J. Phys. Chem. C 2008, 112, 8079–8084
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Comparison of Time and Frequency Domain Methods for Luminescence Lifetime Measurements† Christina M. McGraw,‡,⊥ Gamal Khalil,‡ and James B. Callis*,‡,§,⊥ Department of Chemistry, UniVersity of Washington, Seattle, Washington 98195 and Department of Bioengineering, UniVersity of Washington, Seattle, Washington 98195 ReceiVed: December 18, 2007; ReVised Manuscript ReceiVed: February 19, 2008
Despite the widespread use of luminescence lifetimes in physical measurements, there seems to be no consensus on the best method for lifetime determination. Researchers in this field split into two camps: those who favor methods that proceed from time-domain data and those who favor methods that proceed from frequencydomain data. System theory provides unique insight into the strengths and limitations of these methods of lifetime determination. First, system theory shows that the impulse response experiment and the frequency sweep experiment can both uniquely specify a shift-invariant linear system. Further, the time and frequency data sets are related by the Fourier transform. A careful comparison of these two methods was carried out with both numerical simulations and experiments using the same sample, light source, and detector. These comparisons were not limited to impulse and sine wave excitation; chirp, square wave, and random excitation sequences were evaluated as well. The results indicate that for the same total excitation energies time-domain sequences with a substantial dark period have the lowest uncertainty in the lifetime parameter estimates. I. Introduction Luminescence measurements are widely used for chemical sensing and imaging applications. The fluorescent or phosphorescent lifetime of a molecule is an intrinsic parameter that can be determined using methods that proceed either in the time domain or in the frequency domain. Conventional time-domain methods excite the luminophor using a short duration pulse of light, and the lifetime is determined from the time-dependent intensity of the sample. Conventional frequency-domain measurements excite the sample with sinusoidally modulated light at a known frequency. The emitted luminescence is intensity modulated at the same frequency but with a phase delay. The lifetime can be determined from both the depth of modulation and the phase delay of the emitted radiation as a function of frequency. System theory shows that the transfer function of the frequency-domain method and the impulse response of the time-domain method are equivalent representations of the luminescence response of a molecule.1 Therefore, in a noisefree environment, the time- and frequency-domain methods of lifetime determination should be equal. Very few studies have been undertaken to compare the two methods of lifetime determination. An experimental comparison of the two methods was performed by Gratton et al. for lifetime imaging applications.3 Although the same laser excitation light source and sample were used in the comparison, different detectors were used for each method. The results of this comparison indicated that the time-domain method was better suited to measurements made at low intensities, while the frequency-domain method was better when the measured luminescent intensity was high. However, the signal-to-noise † Part of the “Larry Dalton Festshcrift”. * To whom correspondence should be addressed. E-mail: callis@ u.washington.edu. ‡ Department of Chemistry, University of Washington. § Department of Bioengineering, University of Washington. ⊥ Current address: Department of Chemistry, University of Otago, Dunedin, New Zealand.
