Transcript Theory off chaotic light

Theory off
chaotic light
and
coherence
Rotem Manor
ID:036235976
It is important to distinguish between tow
types of light source:
• Chaotic source: gas discharge lamp, where the different atoms are
excited by an electrical discharge and emit their radiation independently of one
another. emission line determined by statistical spread in atomic velocities and
random occurrence of collisions.
• Laser: high intensity ,coherence
There are three mechanisms witch are mainly
responsible for the spread in frequency of
emitted light.
The decay process it self:
leads to broadening of an absorption line of
Lorentzian frequency distribution (with a width Γ).
Collision broadening: form the atomic motion.
Spread in atomic velocities: leads to Doppler spread in emitted
frequencies.
Note: it is convenient to neglect the other linewidth contributions while
treating a particular broadening process.
Doppler broadening
Atom in exited energy level E2 ,
Has velocity v2 . Emits photon of energy  and falls to
a lower energy level E1 .
The photon has a momentum k where k   / c and is
emission causes a recoil of the atom to a new velocity v1
Momentum :
Mv2  Mv1  k
1
1
2
2
E2  Mv2  E1  Mv1  
Energy:
2
2
Let  0 be the frequency of light which would be emitted
if the atom had zero velocity before and after the
emission.
0  E2  E1
It gives us:
 2k 2
0    v2 k 
2M
We’ll take k to be on the z-axis
v2z
h 2
0   

c
2Mc 2
Typical orders of magnitude:

9
v  10 ;

10
2 Mc 2
5
z
2
Hence

0
1 v
z
2
 0 (1 
v2z
c
)
c
There for the frequency of emitted light suffers a Doppler
shift.
The Maxellian velocity distribution define the frequency
distribution
2
(



)
1
1
c
z 2
z
2
0
exp(  M (v ) )dv  exp(  Mc
) d
2
2
2
0
0
where

Probability function:
FG ( ) 
where
1
2 2

e
1
k BT
( 0 ) 2
2 2
0 2

Mc 2
Collision broadening
We focus attention on the same pair of atomic states that we have
used before, but we now ignore the other linewidth( assumption: the
radiative lifetime of an atom is long compared to time between
collisions).
Consider a particular exited atom radiating light of frequency. When
it suffers a collision his energy levels are shifted by the force of
interaction between the two colliding atoms. After the collision the
wave frequency 0 is resumed with all the other characteristics,
except from the phase of the wave, that is unrelated to the one
before the collision.
If the duration of the collision is sufficiently brief, it is possible to
ignore any radiation emitted during the collision while 0 is shifted.
The collision broadening effect can then be adequately represented
by a model in witch each exited atom always radiates at frequency 0
, but with random changes in the phase of radiated wave each time
a collision occurs.
The wave train radiated by single
atom.
Vertical line represent a collision.
The Phase of The wave train
radiation
According to the kinetic theory of
gases, the probability that an atom
has period of free flight lasting a
length of time between t and t+dt
is
t
p(t )dt 
Hence
0 
1
0
V
4d 2 N

e
0
dt
M

If we consider one period of free flight of an atom, The field can be written in a
complex form:
E (t )  E0 exp( i0t  i )
(t0  t  t0   )
And represent as a Fourier integral
1
E ( ) 
2
t0 
E
0
exp( i0t  i  it )
t0
E0
exp{i (  0 ) }  1

exp{i (  0 )t0  i}
2
i (  0 )
Thus the cycle averaged intensity is
2
E
sin
{(0   ) / 2}
2  0 
I ( ) | E ( ) |   
2

(



)
 
0
2
At any instant of time the total intensity of radiation is made up of
contribution from large number of exited atoms. The probability of
the free flight time is given( in slide 8 ), so it’s necessary to
integrate the intensity with the probability over time for getting the
total intensity.

2
sin
{(0   ) / 2}
I total ( ) | E ( ) |2  
exp(  /  0 )d
2
0 0
(0   )
1
1/ 2

(0   ) 2  (1 /  0 ) 2
realistic values for gas in
room temperature and
pressure 105 Pa .
 0  3 10 11 s
0  3 1015 Hz
0 0  9 10 4
So average atom doing about 15,000 periods of oscillations before a
collision, And the collision linewidth is about 100 times the natural
radiative linewidth( Г )
Composite emission lineshape
It is interesting to note that the Doppler width( slide 6 ) is approximately
equal to the collision linewidth for this parameter values.
In this case it is necessary to determine the composite lineshape of all
the dominant processes. A combination of two line broadening
mechanisms with individually generate normalized lineshape functions
f(ω) and g(ω) is

