1. Photon noise in holography
By Daniel Marks, Oct

2. 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

3. Over N intervals, the probability of P pops is:
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
rDt
rDt
rDt
rDt
Interval Interval 2 Interval 3 Interval ……………… Interval N
Binomial distribution

4. The binomial limit
Total time is T=NDt
Binomial distribution
As N approaches infinity

5. The binomial limit rT is average number of pops Poisson distribution
over time interval T
Poisson distribution
Figure shamelessly
lifted from Wikipedia

6. 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 =
variance

7. 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.

8. A simple example, one interferometric measurement.
object to back scatter from
Michelson interferometer
reference
Reference power IR
Signal power IS

9. The interferometric advantage
is constant
changes
For
and

10. The interferometric advantage continued
Number of signal photons
A is area of detector, Dt is integration time,
hn is photon energy
Number of reference photons
Variance in number of detected photons

11. SNR of interferometric detection
Signal photons =
Photon noise variance =
… but this is the +/- the number of
signal photons, independent
of reference power. =

12. 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.

13. 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

14. Definitions of variables
Object consists of N points in space h3 h1
The scattering amplitudes of these points are hi to form a vector h. h2 h4 S1
Likewise, the detected fields are a vector S with elements Sj S2 S3 S4

15. 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
Or in vector notation

16. Photon noise of the detected field Sj
Photon noise is primarily due to the reference beam
is impedance
of free space
Average # of photons on detector j
average and variance
number of photons
(Poisson process)
Independent of
reference power

17. Finding the covariance of the potential h
result

18. What if H is unitary?
Unitary H means…
Examples of unitary transformations (up to a constant): Fraunhofer (far-field) diffraction, full-rank Fresnel diffraction matrix, identity matrix
C is proportionality
constant in unitary
operator
…therefore for unitary transformations photon noise is uncorrelated at the scatterer, and dependent only on the total intensity incident on the scatterer

19. Fresnel diffraction
are locations of scatterers regularly spaced
are locations of detected field regularly spaced
discrete Fourier transform matrix