Estimating Detection Limits in Ultratrace Analysis. Part III: Monitoring Atoms and Molecules with Laser-Induced Fluorescence

feature articles

Estimating Detection Limits in Ultratrace Analysis. Part IIh Monitoring Atoms and Molecules with Laser-Induced Fluorescence*

C H R I S T O P H E R L. S T E V E N S O N t and J A M E S D. W I N E F O R D N E R ~ Department of Chemistry, University of Florida, Gainesville, Florida 32611-2046

Laser spectroscopic methods which are sensitive enough to detect single atoms or molecules are termed single-atom detection (SAD) techniques. The detection theory of SAD methods, presented in the second paper in this series and summarized here, allows for the evaluation and comparison of these techniques. In this paper, SAD theory is applied to one task in particular: monitoring analyte atoms or molecules as they flow through a continuous-wave laser beam using fluorescence detection (CW- LIF). Several possible signal-processing methods are described and eval- uated through the use of SAD theory and Monte Carlo simulations. The relative merits of these methods for SAD by CW-LIF are illustrated.

Index Heading: Detection limits.

INTRODUCTION

The use of lasers in analytical spectroscopy has fre- quently lead to methods of very high sensitivity and selectivity. This combination can result in techniques which can detect individual atoms or molecules as they interact with the laser beam; thus many laser-based techniques may be capable of single-atom detection (SAD), where the term "atom" here refers to any possible form of the analyte species (atom, molecule, ion, or radical).

As the number of reports of SAD in the literature continues to grow, the importance of, and potential applications for, methods capable of SAD has become ap- parent2 ~ In physical chemistry, these applications might include the observation and measurement of transport phenomena of individual species; the observation of gas- phase reactions of individual atoms or molecules; ~ the study of spectroscopic properties of individual atoms/ molecules in various environments (information which is usually available only as an ensemble average); 6-~° the sorting or "tagging" of atoms; ~ and the observation of the clustering of analyte species2 ~ Some more conventional analytical applications might include the following: the detection of very low bulk concentration of an-

Received 12 September 1991. * This research supported by DOE-DEOFG05-88ER13881 with partial

support by Eastman Kodak (Rochester, NY). t Current address: Advanced Monitoring Development Group, Health

and Safety Research Division, Oak Ridge National Laboratories, Oak Ridge, TN 37831-6101. Author to whom correspondence should be sent.

alyte, where the limiting "noise" on the signal is due to the statistical appearance of individual atoms in the laser; the detection of very rare isotopes; 11-13 and the detection of low concentrations of analyte in difficult ma- trices through extensive sample dilution. Finally, in a related application, it has been suggested that an SAD technique based on molecular fluorescence can be used in the human genome project2 4

This three-part series of papers is concerned with laser-based "ultratrace" analysis, techniques which are so sensitive that it may be possible to detect very low numbers of analyte atoms in the laser beam. In the second part of this series, 15 the definition for a true SAD method was given: SAD is possible only when each and every atom which interacts with the laser during a measurement can be distinguished from the background noise. Some of the relevant concepts of SAD techniques will now be summarized; the original paper should be consulted for details.

A typical SAD experiment for a flowing system, as treated in this paper, will now be described; relevant terms and their definitions are collected in Table I for reference. Analyte atoms flow past a region in which it is possible to interact with the laser beam and produce detected photons; this region is known as the probe volume, Vp. The total number of detected events, It, during the measurement time, Tin, is counted. The number of atoms probed by the laser during the measurement is denoted Np. Each of the Np atoms has a certain nonzero interaction time with the laser, t;, as well as a value for ¢~, the mean flux of detected events during the atom's interaction with the laser. Thus, the mean signal of a given atom passing through the laser beam is given by

{, = ¢,t, (1)

where [, = mean number of events due to an atom passing through Vp. The value of t~ and ¢, may vary for each of the Np atoms (depending, for example, on the path of a particular atom through Vp); the total mean signal per atom, [,, for the measurement is given by

L = ~,~, (2)

where ~, = mean value of ~, for atoms which can enter

Volume 46, Number 5, 1992 0003-7028/92/4605-071552.00/0 APPLIED SPECTROSCOPY 71,5 © 1992 Society for Applied Spectroscopy

TABLE I. List of some symbols used in SAD model and their defini- tions2

Symbols Definitions

I b

i°

L

N~

N~

RSD~ RSD~ T~ t~

t, y,

yo

X, z,

~d

ttb

O't 2

Ti

Tr

Number of background counts observed in a measurement (counts).

Number of signal counts from a particular atom (with given Cs and t~ values) during a measurement (counts/ atom).

Number of signal counts from any atom entering laser during a measurement (counts/atom).

Total number of counts (background and signal) observed during a measurement (counts).

Number of atoms which interact with the laser during a measurement.

Number of atoms measured from It for a measurement (i.e., best estimate of Np).

