Photometric Error Analysis IX: Optimum Use of Photomultipliers

Photometric Error Analysis. IX: Optimum Use

of Photomultipliers

A. T. Young

A critical study of photomultipliers leads to the following conclusions: (1) the dark current observedin tubes with alkali-antimonide cathodes at room temperature is primarily due to gas ions, not thermionicemission; (2) deviation from idealized (simple Poisson) pulse-height distributions is primarily due tosecondary electron loss, particularly in venetian blind multipliers; (3) pulse-counting provides bettersignal-to-noise ratio than any other simple detection scheme, and is not far from optimum detection inmost cases. However, dc methods can approach pulse-counting quite closely if digital readout is used.A convenient method for determining optimum discriminator levels is presented, with examples.

1. Introduction

Recently, several authors'-' have discussed thesignal-to-noise ratios of photomultipliers usedin differentmodes of operation. None of these discussions hasconsidered all modes of operation, and most have beenempirical rather than theoretical. The purpose of thispaper is to give a general treatment of the S/N problem,valid for all photomultipliers.

The principal methods of detection that have beenused are dc current or voltage measurement, chargeintegration, synchronous detection, pulse counting,and the shot-noise method of Pao et al.

4 '5 We noteat the outset that dc and charge integration methods areequivalent. Chopping, followed by synchronous (i.e.,phase-sensitive or "lock-in") detection is similar to dcdetection, except that (a) chopping throws away halfof the incoming signal power, and (b) the final bandpassis shifted from zero frequency (de) to the choppingfrequency. Thus, in the absence of frequency-de-pendent noise, the S/N ratio for synchronous detectionshould be just 1/2 ' that of dc detection, for the sameobservation time.

A brief digression into 1/f noise seems called for atthis point. Pao and Griffiths5 claim to have found a 1/f-like component in both illuminated and dark anodecurrent at frequencies below a few kilohertz. However,the extensive measurements of Mikesell6 showed no 1/f

The author is with the Jet Propulsion Laboratory, CaliforniaInstitute of Technology, Pasadena, California 91103.

Received 30 April 1969.

noise above 0.1 Hz for several different photomulti-pliers; my own unpublished measurements with RCA1P21's and EMI 6256's agree with this statement.On the other hand, Pao et al.

4 reported their tube wasan EMI 9558, and nonrandom dark noise has been foundin this type by Rodman and Smith,7 Gadsden,' andothers. However, such noise was absent from thephotocurrents measured by these investigators. Itseems likely that 1/f noise is not generally important inphotomultipliers, and that any 1/f noise observed byPao et al. originated in amplifiers or in poorly regulatedpower supplies.

If we can ignore 11f noise, we may regard dc, charge-integration, and synchronous detection techniques asequivalent (apart from the 50% chopping loss of signalin synchronous detection). We are therefore reducedto three basic techniques: d, pulse counting, and theshot-noise-power method which Pao and Griffiths'call the ac method. We shall call the latter "the shot-power method," as pulse counting and synchronousdetection also involve ac techniques.

We now consider the nature of the currents deliveredby a photomultiplier, and see how these are treated bythe various detection schemes.

II. What Comes Out of a Photomultiplier?

If enough collecting potential is used to saturate theanode current, a photomultiplier looks very much like apulsed current source. In dc detection, we measure theaverage value of this current; in pulse counting, allpulses over a threshold height or within some range ofheights are counted; in shot-power detection, the meansquare of the current is measured.

If the bandwidth of the circuitry following the photo-multiplier is small compared to that associated with thepulses (roughly, the reciprocal of the pulse width), and

December 1969 / Vol. 8, No. 12 / APPLIED OPTICS 2431

if the pulse rate is so slow that pulse overlaps can beneglected, we can regard each pulse as a delta functionwhose height is the integrated pulse charge. But if veryfast counting equipment is used (or if pulse overlaps aresignificant), the pulse shape influences the apparentdistribution of pulse heights. Any spread in pulse risetimes or widths may then act to smear out the apparentheight distribution. Since pulse shape depends oninterstage electron transit times, which are irrelevantto the pulse's origin, smearing due to variable pulseshape only degrades the information of interest.Hence, it is desirable to avoid using counting equipmentthat can resolve individual pulse shapes; if highercounting rates and short time resolution are requiredfor other reasons, a photomultiplier with narrowerpulses should be used. We shall assume this has beendone, and take "pulse height" to mean "pulse charge."

Unfortunately, the pulses are superimposed on abackground due to leakage. Since interelectrode poten-tials are typically 102-103 V, and dc currents as smallas 10-'2 A may be important, leakage resistances ashigh as 101" Q may be significant. Leakage is generallydue to a film of moisture, often caused by finger marks,on the tube base or socket. Some tube types are in-herently more leaky than others: the RCA P21/931-A is very bad in this respect,9 as the base and socketare both phenolic (a very leaky material at highhumidity) and the anode and cathode pins are juxta-posed. ITT tubes tend to have internal leakageproblems. On the other hand, the glass and Teflonof EMNJI tubes and sockets, and the socketless construc-tion of EIR (Ascop) tubes, are very resistant toleakage.

In spite of variations, the leakage current of anygood tube can be made negligible, even for dc work, ifthe tube and socket are kept quite clean and dry.Silica gel can be used as a drying agent, but should bekept where the tube cannot "see" it, as it has beenknown to emit light; it may be thermo- or tribolumi-nescent, or contain radioactive contaminants.

Leakage noise, if present, may have a 1/f character.A similar noise source can be created by potentialgradients in the tube envelope; Krall" has found that"potentials as low as 50 V when applied to a dummyfaceplate 30 mils thick were visible to a referencephotomultiplier." This problem is usually avoided bysurrounding the cathode end of the tube with an electro-static shield at cathode potential. It is importantnot to run the shield positive with respect to thecathode, or electrolysis of the glass and subsequentcathode poisoning can occur.' It is also importantnot to extend the shield too near the base, to avoid highfields ad the resulting electroluminescence and/orcorona.

If the tube is properly mounted, the fluctuations inthe anode current consist entirely of a random series ofpulses of different heights. Then the answer to thequestion "What comes out of a photomultiplier?" is"pulses." Our main point is that each detectionscheme weights pulses of a given height in a particular

way; the problem is to achieve the best weightingscheme.

111. Pulse-Height Spectra

Before discussing the weighting functions and theireffects, let us see where the pulses come from. Someare due to photons entering the tube window (signalpulses), and some are produced spontaneously when theexternal light source is cut off (dark pulses). It isalways assumed that the output of the tube, whenilluminated, is a superposition of signal pulses and darkpulses-i.e., that the presence of the signal pulses doesnot affect the dark pulses. We shall examine thisassumption later on; for now, we accept it.

A. Signal Pulses

The signal pulses primarily are due to photoelectronsproduced at the cathode and multiplied by the succes-sive dynodes. However, there is not a one-to-onecorrespondence between cathode photoelectrons andsignal pulses. On the one hand, some photoelectronsare lost, either by striking surfaces other than the activedynode area, or by the nonzero probability that nosecondary electrons will be produced at a dynode.Often, as many as 10%-20% of the cathode photoelec-trons are lost.8 12 Therefore, the effective quantum effi-ciency of a photomultiplier must be smaller than thenominal cathode quantum efficiency by this factor. Onthe other hand, some light reaches the dynodes, whichare also photoemissive. The fraction of signal pulses dueto dynode photoemission is quite variable with wave-length, owing to the optical properties of the photo-cathode; it may typically be about 10%. This repre-sents gain in effective quantum efficiency over thenominal cathode efficiency, since this latter is usuallymeasured with all dynodes tied to the anode. An un-desirable side effect of dynode photoemission is theintroduction of a polarization sensitivity (typically afew percent) for end-on tubes.

The pulse height distribution expected from cathodephotoelectrons has been discussed by many authors.The usual assumption is that the secondary emissionprocess obeys Poisson statistics, with a fixed distributionfunction for each dynode."3 In this case, the pulseheight spectrum resembles a Poisson distribution cor-responding to the average multiplication at the firstdynode, but with longer tails. Such distributions havebeen observed in the RCA 1P21,' 4 the ITT tubes,"and various electron multipliers.'3 However, the EMIvenetian blind tubes" and multipliers give a morenearly exponential spectrum, with no maximum (or atbest a very flat one). Prescott 2 has shown that ageneralized distribution function describes all thetheoretical and observed distributions very well, ap-proaching a Poisson distribution in one limit and anexponential in another.

In fact, the data of Sharpe4 show reduced efficienciesof venetian blind (VB) multipliers compared to box-and-grid (BG) units at the same interstage voltages.For example, an eleven stage VB multiplier givestypically five to ten times less gain than an eleven stage

2432 APPLIED OPTICS / Vol. 8, No. 12 / December 1969

BG multiplier, which corresponds to 0.86-0.81 as muchgain per stage. Thus, the overall gain data are con-sistent with the time distribution8 and height distribu-tion'2 of the pulses in suggesting about a 15%--20% lossof electrons in VB multiplier structures. Apparently,this loss is the cause of the quasiexponential heightdistributions of VB multipliers.

