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Photon noise in holography
By Daniel Marks, Oct 28 2009
The Poisson “popcorn” Process
Photons arriving at a detector have
similar statistics to popcorn popping.
At each time instant, there is an independent
probability of a kernel popping.
Time
The Poisson Process
For a small time interval Dt, there is a
probability rDt of a kernel popping.
Over N intervals, the probability of P pops is:
rDt
Interval 1
rDt
rDt
rDt
Interval 2
Interval 3
Interval 4
rDt
………………..
Interval N
N!
(rDt ) P (1  rDt ) N  P Binomial distribution
P!( N  P)!
The binomial limit
Total time is T=NDt
P
Binomial distribution
N!
 rT   rT 
  1 

P!( N  P)!  N  
N 
N P
As N approaches infinity
 rT 
1  
N

N P
 rT
 1  ( N  P)
N
 ( N  P)( N  P  1)  rT 

   ...
2

N
1
2
 1  rT  rT   ...  e  rT
2!
2
The binomial limit
N!
N ( N  1)( N  2)...  rT 
N  rT 
 rT 


 
 
 
P!( N  P)!  N 
P!( N  P)( N  P  1)...  N 
P!  N 
P
Poisson distribution
Figure shamelessly
lifted from Wikipedia
P
rT P exp( rT )
P!
P
rT is average
number of pops
over time interval T
P
Most important facts we use about Poisson distribution
As the average number of events rT gets large,
Poisson approaches a Gaussian distribution.
rT is mean of distribution
rT is also the variance of the distribution!
mean
Signal to noise ratio of Poisson process =
 rT
variance
How do we analyze the photon noise in optical systems?
Important rule of thumb for quantum processes:
PHOTON NOISE OCCURS AT DETECTION,
NOT AT THE SOURCE.
We don’t know how many photons are emitted,
only how many we receive.
We start at the detector and work backwards
to find the mean/variance of unknown quantities.
A simple example, one interferometric measurement.
object to back
scatter from
Michelson
interferometer
reference
Reference power IR
Signal power IS
The interferometric advantage
I  I R  I S  2 I R I S cos 
IR
IS
For I R  I S
hv
I  I R  2 I R I S cos 
and I R  2 I R I S cos 
is constant
changes
The interferometric advantage continued
2 I R I S cos 2 
Number of signal photons
ADt
hv
A is area of detector, Dt is integration time,
hn is photon energy
I
Number of reference photons R ADt
hv
Variance in number of detected photons

2 I R I S cos  
IR
ADt
I R 
 ADt 
hv
hv


SNR of interferometric detection
Signal photons
Photon noise variance
=
=
I S cos 2 
2
ADt
hv
2I S
ADt
hv
2 I R I S cos 2 
ADt
hv
=
cos 2  
IR
ADt
hv
1
2
… but this is the +/- the number of
signal photons, independent
of reference power.
The interferometric advantage
SNR achieves photon noise limit.
This can be achieved without
photon counting detectors! (e.g. photomultiplier)
This is what enables holography, optical coherence
tomography, etc. to use conventional detectors.
Reference power can be adjusted so thermal
noise becomes small compared to photon noise.
Holography and photon noise
An abstract model of holography…
Object consists of N
points in space
Interference
pattern
Incident
wavefront,
amplitude E0
Reference
field ER
Definitions of variables
h3
h1
h2
S1
S2
S3
S4
h4
Object consists of N
points in space
The scattering
amplitudes of these
points are hi to form a
vector h.
Likewise, the
detected fields are a
vector S with
elements Sj
The optical system
The optical system relates the scattering
amplitudes hi to the detected fields Sj.
The optical system is modeled by a matrix
Hij such that
N
S j  E0 hi H ij
i 1
Or in vector notation
S  E0 Hη
Photon noise of the detected field Sj
Photon noise is primarily due to the reference beam
I j
h
2
2
ER  S j 
1
I R  hER2
2
p  Var p 
h
2

ER  2 Re hER S j
2
*

h is impedance
of free space
Average # of photons on detector j
I R ADt ADt 2

hE R
hv
2hv
2
average and variance
number of photons
(Poisson process)
I R hv
hv
 hv 
Var I j  

hER2
 Var p 
ADt 2 ADt
 ADt 
2
 1 
hv
Independent of
 Var I j 
Var S j  
2hADt
reference power
 hE R 
Finding the covariance of the potential h
hv
Cov S  E[SS ] 
I
2hADt
H
1
S  E0 Hη
E0 H 1S  η
2
Cov η  E[ηη ]  E0 E[H 1SS† (H 1 )† ]
†
hv
1
1 †
Cov η 
E
[
H
(
H
) ]
2
2hE0 ADt
1
I 0  hE02
2
hv
Cov η 
H 1 (H 1 ) †
4 I 0 ADt
result
What if H is unitary?
Unitary H means…
HH †  H† H  I
H 1  H†
Examples of unitary transformations (up to a
constant): Fraunhofer (far-field) diffraction, full-rank
Fresnel diffraction matrix, identity matrix
C is proportionality
hv
hv
1
1 †
2
Cov η 
H (H ) 
C  constant in unitary
4 I 0 ADt
4 I 0 ADt
operator
…therefore for unitary transformations photon noise is
uncorrelated at the scatterer, and dependent only on
the total intensity incident on the scatterer
Fresnel diffraction
ri
r' j
are locations of scatterers regularly spaced
are locations of detected field regularly
spaced
2
ik
  ik
H ij   exp 
ri  r ' j 
z
 z

2
ik
  ik 2 
  ik
  2ik

H ij   exp 
ri  exp 
r ' j  exp 
ri  r ' j 
z
 z

 z

 z

discrete Fourier
transform matrix