Precision of counting Np atoms during a measurement. Signal precision due to Np atoms during a measurement. Duration of a single measurement (s). Interaction time of a given atom with the laser beam in

the probe volume (s). Residence time of a given atom in the probe volume (s). Volume probed by the laser beam which can give rise to

counts at the detector (m3). Volume of analyte which flows through Vp during a mea-

surement (m~). Signal detection limit (counts). Guaranteed detection limit in signal domain (counts). Probabili ty of false positive (type I error) during a mea-

surement. Probability of false negative (type II error) during a mea-

surement. Detection efficiency, the probability tha t an atom which

enters Vp will give rise to a signal detectable above the noise.

Mean flux of background counts (counts/s). Mean flux of signal counts due to a given atom in Vp

(counts/s). Mean signal flux of any atom which can enter Vp (counts/

s). Mean number of background counts during a measure-

ment (counts). Signal variance of I~ during a measurement (counts2). Mean interaction time of an atom with the laser in Vp (s). Mean residence time of an atom in Vp (s).

a Table taken from Ref. 15.

Vp, and ri = mean value of ti for atoms which can ent~" yp.

The signal It is comprised of two parts, due to the contribution of the signal due to Np atoms, and that of the background, Ib. A signal due to the analyte atoms can be said to be detected only if It > Xd, where X~ is the signal detection limit. The value of X~ is set by the probability distribution of Ib and the desired value of a, the probability of type I error (false positive).

Given the above framework for an SAD experiment, the requirement for SAD as defined can be stated as:

E~ >- 1 - fl (3)

where fi = the probability of type II error (false negative), and e, = the detection efficiency of an atom, which interacts with the laser, above the background noise. The detection efficiency is defined by the probability that any given atom will be detected above the background:

e, =- P(I , + Ib >- Xd). (4)

The acceptable values of a and fl for an SAD technique

must be decided before the experiment, and would usually be governed by the application.

Besides serving as a criterion to achieve true SAD, the value of E~ can be used to evaluate and compare the detection power of methods as they approach this ulti- mate limit in sensitivity. We will assume a shot noise limit on the background noise throughout this paper. Thus, the value of e~ for a given technique will depend on a number of factors:

1. The value of Cb, the mean flux of background counts during the measurement, and the probability distribution of lb.

2. Tin, the total measurement time for a single measurement.

3. The probability distribution and mean value of Is.

One of the most promising methods of achieving SAD is by using continuous lasers with nondestructive LIF detection (CW-LIF). This method has been used in the past to detect single molecules 1G-2° and atoms 21-2~ as they flow through the laser. The evaluation of the detection capabilities of CW-LIF near SAD conditions must be carefully considered. The purpose of this paper, the final installment in the three-part series, applies the theory of SAD developed previously 15 to the specific problem of monitoring the atoms which pass through the laser beam by using CW-LIF. Some simple signal-processing methods will be applied to this task, and SAD theory and computer simulations will be used to compare and evaluate these methods, as well as to show clearly the conditions necessary to achieve true SAD.

T H E O R Y : CONTINUOUS MONITORING OF ATOMS WITH LIF

S c o p e of an S A D M e t h o d . The requirements for an SAD method have been summarized; as an illustration of what an SAD method might signify in a more conventional analytical situation, let us assume that the flow of atoms of a possible SAD technique can be directed so that every atom in a given sample interacts with the laser. If this technique is capable of SAD, does this mean that the method possesses infinite detection capabilities? In other words, can any concentration of analyte in the sample be detected?

The general SAD model summarized above presents the requirements for an SAD method for a given measurement time, Tin. The meaning and limitation of this requirement should be very clear: a method which is truly capable of SAD is capable of detecting above the background noise every atom which crosses the laser during Tin. Thus, in partial answer to the above question, it is not possible to simply state that a given technique is an "SAD method" without stating the conditions under which this is possible--most particularly all experimental parameters which might affect e, (and/or X~). A simple numerical example with SAD by pulsed LIF will illustrate this point and serve to answer the above ques- tions.§

§ It will be assumed that the number of detected events (i.e., photoelectrons above a discriminator level) can be counted unambiguously. In reality, this achievement would not be trivial with typical pulsed dye laser experiments.

716 Volume 46, Number 5, 1992

Let us imagine the interaction of an atomic beam with a pulsed dye laser, where the following conditions apply: each atom interacts with the laser for one laser pulse only, ~ is equivalent for every atom which enters Vp, and the detection is limited by background shot noise. In such a situation, it is reasonable to assume that I~ is a Poisson variable. Let us further imagine that during a single laser pulse it is found that the mean background, ttb, is 0.25 counts; i.e., there is an average of one count every four laser pulses due to the background. During a single laser pulse (Tin is the pulse duration), we can say that Xd = 3 counts (a = 0.00216) and Xg = 10.8 counts (/~ = 0.00143), where X~ = I~ + tt b, the value off~ necessary to achieve SAD. Thus, we see that if i~ = 10.55 counts/ atom then SAD is possible during a single laser pulse. This means that, if Np > 1 atom, it is always possible (with probability 1 - ~) to detect a signal from the atom(s) above the background during one laser pulse.

