PHOTOMULTIPLIERS Applied to photocathode and first dynode, as well as across successive dynodes, one typically finds a voltage of 100–200 V, which is adequate for electrons to hit the next SILVANO DONATI University of Pavia electrode with enough energy to be multiplied, and to en- Pavia, Italy sure high collection efficiency. When a primary photoelectron hits the first dynode, it produces the emission of g electrons because of secondary emission. Secondary electrons leaving the dynode are ac- Despite being invented more than 80 years ago, the pho- 2 tomultiplier is still unrivaled, even by the most recent celerated to the next dynode, producing g electrons. The semiconductor photodetectors, in its ability to detect weak process continues up to the last dynode, so that for n dy- nodes, we find gn electrons at the anode, which is the gain optical signals down to the single-photon level. The pho- n tomultiplier tube (PMT) readily provides a wide dynamic of the dynode chain, G g , a very high quantity even for small g: For example, for¼ g 3.5, with n 8–14 dynodes, range and an excellent linearity, with a good speed of re- ¼ 4 7 ¼ the gain goes up to G 2.2.10 y4.10 , respectively. sponse and a very low excess noise. PMTs are made avail- ¼ able from manufacturers in a variety of formats (from At a voltage Vak 100y200 V between electrodes, the time-of-flight between¼ dynodes separated by d can be o1 cm outlines to large sensitive areas, up to 50 [cm] 50 cm2). Also, an ample choice of spectral responses ex- calculated from expression: ists, ranging from deep ultraviolet (UVC, down to 100 nm) to near infrared (NIR, up to 1700 nm 1=2 1=2 lE lE td d 2m=eVak 33:7 ns d cm Vak À ; with the new NEA photocathodes). ¼ ð Þ ¼ ½ Š Á ½ Šð Þ A disadvantage of PMTs is the relatively bulky and and the typical result is tdE3ns. Then, after n 10 dy- fragile structure; however, this is mitigated in the most ¼ nodes, we get a total transit time Tdel ntd 30 ns, which recent design incorporating an integrated multiplier chain ¼ ¼ based on the MCP (multichannel plates) technology. In is the delay of the SER (single-electron response), the cur- addition, the high voltage required to supply the tube is a rent pulse obtained at the anode following a single photo- hindrance, but units with a built-in voltage multiplier of- electron released at the photocathode. Another even more fer an adequate solution. important quantity is the duration of the SER, that comes Thus, in a large number of applications to instrumen- about from the transit time spread, Dtd, because of non- tation, PMTs offer the most viable and satisfactory solu- idealities of the geometry. A typical value is tpul- tion to measurements on light signals. Just to quote a few, seE2y3 ns in an 8–12-stage PMT. The peak current of the SER is found (approximately) PMTs are recognized as the working horse for measure- n by dividing the total charge eg by the duration Dtd. Thus, ments of fast-pulsed waveforms, energy spectrometry, de- n 19 7 9 we get: ip eg =Dtd 1:6 10À 4 10 =2 10À 3:2 mA; cay-times, time-of-flight, particle and radiation energy, ¼ ¼ Á Á Á Á ¼ dating with radionuclides, and photon counting. a current large enough for an easy handling by electronic This article is structured as follow. After a short intro- circuits. Thus, the individual current pulses can be duction to PMT operation (Sect. 1), in Sect. 2 we analyze watched at a PMT output by an oscilloscope (as shown in the elements of the PMT — photocathode, electron optics, Fig. 2) are just the single photons that have been detected and multiplying chain and related parameters. Section 3 by our sensor. is devoted to summarize the theoretical results of PMT response. Then, we discuss specifications and character- istics of PMTs (Sect. 4). Finally, in Section 5 we review the 2. BASIC ELEMENTS OF THE PMT most widespread applications of PMTs to measurements and the performance we can obtain. In this section, the principle of operation and performance of elements constituting the PMT is described, then the most commonly employed structures of the PMT as well as some recent variants of them are reviewed. 1. OVERVIEW OF PMT OPERATION

The basic elements and functions in a PMT can be de- 2.1. Photocathode scribed with reference to Fig. 1 as the following (1): The photocathode is the thin film on which incoming pho- tons impinge, at the entrance window of the PMT. By the photoelectric effect, the photons energy is traded for an * Photocathode (either in transmission or in reflection) electron-hole pair, and electrons (or photoelectrons) are to convert the photon flux of the incoming optical sig- emitted in the vacuum where they become available for nal to a photoelectron flux. multiplication at the dynode chain. * Electron optics, to collect and focus the photoelec- According to the incidence surface, reflection (or opa- trons on the first dynode of the multiplier chain. que) photocathodes are distinguished when electrons * Dynode chain, to provide a high gain of multiplication come out from the same side of photons incidence; and in the number of photoelectrons. transmission (or semitransparent) photocathodes are dis- * Anode, to collect the amplified charge and serve as tinguished when electrons are emitted from the surface the output electrode. side opposite to it (2). 1

Wiley Encyclopedia of Biomedical Engineering, Copyright & 2006 John Wiley & Sons, Inc. 2 PHOTOMULTIPLIERS

MULTIPLIER CHAIN

2 4 6 8 10 12

OUTPUT

1 3 5 7 9 11

1st DYNODE ANODE Figure 1. A typical photomultiplier (PMT) for PHOTOCATHODE instrumentation (from Ref.1, by courtesy of ELECTRON OPTICS Prentice-Hall).

eV, we get the photoemission threshold wavelength lt as

lt hc= Eg EA or; lt mm 1:24=E eV : ¼ ð þ Þ ½ Š ¼ ½ Š

The second condition deals with the path traveled by the electron to reach the surface. The path is a random zigzag because of scattering within the atomic lattice. Scattering can be quasi-elastic, which causes a small energy loss (cases b and d in Fig. 3) and a long free path Ldif, or it is anelastic (or ionizing), with a large loss (4Eg) and a short free path Ldif as the electron is above Ep (case c material Fig. 3). For a high escape probability, it is re- quired that Ldif is larger than the absorption length in the, Figure 2. Single-electron-response (SER) current pulse at the or Ldif41/a. anode of a 12-dynode PMT, as seen at the sampling oscilloscope Last, to obtain a wide spectral range of response, the (time scale: 1 ns/div, amplitude scale: 0.5 mA/div). pair level Ep must always be higher than the vacuum level, or EpcEA The picture of band diagram (Fig. 3) can be refined to Photoemission is a crucial phenomenon in PMT. It can include surface states, which are because of unintentional be described as the sequence of (1) photon absorption, and doping or to defects in the first atomic, and are responsible generation of an electron-hole pair; (2) diffusion of the of band bending at the surface. Bending is upward if the electron to the surface; and (3) emission of the electron in defects are p-type on a n-type substrate and is downward the vacuum. if the defects are n-type on a p-type substrate (Fig. 4). In In a simple picture, the energy levels of a reflection the last case, we get a NEA (negative electron affinity) photocathode can be represented as in Fig. 3 (1). Conduc- photocathode, in which the barrier EA at the surface is tion and valence bands are separated by the energy gap virtually lowered to zero or negative. Photogenerated elec- Eg, the vacuum level is the minimum electron energy to trons can tunnel from conduction band to vacuum if the leave the material, whereas the pair production Ep is the barrier thickness is small (a few atomic layers). Thus, in level at which ionization yields new electron-hole pairs. all photocathodes, the final cesiation plays the important The work function and the activation energy are indicated role of helping saturate defects and lower the surface bar- by EW and EA. rier. NEA photocathodes have been demonstrated in GaAs Internal to the photocathode, optical power decays ex- (gallium arsenide) treated with cesium oxide, GaAs:Cs2O, ponentially according to the Lambert–Beer law: P(z) P0 with response up to 900 nm, and in In1 xGaxAs:Cs (in- ¼ À exp-az. Here, a a(l) is an absorption coefficient specific of dium gallium arsenide), with photoresponse up to ¼ the material, and Labs 1/a is the attenuation length rep- 1700 nm (3). ¼ resenting the 1/e depth of penetration. In conclusion, desirable properties for photocathode For the photoelectron to be emitted, two conditions materials are: (1) work function E E E comparable W ¼ g þ A must be satisfied. First, the energy condition, by which with photon energy hnin the spectral range of interest; (2) the generated electron must be raised to at least at the absorption length 1/a less or comparable to the scattering vacuum energy level to be able overcome the potential length Lsc (for high Z); (3) the largest possible pair-pro- barrier and leave the material, must be satisfied. In terms duction energy Ep (for wide spectral response). of photon energy hn, the energy condition is hnZEg EA, Early photocathodes were metallic, with work func- þ and letting n c/l, and expressing l in mm and energy in tions of several eV’s and thresholds in the UV. In the ¼ 1930s, with the discovery of alkaline antimonide, effi- ciency of a few percent in the visible was obtained. Addi- tional progress in the 1960s, with bi- and tri-alkaline PHOTOMULTIPLIERS 3

inelastic scatter Ep pair-production elastic scatters level vacuum level Figure 3. Energy levels in the photocathode. Conduction and valence bands are separated by c1 the energy gap E , and E is the activation en- electron g A c EA ergy to reach the vacuum level. Condition hnZEg EA holds for the photons energy to pro- c2 þ conduction band duce a photoelectron. Not all the generated elec- trons can diffuse to the surface and escape. b hν d EW Electron d is emitted because generated not a far from the surface and with enough energy Eg hν to withstand elastic scattering, other electrons Fermi level have too small an energy (like a and b) or are valence band generated too deep in the material. Electron c from an energetic photon is raised above the hole pair-production energy, where a fast inelastic scatter that produces a new electron follows (c2), but neither c1 nor c2 can be emitted. (Adapted from Ref. 1, with permission of the Publisher).

