Critical Comparison of Photon Counting and Direct Current Measurement Techniques for Quantitative Spectrometric Methods J. D. Ingle, Jr.,I and S. R. Crouch Department of Chemistry, Michigan State University, East Lansing, Mich. 48823
A com arison of signal-to-noise ratio (SNR) expressions t r spectrometric measurements using photon counting and direct current techniques reveals that photon counting techniques provide SNR’s from 5 to 22% higher under equivalent conditions if measurements are shot noise limited. If measurements are limited by source or background flicker noise, both techniques provide equivalent SNR’s. However, dc techniques are capable of measuring light levels several orders of magnitude greater than can be measured without pulse pileup in photon counting. Practical implications for analytical spectrometric methods are discussed, and criteria are presented for choosing the detection technique providing the highest SN R. PHOTON COUNTING has been proposed as a measurement technique for several analytical spectrometric applications (1-5). These are several reasons why photon counting techniques appear to have inherent advantages over the conventional dc method of measuring light intensity. First, radiant flux information is processed in a discrete manner, reducing the number of domain conversions (6) necessary to obtain a number related to the incident light level. Second, the processing of information by digital circuity makes photon counting detection less susceptible to long term drift and llfnoise which often limits analog systems. This feature enables the spectrometrist to utilize much longer averaging times than would be feasible with dc detection where low frequency noise components can predominate at narrow measurement system bandwidths. Other advantages of the photon counting technique include the ability to discriminate against photomultiplier dark current originating down the dynode chain and the virtual elimination of the reading error which can limit analog systems (1). Despite these attractive features, there are drawbacks to photon counting, particularly at the moderate to high light levels encountered in many types of analytical spectrometry. Pulse overlap limits photon counting to relatively low light levels unless deadtime compensation or mathematical correction techniques are used (4,7). Light levels which result in an anode pulse rate of 108 Hz are at the upper limit of photon counting systems using present state of the art electronic cirPresent address, Department of Chemistry, Oregon State University, Corvallis, Ore. 97331. ~
(1) M. L. Franklin, G. Horlick, and H. V. Malmstadt, ANAL.
CHEM.,41,2 (1969). (2) L. J. Cline, K. P. Li, and H. V. Malmstadt, Abstracts, 7th National Meeting, Society for Applied Spectroscopy, Chicago, Ill., May 1968, p 43. (3) E. H. Piepmeier, P. E. Braun, and R. R. Rhodes, ANAL.CHEM., 40, 1667 (1968). (4) K. C. Ash and E. H. Piepmeier, ibid., 43,26 (1971). ( 5 ) D. 0. Cooke, R. M. Dagnall, B. L. Sharp, and T. S. West, Spectrosc. Letr., 4, 91 (1971). ( 6 ) C . G. Enke, ANAL.CHEM.,43 (l), 69A (1971). (7) J. D. Ingle, Jr., and S . R. Crouch, ibid., 44,777 (1972).
cuity. At normal photomultiplier gains of 106-106,this corresponds to an anodic current of only lO-*-lO-’ A, which is several orders of magnitude less than the maximum anodic current for linear operation. Thus using a photomultiplier in the dc mode allows the measurement of light intensities several orders of magnitude greater than can be measured with linearity in the photon counting mode. In shot noise limited operation, where the signal to noise ratio (SNR) increases with the square root of the incident photon flux, use of photon counting may lead to decreased SNR’s if the light level is purposely reduced for good linearity. In many analytical spectrometric applications, measurements must be made in the presence of background radiation. Thus, even when signal strengths are low, the presence of large amounts of background may cause pulse rates to be higher than can be measured without pileup. Another drawback to photon counting at present is the complexity required for processing the data to obtain numerical readout in the appropriate units. Such operations as taking the logarithm, the ratio, or the reciprocal are common in spectrometric applications and require computer data processing for photon counting with a single beam spectrometer unless a conversion back to the analog domain is performed with its consequent errors and nonlinearities. This latter drawback is rapidly becoming obsolete with the increased availability of small laboratory computers. To help put the advantages and possible drawbacks of the photon counting technique into perspective for analytical spectrometry, this report compares SNR expressions ’for the two techniques at the light levels commonly employed. SNR expressions are developed for both photon counting and dc methods and three predominant types of variance are identi-. fied. Both methods are compared under equivalent conditions. Practical implications of the comparison for the signal and background radiation levels normally encountered in analytical spectrometry are discussed. READOUT SIGNAL EXPRESSIONS
To compare photon counting and dc techniques, a single beam spectrometric system is considered. A radiant source and a photomultiplier transducer are common to both techniques. The source is considered to provide an incident signal photon arrival rate of rs sec-1 at the photocathode of the photomultiplier tube. A photon arrival rate rb may also be incident on the photocathode due to unwanted background radiation. The incident flux may come from a complex system as in flame atomic absorption, where the radiation incident on the photocathode may be radiation from a primary source (hollow cathode or continuum source) passed through a flame containing atomic vapor and through a monochromator, or from a simpler source as in molecular absorption, where the incident flux may come from a tungsten or deuterium lamp passed through a monochromator and sample cell. Explicit expressions for r, and rb for absorption, ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
785