F ( ) 
 f (w) g (  
0
 w)dw

Where 0 is the common central frequency component of the
distributions.
Clearly if the two sources of broadening are
Lorentzians
And for Gaussians
total   f  g
total   f   g
2
2
2
Time dependence of the chaotic light beam
It is clear from the above discussions that the frequency spread of the
emitted light is governed by the same physical parameters as the time
dependence of emitting source atom.
The wave train emitted by a single atom( number 1 ) is
E1 (t )  E0 exp( i0t  i1 )
The total emitted wave is represented by a sum of this terms
E (t )   En E0 exp( i0t ) exp( in )
n
n
 E0 exp( i0t )a(t ) exp( i (t ))
For simplicity we will assume that the observed light has a fixed
polarization so that the electric field can be added algebraically
The amplitude a(t) is illustrated in this diagram
Im[ a(t )]
Re[ a(t )]
a(t ) and  (t ) are different and different instate of time.
How ever it is not possible in practice to resolve the
oscillation in E(t) witch occur at the frequency of the
carrier wave.
A good experimental resolving time is of the order 10 9
(six order magnitude too long to detect the oscillation of
the carrier wave).
The real electric field has zero cycle average and the
beam intensity in free space is
1
1
2
2
2
I (t )   0 c | E (t ) |   0 cE0 a(t )
2
2
I (t )
0
This is a computer simulation of a
collision-broadened light source.
It is seen that substantial
changes in intensity and
phase can occur over time
span  0 , but these quantities
are reasonably constant over
 (t )
time intervals
t   0 .
It gives us two new
parameters
•
 c - coherence time,
• c   c c - coherence length.
0
Intensity fluctuation of chaotic light
Intensity fluctuation can be measured experimentally
only with a detector whose response time is short
compared to the coherence time  c . So a normal
detector is measuring averages.
a(t ) 2  |  exp( in ) |2  
n
(the bar denotes a long tome average
for a time long compared to  )
c
So the long time average intensity is
1
2
I (t )   0 cE0 
2
It is seen that in any instant of time the movement of a(t)
is in depend on stepsin random direction like in the
‘random walk’ problem, so let p[a(t)] be the probability of
the end point of a(t).
 a(t ) 2 

p[a(t )] 
exp  

  
1
Probability distribution for the amplitude and
the phase of electric field of chaotic light

 I (t ) 

p[ I (t )] 
exp  

 I 
1
If we’ll calculate:

I (t )
n
1
  I (t ) n exp(  I (t ) / I )dI (t )  n! I n
I 0
It’s means that the fluctuation are
I
2
 I
2
I
.
Despite the very large fluctuation compared to the intensity,
the coherence time  c is usually so short that it is difficult to
observe the fluctuation in experimentally and their influence
is often small.
Young’s interface fringes
The model experiment ignores complications arising from the finite
source diameter and consequent lack of parallelism in the beam
witch illuminates the first screen.
Let E(r,t) be total electric field of radiation at position r on
the observation screen at time t.
E(r, t )  u1E(r1t1 )  u2 E(r2t2 )
Where
t1  t  (s1 / c),
t2  t  (s2 / c),
And u1 ,u2 are inversely proportional to s1 , s2 respectively,
and depend on the geometry of the experiment.
The intensity of the light in the position r averaged on over
a cycle of oscillation is
1
I (rt )   0 c | E (rt ) |2
2
1
  0 c{| u1 |2 | E (r1t1 ) |2  | u2 |2 | E (r2t 2 ) |2 
2
*
*
2u1 u2 Re E (r1t1 ) E (r2t 2 ) }
The fringes in Young’s interface experiment are normally
can be observed by the naked eye. In this case the
recording time is long compared to  c and there for it is
necessary to average I (rt ) .
I  I (rt )
T
1
1
  0c  dt{| u1 |2 | E (r1t ) |2  | u2 |2 | E (r2t  t21 ) |2 
2
T0
2u u Re E * (r1t ) E (r2t  t 21 ) }
*
1 2
Where
t21  t2  t1
The first two terms represents the intensities caused by
each pinholes in the absence of others. The third one is
called first-order correlation function of the field, and it’s
clearly depend on t 21 (the deference between the times at
witch the fields are measured).
Evaluation of the first order correlation function
Let’s calculate the correlation function of a Lorentzian
distribution light source.
The light witch strikes the first screen in Young experiment is
assumed to be propagating in the z direction, the optical cavity
then is one dimensional case and the spacing between the
wave vector is
L – length of cavity
k   / L
We’ll use Fourier sum of the normal-modes contribution to the
light source
Where
E ( zt )   Ek exp( ikz  ik t )
k  ck
The number of different normal-modes contributes significantly
to the electric field is depend on the length L and the coherence
length.
L  c
If we’ll take
   c / L
On the other hand linewidth of the
assumed Lorentzian emission line
can be taken as 2 where   1 /  c
    / 
It that the only significant mode is
k0  0 / c
2I
 | Ek0 | 
 oc
2
Now will take
L  c
    / 
It mean that the distribution is

