## Abstract

We introduce a novel single-photon sensitive photodetection method of analog single-photon counting (SPC) for the application of high-speed scanning microscopy that requires high measurement speed and wide dynamic range for the photodetector. This scheme is based on analog electronic circuits which can perform proper differentiation and integration operations before and after discrimination of the analog signal from the photomultiplier tube (PMT), respectively. In spite of its simpler implementation, our analog SPC scheme exhibits good sensitivity and operation stability. Related with the dynamic range, the maximum count rate of our analog SPC is significantly improved due to the fast operation of the analog circuitry. This characteristic of the higher counting rate makes this scheme very suitable for high-speed scanning microscopy. It has also been demonstrated that the afterpulsing problem of an analog-mode PMT is the major noise source that degrades the image quality in the application of scanning microscopy, and our SPC scheme successfully neutralizes this kind of impulse noises to obtain a nearly shot-noise-limited imaging performance.

© 2008 Optical Society of America

## 1. Introduction

Highly sensitive photodetection methods are critical in a lot of applications where the intensities of optical signals are very weak. In order to realize a highly sensitive detection system with *single-photon sensitivity*, high-gain photodetectors such as photomultiplier tubes (PMTs) [1], Geiger-mode avalanche photodiodes (G-APDs) [2–4] and superconductor-based single-photon counters [5] have been utilized. These detectors are not only able to measure extremely weak light intensities of a few photons but also have wide bandwidths to track fast varying signal changes. In many applications, PMTs are the most popular detectors among them because of the low cost, high speed and wide coverage of wavelengths. However, one of the major drawbacks of a PMT is that its output is not exactly proportional to the intensity of input light or the number of input photons. For a typical metal-dynode PMT, the number of output electrons generated by a single photon exhibits random fluctuations over repeated measurements, which originates from the statistical nature of electron multiplication processes inside a PMT. A single-photon counting (SPC) measurement scheme that utilizes a digital pulse counting method has been used to clean up the output variation of a PMT [1, 6]. SPC measurement scheme provides ultimate accuracy and precision as well as performances immune to the other problems of a PMT like thermal noise, signal distortion by external magnetic fields, gain variation induced by the operation condition etc. Because of these advantages, the SPC photodetection scheme is regarded as an ideal method for low light signal detection and has been widely adopted especially in many spectroscopic applications. However, the dynamic range of the measurement or the maximum counting rates of the conventional SPC method is relatively low; the photon counting rate is more or less limited in the order of ~10^{7} counts per second. In spite of its many attractive features, slow measurement speed has been the major obstacle in some applications.

Microscopy is one of the fields in which SPC has not played an effective role. Various kinds of scanning confocal microscopy techniques such as confocal fluorescence microscopy, multi-photon excitation (MPE) microscopy, coherent anti-Stokes Raman scattering (CARS) microscopy and second-harmonic generation (SHG) microscopy have been proposed and used in many biological imaging applications [7]. In those microscopy techniques, highly sensitive photodetectors are preferred not only because of their inherently weak signals for nonlinear microscopy schemes, but also because of the minimization of photo-bleaches or photo-damages induced by very strong irradiation of excitation laser light. Another most important requirement for a photodetector in a scanning microscopy system is the measurement speed. Two- or three-dimensional images in such microscope systems can be obtained either by moving a sample with a 2- or 3-axis translation stage or by scanning an optical beam with a beam deflector which is usually made up of two galvo-mirror scanners. Sometimes, SPC has been used for low-speed sample-scanning microscopes. However, a beam scanning method is more preferred in high-speed imaging applications, where an image refresh rate of >1 frame per second is required. In such a high-speed beam scanning system, the pixel dwell time is less than 10 microseconds and requires wide a bandwidth above 50 kHz for the photodetector. The maximum count rates of commercially available SPC instruments are usually between ~10 and ~100 Mcps (Mega-counts per second), which are insufficient for high-SNR imaging for a given pixel dwell time less than 10 microseconds or pixel rate above 100 kHz. Most of them do not even support measurement rates more than 10 kHz (less than 100 µs in integration time) and can not acquire all the signals for an image e.g. composed of 100,000 pixels within 10 seconds.