Table I. Definitions of Dark Current Symbols P o d = jotc iotd i,l i,, i., Rod = Rote Ratd Rm Rar iateor Rate = anodic current, A, or average rate of arrival of anode pulses, sec-I, due to thermal emission at the cathode iatdor Rata = anodic current, A, or average rate of arrival of anode pulses, sec-1, due to thermal emission at the dynodes iaI = leakage dark current at the anode, A i,, or R,, = anodic current, A, or average arrival rate of anode pulses, sec-1, due to cold field emission i,, or R,, = anodic current, A, or average arrival rate of anode pulses, sec-1, due to radioactivity (Cerenkov photons, glass fluorescence, cosmic rays) Rlad = Rad - Rat, = average dark current pulse rate at the anode from sources other than thermionic emission at the photocathode, sec-l i*,d = i,a - iat, = anodic dark current of pulsed nature from sources other than thermionic emission at the photocathode, A i.a = i l a d - i,l = total anodic dark current of pulsed nature, A
+ + + + + + +
emission, and luminescence spectrometry can be derived from the equations of Winefordner and coworkers (8-13). In the dc mode, the output of the photomultiplier is considered to be connected directly to the summing point of a high quality operational amplifier (OA) current to voltage converter. A high resolution readout device such as a digital voltmeter or expanded scale recorder, is employed to measure the voltage output of the operational amplifier. For photon counting, the output pulses from the photomultiplier flow through a load resistor which is the input to a fast pulse amplifier and discriminator. A high speed counter is the readout device. Other counting systems have been described (7, 14-16) with minor variations, but that considered here is most common. Current and Pulse Output Expressions for Photomultiplier. For dc measurements, the photoanodic signal current ias from the photomultiplier is given by ias =
r,qK(X)me = qmites = mics
(1)
where K(X) = quantum efficiency of photocathode, dimension-
less collection efficiency of first dynode (17), dimensionless m = current gain of photomultiplier tube, dimensionless e = charge of electron = 1.6 x 10-19, coulomb its' = photocathodic signal current, A iCs = effective photocathodic signal current reaching first dynode and causing emission, A
17
=
=
(8) J. J. Cetorelli, W. J. McCarthy, and J. D. Winefordner, J. Chem. Educ., 45,98 (1968). (9) W. J. McCarthy and J. D. Winefordner, ibid., 44, 136 (1967). (10) M. L. Parsons, W. J. McCarthy, and J. D. Winefordner, ibid., p 214. (11) J. D.Winefordner, V. Svoboda, and L. J. Cline, CRC,Crit. Rev. Anal. Chem., 1,233 (1970). (12) J. J. Cetorelli and J. D. Winefordner, Talanta, 14,705 (1967). (13) P. A. St. John, W. J. McCarthy, and J. D. Winefordner, ANAL. CHEM., 38,1828(1966). (14) F.Robben, Appl. Opt., 10,776 (1971). (15) R. Foord, R. Jones, C. J. Oliver, and E. R. Pike, ibid., 8, 1975 (1969). (16) R. H.Eather and P. L. Reasoner, ibid., p 227. (17) A. T.Young and R. E. Schild, ibid., 10, 1668 (1971). 786
ANALYTICAL CHEMISTRY, VOL. 44,NO. 4,APRIL 1972
For photon counting measurements, the average rate of arrival of signal photoelectron pulses at the photomultiplier anode Raa is Ras = r,qK(X) (2) The photoanodic current due to background radiation, iab, and the photoelectron pulse rate due to background, Rab,are given by Equations 1 and 2, respectively, if rb is substituted for r,. The total current at the anode, iat,and the total average arrival rate of pulses at the anode, R,,, are given by Equations 3 and 4,respectively.
+ + iao Ras f Rad + Rab
iat = ius
Rat =
?ad
(3) (4)
where i'ad is the dark current at the anode in amperes and Rad is the dark current pulse rate at the anode in sec-l. Both dark current terms are defined in Table I. The photomultiplier is assumed to be operated under conditions where regenerative effects and after pulsing (light feedback or gas ionization) are negligible. DC Readout. In dc detection, the anodic current is converted to voltage by the operational amplifier current to voltage converter. Equations 5 and 6 give the resulting output signal voltage, E,, and total output voltage E t .
Et
= (ias
f
iad
E, = iasRf
(5)
+ iao)Rf = ES + Ed f Eb
(6)
where R I is the feedback resistance of the OA in ohms and Ed and Eb are output voltages due to dark current and background radiation, respectively. Often the output voltage of the OA is further modified before display on a readout device such as an oscilloscope, servo recorder, or meter. Such modification may include voltage amplification, log amplification, and voltage or current subtraction to suppress dark current or background and permit scale expansion. Thus the final readout signal may be some function of Et and electronic parameters. To be general, E t is considered here to be the final readout voltage from which E, is to be extracted. Photon Counting Readout. The effects of the counting circuit components (the RC load, the pulse amplifier, and the discriminator-counter) on the final numerical readout in photon counting systems have been previously discussed (7). The relationship between the observed photoelectron pulse rate and the anode photoelectron pulse rate was shown to depend on the discriminator coefficient, A I (the fraction of single photoelectron pulses passed by the discriminator), the resolving or dead time of the counting circuitry, 7,and the particular component of the counting circuitry that limits the dead time of the system. In the absence of pulse overlap, the observed number of counts for signal and background photoelectron pulses, No, and Nob2respectively, are given by Nos
=
Nob =
Rest
=
AiRast
Robt = A&bt
(7) (8)
where Ro, = observed rate of signal pulses, sec-' Rob = observed rate of background pulses, sec-I t = counting time, sec
The observed number of dark current pulses, Nod, is given by Nod
=
Rod
=
(A1Ratc
AdR*ad)t
(9)
where Rod = observed rate of dark current pulses, sec-l Ad = discriminator coefficient for R*ad (fraction of R*ad passed by discriminator), dimensionless When pulse overlap is significant, the total anodic photoelectron pulse rate, Rap= Ras &, must be considered since a large amount of background may cause pulse overlap at small signal pulse rates. The relationships between R a pand the total observed photoelectron count, No, = No, Nob,for cases in which pulse overlap is significant (Rapr> l e a )have been presented (7).