+ - + - - + - + p p n p p n (or n n)

tunnel escape vacuum Eg level Eg EA Figure 4. Band bending at the photocathode surface: a p-type surface layer atop on an n- type substrate curves the bands upward, hinder- p-GaAs ing emission. An n-type layer on a p-type sub- p-GaAs or strate curves the bandwidths downward, easing or p-InGaAs the emission of the electron from the conduction p-InGaAs band to the vacuum by tunneling. The effect, Cs Cs2O called NEA (negative electron affinity), is taken (<1nm) (-1nm) to advantage in cesiated photocathodes. antimonide, extended thresholds to near IR (800–900 nm). power P:s(l) I/P. One can also use the quantum efficiency ¼ In the 1970s, the NEA concept was introduced, and long- Z Z(l), given by the ratio F/F’ of the photoemitted elec- wavelength response improved marginally. In recent trons¼ rate, to received photons rate. Writing the detected years, the practicable threshold for PMTs has reached current as I eF and the radiant power as P F’hn the ¼ ¼ the 1700 nm of the third window of optical fibers. ratio a I/P follows as: ¼ The absorption condition favors materials with large a on a wide spectral range, such as semiconductors with a s l Ze=hn Zl= hc=e Z l=1:24 A=W : direct-bandgap structure, whereas in indirect-bandgap ð Þ ¼ ¼ ð Þ ¼ ð Þ ½ Š materials, like Si and Ge, a lattice vibration phonon is re- quired to conserve momentum in the photon absorption. Then, absorption has a much smaller probability per unit 2.1.1. Properties of Common Photocathodes. The most length (or a) and efficiency is very low. important group of materials is the alkaline metals anti- In applications, the photocathode response is charac- monide. In single-metal antimonides such as Cs Sb, K Sb terized by the spectral sensitivity s s(l), defined as the 3 3 and Na Sb, both peak quantum efficiency and ratio of the photoemitted current I to¼the incident radiant 3 Zmax threshold lt increase with the atomic weight (i.e., in the 4 PHOTOMULTIPLIERS order Li-Na-K-Rb-Cs) where as pair-production threshold typical mixture is made by one part of cesium chromate Ep decreases (and the UV response is reduced). (CsCrO4) and two parts of Si or Al as the reducing agent. Cesium antimonide Cs3Sb has a spectral response (des- The mixture is brought to 7001C by radio-frequency heat- ignated as S-11) close to the eye when deposited on a boro- ing. Deposition rate is controlled by the temperature and silicate-glass window. On fused silica, it goes deep in the duration of the process. UV, to 160 nm (S-4, S-5). Also extended to UV are: cesium For multicomponent photocathodes, the metal is made telluride (Cs2Te) with cutoff at 180 nm and cesium iodide to react with other components, by heating the glass en- (CsI) reaching 120 nm on LiF or MgF windows. These velope to 140–1801. The process is controlled by visual in- photocathodes with a threshold lt smaller than visible are spection (aspect and color) and by measuring the called solar-blind and generally withstand (as opposed to photocurrent in the final steps of fabrication. all other photocathodes) natural illumination without damage. 2.1.2.1. Monoalkali Photocathodes. The prototype is Bi-alkaline photocathodes (CsK2Sb, Na2KSb, CsRb2Sb) CsSb, the best known photocathode produced since the have a little bit wider photoresponses, and low dark cur- 1930s. As resistivity is high, a sublayer of SnO or ITO rents (especially Na2KSb) that make them attractive for (indium-tin oxide) is first formed as the entrance window. single-photon counting. First, a film of antimonium is the deposited, whose thick- Tri-alkaline photocathodes (or multi-alkali) Na2KSb:Cs ness is tuned to an optimal value of E6 nm (E20 atomic has a still larger Zmax, and a response extended up to layers) by looking at a decrease of optical transmission, to E800 nm and to E850 nm in the so-called S-25 or S-20 E75% of the initial value. Then, Cs is evaporated as the extended-red-multi-alkali (ERMA). Using a fused silica envelope is kept at 1301C to promote reaction with Sb. window, it reaches 160 nm in the UV. For the wide spectral Evidence of the reaction is the appearance changing from range, the S-25 photocathode is very commonly used in metallic to reddish. The photoresponse gradually in- spectrometry and wide-band applications. creases up to a peak beyond which irreversible decrease Although sensitivity is poor (Zo1%), the old silver-ce- occurs. At the peak, the evaporation of cesium is blocked sium-oxide Ag-O-Cs (S-1 response) is still important in and, on cooling, a small quantity of oxygen is admitted for low-cost applications of IR up to 1100 nm. Last, NEA pho- a final improvement of photoresponse near the threshold. tocathodes in GaAs and ternary compounds offer an im- provement with respect to tri-alkaline in the near IR up to 2.1.2.2. Other Monoalkali Photocathodes. Photocath- about 900 nm. odes prepared like Cs Sb are the antimonide of Rb, K, A conductive underlayer is necessary in transmission 3 Na, and Li, whose work functions of, respectively, 2.09, photocathodes, especially those with high resistivity like 2.22, 2.29, and 2.50 eV, give a response progressively Na KSb. The ohmic-contact layer shall be transparent, 2 shifted toward the blue and UV, with gradually decreas- and even very thin metal layers cannot be used (with the ing quantum efficiencies. exception of W in the UV). Tin oxide and indium oxide (or ITO, indium-tin oxide) are the best suited for ohmic-contact and yield a satisfac- 2.1.2.3. Bi- and Tri-Alkali Photocathodes. These photo- tory resistance (typically 1 kO) with transmission of cathodes are more critical to fabricate, and call for an ac- E95%. curate control of composition to achieve the best Figure 5 shows the spectral responses of popular pho- sensitivity. In the bi-alkali Na2KSb, sodium and potas- tocathodes in different materials. In the visible and near- sium are in 1:2 ratio, with a slight excess of Sb for the p- infrared, alkaline photocathodes are best suited because of type doping. The Na2KSb photocathode is prepared by re- the high spectral sensitivity (up to 100 mA/W peak) and acting K with Sb to build up a layer of K3Sb. Then, Na quantum efficiency (up to 40%). Transmission and reflec- vapor is admitted and heated as the crystalline structure tion photocathodes have nearly the same shape of spectral changes from hexagonal to cubic. response, but with E20–50% larger s(l) and a more ex- Adding small quantities of oxygen or cesium to the bi- tended long-l response in reflection types. In contrast, alkali layer, the work function is further lowered and the transmission photocathodes have the access window coin- threshold is improved. Thus, the spectral responses of S- cident with the photosensitive surface and can collect 20 and S-20 ERMA are process variants compared with more light (i.e., use an objective lens with larger numer- the Na2KSb rather than composition variants. ical aperture) (1,2,4). 2.1.2.4. Cesiated Silver-Oxide Photocathode. This very 2.1.2. Photocathode Technology. Photocathode technol- old and still used photocathode is obtained by first evap- ogy is a thin-film, high-vacuum process (1,4). Because of orating a thin layer of Ag, typically 10-20 nm (50% optical alkali metals reactivity to oxygen, the photocathode is transmission), and oxidizing it by sputtering with oxygen 6 fabricated under a high vacuum (typ.10 À torr) condition. ions in a low-pressure discharge until the transmissions The glass envelope is fabricated first, outgassing it at returns to 100% (Ag2O is transparent). Finally, the layer is 4001C for several hours to eliminate adsorbed gas from the cesiated at 1301C until the response is maximized. The walls. Glasses free from Pb, Ni, W are used, because these photoemission mechanism appears to be promoted by metals react with Cs and produce compounds of low work Cs2O for the peak in the blue, and to metallic Ag for the function that increase the dark current. Alkali-metal va- IR peak, with tunneling from Ag to Cs2O, with possible pors are produced by reduction of a salt of the metal. A nanoparticle effects. PHOTOMULTIPLIERS 5

η= 50% 25 100 10 8 5

5 NaKSb (S-24) GaAs 2.5 :Cs CsNaKSb ) 2 (S-20) W

/ 1

A CsTe m (

10 σ

8 0.5 Y T

I CsSb 5 V I (S-11) 0.25 T

I CsNaKSb S Cs I (S-20ERMA) N E

S 2 0.1 L A

R 1 T

C 8 E 0.05 P AgOCs S

5 (S-1) InGaAs :Cs 0.025 2 0.15 1.0 8 400 500 600 700 800 1000 1200 1500 1800 n o i 5 s e a

t WAVELENGTH (nm)

s λ s c i 2 i a s l window i s c 3 F a m i l s l s g

i s g a s

d n 2 l M o e a g V r r s t U o 3-mm u

f 0.1 b

100 150 200 300 Figure 5. Spectral sensitivity s(l) vs. wavelength of reflection photocathodes. Lines of constant quantum efficiency Z are also shown. The EIA standard designations are indicated in brackets. Inset shows the cutoff of common window materials (adapted from Ref. 1, with permission of Prentice-Hall).