The evidence indicates that secondary emission is aPoisson process, but that dynode multiplication (owinglargely to imperfect collection of secondaries, especiallyin venetian blind systems) is not. We shall assumethat the distribution of signal pulse heights lies some-where between a Poisson and an exponential distribu-tion.

B. Dark Pulses

The dark pulse distribution is complex, as severaldistinct phenomena contribute to it. The textbookassumption is that it is due primarily to thermionicemission from the cathode, based on the quasiexponen-tial dependence of anode dark current on temperature.We would expect this component of dark current tohave a pulse height distribution similar to that of thesignal pulses.* In fact, such a component is usuallyprominent in the dark pulses of tubes with S-1 cathodes,but not in those with alkali-antimonide cathodes,except for the ITT type. In many tubes, the darkpulse spectrum is still concave upward at the peak ofthe signal pulse distribution. In such cases, cathodethermionic emission is negligible, and the bulk of thedark pulses have other origins. We enumerate these,starting with the smallest pulses.

1. System Noise

At the very bottom, we must be sure that all pulsesmeasured really originate in the photomultiplier andare not due to amplifier noise, or to Johnson noise inthe load resistor. This subject is discussed quantita-tively by Tusting et a.,11 who show that this noisesource can be made negligible. A good practical testin making pulse measurements is to shut off the high-voltage supply that feeds the dynodes, and see whetherany pulses are detectable with the rest of the apparatusoperating normally. The photomultiplier anode shouldnot be disconnected from the amplifier, as the systemnoise depends on the impedance connected to the ampli-fier input.

* Thermionic electrons leave the cathode with energies on theorder of kT 0.03 eV; photoelectrons have energies up to a fewvolts, depending on the photon energy. Since the collectionefficiency and multiplication at the first dynode depend on initialelectron energy, the pulse-height spectra differ slightly fordifferent colors of light 0 and for thermionic electrons. This alsocauses an apparent dependence of spectral response on detectiontechniques, and on discriminator settings in pulse counting.ITT has used a weakly back-biased grid to suppress thermionicelectrons from an S-1 cathode in experimental tubes.

2. Ion PulsesBoth the light and dark spectra of any photomulti-

plier show a rapid rise at very small pulse heights. Bysetting the cathode a few volts positive with respect tothe first dynode, one can readily show that most ofthese dark pulses come from the dynodes. 1 Again, theusual explanation is thermionic emission; but then whyshould this component also appear in the signal pulsedistribution and in tubes with high work-functiondynodes? Furthermore, if this were a spontaneousprocess such as thermionic or field emission, we wouldexpect similar numbers of pulses to originate at eachdynode. This would produce a distribution of pulseheights nearly proportional to 1/h; the observed drop-off is much steeper than this, but not exponential.

Let us instead suppose these dynode pulses are pro-duced by a mechanism proportional to the current ateach dynode. The current incident on the kcth dynodeis roughly

(1)

where io is the cathode current and g is the stage gain(typically about 4). If the number of dynode pulses is

nk = Cik = aCio(k-l), (2)

and these pulses are confined to an interval of pulseheight

Ah, ~ h = og(-m-k), (3)

where m is the total number of stages in the tube, wehave the distribution function

nd(k) nk/Ahk (a1#)iog(1k-m-1) (4)

assuming that the total current due to the "induced"electrons is small compared to the amplified cathodecurrent at each stage. Rewriting this in terms of pulseheight (h) instead of stage (k) we find

nd(h) = (/)3)iog-/h2. (5)

We note that the induced pulses are proportional to thecathode current io; therefore, the appearance of theinduced pulses as a fixed fraction of the signal pulse-height distribution is perfectly natural. We also notethat this part of the spectrum should have slope - 2on a plot of log [n(h) ] against log h. This is confirmed,in Fig. 1, for a 1P21.

The proportionality to h-2 in Fig. 1 extends to pulsesroughly four times the height of single photoelectronpulses. This indicates that the largest pulses in thiscomponent of the dark current are due to ion bombard-ment of the cathode. We therefore suggest that thiscomponent of dark current is due to the ionization ofgas by energetic electrons. The largest are producedby positive ions generated near DI, each of whichstrikes the cathode and liberates about four secondaryelectrons.'9 We can, in fact, distinguish peaks in thedark pulse distribution at intervals of about a factor offour in pulse height. The factor of four appears be-cause, at the interelectrode potential of 95 V used here,that is the average number of secondary electrons pro-


ik = ig(k-l),

CHANNEL NUMBER

3 10 100

CHANNEL NUMBER

1000

0.1 0.2 0.5 1.0 2.0 5.0

NORMALIZED PULSE HEIGHT

Zz'01

0.1 0.2 0.5 1.0 2.0 5.0


Fig. 1. Pulse-height distributions measured in an RCA P21(nine "squirrel-cage" dynodes). The filled and open circlesrepresent data measured at different analyzer gain settings. Asolid curve indicates the course of the data where individual pointsare too close together to plot on this scale. In some cases, themeans of several adjacent channels are plotted instead of indi-vidual channel data, to reduce scatter. All mean points areindependent rather than running means, so the scatter in the figuretruly represents the uncertainty of the data. The light line in(a) has slope -2. In the lower scale, heights are normalized to

mean signal pulse height.

duced by an electron or gas ion accelerated betweenstages.

Morton et al."5 have identified both the afterpulses ofthe signal spectrum and type B dark pulses with positivehydrogen ions striking the cathode, as was suggestedearlier by Barton et al.6 Morton et al."0 attribute thehydrogen to the chemical reduction of residual watervapor by the cesium introduced in activation of thecathode and dynodes. A relatively large amount ofwater may be expected in the 1P21 and similar tubes inwhich the antimony (required to form the cesiumantimonide photoemitter and secondary emitter) isplated onto the electrodes before assembly. In suchtubes, exposure to the air allows water to be absorbedby electrode surfaces. We would not expect to findas much water, and hence as many ions, in the ELVIIvenetian blind tubes, where the antimony is depositedduring the bake out and final pumping. Indeed, wesee in Fig. 2 that such tubes show little evidence ofions; the small dark pulses have more nearly the h-'dependence expected for a constant (thermionic orfield) emission from each dynode, and there is noprominent peak in the dark pulse spectrum at pulseheights about four times a typical cathode photoelectronheight. Pagano et al.2 0 have also found that the EMI9502 "shows no signs whatsoever of afterpulsing."Both tubes exhibit large ion pulses in the signal pulsedistribution by a change of slope at normalized pulse

heights of 4 or 5. However, afterpulses are excluded bythe dead time of the TIC 401D pulse height analyzerused to compile these data.

Ion pulses are appreciably reduced by cooling thetube, as is shown nicely by Fig. S of Barton et al.'6

Evidently more of the residual gas is adsorbed on solidsurfaces as the temperature is reduced. The largepulses produced by ions striking the cathode are reducedby reducing the K-D1 potential difference, as is shownby Barton et al.'6 (their Fig. 7) and Pagano et al.20

Bhaumik et al.2 ' have circumvented ion pulses in a1P21 by pulsing the tube voltages for a time (20nsec) shorter than the ion transit time (-few usec),but this technique is not generally applicable.

3. Light

Although the dark spectra of Figs. 1 and 2 showappreciable single-electron components, these are notwholly due to thermionic emission. Part of the single-electron component is due to photoemission from lightgenerated by cosmic rays2 2 -2 4 and radioactivity in thetube 24-2 and in the windows of its housing. Theseare discussed below. If there are sharp edges in theelectrostatic shield, corona may be a problem.

Electroluminescence of the glass and dynode glow'0

have already been mentioned. These mechanisms areat most only weakly temperature dependent, and areusually made worse, not better, by cooling, owing tothe quenching of fluorescence at higher temperatures.The non thermal dark current observed in S-20 tubes76

probably is due to such fluorescence, as one wouldexpect luminescence from cesium antimonide dynodes tobe strong near the absorption edge (8000 A), whereS-20 cathodes have appreciable response but cesiumantimonide (S-4, -11, etc.) cathodes do not. Thetemporary increase in dark current usually observedafter exposure to light may be due in part to phos-phorescence; it decays more slowly at lower tempera-ture:

,0

10

100

10-

S .

CHANNEL NUMBER

10 100 1000I -I

DARKEMI 950aNo. 71601 I400 V

SYSTEM NOISE

NRA~T NOPLEHIH

-- I

.1 1 0 1 .


100

10'

1

Z ,

5 10

o-,

0-2

0-3

CHANNEL NUMBER

10 lOT 1000

LIGHT S IGNAL .

CHIANNIEL NUMBER

I O 1 00 1 000

l - I I

GAMMA- RAY SIGNAL

No. 7160_ \\ 1400

v 400 min

v

_I

EMI 95028 S I

N. 7160 " I- 1400 V I

I B.C I,, °V I . . I . . I-3~~~~~~~U ~~~V

S) I

0.1 1.0 10.0 0.' 1.5 10.0

NORMALIZED PULSE HEGV,

Fig. 2. Pulse-height distributions measured in an EMI 9502 B(thirteen venetian blind dynodes), plotted as in Fig. 1. Theamplifier noise shown in (a) was measured with the tube voltagereduced to 500 V, and is approximately three decades lower than

the dark pulse rate at 1400 V.


