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Optimum pulse repetition rates for single photon counting experiments
Journal of Luminescence 29 (1984) 491—496 North-Holland, Amsterdam
491
OPTIMUM PULSE REPETITION RATES FOR SINGLE PHOTON COUNTING EXPERIMENTS Hans P. GOOD
*,
Alan J. KALLIR and Urs P. WILD
Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH - Zentrum, CH -8092 Zurich, Switzerland
Received 1 March 1984
In the first part of this paper, the optimum pulse repetition rate (PRR) for fluorescence lifetime measurements based on time correlated single photon counting is obtained by minimizing the variance in the lifetime estimates, assuming a single exponential decay. This allows us, in the second part, to calculate the optimum pulse repetition rate for systems possessing a set of discrete repetition rates; such as synchronously pumped mode locked cavity dumped laser systems.
1. Introduction Fluorescence lifetime measurements are usually performed by a repetitive excitation of a fluorescent sample with a short light pulse and measuring the resulting decay [1]. Small discharge lamps with a pulse width of some ns and pulse repetition rates (PRRs) of some 10 kHz have been extensively used in the past. The mode locked ion laser has more recently been applied and provides shorter pulses together with much higher PRRs. Combining a mode locked ion laser with a synchronously pumped dye laser results in a tuneable light source which has even shorter pulse duration. With such a device a pulse width of a few picoseconds is achieved. The PPR is in the order of 100 MHz. Adding a cavity dumper allows one to reduce this rate by integral factors. Furthermore, one can add an angle tuned frequency doubling crystal to produce UV light pulses, which are in the right wavelength region to excite most molecules of interest. Deconvolution and iterative convolution techniques have been previously applied to correct for the finite width of the excitation pulse and the apparatus response function. With these new laser systems the excitation pulse is much shorter (< 10 ps) than the response time of the electronics. Deconvolution and
*
Present Address: Balzers AG, FL-9496 Baizers, Fuerstentum Lichtenstein
0022-2313/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
492
H.P. Good et al.
/
Optimum pulse repetition rate for lifetime measurements
iterative convolution techniques are therefore only used to correct the apparatus response function [2,3]. Such a synchronously pumped, mode locked, cavity dumped, frequency doubled, dye laser system has been recently realized in our laboratory. The excitation pulse width FWHM (Full-Width-at-Half-Maximum) was always less than 10 Ps when measured with an optical picosecond correlator (Spectra Physics Model 409). These pulses result [4] in an apparatus determined excitation pulse width of: 140 ps, when measured with our single photon counting apparatus and 200 Ps, when measured with a fast photodiode and sampling oscilloscope [Spectra Physics 403 + Tektronix 7313 with S4 head respectively]. A set of typical decay curves using different excitation rates is shown in fig. 1. It is evident that if the lifetime is short with respect to the repetition period the decay curves are well separated and the usual methods of data analysis may be applied. However, if the lifetime is in the order or longer than the repetition period overlapping occurs. Since the experiment is periodic one is naturally lead to a Fourier analysis of the data. The overlapping of the decay curves is automatically taken into account in the Fourier analysis, eliminating problems associated with the “decay tails”. Such an analysis has been described in detail elsewhere [1,3].
RAN DATA
A
EXCITATION
FLUORESCENCE
B
EXCITATION
FLUDRE5C~NCE
c
EXCITATION
FLUORESCENCE
Fig. 1. Typical fluorescence decay measurements of Chrysen (300 K): (A) PRR = 76.6 kHz; (B) PRR = 3.83 MHz; (C) PRR = 76.6 MHz.
H.P. Good et a!.