2.1.2.5. NEA Photocathodes. The most recent (1980) 2.2. The Electron Optics photocathodes, still progressing to extend single-photon The electron optics is placed between photocathode and techniques in the IR. Different from common photocath- first dynode (Fig. 1), in form of one or more electrodes odes, they are based on single-crystal materials grown properly shaped and kept at increasing voltages (2,4). Its with the modern technologies of semiconductors. scope is to collect all of the photoelectrons leaving the Best known are gallium arsenide (GaAs) and the ter- photocathode and focus them on the much smaller first nary compounds GaxIn1 xAs, which allow tuning of the À dynode surface, with high efficiency and the smallest time bandgap energy through the control of composition x of spread. The collection efficiency Zc is important because it the constituents in the III-V crystal lattice cell. All photo- combines with the quantum efficiency of the photo- cathodes are cesiated to take advantage of band bending Zph cathode to give the effective electron/photon ratio . at the surface, thus the NEA name. Substrate for the pho- ZphZc The time spread D d of the photocathode to first dynode tocathode is a relatively thick wafer of the material, thus t flight is the most important factor affecting the speed of all NEA photocathodes are of the reflection type, although response. no reason against transmission types exists. With the recent progress in ternary compounds grown on InP substrates and of bandgap engineering technolo- 2.3. The Electron Multiplier gies, it appears feasible to develop long-l photocathodes In secondary emission, a primary electron of some 100– with responses up to the middle infrared. Indeed, a pho- 200 eV energy excites the crystal lattice, raising electrons tocathodes with 1700-nm cutoff are already available, and to high-energy states favorable for emission. As a result of further work should be able to provide substantial exten- the relatively high energy, ionization is multiple and pro- sion of the IR coverage. duces several electron-hole pairs. For a secondary electron to be emitted, it shall have an energy EsZEg EA (1–2 eV) sufficient to overcome the po- tential barrier þat the surface, and a diffusion length larger 6 PHOTOMULTIPLIERS

50 good collection efficiency and compactness. Its limitation is the maximum number of dynodes, r10. 30 p-Ga P (Cs) More difficult to assemble is the venetian-blind struc- 20 ture (c) that, with a large area and a good collection effi- ciency even without an electron-optics, is fairly compact Cs Sb 10 3 but not the fastest. The linear-chains (d and e), also called g

n i 8 Ag Mg O box-and-grid, are often preferred as a good compromise a g (Sb) between ease of fabrication, access to the photocathode, 5 4 Cu Be O good collection area, and overall size. 3 (Cs) In all PMTs, the anode electrode is not critical and can be simply made by a thin metal plate. In front of it, one 2 may place a fine grid acting as an electrostatic screen be- tween last dynode and anode, so that the SER duration is 1 shortened. In addition, to avoid extra frequency cutoff 50 100 200 500 1000 2000 primary energy (eV) from parasitics, in fast PMTs, the anode is made part of a coaxial-like structure ending in the output connector Figure 6. Secondary emission gain of some dynode material as a matched to 50 . function of the energy of the impinging primary electron (or in- O terdynode voltage) 2.4.1. Photocathode-Related Parameters. 2.4.1.1. Spectral Response. Spectral response is speci- fied by the spectral sensitivity, ( ) I/P, ratio of pho- than the depth of generation from the surface, much alike s s l togenerated current I to incident optical¼ ¼power P, or by the in photoemission. quantum efficiency ( ), ratio of emitted photoelectrons Should the primary energy E be fully used, we get a Z Z l to collected photons. ¼ number of N E/(E E ), typically 50–100, which is max g A Usually, the individual ( ) of a device is not supplied never approached¼ because:þ (1) several electrons are pro- s l by manufacturers, and reference is made to average spec- duced with a too low energy (between E and E E ) or g g A tral curve as Fig. 5. Sometimes, devices are supplied with little excess energy; (2) energy is wasted in quasi-elasticþ the individual luminous sensitivity I/P (in A/lm) for a scattering with the lattice. As the energy E of the primary sL L standard source (typically the W ribbon-lamp¼ at 2850 K). electron is increased, the point of generation in the mate- Of course, the luminous sensitivity cannot be traced ex- rial is deeper, and when depth approaches diffusion actly to ( ), but is useful as a relative scale-factor to com- length, the probability of emission starts decreasing and s l pare different devices. a maximum of the multiplication gain is found. As a result of the random aspect of the multiplication process, all the above pictures apply to the mean values of 2.4.1.2. Temperature Coefficient of the Spectral Sensitiv- This quantity is expressed by the relative variation quantities. For example, the gain g has the meaning of an ity. per unit temperature difference, or (1/ )d /dT. Exper- average number of secondary obtained at the output for a a s imentally, is negative (typically ¼0.3–0.5%/ C) except one primary electron on the dynode. The probability dis- as 1 near the photoelectric threshold, whereÀ it becomes posi- tribution p(n) of the number n of secondary generated by a tive and increases fast. A wavelength is then found at primary electron is found experimentally to follow the ls0 n g which a 0, and this l 0 is slightly above the wavelength Poisson distribution p(n) g e À /n!, from which the s ¼ s ¼ of peak response. mean value /nS and variance s2 Dn2 are /nS g 2 ¼ ¼ and s g. Of great importance in low-level The¼secondary emission gain of commonly used dynode 2.4.1.3. Dark Current. and counting applications of PMTs, the total dark current materials is plotted in Fig. 6 as a function of primary en- is composed of several contributions: (1) dispersion cur- ergy (or interdynode voltage). As one can see, CsSb has a rent through the insulating structure; (2) dark current of fairly high gain at low voltages but withstands only electrodes other than the photocathode; and (3) photocath- o801C and has a moderate current density (E200 mA/ ode dark current. If the first two terms are properly con- cm2). The cesiated oxides of copper-beryllium and Ag-Mg trolled and made negligible, the ultimate limit of dark alloys are more resistant to temperature and have a high current is set by the dark photoelectrons. Their current output current density (4100 mA/cm2). Last, cesiated gal- density is given by Richardson’s law: J (4 em/h3) (kT)2 lium phosphide has a NEA band structure (as in Fig. 4) d p exp( E /kT), where E is the work function.¼ and a diffusion length of some mm (instead of E20 nm of À W W As EWEhn near threshold lt, one can also write Jd the other materials), which combine to yield a high gain 3 2 ¼ (4pem/h )(kT) exp( hc/ltkT), and this equation shows per unit voltage and a much higher saturation level. À that dark current increases with increasing threshold lt. Therefore, in photon-counting applications, one will choose the spectral response the least IR-extended to min- 2.4. Common Photomultiplier Structures imize Jd. Figure 8 is a plot of dark current in common In Fig. 7, (2) a few preferred structures for the PMT are PMTs along with the theoretical values from the above reported. With reflection photocathodes, the squirrel-cage equations. As one can see from the diagram, cooling sub- structure (a) is a popular choice and is still used for the stantially reduces dark current. PHOTOMULTIPLIERS 7

Dynode chain

3 1

Grid 4 2 5 hv 6 0

8 Photocathode Screen 7 9

Envelope Anode (a) Dynode chain Photocathode Conductive layer hv Grid Electrode hv Anode Conductive Photocathode layer Anode Electrode (c) (b)

Photocathode Anode 7 6 3 2 9 8 5 4 1 hv

Grid 1st dynode

(d) Anode 2 10 8 6 4 11 Photocathode 9 7 5 3 hv Dynode chain 1 Figure 7. PMT structures: (a) squirrel-cage for reflection photocathodes and (b) for transmis- Electrodes Focus sion photocathodes; (c) venetian-blind dynodes; (d) box and grid; (e) linear dynode-chain (adapted from Ref. 2, by courtesy of Burle Tech- (e) nologies).