L0

101z

z

zNU

.1 I IDARKIP21950 V

I)I I I I

-110

102

10-2

ttZI

ilZ-1

!5

,0-2

Gamma rays are particularly troublesome because of1 10 100 1000 I

McDONALD- EMI 6256B

FACE UP

-3'C* 700 V

_ \0o 7400V

10 100 1000 10 100I I

DIFFERENCE,(McDONALD -(MINE)

EMI 6256B,b FACE UP3'C

x * 1700V0 e1400V

_ .o

0 0

X '.J

O%

b)I 0n

Fig. 3. Dark pulse distributions measured in a face up EMI6256 B (13 venetian blind dynodes in (a) McDonald Observatory(2050 m above sea level) and (b) the IMC potash mine nearCarlsbad, New Mexico (240 m below ground). The difference(c) is due to cosmic rays. Mean points are used to reduce scat-ter, as in Figs. 1 and 2. Open circles: data taken wih 1400 Voverall; closed circles: data taken at 1700 V overall. The1700 V data have been translated up and to the left by a factor

of about six to allow for the consequent gain change.

Any phosphorescence induced in the dynode surfacesby electron bombardment will contribute to the h 2

component of dark pulses, identified with ions in theprevious subsection. Recombination radiation fromthe gas'0 will behave the same way. However, suchlight emission cannot produce the large type B darkand afterpulses; for luminescence of the first dynodeshould only produce single-electron emission from thecathode.

4. Radioactivity and Gamma RaysThe glass envelopes of most photomultipliers contain

enough potassium 40 to produce one or more pulses persecond.26 The sensitivity to radioactivity is greatest inthe window, where any light produced is stronglycoupled to the cathode. Other materials in the tube,or its housing, may also be radioactive. Tubes withpure silica windows are less radioactive, but are moresensitive to cosmic rays2 2 and external radioactivity. 2"Tubes with more red response are also more sensitive tosuch noise.2 5

However, the gamma ray response is not due entirelyto light. Figure 2 shows the response of an EMI 9502to bremstrahlung from a Sr-90 source placed outside thephotomultiplier chamber. The pulse distribution con-tains more very large and very small pulses than thatdue to light, and relatively more very large pulses thanthe dark pulse distribution. Evidently, both multiplephoton and ion or phosphorescent pulses are involved.Direct production of electrons at the dynodes is alsopossible.

1000 their great penetrating power.lead shielding were required tobelow the normal dark current.

In this case, 2-3 cm ofreduce the added noise

5. Cosmic Rays

These are even more penetrating, and shielding is notusually practical. The noise consists of two distinctparts: (a) large pulses, typically 10-100 times largerthan single-electron pulses, due to Cerenkov radia-tion,2 2 24 and (b) many times more small pulses.2 4

The effects of cosmic rays (CR) can be demonstratedmost clearly by shielding the tube from cosmic rays.This has been done by measuring the dark pulse spec-trum 800 ft below ground. The experiment was firstrun on an EMI 6256 at McDonald Observatory, andthen repeated in a potash mine run by the InternationalMinerals and Chemical Corporation near Carlsbad,New Mexico. The equipment was run on regulatedpower, and the photomultiplier was stabilized at -3 0 Cduring both runs. Figure 3 shows the dark pulsedistributions above and below ground, and the differencespectra. A clear minimum is evident between theCerenkov and the smaller pulses, as is to be expected.2 2

The data of Fig. 3 were taken with the tube facing up-ward; Fig. 4 shows similar data for the tube axis hori-zontal. The Cerenkov pulses are less steeply peaked inthis case, as is to be expected from the angular distribu-tion of CR mesons at ground level.22

Figure 3 contains data taken with overall voltages of1400 V and 1700 V. From measurements of the gain-voltage relation for the 6256, we expect a factor of sixgain change between these two conditions, and the datafit together quite well when this factor is used to com-bine the two sets. We note that, at the higher voltage,the h-2 component of the CR (difference) spectrum is a

01

100

D I -

:10

Z 1o2

10

lo-3

a ) I II b) I

10 100 1000 10 100 B

CHANNEL NUMBER

DIFFERENCE,(McDONALD) -(M I NE)

0

* a aaI

0

S1

0

1 l ± 1 I000 10 100 1000

Fig. 4. Same as Fig. 3, but tube axis horizontal.were all taken at 1400 V overall.

These data

I IMINE

EMI 6256BFACE UP-3'C* 1700 V

- 0 O 1400V


e8

zZ

0

100

10-3

10-4

l- l_ --

- D D z z % a a a a |

CHANNEL NUMBER 1400 V CHANNEL NUMBER 1400 V CHANNEL NUMBER 1400 V)

so

I I

r

0

102

I10

°1 1 0°

10-2

CHANNEL NUMBER

0 10 100

.

1000

DECREASE IN DARKCOUNT IN 13.5 hrAT McDONALD

62561400 V -FACE UP_3'C

-3 C DARKPULSESIN MINE

Fig. 5. Time-dependent compoloelot of dark pulses. The points

represent the difference between two us separated by half a day.The line is the intrinsic (cosmic ray free) dark distribution, mea-

sured in the mine (see Figs. :3 ad 4). Cosmic rays were presentduring both runs, but the difTerence (the time-dependent com-

ponent) shows Io cosmic ray component.

little stronger and extends to slightly larger pulseheights. This is just what we should expect if thiscomponent is due to ions; we therefore identify the non-Cerenkov component of the cosmic ray pulses as ionpulses, produced just as are the normal ion pulses in thedark and light signal distributions. We also remarkthat the good fit of the data at different voltages sug-gests that field emission is negligible.

6. Time-dependent Dark Noise

The gradual decline of dark current to an equilibriumvalue several hours or days after first applying voltageto a photomultiplier is well known. Also, in manycases the dark level is raised temporarily after exposureto light. The possible role of dynode luminescence wasmentioned above.

Figure 5 shows the difference between two dark pulsedistributions of the same 6256 (see also Figs. 3 and 4),measured for 400 min: (a) 12 h after initial turnon ofthe equipment, and (b) 1:3.5 h later yet. This differ-ence shows exactly the same form as the dark pulsedistribution observed in the mine, where cosmic raypulses are absent. We conclude that the variable partof the dark current is caused by the same mechanismthat produces most of the intrinsic (i.e., non-CR) darkcurrent. Since, in these experiments, channel 100 cor-responds to the smallest Cerenkov pulses or about 30photoelectrons per pulse, we are seeing the ion pulsesoriginating at the cathode in Fig. 5.

Very similar data have been published by Gadsden'in his Fig. 9, which shows initial and final dark pulsedistributions for a 9558:, below the temperature-in-

variant tail of large pulses, which must be mostly CRpulses, the initial and final curves (3.5 h later) areparallel. Gadsden's data show a drop by a factor ofabout five; Sharpe2" found a similar fall in dark currentwithin 16 h after the first 10 min of operation. Kral]IOshowed a very similar behavior-a uniform depressionof the dark pulse distribution at all pulse heights-in aphotomultiplier exposed to uv radiation, and demon-strated that the effect is not primarily due to faceplatephosphorescence. As Krall's data show the time-vary-ing component extends to normalized heights of at least10, ions rather than dynode luminescence appear tobe the cause.

Figure 6 shows the difference in dark pulse distribu-tions measured at +C and -3 0 C at McDonaldObservatory. This difference spectrum is rather un-certain owing to an appreciable adjustment that mustbe made to compensate for the temperature-dependentchange in dynode gain. However, it is clear that thetemperature-dependent component, like the time-de-pendent component shown in Fig. 5, has the same gen-eral form as the ion-produced dark pulses observed inthe mine [Figs. 3(b) and 4(b) ].

The smallest pulses in Fig. 6 have normalized heightsof about 3. However, Barton et al.'6 show, in their Fig.4, that the entire dark distribution, from normalizedheights of about 1-10, decreases rather uniformly oncooling. A similar effect was produced by cooling onlythe stem of the tube and not the cathode. This'sup-ports the view that most of the thermal dark pulses aredue to ions, and that thermionic emission is negligible.

We should bear in mind that several ionic species maybe present. Thus, although hydrogen ions appear to

CHANNEL NUMBER (AT -3°C)

10 100 1000105 I

9 DARK PULSE DIFFERENCE,(+8C) - (-30C)AT McDONALD

Z 3103

-3°C DARKO \ PULSES

0 IN MINE

IO'

101

Fig. 6. Temnperature-dependent component of dark currentin the same tube; (see also Figs. 3-5). The arrows indicatemean points too low to plot here. The scatter is large, but thedata indicate that this component also has the shape of the in-trinsic dark noise measured in the mine (indicated by the line).