/
Optimum pulse repetition rate for lifetime measurements
493
The problem addressed here is not isolated to the measurement of fluorescent lifetimes with laser, synchrotron or flashlamp excitation but common to any repetitive pulse measurement. Synchrotron excitation sources typically have a variable PRR in the range of 1—10 MHz with a pulse FWHM of 0.1—I ns [5] whereas a flashlamp system possesses a PRR range from 60 kHz down to 1 kHz and a pulse FWHM of about 1—2.5 ns [6]. In order to perform a mathematical analysis we assume an idealized excitation source with the following properties: (1) it has a continuously variable pulse repetition rate (PRR) (2) the energy per excitation pulse is constant and independent of the PRR. (3) the excitation pulse width may be neglected. To simplify the mathematics we assume that the impulse response function is a single exponential. The results of our deliberations may, however, be used as a basis for discussions concerning multicomponent decays. It is intuitively clear, that, as long as no overlapping occurs an increase in the PRR increases the number of “experiments” which may be performed in a given time T (fig. 1A). The signal-to-noise ratio is consequently improved by SQRT(PRR*T). A further increase in the PRR will result in overlapping response curves. If the overlap is too severe (fig. 1C) the data analysis can only deliver results with a large error. In the first part of this paper the optimum PRR is obtained by minimizing the variance in the lifetime estimates. We will work in dimensionless units and will assign the basic repetition period to 1. In these units a monoexponential decay which delivers a minimum error in the lifetime estimate, has a dimensionless lifetime x01,~. In commercially available systems the PRR are often switch selectable and, therefore, not continuously, but discretely variable. This also applies to mode locked, cavity dumped, lasers, where the cavity dumper acts as a “pulse picker” to divide the fundamental mode locked PRR by integral numbers [7,8]. Contrary to the continuously variable PRR, one selectable PRR will be Table I Repetition rate
Pulse separation
k
Optimal range
76.6 MHz 3.83 MHz 766 KHz 383 KHz 76.6 kHz 7.66 kHz 38.3 3.83kHz 766 Hz 383 Hz
13.1 ns 262 ns 1.319s 2.62us 13.1 tss 131 26.2 ~ss 262 ~zs 1.31 ms 2.62ms
1 20 100 200
r
3 10~ 2>< i0 2X104 iø~ 2x105
16.6 ns 16.6 ns < T 160 ns 160 ns
494
H.P. Good et al.
/
Optimum pulse repetition rate for lifetime measurements
optimal for a given range of lifetimes. In the second part of this paper these lifetime regions are calculated and illustrated by an example. Typical PRRs for such devices are listed in table 1. Recently Van Hock et. al [9] used an external electro-optic modulator as a pulse picker for mode locked lasers. Using this system in the dual pass configuration extinction ratios as high as 1 to 2 x i0~ could be achieved.
2. Continuously variable pulse repetition rate Consider a system with a variable PRR, v, and a fluorescent sample with fixed lifetime, T, which is to be measured during a given time, T. Each excitation pulse is, as mentioned before, assumed to have constant energy, independent of the PRR, v. The optimization is carried out in dimensionless units. The basic repetition interval (0,1) corresponds to a time interval of 1/v. The dimensionless lifetime of the monoexponential decay is given by x VT. All the experimental data are transformed into these dimensionless units. The data analysis provides an estimate of x. Such estimates have been easily obtained using the Fourier transform method or the iterative convolution method [1,3]. The relative standard error in x, is according to [3], given by =
(1) where NT is the total number of photons accumulated in the basic repetition period and is given by (2)
NT=PVT,
where P is the probability of counting a photon in a single interval. P is typically less than 0.01. ~(x) is a function that depends on the method of data evaluation used. ~(x) may be considered as a quality index for fitting techniques. We have previously compared ~(x) for the two most commonly used methods of analysis: the Fourier transform fitting and iterative convolution technique. Both methods have been found to have the same quality index [3]. Inserting eq. (2) into eq. (1) one finds ~x/x
2
=
~(x)x_1/2(PT/TY~
factor
=
x ~(x)x1~2.
(3)
The “factor” has an intuitive meaning; it represents the product of the total number T/T of the periods observed with the probability P of accumulating one count. In (3) we have shown that ~(x) is given by
exp[—1/x] x 2 — (I —exp[—1/x])
2
—1/2
(4)
H.P. Good et al.
/
495
Optimum pulse repetition rate for lifetime measurements
~(x)
1:0
x,xo~O4x,VT
2
=
Fig. 2. The normalized relative standard error ~x/x(PT/r)U = 0.087; x 0~,, = 0.261; x2 = 0.900.
~(x)x1”2
as a function of x:
The function 2—
X ~=factorXx~2
exp[—1/x] (1—exp[—1/x])
x
2
—1/2
has a unique minimum, at x 0~1 0.26 (see fig. 2) For all x values between x1 0.087 and x2 0.90 the normalized relative error is less than twice its minimum value. In the first case, the monoexponential has decayed, within the repetition period, to 10—~% of its initial value, whilst in the latter case it subsides to 33%. These results clearly support the intuition that the experimentors’ results should contain some overlap as long as it is not excessive. The‘r)1. optimum range 1 and (1.1 of pulse repetition rates thus lies between (11.5 T) =
=
3. Fixed discrete pulse repetition rates In the previous section we found the optimum pulse repetition rate for a given lifetime T. In this section we consider the directly relevant case of the optimum lifetime range for fixed discrete pulse repetition rates. From eq. (3) one can easily derive ~/T=
(k/PTv
2~(v 0)~
(6)
0T/k),
where v~is the maximum excitation rate of the apparatus and k an integer constant by which the maximum rate may be divided (see table 1). In fig. 3 the curves .~Tk/T are plotted for typical experimental situations. We assume a value of 0.01 for P a total accumulation time T of 100 s and a maximum repetition rate p0 of 76.6 MHz. The ~Tk/T functions for k 1, 20, 100 and 200 are displayed, on a semi-log scale. The minima of the ~Tk/T =
496
H.P. Good et al.