2.4.1.4. Uniformity of Response. In different portions of especially in cesiated types. When the tube is turned off, the photocathode and in image tubes, the s(l) may spread excessive illumination may cause fatigue, with a sensitiv- up to 75% at the peak of response, and up to 720% near ity loss and a dark current increase, which, however, re- lt. Near the edge of the photocathode surface, the unifor- covers in 10–100 min after returning to the dark condition. mity is worse than at the center. Residual gas in the tube envelope is another important factor of progressive degradation. Photoelectrons in tran- 2.4.1.5. Linearity and Saturation. About compliance to sit to positive electrodes ionize the residual gas atoms, produce positive ions that are accelerated toward the pho- the ideal linear I sP relation, no experimental evidence ¼ tocathode, and hit it with high kinetic energy, sufficient to exists of appreciable linearity error (say, 40.1%) on a dy- namic range from the dark current (pA or fA per cm2) up pull away clusters of atoms. This process is the same as 2 sputtering, or cathode evaporation. Thus, fabrication of a to about 10 mA/cm . At large input power, photocurrent 7 2 PMT starts with a well-outgassed envelope ( 10 À torr). will saturate, at E100 mA/cm , because of charge storage E effects. When adequately fabricated and correctly used, photo- cathodes may reach very respectable lifetimes, and MTTF (mean time to failure) of 5 104 hours (working) and of 2.4.1.6. Fatigue and Degradation. Photoemission is a 3 105 (nonworking) are currentlyÁ obtained. nondestructive process in itself, so photocathode and dy- Á nodes have a potentially unlimited lifetime. However, sev- eral factors lead to fatigue and degradations. Excessive 2.4.2. Special PMT Structures. The basic PMT structure illumination is a source of catastrophic degradation, be- with photocathode, electron optics, and dynode chain de- cause power dissipation and heating of photocathode or scribed in the previous section is by far the most common dynodes (even for few minutes) reactivate the diffusion in commercial devices. However, through the years, sev- process with an irreversible alteration of the structure, eral variant special types have been developed to improve 8 PHOTOMULTIPLIERS

1p ) 2

106 ) InGaAs -2 cm (A/cm

S-20 -1

1f

DENSITY 103

S-24

CURRENT 1a

1 dark photoelectron rate (s DARK

-40 -20 0 20 40 TEMPERATURE (°C)

λt (nm) 1100 1000 900 800 700

O )

2 1p O S-1 )

O 106 -2 InGaAs cm

X -1 X O 1f O O S-20 O 103 O

S-24 X 1a X X X 1 dark photoelectron rate (s Figure 8. (top) Dark current density and dark DARK CURRENT DENSITY (A/cm X photoelectron rate of some PMTs as a function of the work energy. (bottom) The temperature 1.0 1.2 1.4 1.6 1.8 dependence of the dark current density (adapted from Ref. 1 by courtesy of Prentice- Eg+EA (eV) Hall). parameters like speed of response, amplitude statistics, or simply reduce overall size. 2-stage focus MCP

2.4.2.1. Miniaturized PMTs. Miniaturized PMTs are not hν anode really different devices, but they include all ancillary cir- cuits (bias, dc/dc converter, etc.) and have overall dimen- output sions comparable with those of a solid-state photodiode circuit, yet with the single photon capability of a PMT. Another class is that incorporating a MCP (microchan- photocatode nel plate) multiplier as the substitute to the dynode chain (Fig. 9). The microchannel is simply a hollow small (0.1- mm diameter.) tube, with the internal walls deposited −VS with a secondary emitter. Electrons entering the micro- voltage divider channel make a zigzag path multiplying themselves at Figure 9. A two-stage MCP photomultiplier. each hit on the walls. Making a plate of several microchannels put side by side, we get a thin (E5 mm) multiplier chain that, from increased number of primary photoelectrons the statistics point to point, is equivalent to as many dynode chains. return to good values, the MCP-PMT is preferred in sev- With respect to a dynode multiplier, now the number of eral applications. hits (or equivalent stages) is not a constant, which wors- The typical structure of a MCP PMT is shown in Fig. 9. ens the statistics considerably. However, as the outline Two or three multiplying stages are used, conjugated to size is so strongly reduced, and considering that at the PHOTOMULTIPLIERS 9 one another by proximity focusing, and to the photocath- and variance of the number N: ode by an eventual electron optics. Compared with a sin- gle-high gain stage, the two-stage is preferred because it N g g g ; h i ¼ 1 2 Á Á Á n screens better the region of high and low electron density, 2 2 preventing positive feedback from ionization of residual sN g1g2 gi gi 1 gn : ¼ ð Á Á Á Þð þ Á Á Á Þ gas. Ionization is a source of feedback because the positive i 1;n X¼ ions reach the multiplier input and generate new elec- trons. These expressions can be justified by simple arguments. Instability then sets in for example, if the PMT gain is It is reasonable that the mean number is the product of 8 8 10 and barely 10 À ions/electron go back and ionize at the mean gain at each dynode. About variance, it is the the input. This phenomenon is aggravated, in a glass mi- sum of n contributions from the dynodes, of the form 2 crochannel PMT, by the tendency of MCP to release, in the (g1g2ygi)(gi 1ygn) . Indeed, at the ith dynode, one finds þ useful life, gas molecules adsorbed on its surface when an average of g1g2ygi 1 electrons arriving from the pre- À bombarded by ions. With a suitable stagger, opposite in ceding dynode, each being multiplied by gi and adding a the adjacent stages (chevron-type, Fig. 9), ions produced in variance contribution gi. The quadratic fluctuation goes the second stage cannot reach the input of the first. So, if multiplied by the squared gain incurred between (i 1)th 4 2 þ the gain per stage is kept low (for example, o10 ), the ion dynode and anode, or by (gi 1ygn) , and the above ex- 2 þ feedback effect becomes negligible. pression for sN follows. MCP PMTs have typical multiplication gains of 104 (1- Taking all the dynode gains as equal except the first, 6 8 stage), 10 (2-stages), and 10 (3-stages). With interelec- that is, g2 g3 y gn g, the equations can also be ¼ ¼ ¼ ¼ trode voltages of about 1000 V, the SER duration is written as: E500 ps with rise times in the range 100–200 ps. A typi- cal value of the multiplier variance is 2 5 20, con- N gn; eAmc h i ¼ siderably worse than in dynode PMTs, but ¼whichÀ can be 2 2 2 n 1 tolerated when the number of detected photoelectrons is s N 1=g1 1 1=g 1=g 1=g À N ¼ h i ð Þ½ þ þ þ Á Á Á þ Š large. 2 Another advantage of the MCP-PMT structure is the N = g1 1 1=g h i ½ ð À ފ electrostatic conjugation of input/output faceplate, be- N 2g=g g 1 : cause electrons emitted from a certain photocathode pixel ¼  h i 1ð À Þ land to a corresponding pixel on the anode. Now, if the anode is segmented into a number of elements separately The quantity: accessible from as many external electrodes, a multian- ode-MCP-PMT, capable of high gain and mltiple readout of e2 g=g g 1 s2 = N 2 A ¼ 1ð À Þ ¼ N h i the different areas of the photocathode, is obtained. is relative variance of the dynode chain and is called the variance of the multiplier. The 1/g1-dependence shows that 2 first dynode is the most important in determining eA, 3. THE PMT RESPONSE, GAIN, AND NOISE where the others add less fluctuation, as their contribu- tion is summarized by a factor g/(g 1) not far from unity In this section, the main responses describing PMT are [it is, for example, g/(g 1) 1.33 forÀ g 4]. reviewed. The treatment is statistical to assess accuracy of As the second step, Àconsider¼ a light¼pulse on the pho- the measurements performed by the PMT. The main re- tocathode containing F photons. Mean and variance of the sults are reported without derivations, but the interested number of electrons NF collected at the anode is found (1) reader may find exhaustive treatment in Refs. 1 and 5. as /NFS /ZFS /NS and the variance is ¼2 2 2 The responses to consider are: sNF2 ZF sN sR2 N NF 1 eA2 = ZF Comparing¼ þwithhthei relative¼ h i ðvarianceþ Þ of the F-photons 2 2 packet, (i:e:; sF= F 1= F ), one can see that the PMT h i ¼ h i 2 2 2 2 2 – Integral (or charge) response. The number N of elec- has a noise figure given by NF [sNF//NFS ]/[sF//FS ] 2 ¼ trons collected at the output, after a photoelectron (1 eA)/Z ¼ þ impinge on the first dynode or a photon is detected by Of the two contributions to NF, one is the 1/Z factor be- the photocathode, is a random variable characterized cause of the photocathode, we choose one with the maxi- 2 2 by its mean value /NS and variance sN. mum Z at the signal wavelength; the other 1 eA summarizes the dynode multiplication statistics addsþ 2 noise (term eA 0.33 for g 4). The multiplication process is described by the Poisson ¼ ¼ statistics. At the ith dynode, an individual primary elec- – Impulse (or current) response. For an electron im- tron is multiplied to m secondary electrons leaving the pinging on the first dynode at time t 0, the anode dynode, and mean and variance of the random number m response is the time-dependent function¼ known as 2 are /mS gi, and sm gi. When the packet of electrons the SER (single-electron response, Fig. 2). More in ¼ ¼ leaving the first dynode is tracked up to the last dynode general, following an illumination F(t)of the photo- and the anode collects the N electrons, we get for the mean cathode, one will get a current i(t) out the anode. As 10 PHOTOMULTIPLIERS

