Photomultiplier (PMT) basic elements

Transcription

1 Photomultiplier (PMT) basic elements MULTIPLIER CHAIN OUTPUT PHOTOCATHODE ELECTRON OPTICS 1st DYNODE ANODE from: Photodetectors, by S.Donati, Prentice Hall 000 1

2 PMT formats PMT formats range from 1/ - dia. tubes to 60 cm and more, window shape may be circular, hemispherical or hexagonal number of dynodes is from 6 to 1 (max 14) with any S-type photocathode to cover the 150 to 100 nm spectral range from: Photodetectors, by S.Donati, Prentice Hall 000

3 Typical values Typically, at 00 V of inter-dynode voltage, the dynode gain is g=4 per stage, the transit time is.5 ns per stage and its time-spread is 0.5 ns For example, with 1-stage PMT we have: - a gain, G = 4 1 = an (unessential) pulse delay,.5x1 = 30 ns - a pulse duration due to spread, τ =0.5x 1= 1.8 ns (the impulse response) - a current peak, I = eg/τ = / = 1.4 ma (well detectable) from: Photodetectors, by S.Donati, Prentice Hall 000 3

4 Single Electron Response (SER) vert scale: 0.5 ma/div; hor. scale: 1 ns/div from: Photodetectors, by S.Donati, Prentice Hall 000 4

5 Secondary multiplication 1 primary E 0 n secondaries For one primary electron impinging on a dynode with energy E 0 we find n secondary electrons emitted. Number is not fixed, varies f rom primary to primary. Probability of having n secondaries is Poisson distributed, p(n) = g n e -g / n! where g is the mean value (or mean gain of the dynode) from: Photodetectors, by S.Donati, Prentice Hall 000 5

6 Dynode Materials p-ga P (Cs) 0 mean gain g Cs Sb 3 Cu-Be O Ag-Mg O E (ev) 0 from: Photodetectors, by S.Donati, Prentice Hall 000 6

7 The electron optics from: Photodetectors, by S.Donati, Prentice Hall 000 7

8 Excursus: Design of the electron optics Given a tentative structure and electrode voltages, one looks for the potential distribution V(r,z): V(r,z)/ z + (1/r) V(r,z)/ r + V(r,z)/ r = 0 (boundary conditions: V(r,z)=V ext, the applied voltages). As n(r,z) = V(r,z) is the electrostatic index of refraction, the ray equation giving the trajectory r=r(z) can then be integrated: V(r,z) r(z)/ z + V(r,z)/ z r(z)/ z + (r/) V(r,z)/ z = 0 (initial conditions: r=r 0 (position) and dr/dz=tan φ 0 (slope) at z=0). Solution yields the output coordinate r=r(l) of the ray. Repeating ray calculations with different tan φ 0, we get the confusion distribution r=r(l,r 0 ) of the electron-optics for entrance at [z=0, r=r 0 ] in the object plane. Many steps of optimization are repeated on the full image, with slight changes the electrodes shapes and voltages so as to minimize aberrations all over the image-field. from: Photodetectors, by S.Donati, Prentice Hall 000 8

9 Common PMT structures squirrel-cage (reflection photo-cathodes) venetian-blind dynodes squirrel-cage (transmission photocathodes) from: Photodetectors, by S.Donati, Prentice Hall 000 9

10 Common PMT structures (cont d) box and grid linear dynode-chain from: Photodetectors, by S.Donati, Prentice Hall

11 Types of PMT responses to be considered Charge (or integral) response:is the number N of electrons collected at the anode for one photoelectron impinging on the first dynode (multiplier chain response) or one photon detected by the photocathode (PMT response). Extension: to a light pulse containing R photons (scintillation counter response). For all statistics, the random variable N is characterized by mean N and variance σ N = [N- N ]. Impulse (or current) response in the time domain: is the timedependent output SER (single electron response) f or an electron at t=0 on the first dynode. We will calculate: the mean SER(t) and variance σ SER (t), the output current i(t) for 1 or R photon/s detected at t=0, and the response to an optical signal I(t) carrying R photons. A variant is the correlation response. Response in the frequency domain: is the transfer function F(ω) of the PMT, given by the mean and noise spectral density s(ω) for dif ferent inputs. Time localization: is about time measurements, i.e., the delay T between a short light pulse detected at t=0 and the output anode pulse, where T is the mean delay and the variance σ T yields the accuracy of time localization supplied by the PMT. SER(t) T N ω t t from: Photodetectors, by S.Donati, Prentice Hall

12 Charge response multiplication at the dynode (fixed input) for 1 impinging electron for K impinging electrons m = n = g, m = K n = Kg, σ n = g σ m = Kσ n = Kg model of dynode multiplication (random input) 1 n { <n> = g σ = g n k m { <k> <m> = <k> <n> σ { k σ = <k> σ m n+ σ <n> k from: Photodetectors, by S.Donati, Prentice Hall 000 1

