## Abstract

We present a new high-speed lifetime measurement scheme of analog mean-delay (AMD) method which is suitable for studying dynamical time-resolved spectroscopy and high-speed fluorescence lifetime imaging microscopy (FLIM). In our lifetime measurement method, the time-domain intensity signal of a fluorescence decay is acquired as an analog waveform. And the lifetime information is extracted from the mean temporal delay of the acquired signal. Since this method does not rely on the single-photon counting technique, the signals of multiple fluorescence photons can be acquired simultaneously. The measurement speed can be increased easily by raising the fluorescence intensity without a photon-rate limit. We have investigated various characteristics of our method in lifetime accuracy and precision as well as measurement speed. It has been found that our method can provide excellent measurement performances in various aspects. We have demonstrated a high-speed measurement with a high photon detection rate of ~10^{8} photons per second with a nearly shot noise-limited photon economy. A fluorescence lifetime of 3.2 ns was accurately determined with a standard deviation of 3% from the data acquired within 17.8 μs at a rate of 56,300 lifetime determinations per second.

©2009 Optical Society of America

## 1. Introduction

Along with the absorption and fluorescence emission spectra, the fluorescence lifetime provides useful spectroscopic information of the fluorescent molecule and its environment. The fluorescence lifetime measurement has become a very useful investigation tool for studying molecular behaviors by taking advantage of this property [1–5]. Many fluorescent molecules are sensitive to specific ions of the local environment so that they can play useful roles of molecular ion sensors. And the fluorescence lifetime can also be used to measure the Förster (or fluorescence) resonance energy transfer (FRET) in a quantitative manner to investigate nanometric phenomena [5–8]. Various applications of the lifetime measurements have been developed based on those principles for studying molecular mechanisms in biology, biophysics and diagnostics. In studying such a complicated biological system, fluorescence lifetime imaging microscopy (FLIM) techniques are preferably used to obtain spatially resolved lifetime information as a lifetime image. FLIM can easily visualize complicated molecular phenomena with a powerful spatial resolving capability. In those applications, it has been shown that the lifetime information has various advantages over the conventional methodologies based on fluorescence intensity in accuracy and specificity.

A lot of methods for fluorescence lifetime measurement have been developed so far to provide a precise and accurate analysis for time-correlated spectroscopy [1–4]. Those methods can be evaluated by three major performance factors: measurement precision, accuracy and speed. The precision of a measurement system is defined as the amount of random errors in the measured values, and is conventionally estimated by the relative standard deviation of the acquired lifetimes. The precision is directly related to the total energy of the signal *i.e.* the power-time product of the detected fluorescence photons. The precision, by itself, is improved by increasing the measurement time or the number of detected photons. Therefore, the precision performance needs to be evaluated in regard with the number of detected photons involved with a single lifetime determination. Better the precision performance is, less the number of photons is required to obtain a certain level of precision. This is very important for most of the practical applications because a florescent molecule can emit only a finite number of photons before being bleached. A lifetime measurement method of a good *photon economy* can obtain a high-precision measurement result from a small number of fluorescence photons efficiently. The absolute precision performance or the photon economy for a lifetime measurement system is conventionally estimated by a figure of merit, *F*, defined as

where *τ*, *Δτ* and *N* are the lifetime, the standard deviation of the measured lifetimes and the number of detected photons involved with a lifetime determination, respectively [1,9,10]. Because of the fundamental shot noise, *F* is always larger than one for all the lifetime measurements in general. The number of photons required for a given signal-to-noise ratio (SNR) is proportional to the square of *F*. Therefore, a sufficiently low figure of merit is indispensable for a sensitive measurement of fluorescence lifetime.

The accuracy performance of a lifetime measurement method is evaluated by the amount of deterministic bias or fixed lifetime acquisition errors that are found continuously even for the acquired lifetimes averaged over multiple measurement results. Inaccuracy is usually visible in the case of measuring short lifetimes. It determines the reliably measurable minimum lifetime of a measurement system. In most cases, this issue is closely related with the temporal resolution of the detection instrument. The accuracy can be improved by using faster detection parts of wider operation bandwidths that may, however, increase the implementation costs. On the other hand, the measurement speed is also an important performance factor of a lifetime measurement system, especially for the FLIM application. Lifetime measurement methods need a certain time to acquire a sufficient amount of signal for a precise lifetime determination. The measurement rate also needs to be evaluated in regard to the required level of precision. Higher the required precision is, longer the signal acquisition time must be.

Lifetime measurement methods can be classified into three major groups by their physical implementation schemes: frequency-domain methods, time-domain methods and single-photon counting (SPC) methods [1,3,9]. The SPC methodology is basically a time-domain approach but uses very different kinds of detection devices to those of the other two groups which are based on acquiring analog signals of fluorescence intensity directly. Analog methods usually exhibit worse accuracy and poorer photon economy compared to the SPC-based methods because of the worse temporal resolution and the lower noise immunity of the analog techniques. For analog methods, the performances can be enhanced by increasing the modulation and detection frequency or adopting wide-band measurement techniques such as using multiple frequencies in the frequency-domain method or using a high-speed digitizer in the time-domain method. But the technical difficulties increase the costs of implementations to achieve such an enhancement.