i(t) is a random waveform, it is characterized as the the practical limit of the minimum time interval on which second-order statistic by mean /i(t)S and variance a detail of the F(t) waveform can be resolved. 2 si (t). As i(t) en(t), we may restrict ourselves to con- Fluctuations of the PMT response are characterized by ¼ 2 sider the number of electrons n(t). sI (t), which, in the rigid-SER approximation, reduces to the convolution of F(t) and the square of the SER, multi- 2 plied by the scale factor (1 eA)/ZFS. This factor has to be The process of interdynode flights is schematized by þ means of the probability density function f(t), where dp compared with the square of the mean /I(t)S scale factor, 2 2 ¼ (i.e.,/ZFS ) yielding for the signal-to-noise ratio (1 eA)// f(t)dt is the differential probability that the time of flight is þ between t and t dt. It is intuitive that the mean value ZFS, once again the same S/N ratio of the charge mea- þ surement. n(t)is /n(t) /nTS f(t), where nT is the random number of ¼ Last, a term fo(t) is found in mean and variance, the secondary electrons, with mean and variance:/nTS g, 2 ¼ time-of-flight between photocathode and first dynode. This snT g. Compounding the flight-and-multiplication statis- tics¼on the n-dynode chain (1), one obtains for the mean quantity can be neglected when F(t) is slower or compa- rable with the SER, but not when resolution or the decon- value: /i(t)S SER(t) g1g2ygnf1(t)*f2(t)*y*fn(t), where * stands for¼convolution¼ integral operation. This volution calculations is pushed to the lower limits. expression is intuitive, as the n-stage cascade of transfer function has an impulse response given by the convolution – Frequency Domain Response. This response, pertain- of individual impulse responses, and a gain given by the ing to the permanent sinusoidal regime, is described product of individual gains. The variance is (1): by the mean transfer function F(o) and the noise spectral density s. Recalling that the transfer func- 2 tion F(o) of a device is the Fourier transform of the sSER2 t g1 . . . gi gi 1 . . . gn ð Þ ¼ þ i 1;n impulse response Ud(t), and that the noise power ¼ X  à spectrum S2(o) di2/do is the Fourier transform of 2 ¼ f 1 t f 2 t f i t f i 1 t f n t : the autocorrelation KU(t), one can use the results of ð Þ Ã ð Þ Ã Á Á Á à ð Þ Ã ½ þ ð Þ Ã Á Á Á à ð ފ the previous section and write the frequency re- This expression is similar to that of the integral re- sponse of the multiplying chain as F(o) {SER(t)}, where stands for Fourier transform.¼ sponse, with weights decreasing with the stage number i, F F each of them being scaled by the gain g in respect to the In the case of a Gaussian waveform, for example, we i 2 2 have F(o) /NSexp( o nst /2 iont0), and the preceding one. ¼ À þ An important simplification of this equation is obtained high-frequency cutoff (3dB point) is at frequency f T 0:83=2pstHn 0:13=stHn. by the assumption of rigid SER, that is, take the generic ¼ ¼ term in the summation as equal to the square of the mean 2 waveform, or f1(t)* *fi(t)*[fi 1(t)* *fn(t)] E[f1(t)*f2(- y y When one detects a mean current I Iph Ib at the t)* *f (t)]2. Then, the varianceþbecomes the nice result: ¼ þ y n first dynode input, in other words the sum of a signal Iph and a dark current Ib, the noise power spectral density 2 2 S2( ) is computed as S2( ) 2e (I I ) (1 2 ) |F( )|2. sSER2 t eA SER t : o o ph d eA o ð Þ  ð Þ Analogously, the current¼from excitationþ þof the photo- cathode is F( ) {f * SER(t) } { SPR(t) }, and The approximation is named rigid SER because it as- o 0 / S E / S for the detection¼ofFa mean light powerF P, the spectral cribes the fluctuations only to the amplitude of the SER, density of noise is S2( ) 2e ( eP/h I ) (1 2 ) |F( )|2. neglecting shape fluctuations. o Z n d eA o As a remark, the PMT¼is close to theþ quantumþ detection One may extend the results to the detection of a light regime in its pass band for Iph0ZId, (i.e., for an input light pulse F(t) incident on the photocatode. If F are the photons power larger than the breakpoint value Pi0Z (hn/Ze)Id). In contained in the pulse, and assume that the number of 2 this case, the signal-to-noise ratio is (S/N) Iph / [2eB(1 photons is Poisson distributed, one finds: 2 2 ¼ 2 þ eA)] P/[2(hn/Z)B(1 eA)] and the noise figure is NF (1 2¼ þ ¼ eA)/Z. I t ZF F t f 0 t SER t and þ ð Þ ¼ ð Þà ð Þà ð Þ The upper frequency fT of dynode-chain PMTs is usu- 2 2 2 ally located at a few hundreds MHz, and that of MCP- s t 1 e ZF F t f 0 t SER t : I ð Þ ð þ AÞ ð Þà ð Þà ð Þ PMTs up to a few GHz, consistent with the SER duration Dt. This performance may look poor, in view of the tens to With these expressions, one can calculate the average hundreds GHz attained by some modern semiconductor waveform of the output current I(t) from the PMT as the photodiodes [e.g., the SAM or UTC structures (1)]. How- convolution of the SER(t) response and the waveform of ever, it has to be noticed that the bandwidth limitation illumination F(t). With a variety of methods, one can also comes just from the dispersion of the multiplying struc- make the inverse calculation, the deconvolution of the ture, not from the photocathode that has a response time SER from the output I(t) to obtain the waveform of illu- as short as E0.5 ps. Thus, by strongly accelerating the mination F(t) presented at the input. Interesting to note, emitted photoelectrons and reading the optical signal by a the accuracy of the inversion is very good for duration of nondispersion method, one is able to build an ultrafast F(t) large or comparable with the SER duration Dt. One cousin of the PMT, the so-called streak-camera tube (1). can also go down to a fraction of Dt reaching E100 ps as Invented in 1935, the streak-camera has been perfected to PHOTOMULTIPLIERS 11 resolve details in optical waveforms down to 0.5–2 ps, yet as: preserving all the features specific of photocathodes, in- cluding single-photon capability and the wide spectral re- 2 2 2 sT1 eA SER t = d=dt SER t ; sponse (deep UV to near IR). ¼ ð Þ ½ ð Þ Š

and for light pulses much faster than the SER, using F(t) – Time Sorting and Measurements. As a result of the d(t), we get: fast response, PMTs are ideally suited for time-of- ¼

flight measurements and temporal localization down 2 2 2 2 s 1 e =R f 0 t SER t = d=dt f 0 t SER t to the subnanosecond scale, with important applica- T ¼ ½ð þ AÞ Š ð Þ Ã ð Þ ð Þ Ã ð Þ tions in biological diagnostics, spectroscopy, and te- È É È Â ÃÉ lemetry. The PMT response is characterized by the As an example, for Gaussian-distributed time-of-flights delay T between a short light pulse detected at t 0 foy,fn, each with a variance st, using a threshold placed at 2 2 2 ¼ the maximum SER slope, sT [(1 eA)/R](n 1)st is ob- and the output anode pulse, where /TS is the mean d ¼ þ þ 2 tained. Thus, time resolution obtained is given by the delay and the variance sT. sTd SER time-width O(n 1)st and decreases as the square þ In principle, temporal localization is performed by a root OR of the mean number of detected photons. Typical decision circuit, which switches on at a time correlated values of the accuracy sTd are 0.3–1 ns for R 1, and 3– 10 ps for R 104. ¼ with the pulse waveform. In a time-of-flight measurement, ¼ a start and a stop time-localization exit, the difference of which is the desired time of flight. – Correlation response. In correlation measurements, The decision circuit is usually a Schmitt discriminator are described the statistical properties of light fields by the correlation function r(t) defined as: with a switching threshold So (Fig. 10), and the sorting r t L t L t t dt, where L(t) is the input signal is generated when the signal crosses So at time To. L t 0 light-signalð Þ ¼ ¼ Àpower1 ð ,Þ assumedð þ Þ random and stationary. If the amplitude fluctuation DS(t) of the signal around R its mean value /S(t)S is represented by an uncertainty band /S(t)S sS and /S(t)S sS, the switching time will Two functions are needed to compute the output corre- þ À likewise exhibit fluctuations DT with an uncertainty band lation from the input light correlation, the autocorrelation represented by To sT and To sT. With the reasonable function associated with the mean single-photon response, þ À hypothesis of small amplitude fluctuations around the /SPR(t)S f0(t)*/SER(t)S: sS ¼ mean /S(t)S, one can apply the statistical principle of the linear regression and calculate sT as the ratio of amplitude r SPR t SPR t t dt; standard deviation sS to mean signal slope: h ið Þ ¼ t 0 ð þ Þ 2 2 2 Z ¼ À1 s T0 s T0 = dS=dt . T S t T0 ðUsingÞ ¼ theð resultsÞ h of the¼ previousi section, one can write and the autocorrelation function associated with the fluc- the accuracy of a time measurement on an SER pulse tuations: (dynode chain response) following single-photon detection 2 rDSPR t rDSPR t eA f 0 t SER t SER t0 :dt: ð Þ ¼ ð Þ þ t 0 ð Þ Ã ð Þ ð Þ Z ¼ À1