Some alternative technologies can provide with better performances in their maximum count rates. Those photodetection schemes may be more useful in the microscopy application. As a solid-state device version of the PMT, an APD can exhibit a sufficiently high gain (~10^{6}) for single-photon sensitivity in the Geiger-mode operation [2]. G-APD modules, also called as Si-PMs (silicon photomultipliers) or single-photon avalanche diode (SPAD) arrays, are composed of a large number of APD cells to increase the photon detection rate. The height of the output pulse of the G-APD corresponds to the number of detected photons without wide random height distributions so that it can be regarded as an *SPC-equivalent* detector by itself. For a single-cell APD in the Geiger mode, the maximum count rate is limited by the *dead time or recovery time*, typically longer than hundreds of nanoseconds. The maximum count rate of a Si-PM can be improved by increasing the number of the cells. Si-PMs of more than 1,000 cells are commercially available and their maximum count rates are estimated more than 100 Mcps. Although APD’s quantum efficiency can be as high as >80% in principle, the G-APD’s photon detection efficiency is reduced by the geometrical fill factor of the active area and probability of successful avalanche. For a multi-cell Si-PM, the net photon detection efficiency is 10~30%, which is just comparable to that of PMTs. The dark count rate per unit detection area is significantly higher than that of PMTs and limits its capability in some applications.

On the other hand, PMTs of different designs and operation principles such as an HPD (Hybrid Photo-detector^{™} of Hamamatsu, Japan) [1] and CPM (Channel Photomultiplier^{™} of PerkinElmer, USA) [8] have been introduced based on the vacuum-tube technology. Those new classes of PMTs exhibit narrower height distributions than those of metal-dynode PMTs. In the other points, they have photodetection performances similar to those of the classic PMTs. On the contrary to using the alternative photodetectors mentioned above, the problem of count rates can be solved by introducing new SPC electronics. J. Soukka *et*
*al*. proposed an SPC scheme utilizing multiple discriminators to increase the maximum count rate of a PMT-based SPC [9]. In their scheme, the multiple discriminators enable so-called *multi-photon discrimination* for a few photons detected simultaneously. However, the enhancement in the maximum count rate is practically limited below ~4 in its principle and requires more complicated electronic implementations in addition to careful optimization of the discrimination levels. In spite of the attractive features of those alternative technologies, they are not frequently used in practical microscopy. This is partially because the conventional PMTs in analog operation mode are believed to be sufficiently good in terms of SNR for high-speed scanning microscopes. In those cases, the shot noise overwhelms the other noise sources including thermal noise and random height fluctuation noise of a PMT [1]. And those new techniques are not believed to provide significantly better performances in this context. According to the best of our knowledge, none has verified the benefits of those SPC or SPC-equivalent photodetection methods in high-speed scanning microscopy. Therefore, a new practical photodetection method should prove its quantitative benefit as well as should solve the inherent disadvantages such as low dynamic range for the SPC scheme in order to be adopted by this application.

In this report, we propose a novel SPC scheme that extends the count rate limit and makes the SPC technique utilizable for high-speed scanning microscopes. Our proposed SPC only uses analog electronics for simplicity and high operation speed, on the contrary to the conventional complex digital SPC electronics. The maximum count rate is significantly improved to support wide dynamic ranges required in scanning microscopy, still preserving the merits of photon counting schemes. We have found that our novel SPC method is very suitable in various aspects in terms of performances, cost and compatibility to the conventional signal interface. We have also investigated the benefit of SPC techniques in imaging applications. By comparing the performance of our SPC with that of the analog-mode PMT, it has been demonstrated that the high-amplitude afterpulse is an important noise source that degrades the image quality significantly for the analog-mode PMT, and it can be successfully suppressed by using our new SPC scheme. It also suggests that the conventional theory of noise is partially unsatisfactory to analyze the noise of PMTs in the imaging applications. These results also imply that the alternative photodetectors like G-APDs are able to provide performances better than the conventional analog-mode PMTs.