13 Charge response (cont d) Applying iteratively these equations we get: dynode mean N variance σ N 1 g 1 g 1 g 1 g g 1 g + g 1 g 3 g 1 g g 3 g 1 g g 3 + g 1 g g 3 + g 1 g g 3 4 g 1 g g 3 g 4 g 1 g g 3 g 4 + g 1 g g 3 g 4 + g 1 g g 3 g 4 + g 1 g g 3 g n-th g 1 g... g n g 1 g... g i [g i+1... g n ] i=1,n Whence : N = g 1 g... g n σ N = N 1 / (g 1 g... g i ) i=1,n from: Photodetectors, by S.Donati, Prentice Hall

14 Charge response (cont d) Letting all dynode gains g equal except the first g 1 we have: σ N = N (1/g 1 ) [1+1/g+1/g /g n-1 ] N /[g 1 (1-1/g)] = N ε A where ε A = g /[g 1 (g -1)] is called the multiplier variance, because it just gives the relative variance of the charge, σ N / N = ε A from: Photodetectors, by S.Donati, Prentice Hall

15 Charge response (cont d) For exactly R photoelectrons arriving to the 1st dynode, N R = g 1 g... g n R = N R σ NR = σ N R = N R ε A R and the relative variance is: σ NR / N R = ε A /R i.e., it improves as 1/R. Last, consider the R photoelectrons be Poissondistributed, as due to F photons detected by the photocathode with quantum eff iciency η, so that R= ηf photoelectron are emitted. from: Photodetectors, by S.Donati, Prentice Hall

16 Charge response (cont d) If F is Poisson distributed, also R= ηf is Poisson distributed, whence σ R = R. The statistical compounding rules are schematized below: R= ηf η 1 { < η F> σ = <ηf> R N { <N> σ N N F F { <F> σ =<F> F { < >=< ηf> <N> N F σ NF = < ηf> σ N R + σ <N> from: Photodetectors, by S.Donati, Prentice Hall

17 Charge response (cont d) By applying the compounding rules, we get: N F = ηf Ν σ NF = ηf σ Ν + σ R N = N F (1+ ε A ) / ηf As the relative variance carried by the packet of photons F is: σ F / F = 1/ F the photomultiplier adds a worsening by a factor: NF = [σ NF / N F ] / [σ F / F ] = (1+ε A )/η NF is called the noise figure and, is nicely low.!.! [for example, ε A =0.33 for g=4 and with η =0.33 we have NF =4] from: Photodetectors, by S.Donati, Prentice Hall

18 Current response i -th DYNODE 1 f(t) (i+1)-th DYNODE n { <n> = g n σ = g n(t) The process of multiplication is now time-dependent and interdynode flights are schematized by a function f(t), the pdf of having a time-of-flight t t+dt when the electron leaves the i-th-dynode at time t=0. The number of secondaries n T again electrons follows the Poisson statistics. Each electrons from the i-th dynode covers the interdynode f light independently Choices for f(t): Gaussian: f(t) = [ (π)σ t ] -1 exp [-(t -t 0 ) /σ t ] or polynomial-exponential: f(t) = [Γ(m+1)σ t m+1 ] -1 t m exp (-t/σ t ) from: Photodetectors, by S.Donati, Prentice Hall

19 Current response (cont d) A random current will be written as I(t) = e N f (t) where e is the electron charge, N is the random total charge, f(t) the time-dependence (for simplicity, we will omit e from now on) from: Photodetectors, by S.Donati, Prentice Hall

20 Current response (cont d) mean current at the (i+1)-th dynode: n(t) = n T f(t) where n T is the (random) total number of secondaries [n T = 0- n(t) dt], with mean and variance: n T = g, σ nt = g, and as f(t) = f i (t) we have: n(t) = g f i (t) To compute the variance, we shall introduce the generatlized concept of covariance from: Photodetectors, by S.Donati, Prentice Hall 000 0

21 Current response (cont d) Covariance function K n (t,t'): K n (t,t') = [n(t)- n(t) ][n(t')- n(t') ] for t=t' the covariance becomes the variance, K n (t,t)=σ i (t); and, if K is dependent on t-t'=τ only, it becomes the correlation function ρ(τ). For a process with a sequence of n T independent electrons with arrival times f(t), one can find that the covariance is: K n (t,t') = [(σ nt / n T ) + n T -1] f(t) δ(t'-t) where δ(t'-t) is the Dirac delta-function. For a Poisson distributed n T, we have: K n (t,t') = g f (t) δ(t'-t) from: Photodetectors, by S.Donati, Prentice Hall 000 1