Thus, this is a true SAD technique: an analyte atom in any given laser pulse can be reliably detected above the background noise during that pulse. Now suppose that a given sample is analyzed and will take 10,000 laser shots to completely flow past the laser. Again, assume that every atom in the sample interacts with the laser for time ti. Is it possible to reliably detect a single atom in the sample?

The answer is no, because the above method can achieve SAD only for Tm equal to the duration of a single laser pulse. When T~ is changed to 10,000 laser shots, the requirements for SAD will change. If Xd = 3 counts is used as a criterion to distinguish the presence of analyte atoms from the background noise, then, from the value of a given earlier for one laser pulse, we see that during 10,000 shots there will be an average of 216 false positives. Obviously it would be impossible to detect a single atom in the sample with this value for Xd.

With the new value of T~, we see that we must choose Xd high enough to ensure that a is at the desired value. When ttb = 0.25 counts/pulse and the background follows a Poisson distribution it can be calculated that

PUb > 7) = 9.734 x 10 -9

so that, for 10,000 shots with Xd = 7 counts, a = 9.73 x 10 -4, an acceptable level. This value of Xd gives Xg = 18 counts (B = 1.04 x 10-3); the requirement to detect a single atom in the sample is that ~ = 17.25 counts/atom, instead of 10.55 counts/atom.

This simple illustration shows that a given SAD method has a certain "scope," a certain value of T~ over which the technique can reliably detect a single atom above the background noise. Beyond this measurement time, the method can no longer detect single atoms with the desired values of a and ~. For example, at sensitivities between 10.55 and 17.25 counts/atom, the above method will be capable of SAD for somewhere between 1 and 10,000 laser pulses. If the sensitivity of the method is high enough, the scope of the SAD method (measurement time over which SAD is possible) may be so long as to be practically infinite. In other words, X~ can be set so high (while still satisfying Eq. 3) that it is extremely unlikely that Ib will ever exceed Xd during a single laser pulse, no matter how many pulses are counted.

Signal Processing Methods. In the majority of the reports of the detection of individual atoms, the S/N ratio for the atoms is small (S/N = 1-3), and the techniques can usually be described as "near-SAD" methods (0.1 < ~d < 1 -- /~). The optimum detection and subsequent evaluation of the method, in terms of the SAD theory summarized earlier, require careful consideration. This section will describe some possible signal processing methods for the continuous detection and monitoring of atoms entering the laser during Tin; two of the methods presented in this section, which are more efficient for processing near-SAD methods, will be applied to computer simulations of SAD by CW-LIF to demonstrate and evaluate their performance in terms of SAD theory.

Most methods based on atomic or molecular fluorescence of bulk analyte in a sample solution, in the simplest case, will integrate the signal for the measurement time, and the sum (or its normalized analog, the average) will be the measurement value. Such an approach can be applied to an SAD method based on CW-LIF. Appli- cation of the SAD theory when using this signal processing method is relatively straightforward: the distribution of Ib from many blank measurements is used to set Xd (chosen for the pre-defined tolerance to false positive detection), the value of e~ is estimated for the method, and Eq. 3 is used to determine whether SAD is possible during Tin.

This method of simple integration over Tm is very inefficient when Tm >> ri, since the mean signal due to a single analyte atom will remain at &sri while the mean blank value will increase linearly with Tin. In addition, there is no information on the exact length of time the atom is in the laser beam; it is only known that the atom entered Vp sometime during Tin. Nevertheless, there may be certain situations in which the simplicity of the method has its advantages. In the analysis of discrete samples which flow quickly through the laser beam (e.g., LIF detection of analyte atoms atomized in a furnace), if the sensitivity is very high, then SAD may be possible when Tm is chosen so that the entire sample is analyzed. In such a situation, of course, the only possible improvement in the method is to increase experimental parameters such as the atomization efficiency or the size of Vp over which SAD is possible; the detection process itself within Vp is already perfect during the analysis time.

Far more efficient, yet conceptually simple, methods for the analysis of continuously flowing analyte solution through Vp are based on the use of a time "window" of length tw, applied repeatedly during T~. Three such methods will be presented here. Their common feature is that only the number of photoelectron counts within the window tw is considered at any one time, and the duration of tw is chosen to be of the magnitude of the atom's mean residence time, r , in Vp or smaller. Thus, the S/N ratio due to single analyte atoms is increased.

Sequential Application of Time Window (tw < r,). It may be that the sensitivity of the CW-LIF method is high enough to detect the presence of a single atom even when tw = rr/10. The S/N due to a single atom decreases (relative to a situation in which tw = rr) as t~ gets smaller; however, if SAD is still possible by Eq. 3, then this is the best method to use. Such situations have only rarely been reported in the literature. T M

APPLIED SPECTROSCOPY 717

detected p h o t o n s

gate o n

- - ~ t w

.J "1

a t o m

e n t e r s

v

t i m e > FIG. 1. Demonstration of the photon burst method. A single detected photoelectron triggers the gate to open for a set time, tw. The number of counts detected within tw, including the trigger count, constitutes the burst signal. Photon bursts due to atoms, such as the one shown, will be larger than those due to background noise for (near-) SAD methods.