ratio (SNR) of the time-domain method was severely limited by saturation of the detector at high intensity values. A more general comparison that focuses on nonimaging applications requires the use of the same detectors so that differences in saturation are not an issue. A cooperative project between nine international research groups led to the repeated lifetime measurement of twenty fluorescence standard/solvent combinations.4 The fluorescence lifetime of each standard was determined using both time and frequency-domain methods. All groups used laser excitation, although the detection schemes varied. The authors found good correlation between the time- and frequency-domain results and no difference in the precision of the two methods. Recent studies have shown that the frequency-domain results can be improved if the sinusoidally modulated excitation source is replaced by a train of delta functions or square waves.5–7 Philip and Carlsson performed a theoretical comparison and optimization of different methods of frequency-domain lifetime measurements.5 The authors found that when the exciting light was sinusoidally modulated the uncertainty in the lifetime estimate was significantly higher than the uncertainty in the lifetime estimate from previously published time-domain studies. However, when the sinusiodally modulated exciting light was replaced with a train of delta functions or narrow square waves, the noise level of the frequency-domain decreased significantly and approached that of the time domain. The authors showed that the traditional frequency-domain measurement based on sinusoid excitation was not optimal. This is consistent with other studies that have shown that frequency-domain methods that combine gated detection with square wave or impulse train excitation reduces the errors caused by short-lived fluorophores and scattered light.6,7 In this study, a careful comparison of the time and frequencydomain methods of lifetime determination was carried out using numerical simulations and experiments. The time- and frequencydomain simulations were limited by photon counting noise. The comparison was not limited to pulsed and sine wave excitation;
10.1021/jp711867u CCC: $40.75 2008 American Chemical Society Published on Web 04/30/2008
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chirp and random excitation time sequences were evaluated as well. The experimental comparison, which was also limited by photon counting noise, was carried out using the same sample, light source, and detector. This was possible using ultraviolet light-emitting diodes (UV LEDs). LEDs are low-cost excitation sources that can be sinusoidally modulated for measurements in frequency domain or modulated in square waveform for measurements in the time domain.8 Because fluorescent and phosphorescent measurements are generally limited by photoncounting noise, the results should be applicable to most detection systems. II. Theory In the analysis of a system using a state variable approach, the system is characterized by a set of first-order differential equations that describe the temporal evolution of the state variables. This collection of state variables describes a system’s dynamics. For the determination of luminescent lifetimes, the state variables and their transition rates can be defined from the time-dependent transitions between the ground and excited states. For experiments in the time domain, the luminophor’s timedependent response to a short pulse of light is monitored. If the luminophor is excited by a δ-function, the observed luminescent response is a single exponential decay
Ip(t) ) Ioe-t/τ + Bo
(1)
where Io is the initial intensity, t is the time, τ is the fluorescent or phosphorescent lifetime, and Bo is the background signal. For experiments in the frequency domain, the luminescence response to sinusoidal excitation is monitored.9 If the exciting light is modulated at circular frequency, ω,
E(t) ) Eo(1 + sin ωt)
(2)
the resulting luminescent emission signal will be
Ip(t) ) Io(1 + m(ω)sin(ωt + φ(ω)))
(3)
where φ(ω) is the phase angle, and m(ω) is the modulation factor. If the luminescent decay is a single exponential, the fluorophore’s lifetime is easily determined from the phase shift or modulation factor
tan φ(ω) ) ωτ 2 2 -1⁄2
m(ω) ) (1 + ω τ )
(4a) (4b)
Together, these two equations define the system transfer function. According to systems theory, the impulse response and the transfer function are equivalent representations of a linear system. In fact, the time- and frequency-domain methods of lifetime determination are related through the Fourier transform.2 The real component of the Fourier transform of an exponential decay gives the modulation factor and the imaginary component gives the phase angle.2 Therefore, these two measurements of lifetime should be equivalent in a noise-free environment. III. Experimental Methods A. Simulations. A numerical simulation was developed to compare the time- and frequency-domain methods of lifetime determination for a luminescent molecule with a 1.000 ms lifetime. The three time-domain sequences used in this experiment were (a) square wave, (b) chirp, and (c) maximum length sequence (MLS), a pseudorandom binary sequence. The square wave excitation is defined as