## 2. Principle of Analog SPC

Photon counting is digital in its concept and gives an output in the form of an integer value that corresponds to the number of detected photons for a given integration period. Digital signals are not only attractive in acquisition precisions but also in storing and interfacing capabilities. Its advantages become pronounced especially when we need to measure very low-level light signals in long integration times. Extremely low-level light signal can be detected by keeping the noise level of a system very low with the help of a digitizing procedure. However, these advantages are not much helpful for the high-speed measurements of wide-bandwidth signals, in which the shot noise of a system becomes a dominant noise source, and the signal-to-noise ratio (SNR) of a system can be approximated as *SNR* ≈*N*/√*N*=√*N*, where N is the number of detected photons. Notice that the shot noise can not be eliminated by photon counting because it is an optical noise. Therefore, integer precision in photon counting is not thought to be necessary in obtaining a good SNR unless the error in counting is very high. Increasing the count rate can be achieved at the sacrifice of counting precision and does not degrade the overall performance.

A typical structure of the conventional digital SPC system is illustrated in Fig. 1. An analog signal from a PMT is fed to a comparator (Comp) or a discriminator. The output waveform of the comparator is a rectangular pulse, and the width of the pulse is determined by the full width of the input analog signal at the threshold level of the comparator. In order to drive the following digital electronics, this rectangular pulse is re-shaped by a mono-stable multi-vibrator also called as a one-shot so that the width is regulated to be above a certain value. This final digital signal is fed into a digital counter for a given integration time until a clock signal resets the digital counter. The output interface is required to synchronize the output signal with the integration clock and can convert it into serial data or an analog voltage signal by using a converter.

The speed of this electronic circuit is normally determined by the one-shot and the counter used in the system. Since the maximum toggle rates for logic devices are usually less than a few hundreds of MHz, the one-shot should make sure that the output pulse has a predetermined duration above ~10 ns, which supports reliable operation of the counter. Thus, the maximum count rate, *M*
_{max} is determined by the duration of the one-shot output, *Δ*
*t*, so that *M*
_{max}=1/*Δ*
*t*. In practice, the effective maximum count rate is limited by the nonlinearity of a SPC system. As the detection of a single photon is a stochastic process, there is always a nonzero probability of *pile-up* or collision of multiple photons. If more than two photons arrive simultaneously within *Δ*
*t*, a SPC system would count the first one only, and ignore the others. The amount of this error is increased proportionally with the increase of measured photon number and appears as the nonlinearity problem of a SPC system [1]. This linearity error of SPC, *ε* can be represented by

where *N* is the actual rate of detected photons, and *M* is the photon rate measured by the SPC system. In most cases of imaging applications, linearity errors within ±20% are acceptable. Or this level of deterministic errors can be compensated almost exactly by the calibrated information of *M*
_{max}. In this paper, the effective maximum count rate, *M*
_{e.m.} is defined by the highest count rate that exhibits a linearity error smaller than 20% for the convenience. According to the best of our knowledge, the highest value of the maximum count rates of commercially available SPCs is 200 Mcps and, hence, the effective maximum count rate is estimated to be 40 Mcps [10]. This means that it is impossible to obtain *M*>400 for an integration time of 10 µs, required minimally for high-speed scanning microscopes. The dynamic range as the ratio of the maximum signal amplitude to the minimum is only 400 in this case. Considering required margins of the dynamic range for practical microscope operations to be ~10, the dynamic range of an image can be <40, and consequently, the peak signal-to-noise ratio is <40^{1/2} due to the shot noise. There is a need of further enhancement of the dynamic range for better quality of images even for the best of the digital SPC.