The application of this method is simply the use of sequential integrations for time tw for the entire measurement time Tin. In choosing the value of Xd for t~, it should be remembered that Xd should be high enough to ensure that the total number of false positives from Tm/tw windows during the measurement will be at the desired tolerance level. The disadvantage of this method is that higher sensitivity is required to achieve SAD than is required by either of the next two methods which use a time window. However, if this sensitivity is available, then the method is very simple to use and can count atoms in real time as they traverse the laser beam. For CW-LIF techniques which are on the borderline of be- coming a true SAD method, and hence do not have a sensitivity high enough for this method, one of the two methods described below can be used.

Photon Burst Method (t~ ~ T,). This method has been reported in the literature as a method of improving S/N when one is recording spectra of single atoms in an atomic beam. 23,25 Figure 1 demonstrates how the "photon burst" method may be applied to CW-LIF. A single photoelectron count triggers on the gate, which is pre-set to count the photoelectrons for the gate open time tw. The number of photoelectrons (including the trigger pulse) constitutes the "signal" during tw. The gate duration is set so that tw ~ rr. The photon burst method can be imple- mented during real time so that bursts at or above a pre- set value (i.e., Xd) can signal the presence of an analyte atom in the laser beam. Alternatively, the number of counts from all the photon bursts can be stored in a computer and analyzed later; only the bursts with a sum equal to or greater than Xd can be considered due to analyte atoms. All the normal requirements given previously for an SAD method apply; the advantage of the "photon burst" method over simple integration is that only the noise during t~ can contribute to false positives; consequently a lower value of Xd can be chosen, and the sensitivity necessary for SAD is not as high.

Sliding Sum Method (t~ ~ T,). The "sliding sum" signal-processing method is an intuitively obvious tech-

nique which has been previously applied to CW-LIF. TM

After the raw data are collected, which consist of the counts as a function of time during Tin, a data transformation is applied in which the value sum of the number of counts over a width tw is assigned to the middle of the time window. The window is then moved one step and the process is repeated. For continuous monitoring of atoms with CW-LIF, the step size should be a fraction of the residence time of the atom within Vp; e.g., At rr/20. Smaller step sizes are of course better, but only up to the point where the signal-processing step becomes too long. The optimum value for the duration of the moving sum is tw ~ % (and no longer than the largest possible residence time).

The sliding sum peak maximum will occur when tw and tr exactly coincide. Assuming tw > tr, then this sum will contain the entire analyte detected signal; this value can be used as a signal of the presence or absence of the analyte atom. The distribution of peak maxima will follow the distribution of It; thus, application of the SAD theory from the previous section is straightforward. When a sliding sum peak maximum equals or exceeds Xd, then the peak is presumed to be due to the presence of an atom in Vp. In a similar manner, peak maxima can be used to evaluate the detection efficiency and determine whether the method is truly SAD.

A problem with using the peak value as an indicator o f / t will occur when the analyte concentration is high enough for problems with peak overlap to occur. Using sliding sum peak heights to count Np during Tm is prac- tical only when there is a very small probability that such an overlap will occur. One method of alleviating this problem is to simply use the integrated peak area of sliding sum peaks as the "signal"; when the residence times of two atoms overlap, the resulting peak will be longer and the area will be the sum of the contributions of the signal due to both atoms. Each photoelectron count in the raw data array results in a contribution of tw/At counts in the resulting sliding sum peak area as the window is moved past the count, where At is the step size,


as before. Thus, a cluster of I~ counts (due to analyte atoms and background noise) will result in a peak area (tw/At)It. The areas from all sliding sum peaks can be divided by this factor before one decides whether the peaks are due to background noise or analyte atom(s).

The advantage of using peak area detection instead of peak heights is that, with no loss of information or detection efficiency, higher concentrations of analyte can be analyzed. The total peak area from overlapping atoms is a measure of the number of atoms which contributed to the peak. The theory of the precision of counting atoms, RSD~, developed previously ~5 can be directly applied in this situation in order to count the number of atoms which contribute to a given sliding sum peak; i.e., it will be possible to precisely count the number of atoms contributing to a given peak (in addition to the total number, Np, which pass through Vp during T~) if the sensitivity is high enough.

EXPERIMENTAL

Monte Carlo simulations of a CW-LIF experiment were based on an experiment in which the analyte species flow through the probe volume and interact with the CW laser. The purpose of the simulations was to apply two signal-processing methods discussed in the previous section, the photon burst and sliding sum methods, to the task of monitoring atoms using CW-LIF. The general simulations were very similar to those presented in the previous paper, 15 which should be consulted for more details. All simulations were carried out on IBM-com- patible personal computers which were equipped with either a 25-MHz 80386 CPU and an 80387 math co- processor chip, or a 20-MHz 80286 CPU with an 80287 co-processor. All programs were written and compiled in Microsoft QuickBASIC (Version 4.5, Microsoft Corp., Redmond, WA). Algorithms to generate random numbers according to either a normal or Poisson distribution were based on guidelines presented by Knuth 2s and are based on QuickBASIC's pseudo-random number gen- erator.