x(t) ) 1 x(t) ) 0
for 0 e t e ε for ε < t e 2 × ε
(5a) (5b)
where ε defines the light-on and light-off periods, 10 ms in this simulation. The chirp, a swept-frequency cosine excitation, is defined as
z(t) ) cos (ω(t)t + φ)
(6)
where ω(t) is the time varying circular frequency
ω(t) ) ω0 + at
(7)
and φ is the phase. The range of frequencies used for the chirp function was 28.5-2489 Hz. An MLS is a pseudorandom sequence of 0’s and 1’s that was calculated using a shift register. One unique property of the MLS is that the cross-correlation of the MLS excitation sequence with the output of the system reproduces the impulse response.10 The simulation was performed using MATLAB (Mathworks, Natick, MA). Phosphorescence response curves were created from the convolution of the various excitation signals with a single exponential decay. Each simulated response contained 1023 data points at evenly spaced time intervals. The number of data points was constrained by the inclusion of the MLS. An MLS has a length of 2N - 1, where N is an integer. For consistency, the length of all time-domain sequences was set to 1023 (210 - 1). Photoelectron counting noise was added to each response curve using random numbers from a Poisson distribution. Background noise, 0.5% of the maximum intensity and described by a Poisson distribution, was also added to each response. The lifetime parameter, τ, the initial intensity, Io, and the background noise level, Bo, for each response were recovered from the simulated data. Data was fit using an iteratively reweighted nonlinear least-squares fit of the luminescent response. Weights were inversely proportional to the fitted values as is consistent with Poisson-limited data.11 For the frequency-domain simulation, the luminophor was modulated with sine wave excitations at 20 frequencies, evenly spaced on the log scale, between 281 and 7911 Hz. The phosphorescence response at each frequency was determined from the convolution of the sine wave excitation and a single exponential decay. The phosphorescence response at each modulation frequency was monitored for 200 ms, although the first 5 ms of each excitation sequence was discarded to allow for the establishment of equilibrium. Background and photon counting noise was added to each response using the method described above. The phase and modulation of the luminophor at each excitation modulation frequency was determined using a simulated phase sensitive detector (PSD). The PSD extracts the frequency-dependent modulation, m(ω), and phase angle, φ(ω), from the observed phosphorescent response using sine and cosine modulated reference signals:
() S1 S2
(8)
m(ω) ) √S21 + S22
(9)
φ(ω) ) tan-1
where S1 is the low-pass filtered product of the observed phosphorescence and a reference cosine signal, and S2 is the low-pass filtered product of the observed phosphorescence and a sinusoidally modulated reference signal.12 A PSD simulator was designed in LabVIEW (National Instruments, Austin, TX) and used to determine the φ(ω) and m(ω) at each modulation
Comparison of Methods for Luminescence Lifetime Measurements
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Figure 1. Simulated excitation waveforms (gray) and 200 phosphorescence responses (black) to square wave excitation (a), MLS excitation (b), and chirp excitation (c). Simulated excitation waveforms and 400 averaged responses to MLS excitation with a dark period (d) and chirp excitation with a dark period (e). Simulated excitation waveforms and 200 averaged responses to a short-pulse square wave excitation (f). The residuals of the fit of the phosphorescence responses are also shown for the square wave (g), MLS (h), chirp (i), MLS with a dark period (j), chirp with a dark period (k), and short-pulse square wave (l).
frequency. A fourth-order Butterworth filter was used for lowpass filtering. The cutoff frequency, which was equal to half the modulation frequency, was chosen to reduce the higher harmonics that can lead to errors in PSD calculations. The PSD values for φ(ω) and m(ω) were plotted versus modulation frequency, and the lifetime was calculated by fitting the data to eqs 4a,b using the Nelder-Mead simplex method of nonlinear minimization. To ensure a fair comparison, the simulated time- and frequency-domain excitation waveforms must have equal total excitation energy over the time course of the experiment. An LED is limited in the average power it can deliver, so many time-domain excitation pulses were averaged to equal the total excitation energy of the 20 modulation frequencies used in the frequency-domain simulation. Specifically, it was necessary to sum 200 of the 20 ms time-domain responses to equal the total excitation energy delivered during the 4000 ms frequencydomain excitation sequence. Each simulation was repeated 50 times to obtain a mean and standard deviation in the estimate of τ, Iο and Bο. B. Experimental Comparison. A system was developed to experimentally compare the time-domain and frequency-domain methods using the same samples, light sources, and detectors. Data acquisition was carried out on a PC equipped with National Instruments data acquisition board (PCI-6110, Austin, TX). The D/A converter was programmed to provide the appropriate excitation sequences to a 395 nm LED (ETG Inc., Los Angeles, CA, ETG-5UV395-30). The exciting light, directed toward a glass cuvette containing palladium tetra(pentafluorophenyl)porphine sample (PdTFPP) in polymethyl methacrylate, was first passed through a 405 nm band-pass filter. Two convex lenses collected and focused the phosphorescent light onto a photomultiplier tube (PMT). The phosphorescence emission detected by the PMT (Hamamatsu, R928, Bridgewater, NJ) was isolated