With these functions, one finds the compounding of cor- relations and the PMT output as:

+σs(t) 2 2 2 r1 t Ze FrDSPR t Ze F 1 g t r SPR t : ð Þ ¼ ð Þ ð Þ þ ð Þ ½ þ ð ފ à h ið Þ -σs(t)

S Also in this case, by a simple deconvolution, one can go 0 down to resolve field correlation on a time scale compara- ble with the SER duration Dt 2 5ns , and, using the ð À Þ last expression, to a small fraction of Dt.

t 4. PMT PERFORMANCES

Now review the main parameters affecting PMT perfor- discriminator output mances. About photocathodes, a wide choice of formats is avail- t able (e.g., tubes with 12, 18, 25, 38, or 50 mm diameter are standard). Special devices can reach 200 and 500 mm di- T -σ T0 T + 0 t 0 σt ameter with hemispherical surfaces or a hexagonal shape Figure 10. Timing of a PMT signal by a threshold-crossing cir- suitable for side-by-side packing with little wasted area. cuit with trigger at a level S0. Special units with unusually large sizes have been fabri- cated for nuclear physics and biology applications. Ex- 12 PHOTOMULTIPLIERS cluding specific considerations, it is always better to select lected applications where the limited dynamic range of the smallest photocathode area as permitted by the appli- output current (100 nA–1 mA) is acceptable. cation at hand, because not only size and cost are mini- SER waveform and related parameters. Parameters mized, but also the dark current and perhaps the time used to describe the SER waveform (Fig. 2) of a fast response (2). PMT (Fig. 12): the delay time td between excitation and Number of dynodes and gain. Usually, the number of the response peak; the time duration Dt or pulse full-width dynodes of the multiplying chain is between 8 and 12. half maximum (or FWHM); the rise time tr, or the time Special units can have n 6 (intense pulse detection) or n interval during which the pulse rises from 10% to 90% of 14 (for VUV and in some¼ obsolete specimens). No ad- its peak value. vantage¼ exists in using a number of dynodes exceeding n 12, which already gives adequate gain for single-photon detection.¼ Nor is it customary to go below n 6, the minimum to n 4 ¼ 8 obtain a gain (g 410 ) large enough to render further 10 amplification unnecessary and making the load-resistance noise negligible. The typical dependence of gain G gn ¼ 7 from supply voltage Val and number of stages n is shown in 10 Fig. 11 for common dynodes. The gain dependence on volt- age is taken to advantage for an easy trimming of the PMT gain, even on a range of a decade or more. 106 Best performance is obtained by the least possible num-

ber of dynodes, so that the PMT works at the highest volt- GAIN G Ga P n=9 age Val, which makes the dynode gain gi be high and the n=12 5 time dispersion sti low, improving the charge and current 10 2 9 responses (the factor eA) as well as the time response 2 n=12 (variance sT). A few PMTs use a GaP:Cs first dynode (Fig. 9 6 6 7) to get high g1 gain and thus smaller multiplier variance 104 2 eA g/g1(g 1). ¼ À Ag MgO In addition, the SER duration and delay decrease and Cs3 Sb benefit the speed of response; the slope dG/dV is less, and al 500 700 1000 2000 3000 the need for a stabilized supply Val to obtain a stable G is relaxed (6,7). SUPPLY VOLTAGE Val (V) 7 PMTs with microchannel plates have gains up to 10 (3- Figure 11. The gain of the dynode chain as a function of supply 2 stage) but with a large variance eA; they are useful in se- voltage for some commonly used dynode materials and number of stages (from Ref. 1, reproduced by courtesy of Prentice-Hall).

100

50 10

) 20 12

100 s n

90 ( 10 10 τd 8 50 5 ∆τ r τ

2 10 0 12

t d 0 τ τ 1.0 r 3 - M 8 CP 0.5 3 - MCP 0.2 500 700 1000 1500 2000 3000

SUPPLY VOLTAGE Val (V) Figure 12. (left) Definition of the SER characteristic time parameters and, (right) the SER rise time tr (dashed lines) and delay time td (solid lines) of typical PMTs, as a function of the total supply voltage, and with the number of dynodes as a parameter. (From Ref. 1, reproduced with permission by Prentice-Hall). PHOTOMULTIPLIERS 13

All these quantities have a dependence of the type t voltage drop across the resistances of preceding stages in- 1/2 ¼ k(Val/n) À indicating the value of choosing a low n and a creases the voltage and, hence, gain. For larger signals, high Val. In a photon-counting application (6), the SER the anode-to-last dynode voltage is appreciably reduced, peak current ip is kept in the mA range, so that the voltage and the anode collection efficiency drops because of space across the 50 O load is hundreds of mV and the signal can charge effects. Thus, el becomes negative and a saturation be treated directly by subsequent circuits. of the output current is finally reached. Figure 12 (right) illustrates the rise and delay times of Typical maximum dc anode current in a PMT is in the some common PMTs with different numbers of stages, as a range of 100 to 500 mA for dynode structures. When com- 1/2 function of the supply voltage. The trend ValÀ is gener- pared with the dark and dispersion anode currents of 0.1– ally well matched; the range of Val spans about a decade, 10 nA (typically), the above values indicate a dynamic with slow specimens (those with n 10) that may have tr range of linearity (in direct current) of 4–5 decades. ¼ 10 ns and td 50 ns, and the fast ones (n 8 and 12) In the pulsed regime, one can allow for much larger ¼ ¼ ¼ with rise time tr down to 0.6–1 ns (better at small n). Units peak currents without incurring excessive dissipation. with MCPs can reach tr 150–300 ps. Of course, the dynode voltages cannot vary during the Linearity, Dynamic Range¼ and Saturation. Ideally, the pulse, which is ensured with a capacitance bypass of the output current Iu is a linear function of the radiant de- last (2–3) dynodes, those supplying the largest currents. tected power P, that is, Iu GsP, where s is the spectral Small capacitors (CE1000 pF typically) are placed in par- ¼ sensitivity and G is the gain. The integral linearity error el allel to R to ensure an effective bypass up to tens of ms. is defined as the relative deviation DIu of the current from In MCP PMTs, the allowable specific dissipation in the 2 the nominal expected value Iu GsP, or el DIu/Iu. microchannel is smaller (E1 mW/cm ) than for discrete ¼ ¼ In the PMT, el is usually very small (i.e.,o0.1%) over a dynodes and the maximum dc currents are but E100 nA/ wide dynamic range (several decades), which is the error cm2. As bypass capacitors cannot be used, peak current is found in the normal region of use of the device, from the limited by the charge stored in the stray capacitance (typ- minimum level comparable with dark current Ib to the ically E1 nC/cm2), with a slow recovery time in the range large-signal regime. of ms. For large signals, first an increase of el is found, then Resolution in Charge and Time Measurements. This 2 the output signal Iu levels off, with a different behavior in accuracy is described by eA g/[g1(g 1)], the relative a direct current (dc) and pulsed-signal regime. variance of the charge distribution¼ at theÀ anode for R 1 In dc, the maximum current density of dynodes is never photoelectron at the input. One can also refer to resolution¼ attained, because it leads to excessive dissipation. If rA of amplitude measurement, defined as the full-width 100 mW/cm2 is the dissipation tolerated by a dynode be- half maximum (FWHM) of the charge distribution. Then, 2 fore damage, for typical values of 200 V and 0.5 cm ,one it is rA 2.36 eA/OR for a Gaussian statistics. ¼ obtain a current limit of 100 mA. It is the last dynode to Dark current in a PMT, at the anode Id/a, is the sum of sustain the largest dissipation and would be destroyed three terms: Id/a GId/ph G0Id/d Val /Ris, where GId/ph is first, because it has the largest current in the chain. the dark current¼of the photocathodeþ þ amplified by the gain The dynode bias voltages are commonly obtained with a G, the dark currents of the dynodes is summarizes by G0Id/ resistive divider, as in Fig. 13. Letting all the resistances d, and the last term is the leak through the insulation re- equal, the dynode current is limited to the current flowing sistance Ris. in the divider, Ip V /(n 1)R. When I approaches I , a In a well-designed structure, R is high enough to ¼ al þ n p is linearity error el is generated. The error is positive for make the last term negligible. Contribution Val/Ris is a small Iu because In is outgoing from the dynode, and the dc component and can be filtered out when one takes only