+
+
NOISE EXPRESSIONS
The noise sources considered here are those instrumental sources which often limit measurement precision. A modern spectrometric system is considered which, after a suitable warmup time, has excellent stability over the measurement time and is free from unidirectional drifts in the radiation source, the photomultiplier transducer, the amplifier, and the readout device. Thus, the errors considered are random and independent so that the total variance is the sum of the variances from each noise source. The sources of variance can be divided into two basic classes, First, there are fundamental noise sources, such as shot and Johnson noise, which cannot be eliminated and for which theoretical equations are available. Second, there are non-fundamental sources of variance (excess noise) such as power supply variations and light source flicker noise, which are caused by imperfect instrumentation or non-ideal behavior. For these latter sources of variance, empirical equations can be developed to aid in the analysis. These various variance sources will be considered individually for their effects on dc and photon countings systems. Quantum Noise. The random time arrival of photoelectrons at the anode gives rise to quantum or photon noise. This noise in the photoelectron current actually arises from a combination of a number of random processes occurring in the spectrometer such as the random nature of photon emission from the primary radiation source, of transmission through the monochromator, of absorption by the sample in the flame or sample cell, of photoelectron emission, and of photoelectron collection by the first dynode. At the anode, the variance of the average number of signal pulses counted in a time interval t due to quantum noise, ( U ~ ~ ~ can , J ~be , expressed from Poisson statistics (7) as (U?v.Jq
=
Rust = rstK(X)t
=
(mRf)zeqi’cs (mRf)2eics mRf2eias -~ - -t t t
(uzE,)q
=
2meRfzAfias
(12)
where Af
noise equivalent bandwidth of the amplifier-readout system, Hz = 1/(2t) = 1/(4RC) for a simple R C low pass filter (18) =
For the remainder of this paper, all variance terms for a dc measurement system will be written in terms of the noise bandwidth of the amplifier-readout system since this is the usual convention. There is also quantum noise caused by the random time arrival of background pulses at the anode. Since these pulses also follow Poisson statistics, the.observed variance in the number of background pulses counted in a time t due to quantum noise, ( U Z ~ ~ ~taking ) ~ , into account the effect of the discriminator, is equal to Nob as given by Equation 8. For dc measurements, the variance in the output voltage due to quantum noise in the background, ( U Z ~ ~ ) is~ ,given by Equation 12 where the corresponding photoanodic current due to background is substituted for ius. The random time arrival of dark current pulses at the anode is another source of quantum noise. Thermionic dark current pulses follow Poisson statistics as do the photoelectrons; however, other sources of dark current have been shown to have higher variances than those predicted from Poiqson statistics (19, 20). Since the contribution of nonthermionic dark current sources to the total dark current pulse rate is often small at room temperature, assuming that all dark current is Poisson will cause little error. Therefore, the observed variance in the number of dark current pulses counted in a time t due to quantum noise, ( u ~ . ~taking ~ ~ ) into ~ , account the effect of the discriminator, is assumed equal to N o d as given by Equation 9. For dc measurements, the expression for the variance of the dark current is more complex. Thermionic electrons from the cathode receive the same gain as the photoelectrons. However, the dark current from other sources, i*ad,may undergo a different average gain so that the final expression for the variance in the dark current output voltage due to quantum noise, (uZE,),,can be written
(10)
Both pulse discrimination and pulse pileup reduce the pulse rate reaching the counter and hence affect the variance. In the absence of pulse overlap, the observed variance in the number of signal pulses counted due to quantum noise, ( U Z ~ , , ) ~ , is equal to No, as given by Equation 7. The observed variances under conditions where pulse overlap is significant have been previously discussed (7). For a dc system, the corresponding variance in the signal voltage due to quantum noise in the signal photoanodic current, (uZEJg is found by multiplying Equation 10 by (meRf/r)2, which yields (uzE,)q
readout system. If an analog readout device such as a meter or recorder is used, or a digital readout device with a measurement or sampling time much less than the time constant of the amplifier-readout system, Equation 11 can be written in terms of the noise bandwidth of the amplifier-readout system as
( 11 )
Equation 11 is applicable in cases where dc charge integration or digital integration is used if the integrating or measurement time is much longer than the time constant of the amplifier-
where y is the effective gain received by i * a d . Because the leakage current is not composed of pulses, it is assumed not to contribute to the dark current quantum noise. Secondary Emission Noise. The statistical nature of secondary emission gives rise to a variance in the gain, and thus produces the pulse height distribution. For a dc measurement, this adds another noise source, and the variance in the gain due to secondary emission, uZmk,for a photomultiplier with equal gain dynode stages described by Polya statistics, has been show‘n to be (7,21,22) (18) D. M. Hunten, “Introduction to Electronics,” Holt, Rinehart, and Winston, New York, N. Y . , 1964, pp 321-335. (19) J. K. Nakamura and S.E. Schwarz, Appl. Opt., 7,1073(1968). (20) J. p. Rodman and H. J. Smith, ibid.5 2, 181 (1963). (21) J. R. Prescott, Nucl. hstrum. Meth., 39, 173 (1966). (22) RcA photomultiplier Manual, Technical series pT-61, RCA Electronics Components, Harrison, N. J., 1970. ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
* 787
u24 =
where 6 = gain of dynode stage in photomultiplier, dimensionless b = distribution shape parameter (7, 22), 0 5 b 5 1, dimensionless The variance in the photocurrent signal voltage due to secondary emission noise, (u2~&,, is where
+
a = a2,,&n2 = (b6 1)/(6 - 1) = relative variance in gain received by ics or icb, dimensionless
Pulse discrimination is a random process which reduces the average pulse rate but does not change the form of the time arrival distribution of pulses. Therefore, the variance in gain due to secondary emission does not directly cause any added variance in photon counting. However, fluctuations in the discriminator level will contribute to excess noise in photon counting as will be discussed later. Secondary emission noise causes a similar variance in the background output voltage, ( u ~ & ~ ,which is given by Equation 15, where the anodic photocurrent due to background is substituted for iaa. Secondary emission noise due to background will also contribute to excess noise in photon counting when a non-ideal discriminator is used. The corresponding variance in the dark current output voltage due to secondary emission, (u*Ed)eec, is (u2,&ec = 2eRy2df(miatc
+ ri*an)
(16)
Here it is assumed that i*aexperiences the same relative variance in gain as ius, iaa, or iatc. Johnson Noise. For the dc system, Johnson noise in the feedback resistor of the OA gives rise to a variance, u2,, in the output voltage of the OA given by u2j =
4kTRyAf
where k is the Boltzmann constant in joule/”K and T i s the absolute temperature of Ry in OK. R,is assumed to be a high quality resistor with negligible current noise. Johnson noise is also present in the load resistor (RL)used for photon counting. The expression for the variance is given by Equation 17, except that R L is substituted for R , and the bandwidth used is the noise equivalent bandwidth of the RC load-pulse amplifier combination in Hz. Johnson noise can have two possible effects on the number of pulses counted in photon counting. First, at low discriminator levels: a pulse originating from Johnson noise might be counted as a true pulse and add to the dark current pulse rate. Second, superposition of Johnson noise on top of a pulse may cause a pulse not normally counted to be above the discriminator level and thus counted. Likewise a similar loss in count due to superposition of Johnson noise could occur. This superposition increases the variance of the gain or broadens the pulse height distribution. Johnson noise will be considered as part of the excess noise discussed later. Amplifier Noise. In the dc mode, the OA current voltage converter contributes another source of noise to the system. The variance in voltage at the amplifier output due to amplifier noise, uzA,can sometimes be calculated from the data supplied by the manufacturer (23) and can be expressed as (23) Analog Dialogue, 3, (l), March 1969, “Noise and Operational
Amplifier Circuits,” Analog Devices, Cambridge, Mass.