by a 650 nm band-pass filter, which rejected stray and scattered light below 630 nm and above 680 nm. The PMT current was converted to a voltage by an operational amplifier operated in transconductance mode. The time constant for the PMT preamp was 9 µs. The oscillating voltage from the preamp was converted to a digital value by the National Instruments A/D converter, and the signal was saved for further analysis using MATLAB. Twenty log-spaced modulation frequencies between 177-6140 Hz were used for the frequency-domain experiment. The phase and modulation at each modulation frequency was determined using the software-based phase sensitive detector described previously. The values were plotted versus modulation frequency, and the lifetime was calculated by fitting the data to eqs 4a,b. A square wave excitation with a 10 ms light on and 10 ms light off period was used for the time-domain experiments. Each response contained 500 data points at evenly spaced time intervals. To ensure that equal total excitation intensities were used for the time- and frequency-domain experiments, 200 square wave impulses were summed before the phosphorescence response was fit to the convolution of a square wave excitation sequence and the impulse response. Each experiment was repeated 20 times to obtain a mean and standard deviation for the estimate of τ. IV. Results And Discussion A. Numerical Simulations. Figure 1a-c shows the summed phosphorescence response of the time-domain excitation sequences and the least-squares fit of simulated data. A comparison of the residuals (Figure 1g-i) and the phosphorescence responses show that the lowest noise levels were observed during the light-off portion of the square wave excitation. Because the system was limited by photon counting noise, the noise level decreased significantly during the light-off period. The low noise level in the dark period led to a better fit of the phosphorescence
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TABLE 1: Parameter Estimates from the Simulated Experiments of the Time-Domain Method of Lifetime Determination EXCITATION
τ (ms)
Iο
Bο
square wave (50% duty cycle) MLS chirp MLS with dark period chirp with dark period square wave (10% duty cycle)
0.99999 ( 2.2e-4 0.99980 ( 8.7e-4 1.00004 ( 5.3e-4 1.00002 ( 2.3e-4 0.99998 ( 2.5e-4 1.00000 ( 1.1e-4
1000.01 ( 0.24 1000.2 ( 1.1 999.98 ( 0.68 999.98 ( 0.25 1000.03 ( 0.27 1000.02 ( 0.12
4.999 ( 0.010 5.01 ( 0.20 4.99 ( 0.19 5.0005 ( 0.0071 5.0001 ( 0.0085 5.0019 ( 0.0076
response and lower standard deviations in the parameter estimates for the square wave excitation. Although the total dark period of each MLS excitation equaled that of the square wave excitation, the MLS light-off periods were too short for the phosphorescence intensity to decrease substantially and reduce the noise level. To confirm that the lifetime estimates for the MLS and chirp excitations were limited by the short dark periods in the conventional sequences, a dark period was added to the these sequences (Figure 1d-e). The number of averaged phosphorescence responses was then doubled to keep the total excitation energies equal. The dark period was also increased for the square wave excitation. In this case, the total excitation energy per pulse was kept the same as the original square wave excitation. The duty cycle of the square wave excitation was decreased from 50 to 10% and the intensity was increased by a factor of five (Figure 1f). The increase in excitation energy was limited to a five-fold increase because increasing an LED’s intensity infinitely would eventually destroy the LED or saturate the excited state. Table 1 gives the resulting lifetime estimates from each excitation sequence. The results show that the error in the lifetime estimate from the square wave excitation is much lower than the error associated with the original MLS and chirp excitation. However, when a dark period was added to the MLS and chirp sequences to minimize the photon counting noise, there was no significant difference in the error of the lifetime
Figure 3. Averaged PdTFPP response (gray) to square wave excitation and the nonlinear least-squares fit (black) (a) and the residuals from the fit (b).
TABLE 2: Lifetime Estimates from the Simulated Experiments of the Frequency-Domain Method of Lifetime Determination τtotal (ms)
τphase (ms)
TABLE 3: Experimental Estimates of Lifetime from the Time-Domain Method of Lifetime Determination time domain
Figure 2. (a) Simulated modulation and phase angle for a phosphorescent molecule with a lifetime of 1.000 ms. (•) phase angle, (o) percent modulation, (s) fit. Residuals from the fit of the modulation (b) and phase (c) data are also shown.