Time-correlated single photon counting (TCSPC) scheme is the most popular SPC-based lifetime measurement method. A time-resolved statistics is acquired by the principle of single-photon counting for fluorescence photons generated by a pulsed excitation source in the TCSPC. Owing to various advantages of the TCSPC, this scheme is widely used for applications that require a high accuracy and good photon economy especially for measuring short lifetimes with weak fluorescence signals [11–15]. The TCSPC scheme uses highly sensitive photodetectors such as a photo-multiplier tube (PMT) or a Geiger-mode avalanche photodiode (G-APD) for the single-photon detection. Since the arrival time of each incoming photon is determined individually by the rising edge of a photo-electronic pulse in the TCSPC, its accuracy is not limited by the finite response time of the photodetection part. However, the TCSPC scheme obtains this improved temporal resolution at the sacrifice of the measurement speed due to its single-photon requirement. If two or more fluorescence photons are detected within a pulse period simultaneously, it will generate counting errors that cause inaccuracy in the measured results. It means that the average number of detected photons within a pulse period needs to be regulated under a certain fraction *e.g.* 0.01, to satisfy the single-photon requirement with a confidence of ~99%. Even for the case of using an excitation light source with an exceptionally high repetition rate of 100 MHz, the average photon counting rate is limited to 1,000,000 fluorescence photons per second. Eq. (1) suggests that more than 1,000 photons are needed to analyze a lifetime with a precision better than 3% (*Δτ*= 0.03∙*τ*) even for an ideal lifetime measurement system with a perfect photon economy (*F*= 1). Therefore, the maximum measurement rate is just 1,000 lifetime determinations per second. For a TCSPC-based scanning microscope, it takes longer than 100 seconds to acquire a simple 2D image that consists of 100,000 pixels in the above estimation. This low signal acquisition rate is a serious obstacle in practically investigating a dynamic biological process or obtaining a three-dimensional (3D) FLIM image.

Several approaches have been developed to achieve higher measurement speeds in the SPC-based lifetime measurement methods. By the help of the recent progress in electronics, the TCSPC instrument has reached nearly the theoretical limit of photon counting rates with a short dead time. The state-of-the-art TCSPC unit now can handle up to ~10^{6} photon counts per second [12,13]. Further enhancements are possible by utilizing multiple TCSPC units for a single measurement system [12,15]. As an alternative approach, the *time-gating SPC* method has been introduced for higher measurement speeds, which can detect more than one photon for a fluorescence decay being separated by intervals larger than the impulse response of the photodetector [10,17]. Because the arrival time of a photon is determined by discrete temporal gates instead of a real delay, the photon economy of this time-gating SPC is not as good as the conventional TCSPC and can vary widely by the ratio of the lifetime to the gate width. All of those SPC-based techniques are unlikely to obtain even higher measurement speeds far beyond 10^{7} detected photon counts per second or 10^{4} lifetime acquisitions per second for *N*≥1,000 to fulfill the speed requirement of the real-time FLIM imaging applications.

Analog lifetime measurement methods can exhibit higher measurement speeds in principle. But there are many technical difficulties to obtain both a high accuracy and good precision for the analog methodologies. The classical phase fluorometer based on a low-cost electric phase comparator usually suffers from a low measurement speed because of the narrow bandwidth of the phase detection part. The photon economy of the phase fluorometry is usually very bad. The figure of merit exceeds *F*=6 under normal operation conditions [1]. The analog time-domain method is more straightforward in its concept and has been popular in many applications. But its accuracy is critically limited by the temporal resolution of the time-domain measurement means. Without using expensive high-speed detection devices such as a streak camera [16], the analog time-domain method is difficult to achieve a preferable performance in accuracy. In spite of this problem, many wide-field FLIM systems are recently introduced using time-resolved imaging devices for implementation simplicity. They use gated image intensifiers based on micro-channel plates [3,4,8,19] or newly developed solid-state lock-in image sensors [20] for the time-domain signal acquisition. The benefit of utilizing such a device is obvious in terms of the high imaging speed that comes from the nature of the parallel signal processing of the imaging device. The operation speeds can be extremely fast and can support video-rate FLIM imaging [19]. However, their performances are still disappointing in various aspects. The accuracy of the gated intensifier-based lifetime measurements is limited by the relatively long response time of the intensifiers, typically larger than half a nanosecond. The photon economy is poor due to both the imperfect analysis methods and the nature of the gated intensifiers. They acquire a gated fluorescence signal by eliminating the other signals that are out of the gating interval. As a consequence, the photon detection efficiency is lowered by the number of gating intervals used in a measurement. Besides, the acquired signal is prone to be distorted by the changes of a specimen caused by the photobleaching effect or the dynamic motion of a sample [1]. Besides, these wide-field techniques are rarely suitable for scanning microscopy applications. Even though a gated image intensifier can be used for 3D imaging in conjunction with disk-scanning confocal microscopes [8], they are hardly suitable for high-resolution multi-dimensional imaging applications. A high-speed single-channel lifetime measurement scheme is highly demanded for those applications, especially for multi-photon excitation (MPE) microscopy and scanning near-field optical microscopy (SNOM) [11,14,16,18] that can not adopt the gated intensifier-based methodology.

In this report, we propose a new high-speed fluorescence lifetime measurement method that exhibits a high accuracy and precision, aiming at the confocal or multi-photon FLIM applications. In our method, an analog fluorescence intensity signal is acquired in the time domain and the lifetime is determined by taking the mean temporal delay of the fluorescence intensity signal. Because our *analog mean-delay (AMD) method* does not rely on the photon-counting technique but uses analog signals, the signals of multiple fluorescence photons can be detected simultaneously without any limit. The measurement speed can be enhanced easily by increasing the fluorescence intensity and can reach the excitation rate in theory. The accuracy of our AMD method is independent of the response time of the photodetector or the electronics used in the system. By the theoretical and experimental investigations, we have also found that our AMD method has an excellent photon economy that is comparable to that of the conventional TCSPC. These results suggest that our AMD method is very suitable for high-speed applications of lifetime measurements for its good performances of accuracy, precision and measurement speed.