PH A 1 2 3 4 5 n−1 n

Rc In−1 I n Iu

R R R R R R R Ip +VAL C C +20

0 (%) l ε −20 Figure 13. Schematic of the dynode voltage divider −40 (top) and linearity error in dc current near satura- I /I tion (bottom) (from Ref. 1, reproduced with permis- 0.01 0.1 1 u p sion of Prentice-Hall). 14 PHOTOMULTIPLIERS single SER pulses. As a result of the different charge dis- rent at Iamax 1mA, the maximum output signal is Va ¼ ¼ tribution compared with the SER, the dynode contribution RcIamax 160 mV. is significantly reduced by validating the output pulses ¼ through a proper window of discrimination. This han- 5.2. Measurement of Fast Waveforms dling, used in single-photon counting, attains the intrin- sic limits of photocathode dark current. The PMT is the best-suited device for the detection of fast Bias circuits are for general-purpose applications, (ns range) optical signals, as it offers both a high gain G where equal resistances are used in the dynode divider and a large bandwidth B, and a quantum-limited perfor- (typically R 0.1–1MO), with bypass capacitors mance at weak signals. The gain-bandwidth product GB of ¼ (CE1000 pF) across the last two or three stages. This the PMT surpasses by several decades that of other types way, the maximum gain is achieved at a given supply of photodetectors, and offers a low-noise figure. These fea- voltage Val. However, if a specific performance is to be op- tures make the PMT ideal in the spectral range of re- timized, voltages shall be distributed differently. sponse even with small quantum efficiencies, in several To get the best amplitude and time resolution, R1 is set applications of physics, biology, and engineering (7). 2 2 2 to 5 times R, so that both eA and st0 are minimized. Dy- To ensure an adequate output signal amplitude Va ¼ namic range, linearity, and saturation are improved, in RcsGP in subsequent circuit handling (for example, be at both dc and pulsed regimes, increasing the final two re- least E100 mV), one needs a larger gain G gn, obtained sistors to 3–6 times R. either with more dynodes or with larger inter¼ dynode volt- A negative voltage applied to the photocathode is pre- ages. ferred to a positive applied to the anode. In this way, the The best speed of response is obtained with the gain G output signal is referred to ground (and not to the high- given by the smallest number n of dynodes, fed at the voltage V ), and dc decoupling is no longer needed. largest allowed voltage. With a G 108 gain, an optical al ¼ In all PMTs, a magnetic screen is commonly used, made mean power P 1 pW provides a photoelectron rate sP/e 5 1 ¼ of soft ferromagnetic material with high permeability mr 3.10 s À (at l 550 nm), and each photoelectron is de- (e.g., mumetal, permalloy) in the form of a tube to be tected¼ as a pulse ¼of peak amplitude Ge/t 5 mA (having d ¼ slipped on the envelope. This item is usually supplied by taken Dt 3 ns). the manufacturer and has the exact size for each device. If the ¼waveform is pulsed and contains R electrons in For high static (dc) magnetic fields, a loss of gain is in- the SER duration, with a saturation level at a peak of I s ¼ curred because electron paths are deflected and the next 100 mA, a dynamic range R IsDt/Ge 20 is obtained. For electrode experienced a collection loss. In ac, the same ef- even stronger pulses, one may¼ reduce¼ G accordingly (at fect shows up as a modulation of the output current. The fixed Is), for example, with R 2000 photoelectrons, a gain typical susceptivity for a unshielded PMT is 2–10%/gauss G 106 will suffice (which requires¼ n 8 dynodes), etc. for axial fields and, 10–50%/gauss for transversal fields. ¼ ¼ Special units designed for immunity to high fields are 5.3. Time Measurements available with a 102 reduction of the above figures. 2 The time accuracy sT primarily depends on the response times associated with SER and f0 and scales as the square- 5. APPLICATIONS OF PMTS root number of detected photoelectrons. Requirements for output structure, gain, and number of dynodes are those In this section, the common applications of PMTs to in- for fast waveforms. In addition, it is useful to reduce the strumentation and related measurement areas, are re- dispersion st0 and keep a high-gain g1, with a higher volt- viewed and some examples of design and the performance age at the first dynode. obtained are illustrated. In a typical fast PMT with D 3 ns and n 12, one can t ¼ ¼ have st0 0.4 ns for the dispersion of the first dynode and ¼ sti 0.9 ns for those from other dynodes. Then, assuming 5.1. Detection of Weak Signals with Moderate Bandwidth ¼ 2 2 1/2 g1 6 and g 4, sT {st0 [g/(g 1)g1]sti} {0.16 In the detection of weak signals in dc or with low-fre- 0.18}¼ 1/2 0.58 ns¼ is the ¼intrinsicþ limitÀ in a time ¼measure-þ quency content, the PMT offers good performance of sen- ment with¼ a single photoelectron (1,5–7). This figure is sitivity and wide spectral range (extended to the extreme typical of what can be obtained with a CFT (constant frac- UV) together with a small dark current and large sensi- tion timing) commercial circuit, while for a fixed-threshold 2 2 1/2 tive area. timing accuracy it would be sT {st0 n[g/(g 1) g1]sti} Tubes with n 6–9 stages are commonly used, and the 1.52 ns. ¼ þ À ¼ ¼ load resistance Rc is kept as high as possible compatible For pulses F(t) faster than the SER, this figure scales with the stray capacitance Ca of anode and output wiring down as OR with an increase in the number of photoelec- to ground, and with the desired bandwidth B: R 1/ trons per pulse R, going down easily to the limits of res- c ¼ [2pCaB]. For example, with Ca 10 pF and B 100 kHz, Rc olution allowed by fast electronic circuits (E10 ps), even ¼ ¼ 160 kO. In this way, the delta-impulse response is a for relatively weak optical pulses with, for example R ¼ 3 ¼ negative exponential with time constant t RcCa ( 1.6 ms 10 . For slow pulses, the intrinsic limit to accuracy sT is ¼ ¼ in our case). The output signal is given by Va Rc Ia Rc determined by the characteristic time t associated with ¼ ¼ F sG P, where P is the detected radiant power and s is the the waveform F(t), still with the dependence tF/OR for the spectral sensitivity. Establishing a limit of anode dc cur- number of photoelectrons per pulse R. PHOTOMULTIPLIERS 15

5.4. Photocounting Techniques formed (not shown in Fig. 14) is that of dumping the in- tegrator after a time T from each pulse, to recover the In photocounting (3,6), SER pulses coming from the pho- rec zero level for the next pulse to be discriminated. tocathode are descriminated from all other pulses (from Requirements for PMTs in a photocounting regime are: dynodes) and because of spurious actions (i.e., ion bom- a SER amplitude in the mA range (to get E100 mV in cir- bardment and radiation contaminants in the envelope-like 7 8 40 cuits), requiring GE10 –10 and therefore nE12 dynodes; K of glass). a high first dynode gain to get a good discrimination effi- By eliminating all these pulses, the residual counting ciency; and a voltage divider adequate to have a short D of rate achieves the very low level inherent in the dark emis- t SER. sion. A simple method, currently used for discrimination, The recovery time T of the integrator determines the is that of measuring the total charge Q at the anode, and rec maximum photon rate acquired for the photocounting as F accept only those events that have the charge Q falling 1/T , hence a dynamic range (in power) P e/T between two thresholds Q and Q properly selected. rec recs 1 2 (with¼ 20 mA/W and Trec 10 ns, it is P 0.8 nW).¼ The A circuit to implement the photocounting technique is s sensitivity¼ performance for ¼weak signals ¼is then deter- shown in Fig. 14. The anode output is passed on to the mined by the dark current I of the photocathode, which counter through a linear gate, enabled by the result of d can go down to fA or aA at ambient temperature depend- charge amplitude discrimination. Charge discrimination ing on the threshold of the material. is best performed on the last dynode current pulse, made The sensitivity of the photocounting technique can be available by deriving a small resistance in series to the now evaluated (1). The total counting N N N , given dynode. The dynode pulse is well correlated to the anode- s d by the sum of signal and dark currents, has¼ a meanþ value pulse charge (by a factor 1 1/gn) and, additionally, it /NS /NsS /NdS . p FT ( dId /e)T, where is leads the anode pulse by theÀ þtime-of-flight t (a few ns). Z Z Z Z n photocathode¼ þquantum¼efficiencyþ, and , are the count- Thus, the integrate-and-dump amplifier has a time t Zp Zd n ing discrimination efficiencies of photon and dark pulses. available to supply the result of discrimination at the en- Both contributions are distributed according to Poisson able input E before the anode pulse arrives at the linear 2 statistics and, therefore, the variance is sN /NsS / gate input. If tn does not suffice, an additional delay tr will ¼ þ NdS. For Id 0 and ZZp 1, the signal-to-noise ratio in- be added by means of a delay line. ¼ ¼ 2 2 2 trinsic to input signal electrons is (S/N)i /NS /sN / After the linear gate, a shaper is used to shorten the ¼ 2 ¼2 NsS, whereas with a finite dark current, (S/N) /NsS / pulse as much as possible to avoid pile-up of successive ¼ [/NsS /NdS] is obtained. pulses, before going to the counter. A last function per- þ