The counter and its output interface in the SPC system described by Fig. 1 can be simplified by using an analog integrator *i*.*e*. a low-pass filter, as long as the output pulses of the discriminator have a constant amplitude and duration. The response of an analog SPC system with respect to a single photon can be determined by the number of electrons in the output port of the low-pass filter. Neglecting additional noises introduced by using the analog electric devices, the only difference between the digital output and the analog output interface is the way of integration. In the digital SPC scheme, the number of photons is added for a well-defined temporal period of a non-overlapping rectangular slot defined by the clock. In the analog SPC, the number of resulted electrons is integrated by a moving window defined by the filter’s impulse response. So, the outputs can be translated to the number of photons in the digital, and the average photon rate in the analog output interface, respectively. The digital approach is more flexible and provides with some useful operation modes like gated integration. However, the analog output interface seems to be more attractive for the microscopy application owing to the system simplicity and direct compatibility with the analog signal interfaces.

The technical challenge in realizing an SPC system is how to regulate the width of the discriminator’s output pulses. For a typical mono-stable multi-vibrator, the durations of the output will exceed the pre-determined value if the input pulses have larger widths than this value. Furthermore, the output durations should be set larger than the propagation delays of the used logic gates for stable operations, and they are usually too long to support high count rates. There is a simpler alternative to the one-shot used in SPC. It can be realized by utilizing the nature of differentiation. Based on this simple principle, we propose a new analog SPC system whose schematic diagram is shown in Fig. 2. Three plots in the lower part of Fig. 2 describe the schematic signal waveforms at each marked point of ⓐ, ⓑ and ⓒ in the circuit diagram shown in the upper part of Fig. 2. When a single photon is detected by a PMT, an analog electric pulse is generated and is amplified by a pre-amplifier (Amp1). The amplified pulse signal is fed to a comparator through and a differentiating capacitor (C). A square pulse is made by the comparator with a proper setting of an external discrimination level. This pulse is amplified by a second amplifier (Amp2) with a narrow bandwidth, which is used as an output interface. As mentioned above, this narrow-bandwidth amplifier can work as a passive integrator because of its low-pass filtering characteristics. In our analog SPC circuit, we have used an inexpensive head-on PMT (R7400, Hamamatsu), a 1-GHz low-noise amplifier (ZFL-1000LN, Mini-Circuits) for Amp1, and an operational amplifier (AD811, Analog Devices) for Amp2. The 3-dB bandwidth of the integration amplifier (Amp 2) is about 1 MHz. For the comparator, we have used a limiting amplifier (SY88993AV, Micrel) that was originally designed for multi-gigabit-per-second telecom receivers. This limiting amplifier has very short rise/fall time of ~60 ps and is used as a high-speed comparator in our case.

The key operation principle of our analog SPC system can be explained by the role of the differentiating capacitor denoted by C in Fig. 2. Without using this capacitor, the random pulse height fluctuations of the PMT output would be converted to pulse-width variations in comparator output. It is well known that a serially connected resistor-capacitor (RC) circuit acts as an analog differentiator for low-frequency components. The condition of differentiation operation is roughly,

where *Δ*
*f* is the bandwidth of the signal fed to the capacitor and *R* is the load resistance of the comparator, which is 50 Ω in our system. As shown in Fig. 2, the waveform at ⓑ is an approximate derivative of the waveform at ⓐ, and this differentiated signal is discriminated by the comparator. The signal bandwidth at ⓐ was measured to be less than 300 MHz in our analog SPC. The capacitance was chosen to be C=7 pF to satisfy the requirement of Eq. (2).

