Chapter 6Performance C riteria forPhotodetectors6.1 IntroductionIn this chap ter we will consider models of detector performance. The per-formance of detectors may be characterized in t erms of their responsivenessin reacting to inciden t photons. The responsive quan tum efficiency may bedefined as the n umber of output even ts per inciden t photon. This definitionis natural and ob vious, but unsatisfactory because it does not account forrandomness in the production of output ev en ts. A detector that requiresmany photons to produce an output event may be better than a multipliertube that produces many photoelectrons per input photon, even though onedetector is less “responsive” than the other. The compar ison has to includethe added randomness of the detection process.The fundamental principles for detector comparison w ere form ulated b yAlbert Rose and published a classic1946 paper1that laid the foundationfor a measure that includes detector randomness. The measure is calledDetectiv e Quan tum Efficiency ( DQE) after R. C la r k J o nes of Polaroid2.TheDQE is e xpressed in terms of the abilit y of a detector to see a signal againsta background of ambien t radiation, both impinging on the dete ctor. The1Albert Rose, “A Unified Approach to the Performance of Photographic Film, Televi-sion Pickup Tubes, and the Human Eye ,” J. SMPE, 47, 273-294 (1946).2R. Clark Jones, “On the Quantum Efficiency of Photographic Negatives,” Photo-graphic Science and Engineering, Vol 2, Number 2, August, 1958. (57-65)103104CHAPTER 6. PERF ORMANCE CRITERIA F OR PHOTODETECTORSquantum fluctuations of the ambien t radiation induces randomness in thedetector response that is in addition to the randomness that is added by thedetector itself. The DQE is essen tially a comp arison of the two sources ofrandomness.Thes e notes will follow the development in Zweig3which w as written inthe form of a tutorial and which illustrates the concep t of D Q E from threeperspectives, the photographic scien tist, the engineer and the perceptual psy-chologist.Let us first present so me example s to intr oduce the topic. We imaginethat a detector is placed in a photon stream with the intention of measuringthe photon intensity. The intensity is the parameter q which represents thea verage value of the photon stream when modeled as a Poisson process. Wewould like to estimate q or its equivalent, but are prevented from doing soaccurately by the random nature of the photon arrivals. The photon streamitself presents both the “signa l” q and the interfering noise in terms of therandom n ess of the process.The average signal strength is S = q and the root-mean-square (rms)value of the noise is σ =√q. The signal-to-no ise ratio (SNR ) is thenSN Rmax=q√q=√q (6.1)This is the maximum SNR that can ever be achiev ed for this measurement.By measuring more photons w e get a better estimate of the mean v alue inaccordance with what we have learned from the law of large numbers.The measurem ent that we actually mak e with a real detector will in-evitably be corrupted by additional noise. The measured SNR of the detec-tor, SN Rmwill nev e r exceed SN Rmax. We defin e the DQE as the ratioDQE =µSN RmSNRmax¶2(6.2)This is a comparison between the theoretical limit and the actual ac hieve-men t, and will hav e a value bet ween zero and one. The DQ E rating can beused for any kind of detector and therefore is v ery useful in system assess-men ts. It need not be used just to measure the strength of a photon stream.One can use DQ E very widely by properly identifying the appropriate signal3H. J. Zweig, “Performance Criteria for Photodetectors,” Photo graphic Science andEngineering, Vol 8, Number 6, Nov-Dec 1964. (305-311).6.2. PHOTON DETECTOR MODEL 105Figure 6.1: A photon detector in whic h X is the n umber of arriving photonsand Y is the detector response.and noise com ponen ts for a given application. Let us dev elop the model forthis measure in more detail.6.2 P ho ton De tector ModelConsider a device that can be exposed to a photon beam for a kno w n in tervalof time. If the size of the aperture is A and the exposure time is τ then theexpected number of photons is q = λAτ where λ is the intensit y of the beamin photons per unit area per unit time. Let X be the number of photon s thatare sampled on a giv en exposure, and let Y = h(X)+Z be the output of thephoton detector when the input is X. The additiv e term Z is noise that isin ternal to the detector. We will treat Z as an independent random variable.The value of λ represen ts the signal and we would like to determine it fromthe observation Y.We will now conside r the simp les t possible examples —n amely, a line ardetector with additive in tern al noise. First we will look at the noiseless caseand then add detector noise.106CHAPTER 6. PERF ORMANCE CRITERIA F OR PHOTODETECTORSEx ample 6.2.1 Noiseless Linear Detector. Suppose that the detectorhas a linear response function h(x)=gx and that Z =0..ThenY = gX. Weknow that the expected value of the detector output is E[Y ]=gE [X]=gλAτ.This me ans that λ = E[Y ]/gAτ, so this gives us an indication of how tomeasure the “signal.” Now, we cannot observe E[Y ], but we c an use the valueof Y that is actually observe d to compute λ = Y/gAτ. as the estimate of λ.The randomness in Y is caused by the random nature of the photon streamitself, an d not any detector noise. The number of photons that arrive in an yexperiment is a Pois son random variable with mean value E[X]=λAτ. Fora Poisson distribution the variance is also σ2X= λAτ . If we could observe Xthen the SN R would beSN RX=µE[X]σX¶=µλAτ√λAτ¶=√λAτ (6.3)Note that SN RXimproves in proportion to√Aτ so that a longer exposureor a larger aperture are better. We presume, however, that we do not havedirect access to X. W e have to base our calculations on λ = Y/gAτ . ButE[λ]=E[Y ]/gAτ. = λ and σλ= σY/gAτ = σX/Aτ =pλ/Aτ so thatSN Rλ=µE[λ]σλ¶=Ãλpλ/Aτ!=√λAτ (6.4)The quality of the estimate of the intensity done by use of Y to calculate λis, in this case, just as good as could be obtained by d irect observation of thephoton count. This is to be expected because this is a noiseless linear detector,which is the ideal system. It is also possible to calculate the SN RYwhich isthe SNR based upon the mean and standard deviation of Y.SN RY=µE[Y ]σY¶=µgλAτg√λAτ¶=√λAτ (6.5)We find that all three m easures