PULSE EIGHT

Fig. 7. Weighting functions for (a) pulse counting; (b) dcdetection, and (c) shot-power detection. The dashed line in (c)

indicates the effect of clipping at height ho.

tion with height. Thus, it is convenient to consider thetwo extreme cases of strong signal (where dark pulsesare negligible) and weak signal (where the signalpulses are almost negligible), for each weighting func-tion. Both of these limits are unrealistic, since in prac-tice the dark pulses always dominate the small pulseend of the distribution, and signal pulses must make anappreciable contribution or the time required to makean observation is unreasonably long. Nevertheless,both cases are helpful in understanding practical situa-tions.

In general, we observe a pulse distribution

p(h) = s(h) + d(h), (6)

be the cause of ordinary afterpulses,I 9 we may imaginethat cesium (or other alkali-metal) ions are generallypresent, and may be responsible for most of the normaldark pulses. Morton et al.'9 point out that large darkpulses occur in tubes which exhibit no afterpulses;however, these may only be Cerenkov pulses fromcosmic rays.

To sum up the evidence, it appears that most of thedark current comes from a single, time-dependent cause.As this includes much larger pulses than can be producedby single electrons leaving the cathode, positive ionsare the most likely explanation. The time dependenceshould then be regarded as a progressive getteringaction, like an ion pump, when voltage is applied. Ifmost of the dark current is due to ions, the strongtemperature dependence is due to adsorption of the gasat low temperatures and release at high temperatures;the work function of about a volt commonly deducedfrom this temperature dependence may represent thebinding energy of the ions to surfaces within the tube.It then seems reasonable that such ions can be removedby visible photons, with energies of a few volts.

Spicer and Wooten27 have already suggested that"even at room temperature, the relative importance ofthermionic emission is questionable except for theAgOCs photocathode (S-i response)." Even an ap-parent one electron component in the dark pulse dis-tribution may be photoemission from light generatedwithin the tube rather than thermionic electrons.

IV. Detection Schemes and Weighting Systems

A. General Relations

We now investigate the effects of the weightingfunctions corresponding to different detection schemes.In pulse counting, all pulses between two heights (orall above a threshold level) are counted equally (weight1) and all others are ignored (weight 0). In dc orcharge integration, each pulse is weighted by its height(charge). In the shot-noise-power method, the meansquare of the instantaneous current is measured, so eachpulse is weighted by the square of its height. Thesethree weighting functions are shown schematically inFig. 7.

Each weighting function gives different results forsignal and dark pulses, because of the different distribu-

where s and d are, respectively, the signal and darkpulse-height distributions, and s(h) is proportional tothe amount of light falling on the tube per unit time.If our weighting function is w(h), the quantity measuredwith the light on for a time t

L is

P X tL = tL X ,w(h) X p(h)dh = (S + D)XtL (7)

where

S f w(h)s(h)dh, (8)

and

D = f w(h)d(h)dh (9)

is the quantity measured with the light off. (We as-sume that the parent distributions are independent oftime, so that P, S, and D refer to unit times.)

Now consider the statistical fluctuation of a measure-ment of duration t. If there are on the average [p(h) Xdh X t] pulses with heights between (h) and (h + dh),the variance in this quantity is uyp X t = p(h) Xdh X t]. If there is no correlation between pulses ofdifferent heights, the variance in the measured quantityis

topl = t la() X o,2 = t J tw(h)} X p(h)dh,

(10)

where the subscript 1 refers to unit time.If the total observation time is t, of which the light

is on for a fraction f and off for a fraction (1-), themeasurement with the light on is [P X ft i (ftap2)'];with the light off, it is {D X (1 - f)t i [( - f) Xt X OD 2]1}. Hence, the variances of P and D arerespectively

RTp,t2 = Tpo'/(ft) (11)

and

7DB2 Y= D,0/Vl - f) X t. (12)

From Eq. (7), we see that the estimated light flux onthe tube is proportional to

S = P -D, (13)


so that

UJSt2 = tPEI' + TD, t' = (ap,12/ft) + [o-D. 2 /(1 - f)tl, (14)

where

rp~o2

= f [w(h)] 2 p(h)dh (15a)

and

TD, 12 = f [w(h)] 2 d(h)dh. (15b)

The value of f which minimizes 0Sg2 is found by

setting the derivative of Eq. (14) with respect to fequal to zero; this gives

(1 - f) 2crp,12 = f2oD.12 (16)

or

[f/(l - f)l = 7P.,/EFD.I-

Hence,

Opal 1Uf= 1S7Pl I+ -D. I I + (0-D.I/ap.I)

(17)

(18)

we must have > 0 to prevent divergence. Physically,the finite number of dynodes would in principle providesuch a cutoff, but this is usually far below amplifiernoise and load resistor noise. As a practical matter, afinite lower threshold is always required in pulse count-ing; the strong signal limit is physically unattainablefor this weighting function.

What lower threshold should be used to achievemaximum signal-to-noise ratio? The signal-to-noiseratio p is

Pt = S/s.t = n(lu) X [aSt 2 + CD, t21

= t X n,(u) X {(1/f)n(lXu) + {(1/f) + [1/(1 - )}X nd(lu) -',(22)

where n, and nd are defined analogously to n, [see alsoEq. (21) ]. Here we have decomposed the sum

np(lu) = n,(1,u) + nd(lu) (23)

and collected the nd terms. We already know fromEq. (18) that

(n + nd) + (nd)(24)

In the weak signal limit, p(h) d( h) so (D,I/apl) 1

and f a; in the strong signal limit, p(h) > d(h) so(aD,1/ apj) 0 and f 1. These results for thelimiting cases are well known. However, because ofthe dependence of 0

D, and p j on w(h), the optimumvalue of f for intermediate cases will depend on theweighting function used.

If the measurement is made against a backgroundlight level, as in astronomical photometry of faint starsagainst the night sky, the background distribution,

b(h) = sB(h) + d(h), (19)

must be substituted for d(h) in Eqs. (6)-(18). HereSB(h) is proportional to the background light. IfsB(h) is larger than the signal s(h), we again have thelow signal value f = , even if s(h) >> d(h), as mayhappen with a cooled photomultiplier. However, inthis case the distributions b(h) and p(h) are similarapart from a scale factor, so that (CB.1/a-p l) is nearlyindependent of w(h); this similarity introduces somefeatures of the strong signal case.

We now examine the effects of different weightingfunctions in detail.

B. Pulse Counting

As has been pointed out above, the small pulse endof d(h) generally is at least as steep as 1/h. Therefore,since the pulse-counting weighting function

(0, h < w,,(h&) = 1, < < (20)

o, h > )

converts the integrals of Eqs. (7)-(10) to the form

np(l,u) = J p(h)dh, (21)

We could substitute Eq. (24) into Eq. (22) and even-tually solve for the optimum discriminator settings and u in the general case. However, in practice it isusually desirable to set f = or f 1; the small lossin p is compensated by the increase of operationalefficiency due to convenience. In general, we selectI and u to maximize p for the weakest signal we wishto measure. We must keep 1 and u fixed for all signallevels, or [due to the dependence of s(h) on wave-length] the effective spectral response will vary.Optimizing for weak signals means we have less thanoptimum signal-to-noise for strong ones, but this is not aproblem since we still have much better signal-to-noiseon strong signals than on weak ones.

Having selected a value for f, which should be forgenuinely weak signals and should be closer to 1 forstronger ones, we must set

ap(U,t)/au = p(ul)/l = 0 (25)

to find optimum discriminator settings u and 1. Thisgives

d(l)/s(l) = d(u)/s(u) = ( - f) + 2(nd/n.)- (26)

For the weak signal limit, both [d(h)/s(h) I and (nd/

n0 ) - a, so the f term is negligible. This gives

d(l)/s(l) = d(u)/s(u) = 2(nd/n,), (27)

independent of f. (However, we already know weshould pick f = in this case.) This condition is noteasy to determine, since nd and n are both functionsof u and 1.

The choice is simplified if we use the fact that bothd(h) and s(h) are, in general, rapidly decreasing func-tions of h. Therefore, we are not far wrong to set

nd(l,u) nd(l, ) = f d(h)dh (28a)


and

nf(l,u) n(l, c) =f s(h)dh.

The functions on the right are simply the usualcurves" or "integral pulse-height distributions."define

R(h) = 2s(h)/d(h)

and

Q(l,u) = n,(l)/nd(lU),

Equation (27) corresponds to

R(1) = R(u) = Q(lu).

not negligible, we should spend more time measuring it;i.e., increase (1 - f). The stronger the signal, the

(28b) smaller we can make Eq. (37); in the limit of no darkpulses, we would have 1 -l 0 and u -i- , counting all

"bias the pulses. This would clearly give the maximumIf we signal-to-noise ratio possible: we cannot do better than

to count every photoelectron equally.Even if our weakest signal is not strong, we may be

(29) able to use Eq. (37) to determine approximate discrimi-nator settings 11 and d, and again use an iterativetechnique to find the best values:

(30)

(31)

Notice that R(h) is twice the ratio of the signal and darkdistributions, and Q(l,u) is the ratio of signal countsto dark counts (sometimes erroneously called thesignal-to-noise ratio). Equation (28) means that Q(l,u)

Q(lcc), so

R(l) = Q(li, cc) (32)

defines a good first approximation to 1. Then

R(ui) = R(l) (33)

defines an approximate upper cutoff, so we can deter-mine a second approximation 12 from

R(12) = Q(12,ul) (34)

and so on. The process converges very rapidly; anexample is given in the Appendix.