The typical impulse response of a PMT for a single-photon input has a long tail due to the slow discharging process of the parasitic capacitance of the PMT. This slow capacitance discharge results in a long exponential decaying tail in the PMT output. Because of this tail, the pulse duration of a discriminator output has a linear dependence on the input pulse height. By using an analog differentiating circuit, we have solved this problem and obtained a well regulated pulse at the output port of a comparator. The output pulse duration is clearly bounded such that it is less dependent on the pulse height fluctuations. The performance of pulse-width regulation effect in our proposed system was tested by measuring the analog electric waveforms with (AC-coupled) and without (DC-coupled) a differentiation capacitor. Figure 3 shows the single-photon responses measured at the input of the comparator in DC-and AC-coupled cases (a), and the calculated output durations as a function of input pulse amplitude for the comparator (b). The output durations for hypothetical PMT responses are also plotted for a purely exponential decay and a rectangular pulse. In Fig. 3(b), the heights (peaks) of the input pulses were normalized by the discrimination level. As seen in Fig. 3(a), AC coupling differentiated the PMT’s output waveforms successfully and make the positive-voltage part bounded within a finite period. Moreover, the dependence of the output duration on the input amplitude is clearly decreased as shown in Fig. 3(b). It is clear that the discharge decay generated by the PMT correlates the output duration with the input amplitude, while the differentiation capacitor reduces this effect by lowering the tail well below the discrimination level.

It is unnecessary to keep the capacitance very small for accomplishing an exact differentiation operation. It is sufficient for the capacitor to make the discharging tail of the single-photon response be negative below the zero level in voltage. Even though using the differentiation capacitor does not clean up the variation of the output duration perfectly, it regulates the duration sufficiently. Of course, the ultimate performance is anticipated for a rectangular pulse response, which may be obtained by using a one-shot. However, our fully analog approach has an advantage of high-speed operation comparing with the case of using a one-shot. Since the speed of the comparator is extremely high, the maximum count rate is limited by the bandwidth of the PMT in our analog SPC, and can be potentially increased to more than 1,000 Mcps by using a high-speed PMT such as an micro-channel plate PMT (MCP-PMT).

## 3. Photon counting performance

Various characteristics of the analog SPC method were investigated in this research, including the optimization of the discrimination level, photon counting precision and the effective maximum count rate. The counting characteristic of an SPC system depends very much on the discrimination level of the comparator used in the photon counting system. And we should find an optimized value for the best performance. In a digital SPC method, its discrimination level needs to be set between the average pulse height and the effective amplitude of noise of the system for both the best detection efficiency and the least erroneous counts [1]. It is usually optimized by analyzing the count rate statistics as a function of normalized average pulse height, which is defined by the ratio of the average pulse height of single-photon responses to the discrimination level. The measurement of count-rate statistics can be obtained, for a fixed light intensity, by adjusting the bias voltage of the PMT that changes the PMT gain or by adjusting the discrimination level for a fixed PMT gain. From the measurement result, so-called *plateau region* can be determined, in which the count rate is the most insensitive to the variation of the pulse height. The stability of operation is maximized at that operation condition so that the SPC output least depends on the extrinsic or intrinsic variations of the PMT gain. For a constant light input, the *stability gain*, *G*
_{stab} of an SPC system can be defined as the ratio of fractional PMT output change to the fractional variation of an SPC output as

where *V*
^{a}
_{p} is the average pulse height or the average peak output voltage of a PMT for a single-photon input, *V*
^{spc} is the average output amplitude of the SPC system. *V*
^{a}
_{p} can be considered as the voltage at the position of ⓐ in Fig. 2, while *V*
^{spc} corresponds to the voltage at ⓒ in our analog SPC system. *Δ*
*V*
^{a}
_{p} represents the variation of *V*
^{a}
_{p}, which results in the corresponding variation of the SPC output *Δ*
*V*
^{spc}
_{p}. Thus, the stability gain of the SPC system measures how much the PMT’s output fluctuation is suppressed by the SPC scheme. The output voltage of a PMT, *V*
^{a}
_{p} in Eq. (3) can be normalized by the discrimination level. For the normalized average pulse height, *V*
^{a}
_{norm} and its variation, *Δ*V^{a}
_{norm}, the stability gain is represented by

This can be evaluated by an experiment for an SPC system.