| Ek |  
(0   ) 2   2
2

- constant of proportionality
1
1
2
I   0c | E ( zt ) |   0c | Ek |2
2
2
k
Since


0
0
  ( L /  ) dk ( L / c) d
If we’ll take the sum to be
k
 0 L 

1
I
dk   0 L
2
2

2 0 (0  k )  
2
We’ll get the intensity
Witch give us the constant 
2I

 | Ek | 
 0 L (0  k ) 2   2
2
Now we’ll continue the calculation of the correlation function
E * ( z1t1 ) E ( z2t2 )   Ek* Ek ' exp{i(kz1  k t1  k ' z2  k 't2 )}
k ,k '
  | Ek |2 exp( ik )
k
Where
  t1  t2  ( z1  z2 ) / c

exp( ik )
2I
*
 E ( z1t2 ) E ( z2t2 ) 
dk
2
2

 0c 0 (0  k )  
For narrow emission line the lower limit on the integral can
be replaced by   without significant change in its value.

exp( ik )
 
dk   exp( i0   |  |)
2
2
(0  k )  

2I
 E ( z1t2 ) E ( z2t2 ) 
exp( i0t   |  |)
 0c
*
Note: We should remember that:
•We used the independent randomness of different normal modes.
•The large number of cavity modes, is a necessary condition for the
replacement of the sum with an integral.
Fringes intensity and first order coherence
Since:
I (r )  I {| u1 |2  | u2 |2 2u1*u2 exp(  |  | cos 0t )
Where
  (s1  s2 ) / c
The fringes visibility at the position r on the second screen is
defined by:
I (r ) max  I (r ) min 2u1*u2 exp(  | s1  s 2 | / c)

I max  I min
| u1 |2  | u2 |2
It’s easy to see that when s1  s2 and u1  u2 the fringes visibility
is unity, and it’s less than unity otherwise.
The degree of first order coherence between the light fields
at the space-time points (r1 , t1 ) and (r2 , t2 ) is denoted by
g (1) (r1t1 , r2t2 ) .
g (1) (r1t1 , r2t 2 )  g12(1) 
| E * (r1t1 ) E (r2t 2 ) |
 | E (r t ) |
11
2
| E (r2t 2 ) |2

1
2
The angle brackets indicate that ensemble average mast be taken when the
field E(rt) is define statistically.
Coherent - g12  1
(1)
(1)
Incoherent - g12
0
Example:
Lorentzian frequency distribution in Young experiment:
g (1) (r1t1 , r2t2 )  exp(  |  |)
(1)
The form of dependence of g12 is illustrated below while
  t1  t2  ( z1  z2 ) / c   c
Since the corresponding coherence length is about 102 it is
possible.
Note:
this calculation is for
a collision model
Motivation for higher order coherence
The classical stable wave E1 (t )  E0 exp( ikz  i0t  i ) provides
another example of coherence properties. The first order correlation
function is determined without any ensemble averaging in this case.
2I
E ( z1t2 ) E ( z2t2 )  E0 exp( i0 ) 
exp( i0 )
 0c
*
2
It means that the first order coherent is g12  1 .
(1)
There are some more examples under witch light have perfect first
order (if the beam is single cavity mode, the filed can be specified
precisely, with no statistical features).
Recent development in optics have gone beyond the domain of
classical theory.
• The laser has coherence properties which can be varied chaotic
sources.
•Experiments have been preformed in with the intensity fluctuation of a
chaotic source are directly measured.
Intensity interface and higher order coherence
Hanbury Brown and Twiss experiment (intensity of the fluctuation)
•The detectors are symmetrically placed with respect to the mirror.
•The half-silvered mirror produces two exactly similar light beams.
For now we’ll ignore the finite response-times of the
detector.
The experiment measures fluctuations in the intensity:
( I ( zt1 )  I )( I ( zt 2 )  I )  I ( zt1 ) I ( zt 2 )  I 2 ,
Since
I ( zt1 )  I ( zt 2 )  I
The correlation function in here is:
  0c 
*
*
I ( z1t1 ) I ( z 2t 2 )  
E
(
z
t
)
E
( z 2t 2 ) E ( z 2t 2 ) E ( z1t1 )