## 2. Theory of operation

Lifetime analysis based on a mean delay is not a totally new concept by itself but has been used in the TCSPC scheme as one of optional data analysis methods [13]. The key point of our AMD method is that the fluorescence lifetime is determined directly from a measured analog signal taking advantage of the deconvolution-like characteristic of the mean delay. The effect of the finite bandwidth of the detection device can be eliminated effectively by this property. Thus the analog fluorescence signal can be acquired by using a relatively narrow-bandwidth photodetector and digitizing electronics without any concern of signal distortion. This solves the problem of implementation costs in two aspects: the costs of a high-resolution photodetector and those of signal processing electronics. In this section, the basic principles and the characteristics of the AMD method are introduced, starting from a classical linear time-invariant system model for an analog temporal waveform of fluorescence intensity.

#### 2.1 Classical deconvolution

The formation of the photo-electron signal of fluorescence intensity can be understood as a linear time-invariant (LTI) process or simply, a linear process. For a fluorescence lifetime measurement, the signal formation consists of two parts: the fluorescence emission and the photodetection process. Each process can be denoted by an integral convolution operation in the linear system analysis. For a linear process, the output waveform *g*(*t*) is represented by the input function *f*(*t*) convolved with the characteristic function of the system, *h*(*t*), known as the impulse response function where *t* is the coordinate variable of time. This well-known relation is expressed as *g*(*t*) = *f*(*t*)⊗*h*(*t*)= ∫^{+∞}
_{-∞}
*f*(*t*′)∙*h*(*t*-*t*′) *dt*′ in an integral form. The final signal of the detected fluorescence response is determined by the two convolution operations in the analog fluorescence signal measurement. Due to the commutative property of the convolution operation, the final electrical signal can be represented by the convolution of three functions: the intensity profile of excitation light as an input function, the exponential decay function of a fluorescence emission and the impulse response of the photodetector as the impulse response functions of the fluorescence emission and photodetection processes, respectively. The acquired photocurrent signal is represented by

where *i _{e}*(

*t*) is the detected photocurrent,

*γ*is the net conversion coefficient of an excitation photon to a detected photoelectron;

*I*(

_{ex}*t*) is the intensity of an excitation pulse; Ψ

_{τ}(

*t*) is the exponential probability decay of fluorescence emission; and

*I*(

_{pd}*t*) is the impulse response of a photodetector. The integral convolution is represented by ⊗. The normalized fluorescence emission rate Ψ

_{τ}is characterized by the fluorescence lifetime

*τ*for a single exponential decay so that it is represented by Ψ

_{τ}(

*t*) =

*exp*(-

*t*/

*τ*)/

*τ*for

*t*≥0 and Ψ

_{τ}(

*t*)=0 for

*t*<0. The fluorescence lifetime

*τ*is the characteristic time constant of the decay function as well as the time average of the function, which is called the mean lifetime. A lifetime measurement method has a way of extracting the lifetime value of

*τ*from the acquired raw signal of

*i*(

_{e}*t*). In general, the fluorescence decay may consist of multiple decays with multiple lifetime components. But the mean lifetime of a single value is usually of the prime interest in most of the applications.

Lifetime measurement can be performed in the modulation frequency domain as well as in the time domain. The frequency-domain methodology may be more practical due to its property of easy deconvolution. Note that the difficulty of extracting an accurate lifetime from an acquired signal arises from the fact that the desired signal of fluorescence emission rate Ψ_{τ}(*t*) is contaminated by the convolution processes: The molecule is excited by an excitation pulse of a non-zero duration and the analog signal is detected by a photodetector of a finite impulse response. Because the convolution process is an analytic process, the inverse process called as *deconvolution* can not be done easily in the time domain. The Fourier analysis of a linear system suggests that such a deconvolution process can be performed with ease in the frequency domain. The signal processing in practice can be done by either Fourier-transforming the acquired time-domain signal or measuring the response directly in the frequency domain as the phase fluorometer does. The deconvolution process requires the knowledge of the system characteristic known as *instrumental response function* (IRF) defined in the time domain or *instrumental transfer function* (ITF) defined in the frequency domain *i.e.* the Fourier conjugate of the IRF. This function of IRF, *i _{irf}*(

*t*) can be measured by using photon emission or reflection phenomena of virtually zero lifetime so that

The signal distortion caused by the system imperfection (*i _{irf}*≠

*δ*(

*t*)) can be compensated in the deconvolutional signal processing by using the knowledge of the IRF for a measurement system. For an analog time-domain method, the actual fluorescence emission rate, Ψ

_{τ}(

*t*) can be retrieved in principle by

where *f* is the frequency reciprocal to *t*, ℑ represents the Fourier transform, and ℑ^{-1} represents the inverse transform, respectively. The major drawback of this approach is clearly observed in Eq. (4). Because the IRF is band-limited in practice, the measured response is divided by zero at the outside of the valid frequency range. And the SNR consequently decreases after the deconvolution process. In terms of implementation costs, this method requires a time-domain acquisition device of a high sampling rate and a high temporal resolution for measuring short lifetimes. And the digital Fourier transform (DFT) is a computationally heavy process and may be improper for real-time calculations.

