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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 ( N P)(N P 1) rT 1 ( N P) ... 2 N N 2 1 2 1 rT rT ... e rT 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 P rT P Poisson distribution Figure shamelessly lifted from Wikipedia P! exp(rT ) 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 I R is constant I S changes For I R I S hv I I R 2 I R I S cos and I R 2 I R I S cos The interferometric advantage continued 2 I R I S cos2 Number of signal photons ADt hv A is area of detector, Dt is integration time, hn is photon energy IR Number of reference photons ADt hv Variance in number of detected photons 2 I R I S cos IR I A D t ADt R hv hv SNR of interferometric detection Signal photons Photon noise variance = = I S cos2 2 ADt hv 2I S ADt hv 2 I R I S cos2 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 E R 2 Re hE R 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 2 Var I j Var p h E R A D t A D t 2 A D t 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 H1S 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 )† 4I 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