On the limited amplitude resolution of multipixel Geiger-mode APDs
aa r X i v : . [ phy s i c s . i n s - d e t ] J un On the limited amplitude resolution of multipixelGeiger-mode APDs
A Stoykov , , Y Musienko , , A Kuznetsov , S Reucroft andJ Swain Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Department of Physics, Northeastern University, Boston, MA 02115, USA Joint Institute for Nuclear Research, 141980 Dubna, Russia On leave from INR, MoscowE-mail: [email protected] (A Stoykov)
Abstract.
The limited number of active pixels in a Geiger-mode AvalanchePhotodiode (G-APD) results not only in a non-linearity but also in an additionalfluctuation of its response. Both these effects are taken into account to calculate theamplitude resolution of an ideal G-APD, which is shown to be finite. As one of theconsequences, the energy resolution of a scintillation detector based on a G-APD isshown to be limited to some minimum value defined by the number of pixels in theG-APD.PACS numbers: 85.60.Gz
Keywords : avalanche photodiode, silicon photomultiplier n the limited amplitude resolution of multipixel Geiger-mode APDs n photoelectronsthe G-APD responds with an output signal of N fired cells, the dependence N = f ( n )saturates due to the limited number m of the cells in the device, and the fact that morethan one photon can enter one cell without changing the response of that cell. Thenon-linearity of a G-APD response is an established experimental fact (see, e.g [2]). Acorrelation between saturation of the G-APD signal and an increase of its dispersionwas also observed [2].In this work we calculate the amplitude resolution of an ideal G-APD (which we callthe ”intrinsic” resolution) and show that it is limited (non zero) due to an additionalstatistical noise in the process of distributing n photoelectrons over m cells. By anideal G-APD we understand a device with the excess noise factor [3] equal to unity. Inparticular, we assume that there is no crosstalk between cells (so it is never the casethat one photon triggers more than one cell) and that each photon incident hits somecell. The considered problem is equivalent to a well-known problem in mathematicalstatistics of distributing (randomly) n balls (photoelectrons) into m urns (cells), seee.g. [4]. The number N of urns containing one or more balls is a random variable, itsexpected value and variance are:¯ N = m [1 − (1 − m − ) n ] σ N = m ( m −
1) (1 − m − ) n + m (1 − m − ) n − m (1 − m − ) n (1)The distribution of N is approximately normal when m, n → ∞ and the ratio α = n/m is bounded [4]: ¯ N = m (1 − e − α ) σ N = m e − α [1 − (1 + α ) e − α ] . (2)In practice the number of cells in a G-APD is usually greater than ∼ R = 2 . · σ/ ¯ A , where A is the amplitude of thedetector response, and ¯ A and σ are the mean value and standard deviation characterizingthe distribution of A (which is assumed to be normal). In case of a non-linear response,the formula has to be modified. The scheme of our consideration is the following: A f −→ A : ( ¯ A, σ ) f − −→ A : ( ¯ A , σ ) . n the limited amplitude resolution of multipixel Geiger-mode APDs A (constant value) the detector responds with a signal of amplitude A . Due to fluctuations induced by the detector, A is statistically distributed. If thefunction f relating ¯ A to A is known, one can reconstruct an estimate of the input signalfrom the mean detector response. The restored amplitude A is no longer a constant,like A , but is also a random value, with mean and standard deviations (at σ ≪ ¯ A ):¯ A = f − ( ¯ A ) ≡ A , σ = (cid:16) f − (cid:17) ′ ¯ A · σ . (3)The parameters of the restored distribution, rather than the measured one, definethe amplitude resolution of a detector: R = 2 . · ( σ / ¯ A ) . (4)The theoretical function ¯ N = f ( n ) is known (2), and this allows us, using relations(3), to reconstruct the signal at the G-APD input ( A is the reconstructed amplitude)from the measured distribution of N (in (3) ¯ A ≡ ¯ N , σ ≡ σ N ):¯ A = n , σ = σ N e α . By substituting ¯ A and σ in (4) for the intrinsic amplitude resolution R of aG-APD we get: R = 2 . · m − / · Φ ( α ) , Φ ( α ) = α − (e α − − α ) / . (5)The function Φ ( α ) is plotted in Figure 1. For any given value of m the value of R increases (the resolution becomes worse) with the ratio n/m = α increasing. () , () ( ) ( ) Figure 1.
The functions Φ ( α ) and Φ( α ) defined in (5) and (6). When a G-APD is used in a scintillation detector the intrinsic resolution of the G-APD should be summed quadratically (this we prove in Monte-Carlo simulations below)with the resolution determined by the statistics of photoelectrons R stat = 1 / √ ¯ n (the n the limited amplitude resolution of multipixel Geiger-mode APDs n relates to the number of photons as ¯ n = P DF · ¯ n ph ,where P DE is the photon detection efficiency [3]). Note, that here we assume that theenergy resolution of the scintillator [6] is equal to zero. The energy resolution of suchan ideal scintillation detector is then: R = 2 . · m − / · Φ( α ) , Φ( α ) = ( α − + Φ ( α ) ) / . (6)The function Φ( α ) (see Figure 1) has minimum value of Φ min = 1 . α =1 . R min = 2 . / √ m .The obtained result on the energy resolution was verified in Monte-Carlosimulations. In the simulations the energy resolution of an ideal scintillation detectorwas calculated as the follows: a) a number of photoelectrons n was randomly generatedaccording to Poisson statistics; b) those n photoelectrons were randomly (uniformly)distributed over m cells resulting in N occupied (fired) cells. c) from the distributionof N the number of photoelectrons (the energy) and its standard deviation werereconstructed using (3), and the energy resolution calculated according to (4). As isseen in Figure 2, the energy resolution obtained in simulations is in agreement with thevalues calculated according with (6).Physically, while a larger mean number of photons implies a smaller fluctuationin that mean, at some point saturation effects dominate (multiple photons enter singlecells) and resolution degrades. This means that some care should be taken in eachapplication to optimize design taking this into account. Summary
The variation of a G-APD response induced by the statistics of distributing n photoelectrons over m G-APD cells causes its amplitude resolution to degrade as theratio n/m increases. This implies, for example, in that the energy resolution of ascintillation detector using a G-APD is limited to R min = 2 . / √ m . References [1] D. Renker, Nucl. Instr. and Meth. A 567 (2006) 48.[2] P. Buzhan, B. Dolgoshein, A. Ilyin et al., ICFA Intstr. Bull. 23 (2001) 28.[3] Y. Musienko, S. Reucroft, J. Swain, Nucl. Instr. and Meth. A 567 (2006) 57.[4] N.L. Johnson and S. Kotz,
Urn models and their application ,John Wiley & Sons, Inc., 1977.[5] W.R. Leo,
Techniques for nuclear and particle physics experiments , Springer, 1987.[6] P. Dorenbos, J.T.M. de Haas and C.W.E. van Eijk, IEEE Trans. Nucl. Science 42 (1995) 2190. n the limited amplitude resolution of multipixel Geiger-mode APDs R ( % ) m = 500( m ) -0.5 R R ( % ) m = 100 Figure 2. (Top) Simulated values (dots) of the energy resolution of an ideal G-APD based scintillation detector as compared with the values (solid line) calculatedaccording with (6). The dashed lines show two contributions to the energy resolution:the statistical term (1 / √ m α ) and the intrinsic resolution of the G-APD ( R ).(Bottom) the same for G-APDs with different number of cells mm