Under the assumption of a single exponential fluorescence decay, the Fourier transform of Eq. (2) yields a simple algebraic relation of the phase components and the lifetime or that of the amplitude components and the lifetime [1]. The phase fluorometry takes advantage of this deconvolutional property, in which the convolution relation changes to a simple relation of the phase components in the frequency domain. In the phase fluorometry methods, the lifetime *τ _{f}* is determined by subtracting the IRF phase from the signal phase as

where *ϕ* and *ϕ _{irf}* denote the phase of the fluorescence signal and that of the IRF with respect to the excitation moment, respectively. For a multi-frequency measurement, the multiple lifetimes obtained as a function of frequency can be averaged to make a single measured value of enhanced precision. So, the effect of the measurement system can be neutralized by measuring the relative phase shift of a fluorescence signal by this manner.

#### 2.2 AMD lifetime determination

The principle of deconvolution suggests an advantage of transforming the acquired signal into a certain domain where the convolution changes to a simple algebra. It can be generalized to the AMD method. A domain of statistical values can play such a role of *deconvolution* domain [23]. The convolution of probability distribution functions (PDFs) corresponds to a summation operation of the corresponding random variables and consequently, that of the corresponding expected values. We could take advantage of this property for a simplified deconvolution and determining the lifetime from a degraded analog signal.

The arrival time of a fluorescence photon can be decomposed into componential time delays which are related to respective physical processes. Figure 1 shows the schematic timing plots of excitation, fluorescence emission and photodetection that occur serially in time for the measured analog signal of a fluorescence lifetime. A photoelectron in the output current of a photodetector, is resulted by a series of processes with various temporal delays: an excitation photon is delivered to a sample with a delay of *t _{ex}* (Excitation Delivery); the photon is absorbed by a fluorescent molecule exciting a molecule’s electron with

*T*(Excitation); the excited electron relaxes to the fluorescent state with

_{ex}*T*(Relaxation); the electron stays in the state before a radiative recombination with

_{re}*T*(Fluorescence Emission); the resultant fluorescence photon is delivered to the photodetector with

_{fl}*t*(Fluorescence Delivery); and the photoelectron finally experiences a temporal spread of

_{fl}*T*inside the photodetector (Photodetection). Note that the delay variables denoted by capital letters such as

_{pd}*T*are random variables and they have their own characteristic PDFs (probability density functions) as functions of time respectively. And the delays denoted by small letters such as

_{ex}*t*are deterministic non-random variables.

_{ex}*T*is a random variable because of the non-zero duration of the excitation pulse. Among a plurality of the excitation photons, only a single photon is absorbed and excites the fluorescent molecule with temporal randomness.

_{ex}*T*is also a random variable, which contains the information of the impulse response of the photodetector.

_{pd}The arrival time of the final photoelectron at the signal acquisition instrument, *T _{e}*, can be expressed by a summation of those delays defined in the above as

Three terms of the right-hand side in Eq. (6) correspond to the three major steps *i.e.* the physical processes of excitation, fluorescence emission and photodetection, respectively. And the PDFs of *T _{e}*,

*T*, (

_{ex}*T*+

_{re}*T*) and

_{fl}*T*correspond to the temporal shapes of the finally detected electric pulse, excitation light pulse, fluorescence decay and the impulse response of the photodetector, respectively. Thus they have one-to-one relationships of

_{pd}*T*to

_{e}*i*(

_{e}*t*),

*T*to I

_{ex}*(*

_{ex}*t*), (

*T*+

_{re}*T*) to Ψ

_{fl}_{τ}(

*t*), and

*T*to

_{pd}*I*(

_{pd}*t*), in Eq. (2) and Eq. (6). Note that the analog signals acquired in practice are not exact PDFs but histograms with random errors. They are not distinguished in this paper for simplicity.

The lifetime can be determined accurately in a mean-delay domain by a deconvolutional process with pre-determined information of the *instrumental response delay* (IRD) that corresponds to the IRF given by Eq. (3). The IRF is measured in keeping the optical and electric paths identical to those of the fluorescence signal acquisitions. For a photoelectron of the IRF, the final delay, *T _{e}^{0}* is represented by

And the mean value of *T _{e}^{0}* will be called the IRD of the measurement system, which contains the mean-delay information of the measurement system. In this paper, the mean value of a random variable

*T*is denoted by 〈

*T*〉. So, the IRD is denoted by 〈

*T*〉. It is well known that the operation of taking an expected value is a linear operation [23]. Thus subtracting the IRD from the mean value of the temporal delays of the detected photoelectrons given by Eq. (6) yields the mean temporal delay of the fluorescence emission as

_{e}^{0}is derived from Eq. (6) and Eq. (7). Here, we have neglected 〈*T _{re}*〉 because the relaxation delay is on the order of picoseconds, much smaller than the fluorescence lifetime that is usually on the order of nanoseconds. The mean delay of fluorescence emission, 〈

*T*〉 is the fluorescence lifetime of

_{fl}*τ*for the case of single exponential decays. For the case of multi-exponential decays, it is the intensity-weighted average of multiple lifetimes. Thus the lifetime is determined by obtaining the mean delay of the fluorescence signal with respect to the IRD of 〈

*T*〉. In an integral form for the time-domain signals of

_{e}^{0}*i*(

_{e}*t*) and

*i*(

_{irf}*t*), Eq. (8) is rewritten as

where *i _{e}*(

*t*) and

*i*(

_{irf}*t*) are the acquired fluorescence signal and the IRF signal, respectively, which are the measured PDFs of

*T*and

_{e}*T*as defined in Eq. (2) and Eq. (3). In Eq. (9), all the integrations are definite integrations for an integration range of (

_{e}^{0}*t*,

_{0}*t*) by which both

_{1}*i*(

_{e}*t*) and

*i*(

_{irf}*t*) are bounded. Our AMD method of fluorescence lifetime measurement determines the lifetime by using Eq. (9) with acquired analog signals of

*i*(

_{e}*t*) and

*i*(

_{irf}*t*). In this method, an accurate fluorescence lifetime is measured in the mean-delay domain where the effect of the IRF can be easily eliminated by calibrating its systematic mean delay of the IRD. The accuracy of the AMD method is no more hampered by the system imperfection (

*T*,

_{ex}*T*≠0) because of this deconvolutional property.