τmodulation (ms)
frequency 0.99969 ( 7.4e-4 0.99989 ( 4.9e-4 0.99949 ( 9.0e-4 domain
τ (ms)
Iο
Bο
1.57007 ( 5.3 × 10-4
0.64896 ( 3.1 × 10-4
0.00476 ( 0.1 × 10-4
estimates compared to the square wave excitation. It is important to note that the square wave excitation is superior to the MLS and chirp sequences with a dark period because the square wave experiment required only half the data acquisition time. The short-pulse square wave excitation had the least uncertainty in its estimate of τ. This excitation sequence had the largest dark period and the least amount of noise. This led directly to the most precise estimate of τ. Figure 2 shows the phase, modulation, least-squares fit, and residuals from one simulation of the frequency-domain method. Table 2 gives the lifetime estimates from the modulation calculation, phase calculation, and τtotal, the average of the phase and modulation values. The error in the lifetime estimate, 0.99969 ( 0.00074 ms, was comparable to the error estimates of the time-domain methods with no dark period (Table 1). As shown with the time-domain sequences, the best results were obtained when phosphorescence was measured with the excitation light off. With the standard frequency-domain method the excitation light source is always on, so the noise level is never as low as during the time-domain sequences with a light-off period. This leads to an increase in the error of the estimate of τ for the frequency-domain method. B. Experimental Results. Figure 3a shows the averaged phosphorescence response of square wave excitation sequence.
Comparison of Methods for Luminescence Lifetime Measurements
J. Phys. Chem. C, Vol. 112, No. 21, 2008 8083 the results from the numerical simulation. In both cases, timedomain excitation sequences with a substantial dark period gave the most precise estimate of luminescent lifetime. A strict comparison of the time- and frequency-domain methods requires an equal distribution of excitation frequencies. The current comparison excites the luminophor with the same integrated intensity but does not ensure that the energy is being delivered at the same frequency components. The frequencydomain method excites the luminophor at 20 log-spaced steps of equal intensity between 177 and 6140 Hz. The frequency components of the square wave excitation are described by a sinc function with decreasing amplitudes between 50 and 12500 Hz. The broad range of excitation frequencies of the square wave excitation allows higher frequency responses to be observed. However, it is also a higher bandwidth system, which will have more noise. The unequal frequency distribution of the methods in this comparison favored the lower bandwidth of the frequencydomain method. However, despite the high-bandwidth disadvantage, the results show that the time-domain method leads to more precise estimates of luminescent lifetime. An alternative way to think about the Fourier synthesis process in the time domain is given by Vinogradov et al.13
Figure 4. (a) Experimental modulation and phase angle for PdTFPP in polymethyl methacrylate. (•) phase angle, (o) percent modulation, (s) fit. Residuals from the fit of the modulation (b) and phase (c) data are also shown.
TABLE 4: Experimental Estimates of Lifetime from the Frequency-Domain Method of Lifetime Determination frequency domain
τtotal (ms)
τmodulation (ms)
τphase (ms)
1.558 ( 188 × 10-4
1.5396 ( 19 × 10-4
1.5765 ( 20 × 10-4
Examination of the residuals from the fit of the square wave excitation (Figure 3b) indicates nonrandom behavior that arises from deviation from a single exponential decay in the phosphorescence response of PdTFPP. However, the standard deviation of the lifetime estimate obtained using square wave excitation is still low: τ ) 1.57007 ( 5.3 × 10-4 ms (Table 3). Figure 4 shows the phase, modulation, least-squares fit, and residuals from one experiment in the frequency domain. The frequency-domain method estimate of τ is 1.558 ( 0.019 ms. As expected from the numerical simulation, the time-domain method gives a more precise estimate of τ than the frequencydomain method. However, the difference between the standard deviations is 2 orders of magnitude, much larger than simulation