11
 2 
2
The electric field can be expanded as superposition of
plane wave.
2
 c
I ( z1t1 ) I ( z2t 2 )   0   Ek*1 Ek*2 Ek3 Ek4 
 2  k1k2 k3k4
 exp{i(k1 z1  1t1  k2 z2  2t2  k3 z3  3t3  k4 z4  4t4 )
Where
i  cki
.
Thus equation can simplifies to
  0c 
I ( z1t1 ) I ( z2t2 )  

 2 
2

| Ek1 |2 | Ek2 |2 [exp{i(1  2 ) }  1]
k1k 2
This equation can be written in terms of first order
correlation function
  0c 
I ( z1t1 ) I ( z 2t 2 )  

 2 
2
*
E ( z1t1 ) E ( z2t 2 )
2
I2
 I ( z1t1 ) I ( z2t 2 )  I 2 {exp( 2 |  |)  1}
Hence ( I ( z t )  I )( I ( z t )  I )  I 2 exp( 2 | t  t |)
11
2 2
1
2
Note:
I ( zt ) 2  2 I 2
since there is two light beams.
This result is not realistic,
Every detector has minimum response time  r . so now
we’ll calculate the average intensity in the response time.
( I ( z1t1 )  I )( I ( z2t2 )  I ) 
1
r
2
r
r
 dt  dt
1
0
2
( I ( z1t1 )  I )( I ( z2t2 )  I )
0
 ( I 2 / 2 2 r ){exp( 2 r )  1  2 r }
2
The properties of a light beam witch are relevant to an
intensity interface can be expressed in terms of an
extension of the coherence concept.
Analogous to the definition of the first order coherence we
define the second order coherence.
g12( 2) 
E * (r1t1 ) E * (r2t2 ) E (r2t 2 ) E (r1t1 )
| E (r1t1 ) |2 | E (r2t 2 ) |2
The light is said to be second order coherent if
simultaneously
(1)
and
g12  1
g12( 2 )  1
If we’ll use the development of the equations from slides
32-33 we’ll get a new definition for the second-order
coherence of chaotic light:
( 2)
(1) 2
g12 | g12 | 1
Note: this calculation is good for a collision and Doppler broadening models
With those definition it’s not possible for chaotic light to be
second-order coherent for any choice of space-time
points.
Examples for a different
frequency distribution
For lorentzian :
( 2)
g12
 exp( 2 |  |)  1
For gaussian :
( 2)
g12
 exp(  2 2 )  1
Although you can notice that for a classical stable wave:
E ( zt )  E0 exp( ikz  i0t  i )
The correlation of intensity is
I ( z1t1 ) I ( z 2t 2 )  I 2
Hence
g12( 2)  1
It’s second-order coherence in all space-time points.
The degree of first- and second- order coherence is
define by the same pattern. This is just two members of
a hierarchy of coherence functions.
It is possible to envisage a general interface experiment
in which the measured result depends on the correlation
of electric fields at an arbitrary number of space-time
points. The results of this experiment is depends upon
the hierarchy of coherence functions with define like this:
g
(n)
(r1t1...rnt n ; rn 1t n 1...r2 nt 2 n ) 
| E * (r1t1 )...E * (rntn ) E (rn 1tn 1 )...E (r2 nt 2 n ) |
*
2
*
| E (r1t1 ) | ... | E (rnt n ) |
2
2
| E (rn 1tn 1 ) | ... | E (r2 nt2 n ) |
2
Reference
http://people.deas.harvard.edu/~jones/ap216/lectures/l
s_3/ls3_u6A/ls3_unit6A.html
The quantum theory of light, Lauden, R 1973
Appendix I
Quantum coherence
For first order coherence for ” Young interference”
The filed operators associated with the modes are
hence if the pine holes are equal:
So it can be written that:
So that
Thus for a n-photon, single mode incident beam, the intensity at an
observation point Q is given by
Or
Second order coherence for Hanbury Brown interference
correlation between photomultiplier currents is proportional to
Thus, the degree of second-order coherence is
Appendix II
Huygens principle: we assert that component fields
radiated by each coherent cell are spherical waves of
the form
.
Appendix III
Free Space Propagation of Coherence Functions
how coherence propagates through space. We start by
assuming that the analytic signal representing the field of
interest satisfies an wave equation of the form
If we multiply through by ,
we see that
Similarly
Therefore the coherence function must satisfy the fourthorder equation
Appendix IIII
Addition calculation for second order coherence
To obtain the root-mean-square deviation in the cycle average of the
intensity, we first calculate
For a collision model it can be written as
Calculation of the equation in slide 36 (relation between first
and second order coherent in collision model).
Again, for the collision- and Doppler-broadening models
Hence: