Photon Detection. Array of Photon Detectors. Response Function. Photon Detection. The goal is a simple model for photon detection.

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1 Photon Detection The goal is a simple model for photon detection. The detector model will have an onset threshold and a saturation level. Photon Detection The photon stream will be treated as a Poisson process. ectures 8 Spring 2002 The ability to detect and measure changes in the photon stream will be analyzed. The performance measures detective quantum efficiency DQE) and noise equivalent power NEP) will be introduced. ecture 8 1 Array of Photon Detectors We will use a simple model with a close-packed array of N detectors, each with area area A, and identical response functions. This simple model will be adequate to demonstrate the basic principles. The expected number of photons at cell k is I k A k T where A k is the area of cell k, I k is the intensity of the photon beam at cell k, and T is the exposure time. Response Function et X be the number of photons that arrive at a detector. et the detector count all photons up to a level. The response function is is Y hx), where { x, x 1 hx), x We will later add an onset threshold to this basic function. ecture 8 2 ecture 8 3

2 The response Y of a detector in the array is a random variable that is measured every T seconds. The number of photons that arrive at a detector element a Poisson random variable X with mean and standard deviation q IAT. The expected value of the response is µ Y E[Y ] P [X k]hk) Insert the model of hk). k e q µ Y + qk e q k k e q µ Y + qk e q k Expand and rearrange. µ Y e q k1 + e q k2 + e q k3 + + e q k There are terms. Term r is of the form e q e q r 1 e q kr r 1 e q 1 ecture 8 4 ecture 8 5 µ Y 1 e q )+ 1 1 e q + Gather terms and rearrange. µ Y e q ) e q ) e q + + ) + The response µ Y as a function of q for a saturation level of 10. Define a function f 1, q) µ Y 1 f 1 q, )e q) ecture 8 6

3 ThresholdT & Saturation evel: S + T 1 0, x < T hx) x T +1, T x<+ T, x + T Calculation of µ Y The equation for the average count now becomes µ Y P [X k]hk) +T 2 µ Y k T +1) qk e q + qk e q kt k+t 1 Modify the function f 1 so that f 1, q) 1 T 1 q k T + T T Response function has the same form: µ Y 1 f 1 q, )e q) ecture 8 8 ecture 8 9 Detector Response µ Y with T 5, 10 Contrast Contrast is determined by the variation in the output when the input changes. The change must be measured at an operating point q 0 : µ Y q 0 + q) µ Y q 0 )+gq 0 ) q and gq 0 ) is the gradient at the operating point. gq 0 ) dµ Y dq e q 0 f 1 df ) 1 qq0 dq qq 0 ecture 8 11

4 f 1, q) 1 T 1 Contrast + T T T Gradient df 1, q) dq 1 T 2 T T +T The derivative of / is 1 /k 1)!. reduced in length by one. Effectively, each term is If we now subtract the derivative from f 1 all that remains is the last term in each sum. f 2, q) f 1 df ) 1 1 dq Solid curve: 10,T 0; dashed curve: 10,T5. These correspond to the response curves in the earlier figures. ecture 8 12 Mean-squared Detector Response E[Y 2 ] h 2 k)p [X k] k 2qk e q + 2qk e q k This can be expressed in terms of a function f 3, q) as E[Y 2 ] 2 1 f 3 e q) where f 3, q) 1 1 q 1+3 k ) 1 n 2 2n +1) n0 ecture 8 15

5 Variance of the Detector Response σ 2 Y E[Y 2 ] µ 2 Y Comparative Noise evel The action of the detector and internally produced noise contribute to the variance of the detector output. 2 [ 1 f 3 e q ) 1 f 1 e q ) 2] The upper figure shows E[Y 2 ] solid) and µ 2 Y dashed) for a quantum detector with 10 and T 0. The lower curve shows the variance, which is the difference between the two curves in the upper figure. The detector noise performance can be described by the comparative noise level. ε g2 σx 2 σy 2 + σ2 n ecture 8 16 ecture 8 17 Comparative Noise evel The variance of the input is σx 2 q. The gain is the response gradient gq) e q f 2, q) The variance of the output in the absence of internal noise is σy 2 [ 2 1 f 3 e q ) 1 f 1 e q ) 2] The comparative noise level is ε g2 σ 2 X σ 2 Y + σ2 n q e q f 2, q) ) 2 2 [ 1 f 3 e q ) 1 f 1 e q ) 2] + σ 2 n The second law of thermodynamics requires that the output noise be at least as large as the noise that is actually present at the input, so that the comparative noise level is always less than unity. ecture 8 18