Simulations used a model in which both ~ and ti were fixed and constant for all atoms which entered Vp; all measurements were limited by background shot noise. Thus, both Is and Ib were Poisson variables, and values of Xd and Xg were calculated through the use of tables of the Poisson probability distribution, 27 by the procedure which was described previously25 All signals were in counts, and arbitrary time units were used throughout. By way of a basis of comparison to past SAD methods, it has been reported previously that "t" r = 10-100 #s for atomic beam experiments 4 and "/ ' r : 1-10 ms for SAD experiments with flowing solutions29,2°

The purpose of the simulation was to generate a raw data file, which simply consisted of the detected photon counts as a function of time. The raw data file consisted of both background and signal counts. The construction of the file proceeded in the manner described below.

The total number of background counts was chosen randomly from a Poisson distribution with mean q~bT~; in all simulations, q~b = 0.05 counts/unit time, and T~ = 8000 time units. After the number of background counts was generated, these counts were randomly distributed in time throughout the T~.

TABLE II. Observed probability of false positives using the photon burst or sliding sum signal-processing method, with t~ = 21 time units. The numbers in the columns are the average of 4000 loops of the simulation with #, = 400 counts and Tm= 8000 time units.

Observed false positives/measurement

Xd (counts) Photon burst Sliding sum 5 3.628 5.383 6 0.693 1.124 7 0.107 0.197 8 0.014 o.029 9 0.001 0.003

The entry of a given atom into Vp was randomly chosen to occur sometime during the measurement with uniform distribution, so long as the atom both entered and exited Vp during Tin. In all cases, tr = 20 time units. The number of signal photons was a Poisson variable of mean ~b~tr; once this number was retrieved, the signal photons were randomly distributed during the atom's residence in Vp.

Once the raw data file was generated according to this procedure, any signal-processing method could be chosen with an appropriate value of tw. The evaluation of these signal-processing methods was carried out by comparing the number of atoms (Np), which actually entered the probe volume, and the number of atoms detected ("positives") at or above the chosen value of Xd. The number of false positives which were expected for various values of Xd were observed with Np = 0; the detection efficiency for given sensitivity values could be determined by comparing the number of times atoms were actually detected (with the expected number of false positives subtracted) and the known total number of atoms analyzed.

RESULTS AND DISCUSSION

S c o p e o f S A D M e t h o d : C h o o s i n g Xd. The scope of an SAD method in the presence of noise is represented by the value of Tm over which it is possible to detect single atoms at the required detection efficiency. When simple integration over Tm is used, the required sensitivity necessary for SAD increases rapidly. For all simulations, Tm = 8000 time units, and t, = 20 time units. Thus, for the use of simple integration, at the intrinsic limit, the necessary sensitivity for SAD is 6.6 counts/atom at the 99.9 % significance level; when #b is increased to only 10 counts (¢b = 0.0125 counts/unit time), then Xd = 22 counts, and a sensitivity of 29 counts/atom is necessary for SAD (99.9 % confidence level). The background level used in all simulations reported here is ~b = 0.05 counts/unit time, or an average of 400 counts during the entire measurement; in this case, Xd = 460 counts, and a sensitivity of approximately 130 counts/atom would be necessary to achieve true SAD with the same confidence if the signal were simply integrated over Tin. Obviously, with Tm >> tr, this method is very inefficient.

Table II shows the dramatic improvement of using either the photon burst or the sliding sum method for signal processing. This table shows the average number of false positives per measurement with 4000 loops of the simulation at Np = 0 atoms (i.e., 4000 "blank" measurements) for the use of the two signal-processing techniques. From the results it would appear that Xd should be set at 9 counts; the observed values of a with Xd at


TABLE III. Distribution of the number of detected photon bursts in simulations of CW-LIF. A burst was "detected" if it contained at least 9 counts. One thousand measurements were taken with Np = 1 atom.

Number of ~ = 13.3 counts/atom ~ = 20 counts/atom "detected"

photon bursts Observed Expected Observed Expected

0 102 53 7 1 1 862 947 867 999 2 36 0 124 0 3 0 0 2 0

this level are sufficiently low for our purposes (it can be calculated that a = 0.00203 for the photon burst method, and a = 0.00372 for the sliding sum method when Xd = 9 counts).

Several observations concerning the table should be made here. First of all, if it were assumed that Tm= tr (as in the pulsed LIF numerical example given earlier) then Xd would be set at 5 counts, and 13.3 counts/atom would give an SAD method. However, increasing the value of Tm to 8000 units would not give an SAD technique at the same significance level, since, with the two methods shown in Table II, Xu must be increased to 9 counts. At this value of Xa, it is anticipated that a sensitivity of 20 counts/atom would give SAD with /3 = 0.00209, assuming that Is follows a Poisson distribution; this value is obviously a great improvement over the sensitivity necessary (i.e., 130 counts/atom) if simple integration over Tm were used. Finally, notice that the probability of false positives is roughly the same for the two methods, although the photon burst technique is slightly superior, giving the same value of Xd for the conditions of the simulations.