_{pd}#### 2.3 Precision of the AMD method

The precision of the AMD method can be analyzed by the basic principles of the statistics. The standard deviation of the measured lifetimes, denoted by *Δτ*, can be obtained from the variance of the mean delays, *Δτ ^{2}*. It is the variance of the expected value of the temporal delay

*τ*for the PDF of fluorescence emission Ψ

_{τ}(

*t*). The variance of an expected value is the variance of the random variable divided by the number of the statistical samples [23]. In denoting the variance of a random variable

*T*with

*σ*

^{2}[

*T*], the variance of a fluorescence lifetime measured with

*N*detected photons can be represented by

since *T _{fl}* has a PDF of an exponential decay distribution Ψ

_{τ}(

*t*) of which variance is the square of the expected value

*i.e. τ*. From Eq. (1) and Eq. (10), the figure of merit for our AMD method is

^{2}*F*= 1 in the case of the ideal condition of exciting the molecules with an impulse-like pulse and detecting the signal with negligibly small noises and timing jitters. Hence, the theoretical performance of the AMD method is the same with that of the TCSPC scheme in their photon economies.

The figure of merit obtainable in practice, however, is significantly higher than one because of various sources of random errors. The random variables of the right-hand side in Eq. (6) yield non-zero variances that influence the total variance of the measured lifetime. There are two major sources of timing jitters for a detected photoelectron signal: the non-zero duration of the excitation pulse related with *T _{ex}* and the characteristic jitter of a photodetector known as a transit time spread (TTS) related with

*T*. The former timing error caused by the finite duration of the excitation light is resulted from the fact that a single photon among a plurality of excitation photons is absorbed by the molecule with randomness. Its partial variance is

_{pd}*σ*[〈

^{2}*T*〉] = Δ

_{ex}*t*

_{ex}^{2}/

*N*where Δ

*t*is the standard-deviation duration of the excitation pulse and

_{ex}*N*is the number of detected photons. On the other hand, the TTS denoted by Δ

*t*is caused by the operation principle of the high-gain photodetector. For a PMT, the multiple electron paths, especially in the first electron-multiplication stage may induce a significant amount of uncertainty of the transit time [24]. The variance of the TTS for a detected fluorescence photon is defined by

_{tts}when the photodetector generates a set of photoelectrons, *T ^{1}_{pd}*,

*T*, ⋯

^{2}_{pd}*T*for a single fluorescence photon. Because there are always a large number of photoelectrons (M>10

^{M}_{pd}^{4}) for a fluorescence photon, the TTS is independent of the impulse response of the photodetector and should be measured for sets of photoelectrons as Eq. (11). The contribution to the total variance

*Δτ*can be derived in the same way as that of Eq. (10). It is given by

^{2}*Δt*

_{tts}^{2}/

*N*for

*N*detected fluorescence photons.

Amplitude noises of a high-gain photodetector also need to be considered to estimate the practical precision performance. As it is difficult to estimate the amount of the amplitude noises exactly, only a rough outline will be discussed in this estimation. Dark counts are usually the most dominant source of amplitude noises for a PMT. Here, the dark counts are defined collectively as all the high-amplitude noise pulses of which amplitudes are comparable to those of a single-photon response. For simplicity, let us assume that the dark counts have the same amplitude as that of the photon count and they are spread uniformly in time. When the total number of dark counts in a pulse period or a measurement period *T _{m}* is denoted by

*N*, the effective number of dark counts found in the integration window is

_{d}*N*=

_{d}^{e}*N*∙

_{d}*ε*in average. Here,

*ε*is the duty ratio of the integration window width

*ΔT*≡ {

_{w}*t*-

_{1}*t*} to the pulse period

_{0}*T*so that

_{m}*ε*=

*ΔT*/

_{w}*T*. We introduce a dark-count ratio

_{m}*R*≡

_{d}*N*/

_{d}*N*that measures the relative noise power. Thus the effective number of the dark counts is

*N*=

_{d}^{e}*N*∙

*R*∙

_{d}*ε*inside the integration window. For the condition of a low dark-count rate

*i.e. R*≪1 or

_{d}*N*≫

*N*, the variance of the measured lifetime is derived as

_{d}$$\phantom{\rule{2.2em}{0ex}}=\frac{\left(N{\sigma}^{2}\left[{T}_{e}\right]+{N}_{d}^{e}{\sigma}^{2}\left[{T}_{d}\right]\right)}{{\left(N+{N}_{d}^{e}\right)}^{2}}$$

$$\phantom{\rule{1.5em}{0ex}}\cong \frac{{\sigma}^{2}\left[{T}_{e}\right]}{N}+\frac{1}{N}(\frac{{R}_{d}\bullet \epsilon}{12}\bullet \Delta {T}_{w}^{2})$$

where *i* and *j* are positive-integer indices; *T _{e}^{i}* or

*T*is the random time delay of a signal photon count;

_{e}*T*or

_{d}^{j}*T*is the random time delay of a noisy dark count; and

_{d}*ΔT*is the time width of the integration window, respectively. The variance of a uniform distribution function is 1/12 of the width

_{w}*ΔT*. In the last line of Eq. (12), the first term of the right-hand side represents the intrinsic variance of the fluorescence photon signal and the second term corresponds to the contribution of the dark counts.