Notice that the result is independent of the signalstrength used to compute R and Q as both sides ofEqs. (29)-(34) are proportional to S.

The first approximation (11) corresponds to Morton'scondition 2 that the slope of the signal pulse integraldistribution be half that of the dark pulse distribution,when the curves are plotted on semilog paper. For,

d[lognd(h,co)l/dh = -d(h)/nd(h,cc)

and similarly for [n0 (h, c ) ], so that Morton's condithat

d[logn,(h, cc )] /dh = 12d[lognd(h, )] /dh

d(lk) = d(Uk) = fl f) + 2 nd(lk-0,Uk-0)8(1k) S(Uk) n.(1k-1,Uk-0

(38)

All of the above assumes pulses are independent.However, both the signal and dark distributions containafterpulses and other induced pulses such as photo-emission from dynode glow, etc. These induced eventsare certainly not statistically independent of the primaryevents. Any correlation between pulses of heights hiand h2 should appear as a cross product term in theintegrand of Eq. (10). Another way of treating thisproblem is to regard an induced pulse as increasingthe weight of the parent pulse.

Let us consider the large afterpulses, 9 which maytypically be about four times the height of a primaryphotoelectron pulse and occur with probability -0.05about 0.3 usec after the primary pulse. If the countingequipment used is relatively slow, the two pulses willnot be resolved and will be treated as one large pulse.Fast counters, on the other hand, will count both pulses.In this case we can regard 0.95 of the n, independentpulses as having weight 1, and 0.05 ni as having weight2, since these are counted twice. The observed countis then

nobs = 0.95ni + 2 X 0.05ni

= 1.05ni, (39)

(35) and the variance [see Eq. (15) ] is

tion E-obo2

= 0.95ni + 22 X 0.05ni

= 1.15ni i 1.l1Ob,- (40)

Thus the observed variance is about 10% larger thanwould be expected from the total count. Such a smalldeviation from ideal counting statistics would be hardto measure.

The large afterpulses could be rejected by the upperlevel discriminator in a fast system. In a slow system,on the other hand, they would appear (unresolved fromtheir parent pulses) as legitimate signal pulses andshould be counted. This comparison shows that thepulse height distributions should be measured with thesame equipment that is used for light detection, sincedifferent systems will measure different pulse heightdistributions from the same tube.

The situation is much worse in the case of cosmic rayafterpulses, since typically about 10 afterpulses may beproduced per cosmic ray.24 In this case nObs 10 nfand obo2 Z 10 2ni = 10nb 0 , if the cosmic rays dominatethe dark pulses. This agrees with Rodman and Smith's

(36)

gives Eq. (32) if h = 1.If a single level discriminator is used instead of a

window counter, u = and = 1.In the case of moderately large signals, we may

suppose that Q is very large, so that we can neglect the(nd/n,) term in Eq. (26). The optimum discriminatorsettings are then given byd(l) _ d(u) _ I-DI(7d(l) - d(U) (1-if), (37)

which is the fraction of the time we have reserved forcounting dark pulses. We know that d(h)/s(h) be-comes very large as h - 0 or h A.- c; therefore, theonly question regarding the applicability of Eq. (37) iswhether the signal is indeed large enough to make(nd/no) << (1 - f), which is already a small quantity.Another way of saying this is that if the dark count is


data on refrigerated S-20 tubes,7 in which u-2 was fiveto ten times higher than the total number of pulsescounted per sample time. Their typical dark countrates of -20 pulses/sec are a few dozen times theexpected cosmic ray rate for a 2-in. (5-cm) tube; theagreement is very good if about half of Rodman andSmith's dark counts were due to cosmic rays, and theother half to spontaneous events originating within thetube.

The above discussion emphasizes that the effectiveweighting functions may be more complicated thanthose of Fig. 7. The increased severity of these prob-lems with tubes having extended red or uv re-sponses7 24'25 points out the importance of not using atube with wider spectral response than is actuallyneeded.

C. Direct Current Methods

In dc or charge-integration photometry,

Wd (h) = h. (41)

Thus the measured quantity per unit time is

P = h X p(h)dh = pp (42)

with variance

a = f h2 X p(h)dh = E2,p- (43)

The signal-to-noise ratio is therefore

Pt = S/Rst = .I, X [st' + EDal2]= (t)1 p,, (l/f)12,, + {(1/f) + [1/(1 - If)l }W2,d I, (44)

where ( , = ) is the first moment of the signal pulsedistribution, and the 2's are the second moments de-fined as in Eq. (43). Note that ui,, is proportional toS for all i.

In the weak signal limit we can ignore M2,, in Eq. (44)and have

PwvRak - S(t/4p2,d)1 (45)

for f = 4. Here (2 .) is the dark current noise;the dark current itself is Al d.

Compared to pulse counting, d photometry dis-criminates against the small dark pulses originating inthe dynodes. Thus for spontaneous dynode emission,d(h) 1/h for hi < ho and the dark current

rho

I h/hdh

is finite. Even the induced dynode pulses 1/h2 givea finite dark current, since each dynode contributedequally to the anode current and there are only a finitenumber of dynodes.

The noise contribution from dynodes is even smaller.Since spontaneous emission from D is amplified times less than cathode emission, the noise powercontributed by Dl emission is 2 (typically 10-20)times less than that due to cathode emission. (Asimilar result has recently been derived by O'Haver and

Winefordner. 28) Even for induced emission, eachdynode contributes gq times less than its predecessor, andthe total rapidly converges. The dark noise is domi-nated by the very large ion and cosmic ray pulses thatoriginate at the cathode; in a cooled tube, the noisecan be almost entirely due to cosmic rays.2 2 Environ-mental radioactivity also produces large pulses andresembles cosmic ray noise in this respect. In com-puting the cosmic ray noise, the heights of all the CRafterpulses2 4 should be added to that of the initialCerenkov pulse; this roughly doubles the CR noiseestimate made in Paper V.22 Thus, although most ofthe dark current usually comes from the dynodes, mostof the dark current noise comes from the cathode. Thedc value of dark current is therefore not a good indicatorof the tube quality for low level work, even if dcleakage is negligible. This is particularly true for arefrigerated end-on tube, where cosmic rays are relativelymore important.2 2

We may note that the use of symmetrical choppingand synchronous rectification automatically sets f =4, since the dark current is observed during the halfcycles when the light is cut off. No separate darkreading is then necessary. However, the value f 2

is inefficient at higher light levels.In the strong signal limit, we can ignore the -2,d

term in Eq. (44) and have

(46)Pstro-g S(ft1A2,,)1f

We can safely set f 1 here, so

P [p1I-/(U2,) (t).(47)

An ideal quantum detector of efficiency q achieves asignal-to-noise ratio

Pid-l = Nqt/o-v0 = NAqt/(Nqt) = (NYqt)1 (48)

in observing a stream of N photons per second. If wenormalize s(h) so that

s(h) = NAqksO(h), (49)

where so(h) is the probability distribution of signalpulses and q, is an effective* cathode quantum efficiency,we see that Eq. (47) becomes

p (7At)' X (qtcAl, 0 2/,2o) (50)

Thus the detective quantum efficiency of dc photometryis just the cathode quantum efficiency times a degrada-tion factor,

Ada = .LEEOB/J.2,SE (51)

which is the ratio of (the square of the mean pulseheight) to (the mean square pulse height). Since thedetective quantum efficiency for pulse counting, in thestrong signal limit, approaches the cathode quantumefficiency qk, Eq. (51) also gives the ratio of dc to pulse-counting quantum efficiencies for strong signals.

* That is, allowin, for lost electrons.


(47)

I

Prescott' 2 has shown that o(h) probably belongs toa family of functions bounded by the Poisson distribu-tion on one hand and by the exponential distributionon the other. Hence, we can use these two limitingforms to place bounds on Ad,.

For the exponential distribution e' we have il = 1and .U2 = 2, so Ad = . This is the basis for the state-ment made in Papers VP' and VII2' that the detectivequantum efficiency of dc photometry is about half thatof pulse counting, for strong signals.