The stability characteristic was evaluated for our analog SPC by measuring the output voltages for various discrimination levels. Figure 4 shows the output voltage of the analog SPC, *V*
^{spc} as a function of the normalized average pulse height, *V*
^{a}
_{norm}. The stability gain was calculated from the data by using Eq. (4). The shapes of these curves are basically similar to those of the digital SPCs. But the slope of the plateau region is significantly higher than those of the typical digital SPCs. The maximum sensitivity gain was found to be 3.6 when *V*
^{a}
_{norm} was around 8. Considering that the digital SPCs usually have sensitivity gains bigger than 10, our analog SPC exhibits a significantly lower stability. But it exhibits an enhanced stability over the analog-mode PMT and must be more resistant to the variation of the PMT gain induced by changes of the bias voltage, external magnetic fields, mechanical vibrations and hysteresis [1]. In the following experiments of this paper, the discrimination level was set for the best stability by using this result.

Stability characteristic found in Fig. 4 does not only show the stability against long-term variations but also directly present the suppression of random pulse height distributions. It can be deduced from the results in Fig. 4 that the output height distribution must become narrower as much as the stability gain of an SPC system is enhanced. In order to investigate height distribution statistics of our analog SPC system, we measured output pulse height histograms by using an oscilloscope. Figure 5 shows the probability density functions (PDFs) as functions of pulse heights for the analog-mode PMT output (a) and the analog SPC output (b) for our system, respectively. In each graph, vertical axis is normalized by the mean pulse height calculated from each PDF. The inset graphs are displayed in semi-logarithmic scales to show count probability at large pulse heights. In Fig. 5, it is clear that the analog SPC system suppresses the height fluctuations successfully. The standard deviations were evaluated to be 0.49 for the analog-mode PMT output and 0.15 for the analog SPC output, respectively, which corresponds to three-fold improvement in suppressing the gain fluctuation. This result is consistent with the sensitivity gain characteristic of Fig. 4.

Note that the output pulse of a PMT for a single photon input sometimes shows extremely high amplitude. In theory, it is known that the amplitude PDF of the single-photon response obeys the Poisson distribution of electronic shot noise. However, the measured PDF curve of the analog-mode PMT output deviates from this theoretical distribution, and shows erroneous extraordinary probabilities at some high pulse heights as is observable in the inset of Fig. 5(a). This can not be explained by the collisions of multiple photons, because average detected photon rate for out experiment condition was estimated to be sufficiently small: ~0.1 Mcps. On the other hand, the small inset graph in Fig. 5(b) shows that our analog SPC scheme can eliminate these high-amplitude noises very effectively. This issue will be discussed further in the next section.

The discrimination capability of our analog SPC was tested by using 635-nm wavelength diode-laser pulses of ~100 ns duration. Figure 6 shows the oscilloscope trace of the analog SPC output captured in the persistence display mode on the left, and its corresponding vertical histogram on the right. The vertical histogram was extracted from the number of hits within the dotted box in the oscilloscope trace. As displayed in the histogram of Fig. 6, the number of photons within the input laser pulse could be resolved clearly up to five. This result successfully verifies the SPC-equivalent performance of our analog SPC system.

The effective maximum count rate and measurement linearity were evaluated by measuring the output of our analog SPC detection system as the intensity of input light is changed. A CW diode laser operating at an output wavelength of 635 nm was used for a light source. The output voltages were normalized properly to translate the output voltage to the numbers of detected photons. The characteristic time-voltage product *i*.*e*. the time-integrated voltage of a single-photon response was carefully measured to be 7.1 mV·µs for our system by using an oscilloscope with a low-intensity light. The output voltage was normalized by this time-voltage product. The measured photon counting rate as a function of the input light power is shown in Fig. 7(a), and the corresponding linearity error as a function of the measured photon rate is plotted in Fig. 7(b). The average power of the source was measured by an optical power meter. The dashed line in Fig. 7(a) represents the ideal linear response, and the dashed line in Fig. 7(b) shows a linear fit curve obtained from the data up to 120 Mcps. In calculating linearity error, the actual photon counting rate was estimated under the assumption that the linearity error is negligibly small for the case of count rates below 1 Mcps. As clearly observable in Fig. 7(b), the linearity error grows proportionally as the counting rate increases. This result suggests the collision of multiple photons causes errors for the analog SPC system in the same way as the case of a digital SPC system, which is represented by Eq. (1). The effective maximum count rate of 20% error was evaluated to be 100 Mcps in our case, which corresponds to a maximum count rate of 500 Mcps. This is 2.5 times higher than the fastest of commercially available digital SPC systems. If the absolute linearity is not required, it would be even possible to operate the analog SPC system above 200 Mcps for wider dynamic range. In very high count rates above 100 Mcps, our analog SPC exhibited larger linearity errors than expected by the theory as seen in Fig. 7(b). This can be explained by the baseline deviation caused by the AC coupling to the discriminator. For a high pulse rate, the relative discrimination level must have shifted to a higher voltage level owing to the baseline of the signal being lowered and this caused a decrease in the output signal amplitude.