788
ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
+
[ R r 2 ( i d 2 (ed21Af
(18)
where i n 2 and en2represent the equivalent input mean square noise current and voltage, respectively, per unit bandwidth of the OA. Amplifier noise may not be white noise. Frequency dependent noise can occur at low frequencies (l/f noise) and possibly at 60 Hz and its harmonics. Because the noise per unit bandwidth is not constant, the noise contribution from the amplifier must be calculated or measured at the particular bandwidth used for measurement. The noise from the amplifier is assumed to be current independent for a solid state operational amplifier. In photon counting, noise from the pulse amplifier either broadens the pulse height distribution or causes spurious pulses to be counted. To see if pulses originate from amplifier or Johnson noise, the high voltage supply to the photomultiplier should be shut off with the rest of the apparatus operating normally (24). Photomultiplier Flicker Noise. Noise generated in the photomultiplier in excess of quantum and secondary emission noise will be designated as photomultiplier flicker noise. Variations in the work function of the photocathode or dynodes or in the photomultiplier gain due to power supply fluctuations are possible causes of photomultiplier flicker noise. Flicker noise with a l/f frequency dependence due to work function fluctuations is predicted to be proportional to the photocathodic current squared (24). It has been shown to be negligible above 0.1 Hz for most photomultipliers (25) and will be assumed negligible here. The relative change in gain with supply voltage can be approximated (26) by Equation 19. dmjm = kq dVPm/V,,
(19)
where number of dynodes in the photomultiplier, dimensionless V,, = cathode to anode supply voltage, V = factor to account for the type of dynode, (26) 0 5 q q 5 1, dimensionless k
=
The variance in the photocurrent output signal voltage due to photomultiplier flicker noise, (u2E,)pm,can be estimated by Equation 20 if the predominant source is fluctuation in the supply voltage. where
x
~
relative variance in gain received by the photocurrent due to supply voltage fluctuations over the measurement bandwidth, sec-1 = u2mo/m2= (kq)2~2v,,/V2pm = , ~variance of photomultiplier gain due to supply voltage fluctuations over the measurement bandwidth, sec-‘ , = variance of photomultiplier voltage supply over the measurement bandwidth, V2sec=
2
~ 2 ~ ,
The background output voltage also undergoes a fluctuation due to photomultiplier flicker, ( ~ 2 ~ , ) , , given by Equation 20 with background photocurrent substituted for signal (24) R. C. Schwantes, H. J. Hannam, and A. Van Der Ziel, J. Appl. Phvs.. 21, 573 (1956). (25) A.’T,Young, Appl. Opt., 8,2431 (1969). (26) “EM1 Photomultiplier Tubes,” Brochure 30M/6-67(PMT), Gencom Division, Varian/EMI, Plainview, N.Y., 1967, pp 3-12.
photocurrent. Likewise, the dark current output voltage experiences a similar variance, ( u ~ E , ) ~as, given by
+
( ~ ~ ~ = ~ 2eRf2x(rniatc ) p m
~i*aa)
(21)
For photon counting, photomultiplier flicker noise due to power supply fluctuations does not change the average rate of photoelectron pulses reaching the anode, but increases the variance of the gain and broadens the pulse height distribution. Photomultiplier flicker will be included as part of the excess noise discussed later. Signal and Background Radiation Flicker Noise. Nonideal variations in the incident photon arrival rate (variations above quantum noise) can arise from many causes depending on the specific spectrometric application. In molecular absorption and fluorescence spectrometry, variations of the average arrival rate of photons can be caused by variations in the primary light source intensity from changes in source temperature, power supply voltage and current fluctuations, arc wander and filament motion. In flame atomic emission spectrometry, fluctuations in the average arrival rate of photons can be caused by fluctuations in flame background, fluctuations in the solution aspiration rate, etc. In flame atomic absorption and fluorescence spectrometry, fluctuations can occur in the primary light source, in scattered light, and in the flame background. Empirical flicker factors can be introduced to account for all fluctuations of either the signal intensity or the background intensity which occur prior to the radiation striking the photocathode. Depending on the source of the signal and background radiation, these may be simple or may be composed of several factors (8-13). Although the treatment here assumes independent signal and background flicker noise sources, as is the case with flame background flicker and hollow cathode flicker in atomic absorption, there may be dependent flicker noises, as occurs in molecular luminescence spectrometry (13). In this latter case, the rms noise will include a cross term whose magnitude depends on the degree of correlation of the two noise sources. In photon counting the variance in the number of signal pulses counted in a time interval t due to signal flicker noise, (uZNe,),, in the absence of pulse overlap is given by ( ~ 2 ~ , ,= ) ~
E2N20s= (A1Ras@)2
(22)
where
E
= =
ur, =
signal radiation flicker factor, dimensionless ur,lr8 standard deviation of signal photon arrival rate due to flicker noise over the measurement time, sec-1
Likewise the variance in the number of background pulses counted due to background radiation flicker noise, ( ~ 2 ~ , , ) , is given by ( ~ 2 N , ~ >= f {'N20b
= (A1Rabrt)2
(23)
where {
=
= uTb=
ground flicker, (uaEe),and Equations 24 and 25
background radiation flicker factor, dimensionless ur,/ro
standard deviation of background photon arrival rate due to flicker noise over measurement time, sec-
If pulse overlap is significant, expressions from reference (7) can be used in Equations 22 and 23. For a dc measurement, the corresponding variances in the signal and background output voltages due to signal and back-
(U2E,)l
( u * E , ) ~respectively, ,
are given by (24)
= (t&SR1)2
( u 2 ~ , )= f
(biuaRf)2
(25)