For the Poisson distribution, we have to pick themean value. We will surely have an overestimate forAd, if we set A, = gi, the gain of the first dynode;for we then neglect the additional broadening of so(h)by multiplication statistics at the following dynodes.Since the Poisson distribution has the property that

RI2

= 2 - J2

E,

so that

P = f h' X p(h)dh,

Orp2 = f h2 p(h)dh,

(57)

(58)

and

pt = (t)tp2 ,8((1/f)A41, + {(1/f) + 1/(1 - f)} Add) 1- (59)

This method discriminates still more strongly againstsmall pulses than d detection, and is more stronglyaffected by large (CR and after) pulses. Dynode noiseis certainly negligible here. The weak-signal signal-to-noise ratio becomes

Pweak - (/2)(t/p4,d)1(52)

we have

Ad. < A'/(lE' + p) = 9E/(YE + 1). (53)

Hence, even for strong signals, pulse counting is moreefficient than d photometry. Tubes such as the1P21/931-A or ITT tubes, which have a more nearlyPoisson signal pulse distribution, will therefore beslightly better for d photometry than the EIvenetian blind tubes. These results agree with Baum'sexperimental data," which showed a d.q.e. advantagefor pulse counting of about 1.3 for a 1P21 in the strongsignal limit, and factors approaching 2.0 for other types.The relatively small advantage of pulse counting at highlight levels agrees with Nakamura and Schwarz'sstatement' that "simple dc measurements are about asgood as pulse counting for light levels above the darkcurrent [equivalent] level."

Afterpulsing is more important for dc work than forpulse counting. If we again assume a 5% rate of after-pulses four times the average signal pulse height, wehave an anode current proportional to

nobs ; 0.95ni + (4 + 1) X 0.05ni = 1.2ni, (54)

where (4 + 1) is the total height of an afterpulsingevent, and n is the number of photoelectron pulses(= Nqt). Ignoring the spread in pulse heights withineach group, we have

Raobs2 0.95ni + (4 + 1)2 X 0.05ni = 2.2ni

1.83nobs- (55)

Thus, a typical rate of afterpulsing can roughlydouble the relative noise power, and halve the d.q.e.This may explain why P21's usually give best dcsignal/noise ratios at relatively low overall voltages(700-800 V): these relatively gassy tubes must berun at low potentials to keep ion production and afterpulsing to a minimum.9

D. Shot-power Detection

We have

wp(h) = h2, (56)

(60)

for f = . The occurrence of cosmic ray pulses > 10times normal pulse sizes should be disasterous here.

At this point, some errors in the noise analysis ofPao et a. 4"5 must be cleared up. They state that"ideally, the detection system should utilize the entireshot-noise power spectrum of the photomultiplier" andthat for sufficiently large bandwidths "there is con-siderably more power in the ac portion of the powerspectrum than in the dc component." This is a redherring; the noise has just as flat a spectrum as thesignal, so changing the bandwidth of the predetectionelectronics changes both signal and noise by the samefactor, leaving p fixed.

Pao and Griffiths' also make some contradictorystatements about 1/f noise. For example, they statethat as the photomultiplier load resistor is increased,"the signal-to-noise power ratio also decreases providingstrong support for the view that the principal reasonfor the observed superiority of [the shot-power method],for moderately strong light signals, is due to effective-ness in discriminating against 1/f noise." They sup-port this statement with their Fig. 6, which purportsto show "shot-noise spectra" in the interval 15-500kHz, for load resistors of 104 S2 and 105 S2. One canreadily calculate that a minimum cable capacitance of100 pF (about 1 m of typical cable) would reduce thesystem bandpass to 160 kHz for a load resistor of 104 ,

and 16 kHz for 105 U. As these figures agree with theirFig. 6, they have apparently measured only theirsystem response function and not 1/f noise in thephotomultiplier. This is further supported by a laterstatement that shot noise dominated 1/f noise above1 kHz. But this later statement vitiates the earlierclaim that the lower p with 105 £1 was due to 1/f noise,as their correlator 4 did not respond below 5 kHz.

They show that, assuming every pulse the sameheight, the signal-to-noise ratio for their method shouldbe the same as for de detection. This assumptionmeans that p(h) is a delta function, all of whose mo-ments yi are unity. This obviously gives their result;just as obviously, it is an unrealistic assumption. Inthe strong signal limit we obtain (forf = 1)

PStrong = [2../(Y4.s)1l (t)2 (61)


or

A8p = I2,RE'//4.so. (62)

For an exponential distribution 2 = 2, 4 12, soA = . For a Poisson distribution with mean g,12 = g2 + D and 4 = g4 + 6 + 7q2 + , so we get

Pp < ( 4 + 2g' + g')/(g4 + 6g' + 7g2 + g) (63)

which is 400/756 0.53 for g = 4.Thus for strong signals, the shot-power method has

about two-thirds the quantum efficiency of de detection,if afterpulses are absent. The effect of our assumedtypical amount of afterpulsing is to give (with the sameassumption as before)

Bob, 0.95ni + (4 + 1)2 X O.O5n = 2.2ni (64)

andJobs

2 0.95n1 + (4 + 1)4 X O.O5ni = 32.2ni

1 4 .6nObs, (65)

so that the noise is much worse in shot-power detectionif afterpulses occur. Since Pao et al. used an EI\I 9558,which should be relatively free of ion pulses, this isprobably not a problem in their measurements.

However, the large ion and cosmic ray pulses shouldbe a severe problem. Pao et al. have assumed thatdark pulses are primarily dynode pulses, and correctlyshow that such (small) pulses contribute much lessnoise in shot-power than in dc detection. But the sameargument shows that the occasional larger-than-averagedark pulses contribute much more noise than in dcdetection. Such pulses are known22 to occur in the9558; small dynode pulses can dominate 4 (and hencethe noise) only if the dark pulse distribution falls offfaster than h-', which is certainly not true for mosttubes. This raises the question: how could Pao et al.have achieved useful signal-to-noise ratio with theirtechnique?

Pao and Griffiths' report photometric linearity forload resistances of 10' S or less. They comment thatthe voltage across the load resistance "consists of verylarge fluctuations and may saturate amplifiers withoutbeing noticed. We find it good practice to amplifyonly slightly before multiplication." The internal gainof their 9558 is not known; however, under typicaloperating conditions gains of a few X 10' are achieved.With -100 pF capacitive loading, typical photoelectronpulses are a few millivolts high at the amplifier inputs.The Keithley 104 amplifiers used by Pao and Griffithsoverload at 1.4 V output. Therefore, if they used the20 db gain setting, ordinary photoelectron pulses wouldbe a few tenths of a volt high at the output, or within asmall factor of saturation. Typical cosmic ray pulseswould surely be clipped by the amplifiers; as thesquaring occurred after this clipping, the effect of largepulse noise would be greatly reduced. In effect, ampli-fier overloading modified the function w(h) to the formshown in Fig. 7(c).

Changing the load resistor in the Pao circuit onlyaffects pulse width (decay time), not pulse height involts (which depends on the shunt capacitance). If

this resistance is large, pulses begin to overlap, whichwidens the pulse height distribution since the overlappedpulses are higher than single pulses. This increases A4

faster than 1122, and hence degrades p, as observed.'The anode currents of a few nanoamperes reported byPao and Griffiths correspond to pulse rates of _104

pulses/see, again assuming a multiplier gain 106.Thus about 1% of the pulses are overlaps for a timeconstant of 10-E see (load resistance 104 2), but anappreciable fraction of overlaps must occur at 105 QThe reported' degradation of p with increasing resis-tance, and nonlinear response for still higher resistors,are thus explained.

At higher light levels, more pulses will overlap.Since a double (overlap) pulse appears four times ashigh when squared, the overlaps cause S to increasefaster than the light level. Thus, the noise powermethod is just as badly affected by unresolved pulsesas is pulse counting, but the deviations from linearityare in opposite directions for the two methods. Onlylinear weighting (dc detection) is unaffected by pulseoverlapping. Of course, there are many other causes ofnonlinear operation, such as voltage divider loading'" 32

and (in cooled tubes) cathode resistivity,3 3 3 4 whichcan be serious even at low light levels.

V. Better Weighting Functions

A. The Optimum Weighting Function

In the case of a noiseless photomultiplier [d(h) - 0],we maximize the signal/noise ratio p by counting allpulses equally. Also, in a noiseless tube with a back-ground light level [b(h) s(h) ], we readily see that allpulses should be counted equally.

In the case of the window counter, the conditionR(u) = R(l) shows that it is not the pulse height orpulse rate that is important, but the probability thata pulse is a signal pulse; if all pulses are equally likelyto be signal pulses, all should be counted equally.

