(t), 2(t) ≠ 0
Case (iv): Random amplitude and
phase fluctuations
x(t)=xo
Error region
Case (v): Suppose,
x(t )  x o e i   (t )
where x o is real and  (t ) is complex and  (t )  x o . It is not obvious what effect does  (t ) have
unless one writes the complex signal  (t ) using the generalized quadratures,
x(t )  x o e i 
Now it is clear that  (t ) and
 t   i   2  e i 
   2 (t ) represent amplitude and phase modulations, respectively, as
shown below.
xo e
+/ 2(t)
(t)
i 

One can write,
   2 t   i 

x(t )  x o e i   t   i   2 e i   x o   t  1  i
e
xo






 x o   t  e
The signal y t  is,


   2 t  
i  

xo


   2 t 

y (t )  xo   t  coso t   

xo




y
t
are now obvious.
The amplitude and phase modulations in


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