~1O~
/
Optimum pulse repetition rate for lifetime measurements
lIT
5
0.1
1
10
100
TinS)
Fig. 3. Absolute error of a fluorescence lifetime measurement as a function of the lifetime, r, for different pulse repetition rates.
1’~2.This is easily understandable in our model, since function increase of with k the total number counts accumulated is reduced by a factor k. As shown in fig. 3 the envelope of these curves defines the best experimental ranges. These ranges are independent of P and T. The optimum pulse repetition rate is easily found using fig. 3 and is also given in table 1.
4. Conclusion The methods discussed in this paper allow the calculation of the optimal pulse repetition rate (PRR) for experiments using single photon counting. The analysis shows that for a single exponential decay and a continuously variable PRR the optimum pulse repetition rate lies between (11T)1 and T1. For discretely variable PRRs examples are displayed in table 1. The calculations may be easily repeated for other experimental parameters.
References [1] UP. Wild, AR. Holzwarth and H.P. Good, Rev. Sci. Inst. 48 (1977) 1621. [21 D.V. O’Connor, W.R. Ware and J.C. Andre, J. Phys. Chem. 83 (1979) 1333. [3] H.P. Good, A.J. Kallir and UP. Wild, in preparation. [4] S. Canonica et al., in preparation. [5] B. Leskovr, CC. Lo, P.R. Hartig and K. Sauer, Rev. Sci. Inst. 47 (1976) 1113. [6] E. Gratton and R.L. Delgado, Rev. Sci. Inst. 50 (1979) 789. [7] K.G. Spears and L.E. Cramer, Rev. Sci. Inst. 49 (1978) 255. [8] V.J. KOster and R.M. Dowben, Rev. Sci. Inst. 49 (1978) 1186. [91A. van Hoek and A.J.W.G. Viser, Rev. Sci. Inst. 52 (1981) 1199.
491
OPTIMUM PULSE REPETITION RATES FOR SINGLE PHOTON COUNTING EXPERIMENTS Hans P. GOOD
*,
Alan J. KALLIR and Urs P. WILD
Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH - Zentrum, CH -8092 Zurich, Switzerland
Received 1 March 1984
In the first part of this paper, the optimum pulse repetition rate (PRR) for fluorescence lifetime measurements based on time correlated single photon counting is obtained by minimizing the variance in the lifetime estimates, assuming a single exponential decay. This allows us, in the second part, to calculate the optimum pulse repetition rate for systems possessing a set of discrete repetition rates; such as synchronously pumped mode locked cavity dumped laser systems.
1. Introduction Fluorescence lifetime measurements are usually performed by a repetitive excitation of a fluorescent sample with a short light pulse and measuring the resulting decay [1]. Small discharge lamps with a pulse width of some ns and pulse repetition rates (PRRs) of some 10 kHz have been extensively used in the past. The mode locked ion laser has more recently been applied and provides shorter pulses together with much higher PRRs. Combining a mode locked ion laser with a synchronously pumped dye laser results in a tuneable light source which has even shorter pulse duration. With such a device a pulse width of a few picoseconds is achieved. The PPR is in the order of 100 MHz. Adding a cavity dumper allows one to reduce this rate by integral factors. Furthermore, one can add an angle tuned frequency doubling crystal to produce UV light pulses, which are in the right wavelength region to excite most molecules of interest. Deconvolution and iterative convolution techniques have been previously applied to correct for the finite width of the excitation pulse and the apparatus response function. With these new laser systems the excitation pulse is much shorter (< 10 ps) than the response time of the electronics. Deconvolution and
*
Present Address: Balzers AG, FL-9496 Baizers, Fuerstentum Lichtenstein
0022-2313/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
492
H.P. Good et al.
