Photon noise in holography. By Daniel Marks, Oct

Transcription

1 Photon noise in holography By Daniel Marks, Oct 8 009

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 The Poisson Process For a small time interval Dt, there is a probability rdt of a kernel popping. Over intervals, the probability of P pops is: rdt rdt rdt rdt rdt nterval nterval nterval 3 nterval 4.. nterval! P!( ( rdt) P)! P ( rdt) P Binomial distribution

4 The binomial limit Total time is T=Dt Binomial distribution! P!( P)! rt P rt P As approaches infinity rt P ( P) rt ( P)( rt...! P ) rt rt e rt...

5 The binomial limit! P!( P)! rt P ( P!( )( )... P)( P )... rt P P P! rt P rt P Poisson distribution exp( rt) P! rt is average number of pops over time interval T 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! ignal to noise ratio of Poisson process = mean variance rt

7 ow do we analyze the photon noise in optical systems? mportant rule of thumb for quantum processes: POTO O OCCU AT DTCTO, OT AT T OUC. 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. Michelson interferometer obect to back scatter from reference eference power ignal power

9 The interferometric advantage cos is constant changes For cos and cos

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

11 of interferometric detection ignal photons Photon noise variance = cos ADt A D t = cos A D t cos = A D t but this is the +/- the number of signal photons, independent of reference power.

12 The interferometric advantage 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. eference power can be adusted so thermal noise becomes small compared to photon noise.

13 olography and photon noise An abstract model of holography Obect consists of points in space nterference pattern ncident wavefront, amplitude 0 eference field

14 Definitions of variables h h 3 4 h 3 h 4 Obect consists of points in space The scattering amplitudes of these points are h i to form a vector h. Likewise, the detected fields are a vector with elements

15 The optical system The optical system relates the scattering amplitudes h i to the detected fields. The optical system is modeled by a matrix i such that 0hii Or in vector notation i 0η

16 Photon noise of the detected field Average # of photons on detector h t A t A p Var p h D D h is impedance of free space average and variance number of photons (Poisson process) Photon noise is primarily due to the reference beam * e h h h t A t A p Var t A Var h D D D t A Var Var D h h ndependent of reference power

17 Finding the covariance of the potential h t A Cov D h ] [ 0η η 0 ] ) ( [ ] [ 0 ηη η Cov ] ) ( [ 0 D η t A Cov h 0 0 h 0 ) ( 4 D η t A Cov result

18 What if is unitary? Unitary means xamples of unitary transformations (up to a constant): Fraunhofer (far-field) diffraction, full-rank Fresnel diffraction matrix, identity matrix Cov η ( ) C 40ADt 40ADt 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 r r' i are locations of scatterers regularly spaced are locations of detected field regularly spaced i ik z exp ik z r i r' i ik z ik exp z r i exp ik z r' exp ik z r i r' discrete Fourier transform matrix