The Photon Burst Method. Simulations of CW-LIF were performed to construct the raw data file; the photon burst method was then used on this array of detected photons during Tin. The value of Np was fixed at 1 atom for every measurement--in other words, one single atom entered and exited V, sometime during Tin. During the course of a measurement, a burst was considered "detected" if there were at least 9 counts in the burst; as seen in Table II, it should be very unlikely that the background noise level would cause a burst to be detected at this level during Tin.

Table III shows the expected and observed results for 1000 measurements at two different sensitivities. The distributions of the number of detected bursts from the 1000 measurements are shown, along with the expected values. The first sensitivity level would result in an SAD method if Tm= tr and Xu = 5 counts, as described previously. The use of this sensitivity in the simulation clearly demonstrates the limited scope of SAD at that sensitivity: even though SAD at this sensitivity is possible with Tm = L, fully 10% of the simulations reported no atoms detected. The second sensitivity corresponds to the value of Xg with Xd = 9 counts (~3 = 0.00209), assuming 1, is a Poisson variable. Thus, this sensitivity would result in near-certain detection of the atoms in all 1000 measurements if the burst values follow a Poisson distribution.

In comparing the expected and observed values, it ap- pears that the value of ,d is somewhat less than expected for both cases, and there are a number of"double counts," in which there are two detected photon bursts rather than the expected one due to Np alone. The expected number of false positives is very low (1-2 for 1000 measurements), as has already been demonstrated. Thus, the

a t o m

e n t e r s

Vp

d e t e c t e d

p h o t o n s

g a t e o n 2~ ,.I

(a)

d e t e c t e d

p h o t o n s

.I g a t e o n -']

II

>

(b)

t i m e > FIG. 2. Possible complication with photon burst signal processing. (a) Normal operation of photon burst, in which a single detected burst contains the entire signal detected due to the analyte atom. (b) Complication: splitting of the signal burst due to background noise count.


T A B L E IV. Distribution of the number of sliding sum peaks in simulations of CW-L IF . A peak was "detected" if the peak height was at least 9 counts. One thousand measurements were taken with Np = 1 atom.

N u m b e r of

"de tec ted" -/s = 13.3 c o u n t s / a t o m [, = 20 c o u n t s / a t o m sl iding s u m

peaks Observed Expec ted Observed Expec ted

0 58 53 1 1 1 942 947 983 999 2 0 0 16 0 3 0 0 0 0

double bursts and decreased Cd must both be due to the presence of the analyte atom in some manner.

Figure 2 shows an explanation for the discrepancy. The upper portion of the figure is a reproduction of Fig. 1 and is typical of the cases when the entire detected signal is contained in the burst. Obviously, in the particular situation shown in Fig. 2a, if the sensitivity is high enough for SAD, then the probability is high that this particular burst would be detected, since tw contains all the detected photons due to the atom. However, a single spurious background count can result in the spitting of the photon burst due to the analyte atom; an example is shown in Fig. 2b. In the situation shown in the figure, the signal due to the atom is spread out over two separate bursts due to the single noise count. In such situations, depending on the number of signal photons emitted and the exact positioning of the time windows, there is a possibility either that both bursts containing the signal are detected or that neither one is. At higher sensitivities, it is more likely that the divided signal will result in a "double count" than in no detected burst at all; on the other hand, at lower sensitivities, there is a greater chance that the analyte atom will not be detected at all, since the signal would be spread over two bursts. The results shown in Table III reflect both these trends.

Thus, the use of the photon burst method comes with complications. The effect of the "double counts" may not be too critical if the analyte concentration is so low that it is unlikely that two analyte atoms enter Vp so close together. In such a situation, upon inspection of the data, it might be assumed that two closely spaced bursts, such as represented in the figure, were due to single atoms (and would be counted once). However, the problem would be expected to become worse with both increased noise, which would cause a greater incidence of burst splitting, and increasing analyte concentration.

Sliding Sum: Peak Height Detection. One thousand measurements with Np fixed at 1 atom were run at the same two sensitivity values that were used for the photon burst method, and sliding sum signal processing was used on the raw data file. Sliding sum peaks with height > Xd were considered to be due to the presence of an atom in the laser. The results and expected values (same as in the photon burst case) are given in Table IV. Unlike in the case of the photon burst method, there seems to be no problem with decreased values for ~d, indicating that peak heights tend to follow a Poisson distribution. In the case of the higher sensitivity, however, there are still a few double counts. In almost all cases, these extra peaks were found to be due to a "shoulder" on the peak due