/
Optimum pulse repetition rate for lifetime measurements
iterative convolution techniques are therefore only used to correct the apparatus response function [2,3]. Such a synchronously pumped, mode locked, cavity dumped, frequency doubled, dye laser system has been recently realized in our laboratory. The excitation pulse width FWHM (Full-Width-at-Half-Maximum) was always less than 10 Ps when measured with an optical picosecond correlator (Spectra Physics Model 409). These pulses result [4] in an apparatus determined excitation pulse width of: 140 ps, when measured with our single photon counting apparatus and 200 Ps, when measured with a fast photodiode and sampling oscilloscope [Spectra Physics 403 + Tektronix 7313 with S4 head respectively]. A set of typical decay curves using different excitation rates is shown in fig. 1. It is evident that if the lifetime is short with respect to the repetition period the decay curves are well separated and the usual methods of data analysis may be applied. However, if the lifetime is in the order or longer than the repetition period overlapping occurs. Since the experiment is periodic one is naturally lead to a Fourier analysis of the data. The overlapping of the decay curves is automatically taken into account in the Fourier analysis, eliminating problems associated with the “decay tails”. Such an analysis has been described in detail elsewhere [1,3].
RAN DATA
A
EXCITATION
FLUORESCENCE
B
EXCITATION
FLUDRE5C~NCE
c
EXCITATION
FLUORESCENCE
Fig. 1. Typical fluorescence decay measurements of Chrysen (300 K): (A) PRR = 76.6 kHz; (B) PRR = 3.83 MHz; (C) PRR = 76.6 MHz.
H.P. Good et a!.
/
Optimum pulse repetition rate for lifetime measurements
493
The problem addressed here is not isolated to the measurement of fluorescent lifetimes with laser, synchrotron or flashlamp excitation but common to any repetitive pulse measurement. Synchrotron excitation sources typically have a variable PRR in the range of 1—10 MHz with a pulse FWHM of 0.1—I ns [5] whereas a flashlamp system possesses a PRR range from 60 kHz down to 1 kHz and a pulse FWHM of about 1—2.5 ns [6]. In order to perform a mathematical analysis we assume an idealized excitation source with the following properties: (1) it has a continuously variable pulse repetition rate (PRR) (2) the energy per excitation pulse is constant and independent of the PRR. (3) the excitation pulse width may be neglected. To simplify the mathematics we assume that the impulse response function is a single exponential. The results of our deliberations may, however, be used as a basis for discussions concerning multicomponent decays. It is intuitively clear, that, as long as no overlapping occurs an increase in the PRR increases the number of “experiments” which may be performed in a given time T (fig. 1A). The signal-to-noise ratio is consequently improved by SQRT(PRR*T). A further increase in the PRR will result in overlapping response curves. If the overlap is too severe (fig. 1C) the data analysis can only deliver results with a large error. In the first part of this paper the optimum PRR is obtained by minimizing the variance in the lifetime estimates. We will work in dimensionless units and will assign the basic repetition period to 1. In these units a monoexponential decay which delivers a minimum error in the lifetime estimate, has a dimensionless lifetime x01,~. In commercially available systems the PRR are often switch selectable and, therefore, not continuously, but discretely variable. This also applies to mode locked, cavity dumped, lasers, where the cavity dumper acts as a “pulse picker” to divide the fundamental mode locked PRR by integral numbers [7,8]. Contrary to the continuously variable PRR, one selectable PRR will be Table I Repetition rate
Pulse separation
k
Optimal range
76.6 MHz 3.83 MHz 766 KHz 383 KHz 76.6 kHz 7.66 kHz 38.3 3.83kHz 766 Hz 383 Hz
13.1 ns 262 ns 1.319s 2.62us 13.1 tss 131 26.2 ~ss 262 ~zs 1.31 ms 2.62ms
1 20 100 200
r
3 10~ 2>< i0 2X104 iø~ 2x105
16.6 ns 16.6 ns < T 160 ns 160 ns
494
H.P. Good et al.
/
Optimum pulse repetition rate for lifetime measurements
optimal for a given range of lifetimes. In the second part of this paper these lifetime regions are calculated and illustrated by an example. Typical PRRs for such devices are listed in table 1. Recently Van Hock et. al [9] used an external electro-optic modulator as a pulse picker for mode locked lasers. Using this system in the dual pass configuration extinction ratios as high as 1 to 2 x i0~ could be achieved.