In some types of spectrometry where background radiation is not a significant factor, the signal radiation flicker factor can be readily measured. For example, in molecular absorption spectrometry, radiation flicker is usually due to fluctuations in the tungsten or deuterium lamp spectral radiance. The flicker factor can be measured by removing the monochromator and monitoring the source directly with a vacuum or solid state photodiode. Under conditions of high photon flux, the relative contributions of shot noise and other variances should be small compared to source flicker, providing an accurate estimate of 5. In other types of spectrometry, the flicker factors are more difficult to measure. In atomic absorption, for example, the contribution of the hollow cathode lamp or continuum source to the total radiation flicker can be measured in a manner similar to that for a molecular absorption source. However, the contributions of flame flicker and background instability can only be measured under conditions identical to the analysis since these factors depend on the wavelength and spectral bandpass settings of the monochromator. It is therefore difficult to obtain an accurate measurement of the total signal and background flicker because any measurement will include contributions from photomultiplier shot noise. Readout Noise. The readout device itself may add variance in the overall measurement. This variance can be attributed either to noise in the circuits of the readout device as in the amplifiers of an oscilloscope or servo recorder, or to the resolution or readability of the readout display. The latter source of variance is not noise in the sense of electrical noise, but does not contribute to the total variance and limit the maximum obtainable SNR. The total variance due to the , be expressed as readout device, u Z Rcan (T2R
= U2Re
+
(26)
U2Rl
where utRd= variance in the readout voltage due to electrical
noise in the readout device measured over Af, Vz dE1 = variance due to resolution of readout device, V z In the case of an analog readout device (scope, meter, recorder), ( ~ is 2a fraction ~ ~ of the total readout scale. If a digital readout device is used, there is no reading error; however, a resolution uncertainty of j=l count in the least significant digit remains. For photon counting there is a readout variance ~ 2 of . 1 count ~ ~ because the timing circuits are not synchronized with the pulses from the discriminator. Other Noise Sources in DC Measurements. If the signal from the current to voltage converter, E t , is further modified before reaching the readout device, additional variance terms must be added to the total variance to take into account noise from voltage and logarithmic amplifiers and noise present on top of any suppression signals used in scale expansion. Other Variance Sources in Photon Counting. If the photon counting circuitry is performing perfectly, the total observed variance in the numerical readout for photoelectron pulses is due to quantum noise, signal and background radiation flicker noise, and readout noise. The non-ideal operation of the comparator contributes another source of variance. Johnson noise and fluctuations in the gain of the discriminator, in the gain of the pulse amplifier, in the gain of the photoANALYTICAL CHEMISTRY, VOL. 44,NO. 4, APRIL 1972
*
789
measurement is made, as of dark current plus background, only one amplifier-readout variance term need be included for the measurement. For photon counting, the total variances in the signal counts, U ~ N . , , the background counts, uZNob, and the dark counts, are given by Equations 30-32.
/a / /
+
/
I
/
+ + (u2Nob>/ +
u’N..
=
( U ’ N ~ , ) ~ (U’N.,,)~
Q’N~
U’Nh
=
(U2Nob)p
u2d’,8
U’Nod.
=
f
(u2Nod)q
U2NR
+
+ +
(U’N.J~Z
(30)
(U’Nobb
(31)
(u2Nad)eZ
(32)
SIGNAL-TO-NOISE RATIO EXPRESSIONS
DC Measurement System. In the absence of background radiation and dark current, the SNR would be the ratio of the desired signal voltage to the square root of its variance. When dark current and background are present, however, two measurements are needed to extract the desired signal voltage E, from the total readout voltage Et. First, Et is measured and then the sum of the background and dark current output voltages (E8 Ed) is measured and subtracted from E,. Therefore, the measured SNR, (SNR)d,, is given by
+
0
/ 16
15
14
12
13
11
10
9
8
-Log ics
Figure 1. Effect of signal shot and flicker noise on the signal-to-noiseratio where
(a)Signal shot noise limited ( b ) =~ 10-4
&,
(c)t = lo-*
=
total variance in E l , V2
(d)S = 10-2
multiplier (uZmk), and in the discriminator reference voltage level will result in a source of variance above that already discussed. This additional source of variance is designated as (a2N,,)eZ and is the variance in the number of signal counts due to excess noise. Likewise for the background, a variance term ( ~ 2 is~introduced ~ ~ ) to~ account ~ for excess noise. The fluctuations due to excess noise are characteristic of l/f noise so that their relative contributions to the variance are greater for longer measurement times. The magnitude of the excess noise should be small for a stable system and dependent on the discriminator level. At low discriminator levels, excess noise variance is small, but is a maximum at discriminator levels where the rate of change of the observed count rate with respect to the discriminator level (dR,,/dA) is at a maximum. The dark pulse count has a similar term for excess variance, ( u ~ ~ , ~ )which , , , differs from that of the signal and background at a given discriminator level. Total Variance. For a dc measurement, the final variances of the signal voltage, uZEs,the background voltage, uZEb, and the dark current voltage, uZEd,are the sums of each individual variance term and are shown in Equations 27-29. U’E. =
+
(u’E,)~
(u2~,)aeo
+
(~’E.)Y
+
(~‘~,)pnt
+
U’AR
(27)
where U’AR
=
UP,
+ uZA+ usR = amplifier-readout variance, Vt
Equations 27-29 are complete expressions assuming separate measurements are made of E,, Eo, and Ed. If a combined 790
ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
Equation 33 is arranged so that 2 amplifier-readout variance terms appear in the final expression, one for the measurement of El and one for the combined measurement of E8 f Ed. The subtraction indicated in Equation 33 may be performed electronically by suppressing out the sum of the background and dark current voltages. However, Equation 33 is equally valid for manual or electronic subtraction since voltage suppression in this case is equivalent to a measurement. There are other useful SNR’s depending upon how the final measurement is carried out. For example, in molecular or atomic absorption spectrometry, the readout may be in transmittance, l/transmittance or absorbance, and a different SNR expression can be written for each case. By arranging the variance terms in Equation 33 into groups which show similar dependencies on the signal photocathodic current, ics, Equation 34 results.
ES
(SNR)dc=
+ b2 +
,Z)l/Z
(34)
where
+
a2 =
(U2E8)q
b2 =
(u’E,),~
c2 =
2(a2Eb
f
(b2E,)sec
U2Ed
+
(.2E,)pm
- CZAR)
The terms in u 2 are proportional to ics, those in b2 are proportional to P,,, while c2 terms are independent of its. If in a specific application, individual variances other than those considered are present or if cross variance terms exist, this useful classification should still be valid. For instance, if there is an interaction between background and signal flicker, the cross terms will be directly proportional to i,, and additive to the above expression for u2.