150

125

100

75

50

25

0 8

Sliding Sum Peak Heights D i s t r i b u t i o n a b o v e X d

" ~ ~ . (p. = 0 , 4 5

= I i I i I i I i I T ~ " ~ - - - ' - ~

10 12 14 16 1B 20 22

Peak Heights FIG. 3. Distribution of detected sliding sum peak heights. One thousand simulations were run under the following conditions: Np = 1 atom; ub = 1.05 counts during tw,; Tm= 8000 time units, and is = 9 counts/ atom (resulting in e~ = 0.544). The theoretical Poisson distribution of the detected photons at or above 9 counts is also shown (represented by the solid line).

to the analyte atom, in which the value fell below Xd and then increased again due to background counts outside of t~. In these cases, it was obvious that the peak was due to only a single atom present in Vp, and the double counts were more an artifact of the signal-processing algorithm used to count "detected" peaks. Such instances can easily be eliminated by visual inspection or by more sophisti- cated processing of the transformed data.

Unlike the photon burst method, the sliding sum peak maxima can be used to form a picture of the signal distribution due to single analyte atoms. If the analyte concentration is kept very low, then it is unlikely that there are two atoms in V~ at any one time. In such a situation, all the sliding sum peak maxima at or above Xd will be due to single atoms. At the peak maximum, tw will be centered so that the window contains all of t , and the distribution of maxima at or above Xd provides insight into the distribution of Is. Figure 3 shows such a distribution for a simulation of a near-SAD CW-LIF method: f8 = 9 counts/atom, ub = 1.05 counts during tw, so that the distribution of peak maxima should have a Poisson distribution with a mean of 10.05 counts. Both the observed and expected distributions are shown in the figure, and they match quite well.

Sliding Sum: Peak Area Detection. For either of the two signal-processing methods demonstrated thus far on CW-LIF, problems occur as the analyte concentration increases to the point where it is likely that two (or more) atoms are in Vp at the same time. In the case of sliding sum processing, the peak height will no longer give an indication of the number of atoms which contribute to the sliding sum peak. Figure 4 illustrates a case in which the residence times of two different atoms within Vp overlap, and shows the resulting sliding sum peak which would occur (note that background noise was eliminated for clarity). It is easily understood from inspection of the figure that the peak height will be a function of the degree


a t o m 1 : I

[ 2 3 c o u n t s I II I I I : = [ [, J ~

t~

a t o m 2:

1 7 c o u n t s

: i I i I

3 5 Q5

== 3O

O u 2 5

N 2 0

~1 15

u 10

* ~

o

,, J ,=l,,r==== t

. e e

o •

o e e

e

o o

• * e e o e

o o • * o

0 * * e o e o e

~ = : = 1 : 1 ~ l ' I ' : : ;

0 l 0 2 0 3 0 4 0 5 0 6 0

T i m e ( a r b i t r a r y u n i t s ) FIG. 4. The effect of overlapping residence times of atoms on the sliding sum peaks. The top portion shows the detection times of emitted photons due to two atoms, along with their residence times in Vp. The bottom portion shows the resulting sliding sum peak; conditions were the same as in Table IV (i, = 20 counts/atom), but Ib = 0 counts for clarity.

of overlap of the two atoms. Equally obvious is the fact that, at bulk concentrations of analyte where there is significant probability of two or more atoms residing in Vp at the same time, the distribution of peak heights will deviate markedly from a Poisson distribution. Thus, even when it is obvious from the fact that the height is greater than 9 counts that Np > 1 atom, the peak height value will not give a good indication of the number of analyte atoms which give rise to the sliding sum peak. However, integrating the entire area of the sliding sum peak allows an estimate of the number of atoms contributing to the peak by the following formula:

where the RND function rounds the expression in the parentheses to the nearest whole number (since the number of atoms must be an integer), and Nm is the best estimate for the number of atoms contributing to the sliding sum peak. The precision of this estimate has been examined in some detail. 1~ The value of Is in the above equation is calculated from the sliding sum peak area by dividing the area by the factor tw/At. In the example shown in the figure, the integrated area is 840 counts, yielding Nm = 2, the correct value.

7 2 2 Volume 46, Number 5, 1992

TABLE V. Comparison of peak height and area detection for sliding sum signal processing as the analyte concentrations increases. N . represents the best estimate of Np; the columns represent the average values of N . for 1000 simulations fixed at the value of Np given in the first column. Sensitivity was 20 counts/atom (all other conditions as for Table iv).

Np (atoms)

Nm (atoms)

Height detection Area detection

25 22.665 25.102 50 40.900 50.046 75 55.602 75.006

100 67.008 99.947

In order to investigate the advantages of using sliding sum peak areas for counting atoms, 1000 CW-LIF simulations were run with increasing values of Np. Table V shows the results from four different analyte levels when either peak height detection or peak areas were used to estimate Np for each simulation after the sliding sum transformation was applied to the raw data. The table reports the average values of Nm, the number of atoms measured in a simulation, for the 1000 "measurements." For peak height detection, Nm was simply equal to the number of peaks with heights at or above Xd (9 counts); no at tempt was made to deduce the number of atoms contributing to a peak from the value of the peak height. The sliding sum peak areas were used to calculate It for use in Eq. 6; ttb was calculated from the known value of ¢b and the total duration of the sliding sum peak. Thus, Eq. 6 is used to calculate the number of atoms contributing to each peak, and this number was summed at the end of the simulation.