_{w}The precision of the AMD method is determined by summing those error sources: the duration of the excitation pulse, the TTS of the photodetector, dark-count effect and the intrinsic random error caused by the fluorescence emission. Summing up all those partial variances, the variance of the measured mean delay is

when error sources other than mentioned in the above are neglected. Therefore, more practical estimation of the figure of merit, *F′* for the AMD method is given by

as a function of fluorescence lifetime *τ* to be measured from Eq. (1) and Eq. (13). This photon economy characteristic is similar to that of the TCSPC. But it may degrade further due to other amplitude noises that are almost absent in the case of single-photon counting. Note that the figure of merit in Eq. (14) depends on ~3/2’th power of the window width *ΔT _{w}*. It is important to take an integration window as small as possible, to obtain a good precision performance of a low figure of merit.

## 3. Experiment

A lifetime measurement system based on our AMD method has been constructed by using low-cost detection electronics in this study. The key feature of our method described theoretically in the previous section is the deconvolutional property of the mean delay-based analysis. The effect of a narrow-bandwidth detection part can be compensated by measuring the IRD with ease and does not degrade the accuracy and precision of the lifetime measurement system. This characteristic of our AMD method permits us to use economic narrow-bandwidth photodetection and data acquisition devices. It is more cost-effective to use a signal digitizer with a lower sampling rate, not only for a reduction in the instrumental costs but also for the convenience of data storage and signal processing. And the signal needs to be low-pass filtered to avoid the aliasing effect when the sampling rate is reduced. In this case, the precision performance may degrade significantly by the wider integration window of *ΔT _{w}* for a given dark-count rate as Eq. (14) suggests. The optimization of the width of the integration needs to be carefully concerned in the implementation of the AMD method.

A schematic diagram of the lifetime measurement system in the AMD method is illustrated in Fig. 2. The measurement system is constructed based on an epi-fluorescence microscope such that the fluorescence photons are collected backward by an objective lens that is simultaneously used for focusing excitation light. In our system, an excitation pulse at a wavelength of 635 nm was generated by a gain-switched semiconductor laser at a repetition rate of 2.7 MHz. The average output power of the laser was 15 μW. And the excitation light was delivered by a single-mode (SM) fiber to the main body of the measurement system.

After passing through a collimator lens, the excitation light passed through a band-pass filter (denoted by laser filter in Fig. 2), an optical beam splitter and an objective lens consecutively before it was launched onto a fluorescence sample. The emitted fluorescence light from the sample was collected by the objective lens and delivered backward to a multimode (MM) fiber via the beam splitter, a long-wavelength pass (LWP) filter, and a fiber-coupling lens. The core of the multimode (MM) fiber acted as a pin-hole and formed a confocal geometry with the core of the single-mode (SM) fiber. The multimode fiber was connected to a PMT. The photoelectronic signal from the PMT was acquired by a signal digitizer after passing through an electric low-pass filter (LPF). In addition, an electrical clock signal from the excitation pulse source (denoted by pulse clock in Fig. 2) was fed to the digitizer to synchronize it with the pulse laser. For the case of IRF measurements, a piece of flat glass plate was used as a sample, which acts as a mirror with a reflectivity of 4%. By removing the LWP filter and attenuating the light to the PMT, the IRF could be acquired successfully without changing the system significantly.

In our measurement system, the duration of the excitation pulse was 0.8 ns in full width at half maxima (FWHM) or 290 ps in standard-deviation duration. The numerical aperture of the objective lens was 0.4. The cut-on wavelength of the LWP filter was 650 nm for filtering out the residual excitation light. The core of the multimode (MM) fiber was 62.5 μm in diameter and formed a relatively loose pin-hole for a high photon collection efficiency. The PMT used in this study was a low-cost head-on type PMT (R7400U-20, Hamamatsu) that uses metal dynodes for electron gains. This PMT was measured to have an impulse response of 1.2 ns in FWHM that corresponds to an electric bandwidth of ~300 MHz. The signal digitizer (DSO6000, Agilent Technologies) used in this study has a maximum sampling rate of 2,000 mega-samples per second (MS/s) but was used intentionally in an operation mode of 100 MS/s for most of our experiments to simulate a low-cost digitizer. The sampling depth was 8 bits and supported 256 amplitude levels. The low-pass filter (LPF) was inserted only for the 100-MS/s sampling operation as an anti-aliasing filter. This filter is designed by the 5th-order Gaussian filter, of which 6-dB bandwidth of electric power transfer was evaluated to be 10 MHz [21]. Thus the electric bandwidth of the photodetection part is switched from ~300 MHz for the 2,000-MS/s case to 10 MHz for the 100-MS/s sampling case. By the Nyquist-Shannon sampling theorem, the full information must have been acquired effectively at those two sampling rates.

The integration range of (*t _{0}*,

*t*) in Eq. (9) is defined by an integration window of a finite width in our signal processing. Shorter the window width is, larger the deterministic error may reside in the obtained mean delay. On the contrary, the amount of random errors is roughly proportional to the window width

_{1}*ΔT*as Eq. (13) and Eq. (14) suggest. Care is needed to find the optimal size of the integration window. Fortunately, the amount of the final deterministic error is negligibly small for the case of narrow-bandwidth acquisitions when the fluorescence lifetime is much smaller than the width of the system impulse response. The deterministic error found in the delay of the fluorescence signal is almost the same as that of the IRF for that case because their shapes are very similar to each other. Thus those deterministic errors are canceled out in the final relative mean delay of the measured lifetime. This issue of optimal window size will be discussed again in the next section.