22 Current response (cont d) δ(t) <n(t)> K (t,t') n n(t) { <n(t)> = g f(t) K (t,t') = g f(t) δ(t'-t) n { n (t) I <n (t)> I K (t,t') I <n(t)> K (t,t') n { m(t) <m(t)> = <n (t)> <n(t)> I K (t,t') =<n (t)>δ(t'-t) m I K n (t,t')+ + K I (t,t') <n(t)><n(t')> Schematization of a random time-dependent multiplication through mean and covariance (top) and rule of composition of signal and delta response from: Photodetectors, by S.Donati, Prentice Hall 000

23 Current response (cont d) Applyi ng iterativ ely the composition rules we have : dynode mean i(t) covariance K i (t,t') 1st g 1 f 1 (t) g 1 f 1 (t)δ(t'-t) nd g 1 g f 1 (t) f (t) g 1 f 1 (t)δ(t'-t) g f (t)δ(t'-t) + g 1 f 1 (t)δ(t'-t) g f (t)f (t') = g 1 g f 1 (t) f (t)δ(t'-t) + g 1 g f 1 (t) [f (t)f (t')] 3rd g 1 g g 3 f 1 (t) f (t) f 3 (t) g 1 g g 3 f 1 (t) f (t) f 3 (t)δ(t'-t) + + g 1 g g 3 f 1 (t) f (t) [f 3 (t)f 3 (t')]+ + g 1 g g 3 f 1 (t) [f (t) f 3 (t)][f (t') f 3 (t')] n-th g 1 g... g n f 1 (t) f (t)... f n (t) g 1 g... g i [g.. i+1 g n ] f 1 (t) f (t)... f i (t) [ f i+1 (t)... f i n(t)][ f i+1 (t')... f n (t')] from: Photodetectors, by S.Donati, Prentice Hall 000 3

24 Current response (cont d) This response is the SER (single-electron-response). Mean value is: SER(t) = g 1 g... g n f 1 (t) f (t)... f n (t) Also, as N = g 1 g... g n : The covariance is: SER(t) = N f 1 (t) f (t)... f n (t) K SER (t,t') = i=1,n g 1 g... g i [g i+1.. g n ] f 1 (t) f (t)... f i (t) and the SER variance (for t=t ) is : [f i+1 (t)... f n (t)][f i+1 (t').. f n (t')] σ SER (t) = i=1,n g 1... g i [g i+1.. g n ] f 1 (t) f (t)... f i (t) [f i+1 (t)... f n (t)] from: Photodetectors, by S.Donati, Prentice Hall 000 4

25 Current response (cont d) Rigid SER approximation: f 1 (t)... f i (t) [f i+1 (t)... f n (t)] [f 1 (t) f (t)... f n (t)] then we have: K SER (t,t') 1/(g 1 g... g i ) SER(t) SER(t') = ε A SER(t) SER(t') σ SER (t) ε A SER(t) An example of the accuracy of the rigid SER approximation for n=1 stages and a Gaussian time-of-fight distribution from: Photodetectors, by S.Donati, Prentice Hall 000 5

26 Current response (cont d) g 1 f1 g f g n f n out N 1 N N n F Φ(t) η f 0 g 1 f1 g 1 f 1 g f n n out N 0 N 1 N N n Noise equivalent circuit of the dynode multiplying chain (top) and of the photomultiplier (bottom). Noisegenerators N i sum to the mean signal a white noise of intensity equal to the mean signal from: Photodetectors, by S.Donati, Prentice Hall 000 6

27 Current response (cont d) Case of a fixed number R of photoelectrons in a single short pulse with L(t) = F Φ(t): I(t) = R f 0 (t) SER(t) K I (t,t') = R f 0 (t)δ(t'-t) * K SER (t,t') R f 0 (t) SER(t) SER(t') ε A σ I (t) R f 0 (t) SER(t) ε A Also in this case, the relative variance of the response, given by: σ I (t)/ I(t) = [ f 0 (t) SER(t) ε A ] / R [f 0 (t) SER(t) ] can be approximated, on the rigid SER approx., to: σ I (t) / I(t) ε A /R from: Photodetectors, by S.Donati, Prentice Hall 000 7

28 Current response (cont d) Some samples of the anode response of the photomultiplier for R= 10, 0 (top), 50, 100 (bottom) photoelectrons emitted at t=0, calculated by a Montecarlo method. Horizontal scale: time in ns; vertical scale: current in relative units from: Photodetectors, by S.Donati, Prentice Hall 000 8

29 Correlation response The correlation function ρ(τ) of a light field L(t) is def ined as: ρ L (τ) = t =0- L(t) L(t+τ)dt If L(t) is a train of photons occurring at random times t k, i.e.: L(t) = hν Σ δ(t-t k ) k=-, + developing ρ L the correlation is found as: ρ L (τ) = (hν) {F δ(τ) + F [1+γ(τ)]} where γ(τ) is the reduced correlation f unction, γ(τ) = [L(t)- L(t) ] [L(t+τ)- L(t+τ) ] / L(t) from: Photodetectors, by S.Donati, Prentice Hall 000 9