/
For many spectrometric applications, simplifications can be made in some of the variance terms. For example, in molecular absorption spectrometry, background radiation is usually negligible and c 2 would consist only of dark current and amplifier-readout variance. A full treatment of the special considerations involved in molecular absorption spectrometry is given elsewhere (27). In other applications, a knowledge of the predominate variance terms will allow considerable simplification of Equation 34. In most analytical spectrometric situations, light levels are high enough that photocathodic currents are greater than 10-le A. Under these conditions Johnson noise will be negligible (28). By selection of a high quality, low-noise operational amplifier, the contribution of amplifier noise can be made negligible with respect to other noise sources. For instance for a good OA, e , = 1 pV(Hz)-lfZ and i, = 1 ~ A ( H Z ) - ~making / ~ , the variance calculated from Equation 25 negligible in most cases. Also for a well regulated photomultiplier power supply, photomultiplier flicker noise will be small. Thus the total variance, and hence the SNR, is usually determined by shot noise from dark current, background and signal, signal and background radiation flicker noise, and readout noise. With these simplifying assumptions, the group variance terms become
+
a2 = (u2.&
(u2~8)8e =c
b2 =
+ +
c 2 = 2[(aZEb),
(azs.)f =
(g2Ea)aeo
(u2~,JaeCU 2 R l =
2meiasRfzAf(l
+4
(fiaaRf)2
+ (uzEb)f+ 4me(iab
/
/ A
3 U 0
a
z
2
8
A
2
1
(35a) (35b)
0 16
15
13
14
12
10
11
9
8
-Log i,
+ + iad)Rf2Af(l+ a)+ 2(&aR# + 2 0 ' ~ (35C) (g2&'d)q
Figure 2. Effect of dark and background current shot noise and background flicker noise on the signal-to-noise ratio (a) Signal shot noise limited
T o obtain Equation 35c, the assumption is made that y = m in Equations 13 and 16, which for most cases is an overestimate of the shot noise of i*ad. If the above group variance terms are substituted into Equation 34 and all terms are divided by m R f ,Equation 36 results
+ = A, r = 0 + = A, r 0 (6) + i,a = 10-14A, r = 0 (e) led + = 10-13A, r = 0 (b)icd
i,b
(c)
icb
icd ied
i,b
(f)i c b
=
=
A >> i c d , r
IO-'
iCs
2eMl
+ a)(ic8 + 2icb + 2icd + Ezizc8+ 2r2iZCa+
where icd
iad = - = effective cathodic dark current, A m
If the readout variance is negligible, Equation 36 indicates that (SNR)d, is basically independent of feedback resistance R f and photomultiplier gain m , although there is a slight dependence of a,icd, its, and ieb on photomultiplier gain. Figures 1 and 2 show plots of log(SNR)d, US. log i,, for different values of background and dark cathodic currents, and signal background flicker factors. In constructing the plots, it is assumed that the noise bandwidth Af = 1 Hz,the secondary emission factor a = 0.275, and u2R/m2Rf2is negligible. The dotted line in both figures indicates the signal shot noise limit ( a z >> bZ c 2 for all its). Figure 1 illustrates the effect of signal radiation flicker and hence the magnitude of b2 or f on the SNR. Figure 2 indicates the effect of the magnitude of c 2 variances (dark current and background shot noise and background radiation flicker noise) on (SNR)d, for different values of icd, icb,and {. The figures clearly illustrate the 3 limiting regions where the (SNR)d, is dependent on its, icsl/z,and independent of i,, and
+
indicate the importance of reducing the signal flicker factor 5 for high signal photocurrents and icd,icb, and { for low signal photocurrents. The two figures can be used in common spectrometric applications to estimate the SNR expected and the dependence of the SNR on the signal photocathodic current since order of magnitude values of the group variance terms can be derived from a few experimental parameters. For molecular absorption spectrometry, the photocathodic background current icb is negligible, and the simplified (SNR)d, depends only on ics, 5, and icd. The relatively high amounts of background radiation in many analytical flame spectrometric applications make the simplified (SNR)d, dependent on the photocathodic background current and the background radiation flicker factor in addition. Photon Counting System. The SNR expressions for photon counting are similar to those for dc measurements, but less complicated because the number of variance terms is smaller. The SNR for a photon counting measurement, (SNR),, including dark current and background is given by Equation 37
(27) J. D. Ingle, Jr., and S.R. Crouch, ANAL.CHEM., in press. (28) J. D. Ingle, Jr., and S. R. Crouch, ibid., 43,1331 (1971). ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
791
where
Not = Nos f Nab g z N , t = (T2No,
f
g2Nob
often considerably higher, signal and background flicker noise
+ Nod = total observed count
f
g2??o,j
-
will often be significant. COMPARISON OF PHOTON COUNTING AND DC MEASUREMENT TECHNIQUES
=
2g2N,
total observed variance in Not Equation 37 can be rewritten by grouping the variance terms according to their dependence on No, as Nos
(SNR), =
(38)
+ y z + z2]1/2
[x2
where x2 = y2 =
z2 =
(U2No,)0
+ 2[U2N0,+
(UZNo,)ez
(f12N.,)f
42N.d
-
U2Ny,1
The term in x2 is proportional to No,, those in y 2 are proportional to NZ0,,while z 2 terms are independent of Nos. Here spurious pulses caused by Johnson noise or amplifier noise are assumed to have been eliminated so that excess noise arises from fluctuations in photomultiplier gain, fluctuations in the pulse amplifier gain, or fluctuations in the discriminator level. Thus it is assumed that these fluctuations are small, and the variance due to excess noise is proportional to NZos. As was true with the dc measurement case, practical considerations can lead to the simplification of Equation 38. For example, the readout variance of 1 count is negligible to 1 if more than 100 counts are accumulated. Also, if nearly all pulses are counted so that A1 is near unity, the number of pulses passed by the discriminator is essentially independent of gain and discriminator level fluctuations, allowing the excess noise terms to be eliminated. Under these conditions, e (SNR)p
- (1
A & t
(SNR)dc -
(
)[A1Ra8t
The main advantages of photon counting usually mentioned are higher signal-to-noise ratio, discrimination against dark current not originating at the photocathode, direct processing of discrete spectral information, elimination of reading error, and system stability against drift. The merits of these advantages compared to dc measurement are discussed below. Photon counting has been shown to be most advantageous in comparison to dc measurements under low light level conditions where the S N R is near unity or less (19). Since the S N R s in most common analytical spectrometric applications are much greater than unity, the advantages of photon counting are not to be expected to be as great as for low light level measurements. SNR Comparison. To compare theoretical SNR's of photon counting with dc techniques, equivalent spectrometric system parameters up to the output of the photomultiplier tube are assumed so that ius = meRa8. Some confusion exists in the literature (14) in correlating the counting time, t , in photon counting to the noise equivalent bandwidth, Af, in a dc system. For equivalent conditions, Af must equal 1/(2t) (18). Of course if an integrating procedure is used in the dc case, the integration time should equal the counting time for equivalent noise bandwidths. If the readout variance, uzR,in the dc system is assumed negligible and l / ( 2 t ) is substituted for Afin the dc system: the ratio of Equation 40 to Equation 36 can be taken to indicate the improvement in S N R expected when using pboton counting.