_{w}The signal processing of mean-delay extraction was performed digitally with a computer by the following procedures. First, the DC level of the acquired signal was adjusted so that the base of the signal becomes zero. Second, a certain number of fluorescence pulses were collected and added to form an averaged fluorescence waveform. This averaging takes more number of detected photons and results in a higher SNR. The total number of detected photons for the averaged waveform will be considered in the following performance analysis. For a signal acquired in a low sampling rate of 100 MS/s, the spline interpolation is applied to increase the number of sampling points to 2,000 MS/s for better precision in the numerical mean-delay calculation. Actually, this numerical interpolation process is not a necessary step in practical implementations. We have observed that sufficiently good performances are obtained without such an interpolation processing. In the third step, an initial mean delay is calculated for an integration window wide enough to cover the whole pulsed waveform. In the fourth step, a secondary mean delay is computed for a new integration window whose center is set to the initial mean delay obtained in the previous step. Here, the width of the new integration window is reduced approximately to the FWHM of the pulse signal. By an iterative manner, the fourth step of getting a mean delay with a re-centered integration window is repeated several times resetting the center of the integration window at the mean delay value obtained in the previous step. The obtained lifetime converges to a final value as the number of iterations increases. This iteration algorithm is helpful in minimizing the effect of the background noise error. Without this, the determined mean delay tends to deviate to the center of the integration window for a positive background noise like dark counts of a PMT that is given in a random manner. Thus the iteration algorithm minimizes the random lifetime error caused by the dark counts. The measurement precision is improved up to a certain level by using a narrower integration window because of the consequent increase in SNR. This is explained by the fact that the signal is concentrated in the vicinity of its center while noises spread over the uniformly in time.

## 4. Results and discussion

The accuracy of our AMD method was evaluated by two different fluorophores: Alexa Fluor 633^{TM} (Invitrogen) and CY5 (Amersham Biosciences). It is known that Alexa Fluor 633 has a relatively long lifetime of 3.2 ns in water and CY5 has a short lifetime of 1.0 ns in phosphate buffered saline (PBS) [22]. In our experiment, those fluorophores were diluted by PBS and were placed on glass plates as sample specimens respectively. Fig. 3 shows the signals acquired at an acquisition rate of 100 MS/s (a), and acquired at 2,000 MS/s (b), respectively, for Alexa Fluor 633. In each graph, a bold blue line represents the detected fluorescence signal and a dashed black line represents the IRF. Both of them have been waveform-averaged for high SNRs. The number of detected photons was approximately 100,000. Herein, the optical power was measured by an optical power meter (8153A, Agilent Technologies). The photon counting rate was calibrated for the PMT with CW irradiation of a laser operating at 635 nm. In this calibration process, the discrimination level for photon counting was set to be 1/3 of the mean amplitude of the pulse peaks for the single-photon response. A detected photon rate was estimated by the optical power in regard to the conversion factor that had been taken by that calibration process.

As seen in Fig. 3(b), a characteristic exponential decay curve is obtained by using a high-speed photodetector for an impulse-like excitation. Because of the finite response of the photodetector, the decay curve is slightly spread, being convolved with the IRF. In contrast, the detailed properties of the exponential decay are completely lost in Fig. 3(a) mainly due to the LPF (low-pass filter) installed after the PMT. The shape of the fluorescence response is almost equal to that of the IRF for this narrow-bandwidth case. However, a sufficient amount of information about the fluorescence lifetime is conveyed by the relative temporal delay. In our AMD method, this temporal delay is measured neglecting the other properties of the detected fluorescence signal as explained.

Our AMD method was applied both to the narrow-bandwidth signals acquired at a signal sampling rate of 100 MS/s and the wide-bandwidth signals acquired at 2,000 MS/s. The number of detected photons was 1.2×10^{3} in average for a single lifetime determination. Each fluorescence waveform was acquired by averaging 48 measured pulses to obtain this number of photons. Since we operated the pulse laser at a repetition rate of 2.7 MHz or a pulse period of 370 ns, it takes 17.8 μs to acquire a signal dataset. In total, the average photon detection rate was 6.8×10^{7} detected photons per second. For the narrow-bandwidth case (100 MS/s), the iterative algorithm was used for mean-delay determination. The number of iterations was set to be 10, which was sufficiently large for convergence. The width of the integration window was 1.24 times the FWHM of the IRF (Δ*T _{w}*=56 ns). For the wide-bandwidth case (2,000 MS/s), a fixed integration window of a sufficiently large width (Δ

*T*≈8∙

_{w}*τ*) was used without the iteration algorithm. The lifetime was determined repeatedly 113 times in the same condition. And the whole measurement was repeated for the other fluorophore, CY5 as well.

The lifetimes obtained by our AMD method were compared with those of the conventional method of the phase fluorometry explained by Eq. (5). The measured IRFs and the fluorescence signals shown in Fig. 3 were Fourier-transformed to obtain frequency-domain responses. The phase components were extracted from them and the fluorescence lifetimes were calculated with the relative phase shifts by using Eq. (5) as functions of frequency. Single lifetime values averaged over a broad frequency range of 10–150 MHz were used as reference data to be compared to the results of our AMD method. Table 1 summarizes the comparison between the results of the AMD method performed at the two different sampling rates and those of phase fluorometry method. Only Small differences between the measured lifetimes of different methods were observed and successfully demonstrate the accuracy of our AMD method. Furthermore, results in Table 1 show that the results calculated from the narrow-bandwidth signals with a 100-MS/s sampling rate are almost same as those obtained from the wide-bandwidth signals taken at 2000 MS/s. This means that the accuracy of lifetime determinations has little dependency on the bandwidth of the measurement system in our AMD method.

