Detector. Detector Model. Estimator. Detective Quantum Efficiency. An input photon stream X is presented to the detector.
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1 Detector An input photon stream X is presented to the detector. Detective Quantum Efficiency Lecture 9 Spring 2002 X is a random variable with a Poisson distribution. P (X k) qk k! e q where q λ The response is an event stream Y. Y h(x)+z Goal: Determine the intensity λ of the input photon stream. Lecture 9 1 Detector Model Estimator An estimator is a rule to calculate the value of a quantity of interest from one or more observations of a random variable. An estimate is a function of a random variable, and is therefore itself a random variable. Y h(x)+z h(x) is a known detector response function Z is an independent random variable with a known probability distribution. Questions: 1. How can we estimate q or λ based on the observation Y? 2. How do we describe the quality of the estimate? Let λ be an estimate of λ. We would like to have E[ λ] λ. This is an unbiased estimator of λ. We would also like to have var[ λ] be as small as possible. The quality of the detector is measured by comparing the quality of the estimate that could be made from X to the quality of the estimate that can be made from Y. Lecture 9 2 Lecture 9 3
2 Detective Quantum Efficiency DQE can be defined in a number of equivalent forms. definition that we will use is The basic Definition: The DQE for a detector is the ratio of the variance of an estimate of ˆλ based on the detector input X to the variance of an estimate of λ based on the detector output Y. Estimate based on X Pretend that X is available and construct an unbiased estimate of λ. The mean value of the input Poisson distribution is E[X] q λ λ E[X] Given an observation X, construct an estimate DQE var[ˆλ] var[ λ] The DQE will always be in the range 0 to 1. ˆλ X This estimator is unbiased, since E[ˆλ] E[X] The variance of the estimator is q λ var[ˆλ] var[x] A 2 τ 2 q A 2 τ 2 λ Lecture 9 4 Lecture 9 5 Estimate based on Y To estimate q or λ based on Y we need an equation of the form q w(y ) To find this function, begin with the inverse function and then invert. From a previous lecture, E[Y ]µ Y (q) L ( 1 f 1 (q, T, S)e q) Given any chosen operating point q 0 we can write an expression for µ Y (q) intermsofµ Y (q 0 ). µ Y (q) µ Y (q 0 )+ dµ Y (q q dq 0 )+ q0 Estimate (continued) This gives us a hint at the estimator equation. Replace µ Y (q) byy and q by the estimate q and drop higher order terms. Y µ Y (q 0 )+ dµ Y ( q q dq 0 ) q0 The derivative is the detector gain at the operating point. Solve for the estimate as a function of Y q q 0 + Y µ Y (q 0 ) w(y ) g The function q w(y ) is an unbiased estimator of q near q q 0. An unbiased estimate of λ is λ q w(y ) g Lecture 9 6 Lecture 9 7
3 The variance of λ is Estimator (continued) var[ λ] var[y] (g) 2 From the results of the last lecture, the variance of a piecewise linear detector is var[y] L 2 [ (1 f 3 e q ) (1 f 1 e q ) 2] The gain is given by where g(q 0 ) dµ Y dq Le q 0f 2 (L, q 0 ) qq0 f 2 (L, q) 1 L 1 q k L k! k0 DQE The DQE of a piecewise linear detector can be calculated by DQE var[ˆλ] var[ λ] (g) 2 λ/ L 2 [ (1 f 3 e q ) (1 f 1 e q ) 2] L 2 [ (1 f 3 e q ) (1 f 1 e q ) 2] This is the same result as that obtained in the last lecture for the comparative noise level when the variance of Z is zero. Lecture 9 8 Lecture 9 9 Example Linear Amplifier Consider a system in which Y gx + Z, wherex is a Poisson r.v. with mean q λ, g is known, and Z is an independent r.v. with zero mean and variance σ 2 Z. An estimate of λ based on X would be The estimate has variance An estimate based on Y is ˆλ X var[ˆλ] var[x] A 2 τ 2 λ Y g λ Linear Amplifier (continued) This estimate has variance The DQE is var[ λ] var[y ] A 2 τ 2 g 2 g2 σ 2 X + σ2 Z A 2 τ 2 g 2 λ + σ2 Z A 2 τ 2 g 2 DQE var[ˆλ] var[ λ] λ + λ σ2 Z A 2 τ 2 g 2 λg 2 λg 2 + σz 2 λ 1 λ + σ2 Z 1+ σ2 z g 2 The DQE can be viewed from either the input or output perspective. The DQE for a linear amplifier is close to unity when σ Z /g is small compared to q λ. Increasing g improves the DQE. Lecture 9 10 Lecture 9 11
4 Example Secondary Emission Each input photon produces an output event with probability η. If q X λ then q Y ηλ and E[Y ]ηλ The output process is still Poisson, so that The gain is g η. var(y )E[Y ]ηλ The ideal estimate is ˆλ X/ so that Then var[ˆλ] var[x]/a 2 τ 2 λ/ DQE var(ˆλ) var( λ) λ/ ηλ (η)2 η DQE and SNR DQE may also be defined as the ratio of the (power) SNR at the detector output to the maximum possible SNR. SNR in N σ n SNR out M σ m DQE (SNR out) 2 (SNR in ) 2 ( M)2 /σ 2 m ( N) 2 /σ 2 n Lecture 9 12 Lecture 9 13 Linear Amplifier Revisited Input SignalE[N] q Input Noiseσ n q Output SignalE[M] gq Output Noiseσ m + σz 2 DQE ( M)2 /σ 2 m ( N) 2 /σ 2 n (gq)2 /( + σ 2 z ) (q/ q) 2 + σ 2 z 1 1+ σ2 z Noisy Detector A photon detector has a dark noise whose variance σd counts/second is proportional to the exposure time and a readout noise whose variance σr 2 10 counts is at a fixed level. The detector area is A 0.3 cm 2 and the detector conversion efficiency is ɛ 0.75 counts/photon. Find the DQE for when the detector is placed in a flux of Φ 3.3 photons/cm 2 /s and exposed for T 10 minutes. which is equivalent to our previous result. Lecture 9 14 Lecture 9 15
5 Input Signal & Noise The input signal is the expected number of photons to the detector. This is S in q ΦAT (3.3)(0.3)(10)(60) 594 photons The standard deviation of the photon stream is the input noise. Output Signal & Noise The output signal is the input signal times the conversion efficiency S out qɛ counts The output noise has three contributors the Poisson noise in the detected counts, the dark noise and the readout noise. The input SNR is N in q 24.4 photons ( ) S 594 N in σ 2 N out qɛ + σ 2 D T + σ2 R The standard deviation of the output noise is The output SNR is then ( ) S N out σ Nout 24.8 counts Lecture 9 16 Lecture 9 17 DQE of Noisy Detector The DQE can be computed by [ ] 2 (S/N)out DQE (S/N) in ( ) Lecture 9 18
Detective Quantum Efficiency
Detective Quantum Efficiency Lecture 9 Spring 2002 Detector An input photon stream X is presented to the detector. X is a random variable with a Poisson distribution. P (X = k) = qk k! e q where q = λaτ
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