1"'
+ a) (its +2icd + 2 i 4 + E2iZcsf 2t2i2,b
4-: f ( A i R ~ sf
AiRatc
+ AdR*ad) -k (AiRadW + 2(AiRabt{)2
(41)
Use of the simplified group variance terms in Equation 38 yields
If the signal and background photocathodic currents are converted to equivalent anodic pulse rates and the assumptions are made that A I = A d = 1 and icd = eRad, Equation 41 can be reduced to (SNR), (SNR)dc (Ras f 2Rab f 2Rad) ( 1 f a) f2R2as 2r2R2ub (Ras ~ R UfD2Rmi) f E2R2usf 2c2R2ub
Plots of log ( S N R ) , us. log No, or log i,, under equivalent noise bandwidth conditions would be quite similar in shape to those shown in Figures 1 and 2 for the dc system. However, Equation 40 holds only for count rates which are low enough that pulse pileup is negligible. Hence curves for photon counting would extend only to a photocathodic current of lo-'* A, which is the present upper limit for even a very fast counting system. Because of the necessity of operating at relatively low light levels where pulse overlap can be neglected, flicker in the signal and background will rarely become dominant noise sources. In some applications, for example molecular absorption spectrometry, where the signal radiation flicker factor for a modern system is usually less than 0.01 and background is negligible, Equation 40 can be further simplified. In flame spectrometry, where flicker factors are
Equation 42 gives a good indication of the S N R improvement to be expected from using photon counting. If flicker noises in the signal or background predominate, the first terms in the numerator and denominator of Equation 41 can be dropped, and the SNR's of the two techniques are equivalent. As was previously mentioned, this is not likely to occur because pulse pileup limits the maximum light level which can be measured in photon counting. At the other extreme, if the flicker noises are negligible, the last two terms in both the numerator and denominator can be dropped, which reveals a net advantage in S N R of for photon counting under shot noise limited conditions. Since the secondary emission factor Q: usually ranges from 0.1 to 0.5 for typical values of dynode gain 6, the S N R for photon counting is better by a factor of 1.05 to 1.22 under these conditions.
the group variance terms become X '
= (u~N,,),=
A1R.J
y 2 = ( u ~ =~ (A1Rast€)2 ~ J ~ z2 =
[(C2Nob)C
f
792
(39b)
+ + AiRatc + AtiR*ud f 2(A1Ruat()~(3%)
(a2Nob)f
2f[A1Rab
(39a)
(g2Nod)@l
=
ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
[
+
+
d G
Discrimination against Dark Current. Results of previous studies concerning the relative merits of using discrimination against dark current as a means to improve the SNR in photon counting systems (19, 29) are inconclusive. Clearly the effect of discrimination against dark current on the SNR depends on the relative magnitude of the dark current compared to the photocurrent and also on the sources of dark current. For a dc measurement, if ius iob is much greater than i a d , the shot noise of the dark current makes little contribution to the total shot noise. Thus, even if photon counting allowed discrimination against all dark current, the improvement in SNR would be negligible. The relative magnitudes of the different kinds of dark current, which are dependent on the size and nature of the photocathodic surface, are important because pulse discrimination will only improve (SNR), as A1 is decreased if the dark current pulse rate decreases much more rapidly than the photoelectron rate. Since htC has the same discriminator coefficient as Rat, if R,,, >> R*adrlittle improvement in SNR will result with pulse discrimination. If R*"d 2 Rut,, then pulse discrimination will significantly reduce R*=d, since A d decreases more rapidly than A1 if R*,d is composed of smaller pulses from down the dynode chain. Note that discrimination reduces the SNR by the discriminator coefficient A1 in the absence of dark current pulses, or if all the dark current pulses originate from cathodic thermal emission. Thus, under these conditions, the SNR advantage of photon counting is lost when A1 5 (1 a)+, Because of this, a small amount of discrimination may improve the SNR if R*,d is significantly reduced. However, too much discrimination will reduce Roasignificantly and reduce the SNR. In a dc measurement, dark current pulses which originate from down the dynode chain are inherently discriminated against since they receive an average gain which is lower than m, the full average gain. Hence, if the major contributor to i*,d is thermionic emission from down the dynode chain, rather than cold field emission or radioactivity, the contribution of i*ad to the total dark current shot noise in a dc measurement is inherently small. In photon counting, all pulses no matter what their origin are weighted equally in the absence of pulse height discrimination. Because of this, the SNR improvement for photon counting with no pulse height discrimination may be less than predicted by the factor. Thus, a small amount of discrimination is usually necessary to bring the level above the amplifier noise and to reduce the observed rate of small dark current pulses. If such factors as cosmic rays or after pulses are a large contributor to i*ad and i*ad is of comparable magnitude to iatc,pulse counting would be advantageous, since the large pulses are counted only once or can be eliminated by upper level discrimination. For most analytical applications of spectrometry, i, i,b is considerably larger than the total dark current and for commonly used photomultipliers, such as the RCA 1P28,therm.ionic dark current from the photocathode i,,, contributes approximately 90 of the dark current at room temperature (30). Thus, pulse discrimination in photon counting appears to offer little SNR advantage for most analytical spectrometric applications. Reading Error and Direct Digital Processing. If the readout device for the dc measurement system utilizes scale expansion and high resolution digital readout, the readout variance can often be made negligibly small. Under these