F DELAY tr LINEAR GATE 50Ω E SHAPER

−VAL DISCR. S1 COUNTER

DISCR. reset INTEGRATE 0−T AND DUMP S2 DISPLAY

90

Ns+Nd 60 COUNTS Nd

30 PHOTON Figure 14. Basic functional scheme of photo- counting (top), and an example of measurement, showing evolution of signal plus dark, dark only, ∆N and result after dark subtraction (bottom). Verti- 0 cal bars on N indicate the 0.5 confidence 1 2 34 D 7 sDn level. (from Ref. 1, reproduced with permission of COUNTING TIME T Prentice-Hall). 16 PHOTOMULTIPLIERS

2 With dark counting rates Id/e of a few electrons/cm s, radiation, particle identification, time of flight, and coin- the minimum measurable radiant power is P Fhn/ cidence detectors. When the PMT is optically coupled to a 18 ¼ Z.ZpE10 À W (at Z.Zp 0.1, near the peak of photocath- scintillator, the combination is called a scintillation detec- ode response). This performance¼ is unsurpassed by other tor (5). photodetectors. The scintillator is a sort of converter that, when excited A refinement is to subtract dark counting Nd with a by energetic radiation (g-rays, fast electrons, neutrons, second measurement performed with N 0 (no light in) etc.), generates a fast light pulse containing a number of s ¼ and lasting as above, a time T. Subtracting the dark gives photons R proportional to the energy Egof the absorbed as a result DN /N1S /N2S /NsS /NdS /NdS quantum, i.e., R kE , with k being a conversion factor of ¼ À ¼ þ À ¼ g /NsS. For the variance of DN, recall that for a differ- the material (typically kE1 photoelectron/keV). ¼ 2 ence it is the sum of variances, so that sDN /NsS 2/ Common scintillators are crystals like NaI:Tl (thal- 2 2 ¼ þ NdS and (S/N) /NsS / [/NsS 2/NdS]. lium-activated sodium iodide), CsI:Tl, BiGeO (bismuth Figure 14 (bottom)¼ illustratesþthe typical outcome of a germanate), or organic and polymeric compounds known photocounting measurement, plotting the signal plus dark as anthracene, stilbene, etc. Scintillators’ emission spec- and the dark alone. Both graphs increase with time T, but trum is centered in the interval of 400–550 nm to match the randomness makes their difference surely positive the common photocathode response (particularly, S-11 and only after a certain time; in the figure, condition / S-20), and their impulse response waveform has a nearly NsS4sDN is reached at the right end of the diagram. exponential decay, with time constants ranging from a few The minimum signal detectable in photocounting with ns (organic) to 100 ns (crystals) (1,5). dark subtraction is obtained by setting the condition S/N The scintillator base is placed in optical contact on the 2 1, thus /NsS /NsS 2/NdS.When the signal trying input window of the PMT, while its walls are treated with ¼ ¼ þ to be recovered is weaker than the dark, that is white paint (TiO2), to help rediffuse photons toward the Ns {2 Nd , /NsSmin O[2/NdS] O(2ZdId T/e) is ob- PMT. From the measurement of the total charge Q ZeGR tained.h i h i ¼ ¼ of the anode pulse, one goes back to the energy of¼the de-

As /NsS ZZp FT, it corresponds to a minimum de- tected quantum as Eg Q/kZeG. Using the integral re- ¼ ¼ 2 tectable photon rate Fmin given by: sponse accuracy, the charge S/N ratio sQ/Q O[(1 eA)/R] ¼ þ coincides with the S/N ratio of the energy sE/Eg, hence an energy resolution E [(1 2 )/R] [(1 2 )/ ]. For Fmin 2ZdI=eT =ZZp; sE gO eA O eA k ¼ ð Þ example, the 1.46 MeV¼ peak ofþ40K is ¼resolvedþ by NaI:Tl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with E 40 keV. In this measurement, it is the low value from which one can see that F improves as the in- s E min of that sets the resolution, whereas the factor 2 , the verse square root of integration (or counting) time T. Let- k eA noise added by the PMT, is small. ting all Zs 1, Fmin becomes the geometrical mean O(Id/ Energy spectrometry measurements (7) are carried out e).(2/T) of ¼the dark pulse rate I /e and of 2/T (i.e., two d with the setup reported in Fig. 15. By integration of the counts in the integration time T). anode current in a charge amplifier, one first gets a qua- Typically, the minimum detectable rate is evaluated as sistep waveform whose amplitude is proportional to the Fmin O(2 0.7 1/T)/0.35 0.7 4.83/OT photon/s. For an ¼ Á Á Á ¼ charge. The integrated pulse is passed to a pulse-shaping 8 h integration period, it would yield 4.83/O28800 0.028 –20 ¼ circuit, to limit pulse width and maximize the pulse ac- photon/s or also Pmin hnFmin 1.3 10 W, which is the ceptance rate without superposition of waveforms. power collected from¼a m 28th¼ magnitudeÁ star with a 2 Finally, the signal is sent to a multichannel analyzer 1m telescope aperture. ¼ (MCA), an instrument that classifies the pulses according About photocounting, it is sometimes claimed that the to their peak amplitude, stores the event in the memory, sPAD (single-photon avalanche photodiode) has the same and displays the resulting distribution. capability of PMTs in detecting and counting single pho- tons. The SPAD is an avalanche photodiode (1) based on a pin structure fabricated in a suitable (EgEhn) semicon- 5.6. Dating with Radionuclides ductor. It is biased a little bit beyond the infinite-gain Energy spectrometry has an application in dating with voltage of the linear-regime avalanche multiplication (1). radionuclides, especially with carbon, which is based on The SPAD can indeed detect single photons, but only in a 14 Geiger-counter mode, that is, with a long dead-time fol- the decay of C, a radioisotope that is produced in the atmosphere by reaction of nitrogen with slow neutrons lowing each detected event and no capability of linear re- sponse. In fact, the response to, say, R 10 simultaneous and secondary components of cosmic rays and is present in photons is the same as for R 1 photon,¼whereas in PMTs, atmospheric carbon dioxide with a concentration confi- dently assumed constant through the last million years. it is ten times as large. The SP¼ AD also has a limited spec- Dating consists of measuring the content of the isotope tral range, difficult to be extended into the UV, and a very 14 minute sensitive area, a tiny fraction of mm2 compared C in organic substances, where it has been initially ab- with cm2 of the smallest PMT. sorbed from atmosphere during their lifetime, and since then has decayed with a half-life of T1/2 5700 years with the emission of 160 keV electrons. ¼ 5.5. Nuclear Radiation Spectrometry The measurement is performed by looking at the rate of In the context of nuclear physics, PMTs are widely used in detection of 160 keV electrons from the specimen, which is a number of measurements, such as energy spectrum of proportional to the residual content of 14C in it. PHOTOMULTIPLIERS 17

SCINTILLATOR CHARGE γ AMPLIFIER A PULSE B SHAPER 50Ω C in −VAL enable MULTICHANNEL DISCRIMINATOR ANALYZER A B DISPLAY

t C

Cs 137

K 40

COUNTS Figure 15. Scheme of energy spectrometry measurements (top) and signal waveforms (middle). The energy spectrum measured with the scintillation detector (bottom) reveals radio- nuclides species (energy signature) and their 662 KeV 1.46 MeV concentrations (counts intensity) (from Ref. 1, AMPLITUDE reproduced with permission of Prentice-Hall).

With an energy spectrometry as in Fig. 1–15, one can 2. Burle Industries, Photomultiplier Handbook, vol. PMT-62. distinguish such electrons from the background of radia- Lancaster, PA: Burle, 1987. tion (which has a different energy distribution), and re- 3. R. L. Bell and W. E. Spicer, Compound photocathodes: a new 3 solve concentrations down to cE10 À of the initial value family of photoemitters with greatly improved performance. Proc. IEEE 1970; 58:1788–1801. c0. As concentration decays according to the simple ex- T/To pression c/c0 2 À , for the age T: T T1/2 log2 (c0/c) is 4. L. M. Biberman, and S. Nudelman, Photoelectronic Imaging obtained, with¼ a practical limit in dating¼ to about 40,000 Devices, vols. 1 and 2. New York: Plenum Press, 1971. years. 5. S. Donati, E. Gatti, and V. Svelto, The statistical behaviour of Also, other radionuclides, such as 87Rb, 232Th, 235U, and the scintillation detector: theory and experiments. In: L. Mart- 40K, are currently used to cover much larger time spans on, ed., Advances in Electronics and Electron Physics, vol. 26. (up to 106–109 years) for use with inorganic samples where New York: Academic Press, 1969, pp.251–304. 14C is absent, again with the same method and measure- 6. S. K. Poultney, Single photon detection and timing: experi- ment setup. ments and techniques. In: L. Marton, ed., Advances in Elec- tronics and Electron Physics, vol. 31. New York: Academic Press, 1972, pp. 39–116. BIBLIOGRAPHY 7. J. P. Boutot, J. Nussli, and D. Vallant, Recent trends in photo- multipliers for nuclear physics, In: L. Marton, ed., Advances in 1. S. Donati, Photodetectors, Upper Saddle River, NJ: Prentice- Electronics and Electron Physics, vol. 31. New York: Academic Hall, 2000. Press, 1983, pp. 223–305.