This suggests that we should take

w(h) = s(h)/p(h). (66)

In the weak signal limit, s - 0 and p - d, so we mayadopt

wo(h) = s(h)/d(h), (67)

where so(h) is defined by Eq. (49).In fact, we can show that this function does maximize

the weak-signal signal-to-noise ratio p. The measuredsignal is

S = wo(h) s(h)dh

=f 8h)x Nq; X .s(h)dh0 d(h)

=Nqk J - d( dh

= n X Wo, (68)


where the number of photoevents is n, = Nqk and Wois the value of the integral. For the weak signal f -

2 and s(h) << d(h), Eq. (14) becomes

St2

= 4

0D 2/t

4 r=- +J wo2(h)d(h)dh

4 S f~h) dh = 4Wo/t. (69)

Because of the choice of wo(h), the factor Wo appearsin Eqs. (68) and (69). The square of the signal-to-noiseratio is then

p2

= S2/0RSt2 = (n,'WO2/4Wo)t = n8

2Wot/4. (70)

Comparison with Eq. (48) shows that the detectivequantum efficiency is

q = Nqk2Wo; (71)

however, a more significant measure of performance isthe noise equivalent input, for which p = 1, if t = 1.This is a photon flux

No = 2/qk(WO)1. (72)

Now let us suppose that a better weighting functionexists. In particular, we imagine that a change Aw forsome small range of pulse heights between (ho - Ah/2)and (ho + Ah/'2) would give a larger value of p. Thenew weighting function

w1(h) = (wo(h), h < ho - h/2wo(h) + Aw, ho - Ah/2 < ho +

(wo(h), h > ho + Ah/2

= wo(h) + Aw(h)

Ah/2

The new value of p2 is then

2 = -S 1 2(AS t2)1

n,2[Wo2 + 2Wo X Aw X s(ho) X Ah + (W)2so2(ho)(Ah)2I

Wo + 2Awso(ho)Ah + (w)2d(ho)Ah

[ Wo + 2wso(ho)Ah + (W)2So2(ho)(Ah)21nWo o + 2Awso(ho)Ah + (Aw)2d(ho)th J. (78)

Since the first-order terms in Aw are equal innumerator and denominator, a small change from wodoes not change p. Hence, wo(h) gives either a maxi-mum or a minimum in p. But the second-order termin the numerator is also of second order in Ah, whereasthe (AW)2 term in the denominator is of first order inAh; hence any small change Aw(h) causes p2 to decrease,and p must be a maximum for the weighting functionwo(h).

We may regard this result as an example of theprinciple of matched filtering, in which a weightingscheme proportional to the probability of success givesoptimum detection.

How much can we hope to improve p by usingoptimum weighting, and how can it be achieved?

B. Comparison of Weighting Functions:an Example

Let us consider a rather idealized tube in which

so(h) = e-h (79)

andd(h) = X h-2

(73)

then gives

SI = w(h) s(h)dh

= n8[Wo + Aw X s(ho) X Ah], (74)

and the variance

(RIsE') = f WE(h)d(h)dht Jo

= f f [wo'(h) + 2w0(h) X Aw(h) + Aw'(h)]d(h)dh.

(75)

We then interchange the order of integration andsummation, and obtain three integrals. The first isjust Wo again; the second is

2 wo(h) X Aw(h) X d(h)dh = 2 | so(h) X Aw(h)dh

= 2Aw X so(ho) X Ah; (76)

and the third is

(80)

for h < h.ax; we suppose h.ax 4 if ion pulses arethe largest dark pulses, and h.ax 10-100 if cosmicray and gamma ray pulses are important.* We adoptthese forms for s(h) and d(h) because they are typicalof many real tubes, and because this choice makes sand d very different in shape and hence produces largechanges for different weighting functions. For tubes(such as the ITT tubes) in which s(h) and d(h) aresimilar, we have essentially the large signal or back-ground situation, in which different weighting functionsproduce rather similar results, and pulse counting givesthe best results.

For pulse counting, we find 1, = ; the successiveapproximations are ul = 5.19; 12 = 0.520; u2 = 5.07;13 = 0.520. The corresponding values of p2 as a func-tion of 1 and u are

Pp(2 co ) = tN'qk2/8e = (tN2'kq/6) X 0.046 (81a)

andppc'(0.5 2 , 5.07) = (tN 2qk2/3) X 0.050. (8b)

I Aw2(h) X d(h)dh = (Aw)2 X d(ho) X (h). (77)0

* In some of the following we shall take hma : c o, where onlya small error is involved.


Thus a window counter, used optimally, would giveabout 10% higher efficiency than a simple discriminator.In order to reject the optimum amount of dark noise,we can only count about 60% of the actual signalpulses; if a wide dynamic range is required, the detec-tive quantum efficiency for strong signals cannot exceed0.6 qk. For weak signals, the noise equivalent input isabout

which is slightly inferior to the value for pulse counting[Eq. (81)].

Now let us look at the results expected for theoptimum weighting function wo(h) = h 2e-h/. Wehave

TV = f so'(h)/ld(h)dh = h...JO O (e- 2h/a h-')dh = 143,

V., = (8e3)1/q, (82)

for the simple discriminator, and about 5% lower forthe optimum window counter.

With d detection, Eq. (45) gives*

pol = Nqk(t/4Sh.ax)1,

so

popt2 = N2q,t/163 (92)

andt(83)

so

NOd = 2(31hm.,)1/qk. (84)

Thus the dc performance for weak signals depends onthe largest dark pulses, and may be either better orworse than pulse counting, depending on whether hmaxis less or greater than 2 e 5.4. Nakamura andSchwarz compared d and pulse counting in an EM\II9558 at -45 0 C, and found only about a factor of twodifference in noise equivalent input. This gives h,ax-20, which seems reasonable since cosmic ray noisedominates under these conditions. 22 As mentionedbefore, q = q/2 for this case at large signals.

For shot-power detection,

Pep = 2Nqk(3t/1hmax')1 (85)

and

-X O.,p = (l/2qk)[(3h,..X'/3)ll. (86)

Here the dependence on hax1 is of prime importance.If amplifier saturation occurs, we may approximatelytake h_. as the value of h at which clipping begins.We note that pdc = Psp if hx = (48)1 6.9. Thus,this method could be superior to d detection in a tubewhich normally produces large cosmic ray dark pulses;this probably accounts, in part, for Pao and Griffiths'data for a 9558.

If we assume all pulses over a height ho are clipped tothe same height, we have

w(hho) = {(h/ho) h < h (87)

This then gives

p'(ho) = (3AN'2q>,t/4ho3) X [I - (ho + )e-1012 (88)

so that

No(hto) = 2(3h/3q)1/[1 -e-hE(h + 1)]. (89)

The minimum of No is near ho = 0.8, which gives

A'.opt = 4(5)1 /qk. (93)

Thus, in the case we have considered, the advantage ofusing optimum weighting is small; the noise equivalentinput is lower by a factor of only (e/2)1 = 1.17 thanin pulse counting.

The price paid for optimum weak signal detection issome decrease in strong signal detective quantumefficiency. In the strong signal limit,

Ust R s,E'/ft = f wo'(h) X s(h)dh/ft

= (h12e-h)2 Nqkehdh/fta2

Nqk J

f13' Jo

Nqk 8

pf3'2 81'

and

S = j' wo(h)s(h)dh

('f s(h)j0 d(h) ~NkohdE

= Nqk X TV, = Vqk/43.

So for f 1,

2 AT 2q

2/163 2= Vqkt X 1-p S/RSE= - q I -SNqi-/181 t32 128

(94)

(96)

The detective quantum efficiency for large signals isthus (81/128) q = 0.633 q. For the model we havechosen, this represents less degradation at high lightlevels than for any of the other detection methods(weighting functions). Thus, optimum weightingshould produce better results at all light levels. Therelatively small improvement in going from pulsecounting to optimum detection is surprising.

p2(0.S) = (N'qk't/3) X 0034,

* Ignoring teams i ea.

(90)

t Equation (93) was derived assuming hmax = co. The errormade is approximately 0.5 hm,2 exp (-2hm.,), which is -0.01.for hm,. = 3 and 0.003 for hma = 4.


C. Realization of w(h)

At very low light levels, the most straightforwardmeans of realizing wo(h) is to record all data with apulse height analyzer, and to find wo(h) and apply itto the observations by a suitable computer analysis ofthe data. This method is worthwhile only if theanalyzer memory can be read out in a time smallcompared to the total time of observation. Althoughthis method is rather inconvenient, it certainly willproduce the best results. Also, the gate time andstorage cycle time of the analyzer may be chosen toignore afterpulses, thereby giving cleaner statistics. Auseful degree of pulse height analysis could even beobtained by using two or three single channel counterssimultaneously, with different thresholds.

In our example, the desired weighting function is(h

2e-h). Since this goes as h2 for small pulses, one

might expect the shot-power method combined withamplifier saturation to give a reasonably good weightingfunction. [The behavior of w(h) for large h need notbe so close to ideal as for small h, because large pulsesare relatively rare. It is the great number of smallpulses that require careful treatment.] However, asEq. (90) shows, the shot-power method is inferior topulse counting, even with amplifier saturation. Wecould perhaps improve the situation by making oneamplifier in the Pao et al.' circuit saturate at a pulseheight hi and the other at h2> hi; this would produceh2 weighting up to hi, linear weighting between hi andh2, and constant weight above h2. If h2 >> hi, thesquare law region becomes negligible; we consider thiscase below.

If we had assumed so(h) to be more nearly a Poissondistribution, we should have given still less weight tothe smallest pulses.

Other methods might be used to manipulate theweighting function. One could use a discriminatorwithout pulse shaping to shift the zero for either sub-sequent charge integration or noise-power detection.This would replace h by (h - ho) in the weightingfunctions, where ho is the discriminator level belowwhich all pulses are rejected (zero weight). A non-linear circuit element, such as a diode used as loadresistor, or a voltage-sensitive capacitor, could be usedto modify the height or width of the pulses prior tonoise power detection.