From the table, the limitations of peak detection as used here are obvious. Using the sliding sum peak areas compensates for the overlap of the atoms' residence time in Vp with only a moderate increase in computation time. Since there is no theoretical decrease in ed when one is using peak areas instead of heights, Table V represents the application of an SAD method over two orders of analyte concentration. Of course, there is no reason why sliding sum peak areas cannot be used for even higher concentrations; however, as concentrations increase further, there will come a point where the use of simple integration of the raw data over Tm would be preferred over sliding sum signal processing.

Compar i son of Signal-Processing Methods. There have been a total of four possible signal-processing methods mentioned thus far in this work which can be applied to SAD methods-- the "simple integration" method (with T~ > tr); and the three time window methods: the sequential time window (with tw ~ tr/20), the photon burst, and the sliding sum (peak or area) methods. It is difficult to state unequivocally which of these is "better," since the performance is highly dependent on the conditions and the requirements of the experiment to be performed.

The process of simply integrating the signal during Tm may be desirable for small discrete samples or for the application of SAD methods to larger analyte concentrations, as has been mentioned. The advantage of simplicity is obvious; in addition, the signal can be processed in real time. However, any desired temporal information

P

F

of the entry/exit times of individual atoms into Vp is restricted by the duration of Tin.

If the sensitivity of the SAD method is very high, then the possibility of sequential application of small time windows over Tm should be investigated. If SAD is possible during tw, then the atoms can be monitored in real time as they cross through Vp. Obviously, this is the best situation one can hope for in an SAD method.

For those methods in which the sensitivity is low enough to recommend increasing tw to about the level of t,, two possible approaches have been addressed in this section, and some of their strengths and weaknesses have been demonstrated. The two main advantages of the photon burst method are the fact that it can be a real-time method 2~ and that it is fairly uncomplicated. However, the application of this method to higher levels of analyte concentration or background noise is complicated by the presence of double counts due to single atoms. In addition, there is some loss in detection efficiency of near- SAD methods due to the "splitting" of an atom's photon burst.

The sliding sum method is a fairly powerful signal- processing method to apply to near-SAD techniques. There is no loss of detection efficiency, and information can be gained about the distribution of the analyte signal due to single atoms. In addition, the use of peak areas allows the application of this method to atom counting, even at relatively high concentrations of analyte. The main disadvantages are that the method is computa- tionally more complex than any of the others--particularly for long measurement t imes--and that the method cannot be used in real time.

Mention should be made of one other signal-processing method which has been used to detect photon bursts due to single atoms: autocorrelation of the raw data collected o v e r Tin. 19'2° Use of an autocorrelator is relatively simple and yet supplies good evidence of SAD if the analyte concentration is low enough to ensure that there is no more than one atom in Vp at a given time. Another advantage of this method is that the autocorrelogram can supply a good estimate of Ti if Tm is long enough and the noise is low enough. However, the application of the autocorrelogram to measuring Nm is not straightforward, 19 and, as in simple integration, the entry time of the atom into V~ during Tm cannot be determined.

There was no intent in this work to attempt to cover all possible signal-processing methods which can be applied to the raw data from an SAD experiment using CW-LIF. There are many more complicated signal-processing schemes or moving filters that might be useful. The intent in this paper was merely to demonstrate that, for all of these, the detection theory presented for the general SAD model ~5 is still applicable in these instances.

CONCLUSIONS

It must be emphasized that SAD techniques, as defined here, can detect single atoms which enter Vp during the measurement; this ability must be differentiated from the conventional concept of detection of bulk analyte concentration/amount. It is interesting to imagine the performance of an SAD method in a more conventional sense--i.e., in terms of bulk detection limits, rather than

N,. Comparison of the detection limit of resonance ionization spectroscopy (RIS) and laser-induced atom fluorescence in an electrothermal atomizer (ETA-LIF) demonstrates that, even though RIS has traditionally been an SAD method, the detection power of the two techniques is very similar. 28,29 The reason for this ap- parent disparity is that the detection power of a technique depends on the overall efficiency of detection, Eo, rather than ed, as explained originally by Alkdemade2 ° The value of eo depends on such factors as duty cycle and atomization efficiency, in addition to ed.

Thus, the performance of a method with respect to Ed and eo must be kept firmly in mind when one is applying an SAD method to conventional analysis. To further illustrate this point, imagine that a given CW-LIF technique can achieve SAD over a certain measurement time using simple integration of the signal during the measurement, limited by background shot noise. Further increasing the measurement time will result in e, < 1 - fl; thus, SAD can no longer be achieved. However, the bulk concentration detection limit will improve with (t)'/2, even though SAD is no longer possible over the measurement time.

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Documents

Estimating Detection Limits in Ultratrace Analysis. Part III: Monitoring Atoms and Molecules with Laser-Induced Fluorescence