30 Correlation response (cont d) The output current correlation ρ I (τ) of I(t), can then be computed and the result is : where ρ I (τ) = (ηe) F ρ SPR (τ) +(ηe) F [1+γ(τ)] * ρ SPR (τ) ρ SPR (τ) = t =0- SPR(t) SPR(t+τ) dt is the autocorrelation associated with the mean waveform SPR(t) =f 0 (t)* SER(t) of the single photon response, and ρ SPR (τ) = ρ SPR (τ) + K SPR (t,t+τ) dt t =0- is the autocorrelation of the SPR fluctuations, contained in the covariance K given by : K SPR (t,t+τ) = ε A f 0 (t) * SER(t) SER(t') from: Photodetectors, by S.Donati, Prentice Hall

31 Correlation response (cont d) ρ SPR ρ SPR SPR To describe the PMT correlation response, two functions are required, ρ( SPR) and ρ SPR t (ns) from: Photodetectors, by S.Donati, Prentice Hall

32 Correlation response (cont d) When the correlation times are much longer, or f 0 (t) is short compared to the SER, a rigid SPR approximation is adequate and the previous equations become: ρ SPR (τ) = (1+ε A ) ρ SPR (τ) ρ I (τ) = (ηe) (1+ε A ) Fρ SPR (τ)+(ηe) F [1+γ(τ)] * ρ SPR (τ) By deconvolution of ρ I (τ) with ρ SPR, light correlations γ(τ) waveforms down to about 10% of SER duration can be recovered from: Photodetectors, by S.Donati, Prentice Hall 000 3

33 Frequency Domain response The transfer function F(ω) is the Fourier transform of the δ response U δ (t), and the noise power spectrum S (ω)= di /dω is the Fourier transform of K U (τ): frequency response: F(ω) = F{ SER(t) }, for a Gaussian waveform : F(ω)= N exp(-ω nσ t /+iωnt 0 ), whence a high frequency cutoff (at -3dB) of f T =0.83/πσ t n= 0.13/σ t n. noise power spectral density: S (ω) = e (I ph +I d ) (1+ε A ) F(ω) In the passband, the PMT is close to the quantum limit for I ph0 I d, i.e. for an input power larger than P i0 (hν/ηe)i d. In this case, the S/N is: (S/N) = I ph / eb(1+ε A ) = P/(hν/η)B(1+ε A ) noise figure: NF =(1+ε A )/η. P i is very small, even for small η, because of the very minute dark currents I d of the photocathode, for ex., f or I d =1 f A and η=0.1, quantum-limited performance is at P P i0 = 0 fw and is over the full frequency band. from: Photodetectors, by S.Donati, Prentice Hall

34 Time Sorting and Measurements S 0 <S(t)> <S(t)>+σ (t) S <S(t)>-σ (t) S discriminator output with threshold S0 Time sorting can be simply performed by a threshold crossing at a suitable level S o and taking T o as the timing information carried by the signal T 0 - σ t T 0 T + σ 0 t t from: Photodetectors, by S.Donati, Prentice Hall

35 Time Sorting and Measurements (cont d) Time variance for crossing of a fixed amplitude threshold: for the time measurement on SER: σ T (T o ) = σ S (T o )/ ds/dt t=to σ T1 = ε A SER(t) /[d/dt SER(t) ] and for the time measurement on a fast pulse: For the time measurement on SER: σ Tδ = [(1+ε A )/R]{f o (t) SER(t) } /{d/dt [f o (t) SER(t) ]} Example: for Gaussian-distributed time-of-flights f o,f 1,.., f n, each with a variance σ t, using a threshold at the maximum slope of the SER we have: σ Tδ = [(1+ε A )/R](n+1)σ t from: Photodetectors, by S.Donati, Prentice Hall

36 Time Sorting and Measurements (cont d) The variance of the SER centroid, as the weighted sum of interdynode-fl ight uncertainties is: σ Tc = σ t0 +(1/g 1 )σ t1 +(1/g 1 g )σ t (1/g 1 g...g n )σ tn and for equal σ ti =σ t : σ T σ t0 +ε A σ t and in the case of R= ηf photoelectrons: σ Tc = [σ t0 +(1/g 1 )σ t1 +(1/g 1 g )σ t (1/g 1 g...g n )σ tn ]/R = [σ t0 +ε A σ t ]/R Written as σ Tc =[1+ε A ]σ t /R the centroid variance is smaller by a f actor (n+1) as compared to the fixed-threshold crossing. from: Photodetectors, by S.Donati, Prentice Hall