Table 1 also shows the precision performance of our method for the narrow-bandwidth acquisition case (100 MS/s). The figure of merit was very good for a long fluorescence lifetime of 3.2 ns (*F* = 1.2), almost reaching the theoretical limit. It is significantly degraded for a short lifetime of 0.9 ns (*F* = 2.5) but shows an acceptable level of precision. On the other hand, we have observed that the figure of merit obtained for CY5 (*τ*= 0.9 ns) was *F* = 1.6 in the case of the wide-bandwidth acquisition (2,000 MS/s) for *ΔT _{w}* = 8 ns. Therefore, it is believed that the long integration window of 56 ns in the 100-MS/s case had caused the increase in

*F*. As Eq. (14) suggests, increasing the integration window width

*ΔT*results in the increase in the figure of merit for a given dark-count rate. It implies that the photon economy might be enhanced by using a photodetector of a low noise count rate.

_{w}The effect of the integration window width was further investigated for the case of 100 MS/s. For an IRF signal, the precision of the mean-delay determinations was estimated by the standard deviation Δ*τ* for various window widths. And the speed of the convergence was also evaluated by introducing *effective number of iterations*, *N _{iter}*. This is defined as the number of iterations required for Δ

*τ*to reach 110% of the final value that was obtained after 20 iterations. Thus

*N*can be understood as a minimum number of iterations for an optimized precision. Fig. 4 shows the effect of the window width

_{iter}*ΔT*for the iterative algorithm of mean-delay determination on the standard deviation

_{w}*Δτ*and the minimum number of iterations

*N*. It is clear that a smaller number of iterations are required for a wide window width. Hence, a wider integration window is preferred in terms of computing speed. Fig. 4 also shows that the precision of calculated mean delay is optimized for a window width which is approximately the width of the IRF. It must be obvious that a significant amount of photon signal is lost for a very narrow window, and additional noise counts are included for an excessively wide window.

_{iter}The signal acquisition is very fast in our AMD method. Collecting a set of data for determining a 3.2-ns fluorescence lifetime with a ±3% random error (±110 ps) was completed in 17.8 μs or 48 pulse periods, in which ~1,200 detected photons were collected. This measurement rate is equivalent to signal acquisitions of a FLIM image composed of 100,000 pixels (~316×316 pixels) in 1.8 seconds. The acquisition speed can be further improved with ease by raising the excitation light intensity that will increase the fluorescence photon rate. For a single-channel TCSPC instrument, it takes longer than 100,000 pulse periods for a single lifetime determination to obtain an equivalent result. Even for 10 times higher excitation pulse rate of 27 MHz that is around the typical pulse rate, 3,700 μs needs to be spent to acquire a single lifetime determination with a TCSPC method. This indicates that the measurement speed of our AMD method is more than 2 orders of magnitude faster than that of the TCSPC scheme.

## 5. Conclusion

In this report, we have introduced a new fluorescence lifetime measurement scheme of the analog mean-delay (AMD) method for high-speed fluorescence lifetime measurements. The basic characteristics of our method have been investigated in both theory and experiment. Owing to the linear property of mean-delay analysis, the fluorescence lifetime can be accurately extracted from an analog signal which is acquired with a photodetection device whose impulse response is much longer than the lifetime to be measured. We have demonstrated that the measurement of this AMD method is so fast that lifetimes can be determined in a rate of ~10^{5} measurements per second. The photon detection rate achieved in the experiment was on the order of ~10^{8} detected photons per second. In theory, the measurement speed can be increased up to the repetition rate of a pulsed excitation light just by increasing the excitation power. This high measurement speed can enable fast image acquisitions in FLIM, which can visualize the fast-varying dynamic features of a biological sample. Even though the practically achievable measurement rate might be limited by the finite power of the excitation source or by the photobleaching effect of fluorophores, the absence of the maximum photon rate would be still beneficial. In a TCSPC-based FLIM system, cautious operating conditions must be satisfied in order to optimize both the photon counting rate and the accuracy performance. We have also shown that the accuracy and precision of our AMD method are comparable with those of the TCSPC method. For a long fluorescence lifetime of a few nanoseconds, the figure of merit can nearly reach the theoretical limit. An additional benefit of our AMD method is that these attractive features are obtained with low-cost electronic components of low bandwidths and sampling rates.

The most promising application of our AMD method would be the scanning FLIM microscopy such as a laser-scanning confocal microscope, multi-photon excitation microscope or a scanning near-field optical microscope (SNOM). The high-speed FLIM techniques based on gated intensifiers are not easily compatible with those scanning microscopes. The slow TCSPC has mainly been used for those applications so far. Our method can increase the imaging speed in these applications to ~1 frame per second with ease. Of course, the measurement speed, herein, means the signal acquisition speed and does not automatically mean that the lifetime is automatically calculated in real time. But the algorithm used in our method is very easy to be computed. There seems to be little technical difficulty for well-developed digital signal processors to compute the mean delays in real time for a relatively low sampling rate of 100 MS/s that corresponds to 100 MB/s in data rate for 8-bit digitization. Although our AMD method can only deal with a simple lifetime analysis of the single-exponential decay, those various advantages make it suitable for most of the FLIM applications.

## Acknowledgments

This work was supported by the Creative Research Initiatives (CRI) Program of Korea Science and Engineering Foundation (KOSEF).

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