2. Continuously variable pulse repetition rate Consider a system with a variable PRR, v, and a fluorescent sample with fixed lifetime, T, which is to be measured during a given time, T. Each excitation pulse is, as mentioned before, assumed to have constant energy, independent of the PRR, v. The optimization is carried out in dimensionless units. The basic repetition interval (0,1) corresponds to a time interval of 1/v. The dimensionless lifetime of the monoexponential decay is given by x VT. All the experimental data are transformed into these dimensionless units. The data analysis provides an estimate of x. Such estimates have been easily obtained using the Fourier transform method or the iterative convolution method [1,3]. The relative standard error in x, is according to [3], given by =
(1) where NT is the total number of photons accumulated in the basic repetition period and is given by (2)
NT=PVT,
where P is the probability of counting a photon in a single interval. P is typically less than 0.01. ~(x) is a function that depends on the method of data evaluation used. ~(x) may be considered as a quality index for fitting techniques. We have previously compared ~(x) for the two most commonly used methods of analysis: the Fourier transform fitting and iterative convolution technique. Both methods have been found to have the same quality index [3]. Inserting eq. (2) into eq. (1) one finds ~x/x
2
=
~(x)x_1/2(PT/TY~
factor
=
x ~(x)x1~2.
(3)
The “factor” has an intuitive meaning; it represents the product of the total number T/T of the periods observed with the probability P of accumulating one count. In (3) we have shown that ~(x) is given by
exp[—1/x] x 2 — (I —exp[—1/x])
2
—1/2
(4)
H.P. Good et al.
/
495
Optimum pulse repetition rate for lifetime measurements
~(x)
1:0
x,xo~O4x,VT
2
=
Fig. 2. The normalized relative standard error ~x/x(PT/r)U = 0.087; x 0~,, = 0.261; x2 = 0.900.
~(x)x1”2
as a function of x:
The function 2—
X ~=factorXx~2
exp[—1/x] (1—exp[—1/x])
x
2
—1/2
has a unique minimum, at x 0~1 0.26 (see fig. 2) For all x values between x1 0.087 and x2 0.90 the normalized relative error is less than twice its minimum value. In the first case, the monoexponential has decayed, within the repetition period, to 10—~% of its initial value, whilst in the latter case it subsides to 33%. These results clearly support the intuition that the experimentors’ results should contain some overlap as long as it is not excessive. The‘r)1. optimum range 1 and (1.1 of pulse repetition rates thus lies between (11.5 T) =
=
3. Fixed discrete pulse repetition rates In the previous section we found the optimum pulse repetition rate for a given lifetime T. In this section we consider the directly relevant case of the optimum lifetime range for fixed discrete pulse repetition rates. From eq. (3) one can easily derive ~/T=
(k/PTv
2~(v 0)~
(6)
0T/k),
where v~is the maximum excitation rate of the apparatus and k an integer constant by which the maximum rate may be divided (see table 1). In fig. 3 the curves .~Tk/T are plotted for typical experimental situations. We assume a value of 0.01 for P a total accumulation time T of 100 s and a maximum repetition rate p0 of 76.6 MHz. The ~Tk/T functions for k 1, 20, 100 and 200 are displayed, on a semi-log scale. The minima of the ~Tk/T =
496
H.P. Good et al.
~1O~
/
Optimum pulse repetition rate for lifetime measurements
lIT
5
0.1
1
10
100
TinS)
Fig. 3. Absolute error of a fluorescence lifetime measurement as a function of the lifetime, r, for different pulse repetition rates.
1’~2.This is easily understandable in our model, since function increase of with k the total number counts accumulated is reduced by a factor k. As shown in fig. 3 the envelope of these curves defines the best experimental ranges. These ranges are independent of P and T. The optimum pulse repetition rate is easily found using fig. 3 and is also given in table 1.
4. Conclusion The methods discussed in this paper allow the calculation of the optimal pulse repetition rate (PRR) for experiments using single photon counting. The analysis shows that for a single exponential decay and a continuously variable PRR the optimum pulse repetition rate lies between (11T)1 and T1. For discretely variable PRRs examples are displayed in table 1. The calculations may be easily repeated for other experimental parameters.
References [1] UP. Wild, AR. Holzwarth and H.P. Good, Rev. Sci. Inst. 48 (1977) 1621. [21 D.V. O’Connor, W.R. Ware and J.C. Andre, J. Phys. Chem. 83 (1979) 1333. [3] H.P. Good, A.J. Kallir and UP. Wild, in preparation. [4] S. Canonica et al., in preparation. [5] B. Leskovr, CC. Lo, P.R. Hartig and K. Sauer, Rev. Sci. Inst. 47 (1976) 1113. [6] E. Gratton and R.L. Delgado, Rev. Sci. Inst. 50 (1979) 789. [7] K.G. Spears and L.E. Cramer, Rev. Sci. Inst. 49 (1978) 255. [8] V.J. KOster and R.M. Dowben, Rev. Sci. Inst. 49 (1978) 1186. [91A. van Hoek and A.J.W.G. Viser, Rev. Sci. Inst. 52 (1981) 1199.