conditions, the absence of reading error in photon counting presents no real advantage, although there is certainly an advantage over the low resolution meter or recorder readouts which are still common in analytical spectrometers. The direct digital processing of data in photon counting is practically, as well as philosophically, an advantage. Although the dc measurement system is assumed to have digital readout, a number of data domain conversions (6) are necessary to obtain the final numerical readout. Therefore conversion errors, nonlinearities and the resolution of the A/D converter or the readout device can limit the ultimate measurement precision, while photon counting measurements show high linearity in the absence of pulse overlap. Measurement System Stability. Probably the greatest advantage of photon counting is the inherent system stability, which allows precise, long time integration to reduce the noise bandwidth. To increase the SNR by a factor of 10, the counting time must be increased by a factor of 100. By increasing the integrating time or reducing the bandwidth in a dc measurement system, the SNR can also be increased. Hower, drift and l/f noise in the gain of the photomultiplier, in the amplifier, and in the readout device or A/D converter can become limiting for small bandwidths. Thus, many of the variance terms that were dropped to obtain Equation 36 may be important at small bandwidths and should be included. Robben (14) has found that his photon counting system was a factor of 5 more stable than a dc system for long integration times (greater than 10 seconds). For photon counting, especially at low discriminator levels, drifts in the pulse heights due to photomultiplier or amplifier gain drifts have little effect on the readout. If deadtime compensation is utilized to extend the linearity in photon counting, A1 is approximately 0.5 (9,and the system will be more susceptible to drift. In many routine analytical spectrometric applications, when SNR's are reasonably high, the increased stability of photon counting is not a significant advantage. However, in some applications where low intensities are being measured near the limit of detection, the ability to increase the measurement time without drift may be of importance. Use of Modulation Techniques. In many analyticdl applications, modulation techniques are used for various reasons including SNR improvement. In flame atomic absorption and fluorescence, modulation of the source is often necessary to reduce the contribution of flame emission. Modulation also allows the use of ac amplification so that amplifier drifts become less important. Modulation techniques can also be used with photon counting by the use of up-down counters synchronized to the modulation frequency (31). Again, however, the unmodulated background radiation must not be so intense that pulse overlap occurs, or nonlinearity will result. When modulation techniques are utilized, many of the same SNR considerations previously discussed are valid, although modulation may suppress certain low frequency noise components. If a modulation system is used which has the same noise equivalent bandwidth as a conventional dc or photon counting system, noise whose power spectrum is white, such as Johnson and shot noise, will not be suppressed. Flicker noises which show greater amplitudes at low frequency can be suppressed by the appropriate modulation technique. In flame spectrometry, mechanical light chopping, pulsing of hollow cathodes, selective modulation of specific
(29) J. Rolfe and S. E. Moore, Appl. Opt., 9, 63 (1970). (30) R. M. Schaffer, RCA Corp., Lancaster, Pa., communication, 1971.
(31) F. T. Arecchi, E. Gatti, and A. Sona, Rea. Sci. Instrum., 31,942 (1966).
+
+
d G
+
personal
ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
793
absorption lines, and modulation of the aerosol stream (32) have been utilized. Such techniques can be used with either analog detection or synchronous photon counting, and the same conclusions discussed above as to the relative S N R s should hold, although photon counting should be of advantage for the detection of weak sources, where the long term stability allows very long measurement times. In molecular absorption spectrometry, double beam ac systems are often used to suppress low frequency source fluctuations and can likewise be used with synchronous photon counting. CONCLUSIONS
Under equivalent conditions, photon counting systems compared to dc measurement systems can provide the advantages of 5-22 higher SNR under shot noise limited conditions, greater stability, and better linearity, if pulse overlap is negligible. The significance of these advantages depends on the magnitude of the SNR. For applications where the dc SNR is inherently high with normally used noise bandwidths (it., 1 Hz), the above advantages are not very significant. However, under conditions where the SNR is small, the 5-22 % SNR advantage of photon counting can be highly significant, and the stability advantage can be used effectively to improve the SNR by increasing the counting time. Whether photon counting will show an improvement over dc techniques in a given spectrometric measurement depends to a large extent on the type of spectrometry (absorption, emission, or luminescence), the magnitude of the signal, the magnitude of the background, and the resolution required. Since the SNR in a given application depends upon many experimental and spectral parameters whose influence may differ from one application to another, it is difficult to generalize. The specific considerations involved in molecular absorption spectrometry are presented in a subsequent paper (27). The spectral experimental conditions necessary to optimize the SNR should be the same for both detection systems. If the (32) M. Marinkovic and T. J. Vickers, ANAL. CHEM., 42, 1613 (1970).
794
ANALYTICAL CHEMISTRY, VOL. 44, NO. 4, APRIL 1972
SNR is optimized (9) within the resolution requirements of the particular application, the choice of the better detection system is relatively simple. If the photoelectron pulse rate is in the region where pulse overlap is negligible [(RaS R,& < photon counting provides superior characteristics. Under conditions of significant pulse overlap, nonlinearity makes photon counting undesirable unless mathematical correction can be used. Once SNR optimization has been carried out, the photocathodic signal current should not be reduced (Le., by decreasing the signal radiation intensity) to bring the pulse rate into the linear range of photon counting. Such practice is unsound because the reduction in SNR, due to the reduction of its, will offset any advantages of photon counting. From the above considerations, photon counting appears to be most useful in analytical applications where signal and background radiation intensities are low and signal-to-noise ratios are small. Such applications may include nonflame atomic fluorescence spectrometry and molecular luminescence techniques. For nonflame methods where peak responses are obtained, the integrating and fast response characteristics of photon counting are advantageous. In flame emission spectrometry, the presence of background may prevent the use of photon counting techniques under conditions of optimum SNR. In atomic and molecular absorption spectrometry, light levels are often high enough under optimum conditions that photon counting is not the advantageous detection technique. However, for certain atomic and molecular absorption applications where very high resolution is desirable, such as atomic absorption with a continuum source and scanning molecular absorption, the low light levels encountered may make photon counting the more attractive measurement system.
+
RECEIVED for review September 24, 1971. Accepted December 21, 1971. Work partially supported by NSF Grant No. GP-18123 and an American Chemical Society, Analytical Division Fellowship sponsored by Perkin-Elmer Corporation.