A particularly simple weighting function that couldbe achieved either by clipping all pulses larger than ho(e.g., by a limiter) followed by dc detection, or by usingvery dissimilar saturation heights for the two amplifiersin Pao and Griffiths' circuit,' is

w(h) = hho, h < ho} (97)

In our hypothetical photomultiplier, this gives

p'(ho) = (N'qk't/3) X (1 -e-h)2/8ho (98)

and

No = (85ho )i[qk(1 - eho)] -1. (99)

The optimum value is ho 1.26, which gives

p2(1.26) = Nqkt X 0051,a

(100)

which is practically the same as the best value for pulsecounting [see also Eq. (81) ].

When any scheme other than linear (dc) weighting isused, the results will depend on whether appreciablepulse overlap occurs, and on whether afterpulses aredetected separately or are merged with their parentpulses. Thus the average pulse rate and predetectionbandwidth must be considered carefully.

VI. Other Considerations for Stable Operation

It is desirable to monitor the dc anode current evenin non-dc detection, as gain drifts on the order of afactor of two may occur at microampere anode currents,due to dynode fatigue 3" Such large signals usuallyproduce a temporary rise in dark noise, also.,"" 6 2

One argument frequently offered2 in favor of pulsecounting is that it is relatively insensitive to gain drifts.This argument assumes that the discriminator level canbe set at a point on s(h) where nearly all signal pulsesare counted, and where ds/d(ln h) is small, so that a smallfractional change in effective discriminator level produces a very small fractional change in n(l,u).These assumptions can only be approximately met forstrong signals, however. In the weak signal limit, weare always fighting the 1/h2 component of the darkpulses, so that a fractional change in 1 produces a com-parable change in nd(lu) - /. Thus pulse countingand dc detection have similar sensitivities to gainchanges.

A major limitation of any nonlinear weighting systemis limited dynamic range; when a significant fraction(say 1%) of pulses overlap, a similar degree of non-linearity results. If we try to extend the range byswitching over to dc methods for strong signals, wemust remember that the apparent spectral responsewill change when we change weighting functions. Thesame problem occurs if we try to avoid saturation andfatigue effects'4 by changing the dynode voltage; theapparent change in red/blue response ratio", 7 canbe several percent. This problem can be almost elimi-nated by keeping the voltage fixed between cathode andfirst dynode.

Both the cathode spectral response and the multipliergain are temperature sensitive"" 8 ; coefficients of 0.5-1.0%/°C are common. All kinds of optical filters'9

show similar effects also, as may monochromators.4 0

Calibration by means of reference light sources is notgenerally an acceptable solution, because these in turnhave similar temperature coefficients,4 1-4' and may alsoshow hysteresis if cycled in temperature. These prob-lems are very severe in field work. Even in the lab-oratory, some care must be given to temperatureregulation.

Finally, although the large (- X 2) variations inresponse across the sensitive area7' 44' 4' are well knownand are usually controlled by imaging the instrumentalpupil on the cathode, the variation with the state of


CHANNEL NUMBER0 100 200 300 400

b) /_ / s I

// \R(h)

O(h u = 33

~~iA 52 - ~ ~ .2= 338"IN1

I

IIII

5000 a

42000 V'4000~ _ ... .,v'a-^v i, J/ R(h) -

3000 _ 'a

2000 L/ O(h 8

I , 'a~~~~~~~~~~~3I I 'N~~~~~~~

Fig. 8. The functions R(h) and Q(h), discussed in the text, forthe 1P21 of Fig. 1. (See Appendix for details.) R(h) has theform of the optimum weighting function. The light pulse dis-

tribution of Fig. (b) is reproduced on a linear scale in (c).

polarization of the light4 6 -48 is not so widely recognized.Both effects are wavelength and temperature depen-dent. In addition, the nonuniformity may be voltagedependent, and the polarization depends on the angleat which the light strikes the cathode.

VII. Conclusion

At high light levels, pulse counting is the mostefficient detection method, but only by a factor of twoor so. At low levels, where dark noise dominates overphoton noise, pulse counting is very nearly the optimummethod and is certainly better than any other method ofcomparable complexity. The advantage of optimumdetection over pulse counting may be typically only afactor 1.4 in efficiency. Thus in most practicalsituations, pulse counting is the best detection method.

Since all methods are so similar in efficiency, otherconsiderations than short term signal/noise may bedecisive. The linearity of d detection at high lightlevels where pulse overlap is important, and the sensi-tivity of wideband equipment to interference and linetransients both favor d over other methods. On theother hand, it appears that some workers have failedto use optimum discriminator settings, or have notincluded a wide enough range of pulse heights to enjoythe real advantage of pulse counting.

One should be aware that the actual uncertainty of ameasurement is usually not limited by the inherentsignal/noise ratio of the detector, but uncontrolledenvironmental parameters or by readout devices. Forexample, the use of a strip chart recorder29 rather thandigital readout increases the time required to reach agiven precision by a factor of three or more-a largerfactor than exists between different detection methods.

The experimental work reported here was done at theUniversity of Texas, Department of Astronomy, with

the assistance of a grant (Army Grant DA-ARO-D-31-124-G 757) from the Advanced Research ProjectsAgency. I thank XII. L. Davis for assistance with thepulse-counting equipment. C. E. Childers, Tom Par-ker, and Stan Brode of International Minerals andChemical Corporation were exceedingly helpful, andmade possible the cosmic-ray experiment described inSection III. Discussions with C. L. Seeger and J. N.Douglas were very helpful.

This is the ninth in a series of papers analyzing errorsources in astronomical photoelectric photometry.Earlier papers are referenced in Paper VI."

This paper presents the results of one phase of re-search carried out at the Jet Propulsion Laboratory,California Institute of Technology, under contract No.NAS 7-100, sponsored by the National Aeronautics andSpace Administration.

Appendix

To demonstrate the methods described in the text,optimum discriminator settings for pulse counting andquantum efficiencies have been calculated for the 1P21whose light and dark pulse distributions are shown inFig. 1.

Figure 8 shows, on a linear scale, the functions R(h)and Q(/h,u) defined in Eqs. (29) and (30). In Fig. 8(a),we take u = , so that the interaction of R(h) withQ(h, -) determines the discriminator setting 1 definedin Eq. (32). In our example, 1 corresponds to channel51 of the 400-channel analyzer used. If only a simplediscriminator were available, 1 would be the bestthreshold setting for weak signals. In this case, 82%of all light signal pulses would be counted, so the detec-tive quantum efficiency for strong signals would be0.82 times the cathode quantum efficiency.

However, our analyzer can be used as a windowcounter, so we can reject all noise pulses larger thansome upper cutoff. Our first approximation u isfound by drawing a horizontal line from the inter-section of R and Q to the right, until it again intersectsR [see also Eq. (33)]. We 'find u = channel 340,which only reduces the strong signal efficiency by0.3%, but increases the weak signal efficiency by 3.2%.

We now recompute the function Q using u as anupper cutoff. The resulting function Q(h,ul) is shownin Fig. 8(b). The new intersections 12 and 2 are atchannels 52 and 338, giving nearly the same efficienciesas 1 and u1 . A third iteration does not change thesevalues, so we adopt them as final.

The strong signal quantum efficiency is 81% of thecathode efficiency for pulse counting, 75% for dc detec-tion, and 81% for optimum weighting. For weaksignals, pulse counting gives 87% the efficiency ofoptimum weighting, and d gives 71% of the optimumefficiency. However, since the deadtime of our instru-ment excluded most afterpulses, we have probably over-estimated the d efficiencies. Nevertheless, this exam-ple shows clearly that different detection schemes giveremarkably similar detectivities, and that pulse count-ing is not far from ideal.


I) /*_* . I

r. - \ R(h)_ ll \ \I K Q(h, of)

I ,,elL - 51 -A >- , = 340

[dl I

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(1969).

Fo{;im;i~ rthsj EM AE: d :9ito ::i iis.{:0/::Lg:::.:ii:::.::r$ :

Some time ago (June 1968) we discussed in this column an-other type of reform that is sometimes pressed nowadays: reformof the calendar. We do not mean the kind of time reform some-times pushed by the more ardent proponents of the SI units.These people feel that only the second is sacred, such units asminutes, hours, days, weeks, months and years are all abomina-tions to be avoided, and replaced by the proper unit of seconds,multiplied by the proper prefixes mega, giga, and so on. Historybooks would have to be rewritten, of course; such events as thecreation of the earth in 4004 B.C. or the coronation of Charles Von Christmas Day in 1500 would then be pin-pointed much moreprecisely in seconds. These advocates that phenomena shouldonly be expressed in fundamental SI units assert that we canbetter discern relationships if derived units are abandoned.In radioactivity, for example, one finds half-lives of nuclei var-iously expressed in seconds, hours, days, or years, and surelythey could all be expressed in seconds. (By the same logic,these proponents of basic units feel that astronomers should ex-press all distances in meters, and avoid such "distances" as lightyears, parsecs, or astronomical units.)


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Photometric Error Analysis IX: Optimum Use of Photomultipliers

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