ECE 228A Fall 2007Daniel J. Blumenthal! 2.1!Lecture 2- Photon Statistics and Basics of Propagation in Dielectric Media!ECE 228A Fall 2007Daniel J. Blumenthal! 2.2!Photodetection!ECE 228A Fall 2007Daniel J. Blumenthal! 2.3!Detection of Optical Signals !! Thermal: Temperature change with photon absorption! Thermoelectric! Pyromagnetic! Pyroelectric! Liquid crystals! Bolometers! Wave Interaction: Exchange energy between waves at different frequencies! Parametric down-conversion! Parametric up-conversion! Parametric amplification! Photon Effects: Generation of photocarriers from photon absorption! Photoconductors! Photoemissive! Photovoltaics!ECE 228A Fall 2007Daniel J. Blumenthal! 2.4!Photon Statistics! Photon sources can in general be characterized as coherent or incoherent†! Coherent: Probability that a photon is generated at time t0 is mutually independent of probability of photons generated at other times (Markov Process)! Poisson Process: Probability of finding n photons in time interval T! Bunching is a trait of the Poisson process! Interarrival time is decaying exponentially distributed!P(n | T ) =(rT )ne−rTn!t!T!Where :!P(n|T) is probability of finding n photons in time interval T!R is mean photon arrival rate (photons/second)!† Can also be a combination of these two types -> partially coherent!ECE 228A Fall 2007Daniel J. Blumenthal! 2.5!Photon Statistics (II)! Narrowband Thermal (Gaussian):! Bose-Einstein Process: Probability of finding n photons in time interval T!P(n) =11 + nbnb1 + nbnt!T!Where :!P(n) = probability of finding n photons given! nb = mean number photons from incoherent source = N0/hv0! N0 = spectral density of source = Popt/B0! Popt = total optical power from source! B0 = source optical bandwidth! T = observation time ≤ 1/B0!ECE 228A Fall 2007Daniel J. Blumenthal! 2.6!Detecting Photons (1)! Any material that can respond to single photons can be used to count photons! Ideal Detector! Generation of a electron-hole pair per absorbed photon results in an instantaneous current pulse !t!T!t!Input photons hν"Photo-generated current i’(t)!IdealDetector!hν!i’(t)!ECE 228A Fall 2007Daniel J. Blumenthal! 2.7!Detecting Photons (2)! Real Detector! Has an inherent “impulse response,” hd(t), due to built in resistance and/or capacitance.! Can be modeled as an RC filter with low pass response !t!T!t!Input photons hν"Observed photo-generated current i(t)!IdealDetector!hν!i’(t)!Hd(ω) = FT {hd(t)}"i(t)!Area = q = hd(t)dt−∞∞∫ECE 228A Fall 2007Daniel J. Blumenthal! 2.8!Detecting Photons (3)! As the average photon rate increases, the observed photo-current starts smoothing out, with a variance around the mean (average) count that is based on the statistics (which tends to Gaussian for large photon arrival rate)! P(i) is the probability function of measuring the current at a certain value at time t.!t!i!2σ!<Idc>!Ideal Detector!i’(t)!Hd(ω) = FT {hd(t)}"i(t)!T!P(i)!ECE 228A Fall 2007Daniel J. Blumenthal! 2.9!Detecting Photons (4)! The detector output current i(t) can be modeled as a discrete “filtered Poisson” process!i(t) = hd(t −τj)j = 1N∑ Where hd(t) is PD impulse response, N is total number e-h pairs generated, τj is the random time the jth photocarrier is generated.! Define: Quantum Efficiency (QE), unitless, as!η=number of photocarriers producednumber of incident photons, 0 ≤η≤ 1 Define: Time varying photon rate parameter (λ(t)) in units of photocarriers/second as!λ(t ) =ηhυPrecvd(t )ECE 228A Fall 2007Daniel J. Blumenthal! 2.10!Detecting Photons (5)! The power incident on a photodetector of area A, in units of Watts, is! P(t) = I(rp,t)dAA∫ where the instantaneous optical intensity at an observation point p is given by! rp I(rp,t) =1Z0E(rp,t)2 The time varying photon rate parameter λ(t) can then be written in terms of P(t)!λ(t ) =ηhνE(t )2Z0ECE 228A Fall 2007Daniel J. Blumenthal! 2.11!Detecting Photons (6)! If we consider an observation interval, over which we are going to average our photon count over! This can be due either to the inherent bandwidth of the detector or (as we will see later) on purpose to match the receiver bandwidth to the data bit rate! Then the number of photocarriers generated over the interval T counted at the jth observation interval!Nj=λj(τ)dτ0T∫ Assuming a coherent source, the conditional inhomogeneous Poisson process describes this photon count during the jth observation interval!P(Nj= N ) =λj(τ)dτ0T∫NN!e−λj(τ)dτ0T∫ECE 228A Fall 2007Daniel J. Blumenthal! 2.12!Detecting Photons (7)! If we assume a constant rate parameter over the time interval T (independent of j), then the photo-generated current can be written as!i(t) =λ(t)qλ(t) =NT Then the photocurrent produced by the photodetector can be written in Amperes, assuming the observation time is normalized to one second!i(t) =λ(t)q =ηqhνPrcvd(t)= ℜPrcvd(t) Where we have defined the detector responsivity as!ℜ =ηqhνReview of The Wave Equation in Dielectric Media!ECE 228A Fall 2007Daniel J. Blumenthal! 2.14!Notation! MKS units! Lower case for time varying quantities! Capitals for the amplitudes of time varying quantities! Complex quantities used to represent amplitude and phase:! (at least in Chapter 1. In Chapter 2, E(x,y,z,t)=Re [E(x,y,z) eiωt]!]Re[)(tiAetaω=ECE 228A Fall 2007Daniel J. Blumenthal! 2.15!Maxwell’s Equations! ∇ × h = i +∂d∂t∇ × e = −∂b∂t∇ • d = 0∇ • b = 0where e and h are the electric and magnetic field vectors!d and b are the electric and magnetic displacement vectors!No free charge.!ECE 228A Fall 2007Daniel J. Blumenthal! 2.16!Constitutive Relations! d =ε0e + pb =µ0(h + m)p and m are the electric and magnetic polarizations of the medium!ε0 and µ0 are the electric and magnetic permeabilities of vacuum!e and h are the electric and magnetic field vectors!d and b are the electric and magnetic displacement vectors!ECE 228A Fall 2007Daniel J. Blumenthal! 2.17!Electric Susceptibility χ (Isotropic)!P =ε0χijEIsotropic Media: χ is a complex number!The real part determines the index (velocity) and