results. This can be explained by examining the estimate of τ from the phase and modulation data separately (Table 4). If the luminescent impulse response is not a single exponential decay, the phase and modulation estimates of τ will not be equal.7 Because the time-domain results indicated the response was not fully described by a single exponential model, the difference between the phase (1.5765 ( 0.0020 ms) and modulation (1.5396 ( 0.0019 ms) estimates of τ are not unexpected. The 40 µs difference between the lifetime estimates from the phase and modulation calculations led to a large increase in uncertainty when the two values were combined for the frequency-domain’s total lifetime estimate. Table 4 shows that the precision of the frequency-domain lifetime estimate improves significantly if the modulation and phase estimates are examined separately. However, the uncertainty is still four times higher than that from the time-domain lifetime estimate (Table 3). This difference in precision for the time-domain and frequency-domain methods is consistent with
V. Conclusions For the same total excitation energy, a numerical comparison showed that the frequency-domain method had a higher uncertainty in the estimate of luminescent lifetime than the timedomain sequences that included a dark period. Because the simulation was limited by photon-counting noise, the noise level decreased significantly when an excitation sequence included a light-off period. The frequency-domain excitation, which had no dark period, had a higher noise level than the time-domain sequences with a substantial dark period. The increased noise of the frequency-domain method led to a higher standard deviation in the lifetime parameter estimate. An experimental comparison of the time- and frequencydomain methods verified the numerical results. The uncertainty of the frequency-domain estimate of τ was twice that of the time-domain method with square wave excitation. However, frequency-domain methods have been developed which incorporate dark periods into the excitation sequence by replacing the sinusoidal excitation with an impulse train6 or a series of square waves.7 Gated detection is used with these methods so the luminescent response is only collected during the dark period. In addition to reducing errors from scattered light and short-lived fluorophores, the SNR should be further increased compared to conventional phase-based fluorometry methods since the photon counting noise is also reduced. Acknowledgment. The authors thank Roy Olund for assistance with the experimental aspects of this research and Peter Dillingham for statistical guidance. This research was supported by the National Science Foundation under Grant 9980069. References and Notes (1) Sugar, I. P. J. Phys. Chem. 1991, 95, 7508–7515. (2) Neal, S. L. J. Phys Chem. A. 1997, 101, 6883–6889. (3) Gratton, E.; Breusegem, S.; Sutin, J.; Ruan, Q.; Barry, N. J. Biomed. Opt. 2003, 8, 381–390. (4) Boens, N.; Qin, W.; Basari, N.; Hofkens, J.; Ameloot, M.; Pouget, J.; Lefe`vre, J. P.; Valeur, B.; Gratton, E.; vandeVen, M.; Silva, N. D., Jr.;
8084 J. Phys. Chem. C, Vol. 112, No. 21, 2008 Engelborghs, Y.; Willaert, K.; Sillen, A.; Rumbles, G.; Phillips, D.; Visser, A. J. W. G.; van Hoek, A.; Lakowicz, J. R.; Malak, H.; Gryczynski, I.; Szabo, A. G.; Krajcarski, D. T.; Tamai, N.; Miura, A. Anal. Chem. 2007, 79, 2137–2149. (5) Philip, J.; Carlsson, K. J. Opt. Soc. Am. A 2003, 20, 368–379. (6) Lakowicz, J. R.; Gryczynski, I.; Gryczynski, Z.; Johnson, M. L. Anal. Biochem. 2000, 277, 74–85. (7) Rowe, H. M.; Chan, S. P.; Demas, J. N.; DeGraff, B. A. Anal. Chem. 2002, 74, 4821–4827. (8) Szmacinski, H.; Chang, Q. Appl. Spectrosc. 2000, 54, 106–109.
McGraw et al. (9) Jameson, D. M.; Gratton, E.; Hall, R. D. Appl. Spectrosc. ReV. 1984, 20, 55–106. (10) Garai, M. Appl. Acoust. 1993, 39, 119–139. (11) Jennrich R. I. An Introduction to Computational Statistics: Regression Analysis; Prentice-Hall: New Jersey, 1995. (12) Gaspar, J.; Chen, S. F.; Gordillo, A.; Hepp, M.; Ferreyra, P.; Marques, C. Microproces. Microsy. 2004, 28, 157–162. (13) Vinogradov, S. A. ; Fernandez-Searra, M. A.; Dugan, B. W.; Wilson, D. F. ReV. Sci. Instrum. 2001, 72, 3396–3406.
JP711867U