The maximum count rate of an analog SPC system is determined by the pulse-width of a PMT output with respect to a single-photon input when the speed of a comparator used in the system is fast enough. While the pile-up occurs in a digital manner for the conventional SPC, the collision of two detected photons gives variable amount of the effect on the final result for our analog SPC. The two pulses are resolvable when the interval between the two is larger than the pulse-width. In this case, the comparator followed by the differentiation capacitor can detect two rising edges and produce two independent output pulses. However, the pulse-width of the second rectangular pulse of the comparator output can be smaller than that of the first one without a sufficiently wide temporal separation of the pulses. Then, the output of the analog SPC corresponds to the number of detected photons less than 2. Because of this complex nature, the actual maximum count rate is difficult to predict for the analog SPC. But it is obvious that the maximum count rate is inversely proportional to the pulse-width of the PMT response. Therefore, the effective maximum count rate, *M*
_{e.m.} can be represented as

where *Δ*
*t* is the pulse-width (FWHM) of the single-photon response, and *γ* is the pile-up factor. The pulse-width measured after the pre-amplifier was 1.6 ns in the full width at half maxima (FWHM) for our case. Thus, *γ* was evaluated to be 0.8 for our analog SPC system. For example, it is expected to obtain an effective maximum count rate more than 400 Mcps if we use a high-speed MCP-PMT of which duration of the single-photon response is less than 400 ps.

## 4. Signal-to-Noise Ratio in imaging

There are various noise sources for a PMT that affect the image quality of a scanning microscope system. The optical and electronic shot noises as well as the thermal noise of a PMT are well understood. They are a sort of the Gaussian noise in nature and can be evaluated by the root-mean-square amplitudes or standard deviations. However, PMT noises caused by the ionization current of residual gases or the high-energy particles are not suitable for this kind of analyses. These noise sources are usually too low in their frequencies of occurrence to affect the root-mean-square values. But their amplitudes are exceptionally high so that they can mess up the desired information.

The noise characteristics of the analog-mode PMT and our analog SPC were investigated and compared experimentally. CW laser light at 635-nm wavelength was used as a light source. One set of signals from the analog-mode PMT output and the other set of signals from the analog SPC system output were acquired in the time-domain by using an oscilloscope (DSO 6032A, Agilent Technologies) operating at an acquisition rate of 5 MS/s (mega-samples per second). Measured data sets were low-pass-filtered by a digital filter implemented in a computer. The 3-dB bandwidth of the full system including this digital filter was measured to be 70 kHz. If the digital filter were replaced by an analog filter, the optimal sampling rate would be approximately 4 times higher than the 3-dB bandwidth, ~280 kS/s. Thus, this experiment simulated a scanning microscopy imaging system of a pixel dwell time, ~4 µs. The lengths of the data were 10 ms, which corresponded to the signals for ~2,500 pixels, although they consisted of 50,000 data points acquired at 5 MS/s.

By using the two sets of data, two images were constructed for better understanding of the influence of these noises in imaging. Even though they are not actual images of an object but those of constant signals, they give us a good opportunity to demonstrate the effect of our analog SPC scheme in an actual imaging application. Figure 8 shows the time-domain waveforms (a), the corresponding images (b), and the vertical histograms (c) from the top to the bottom, for the analog SPC (Left column) and the analog-mode PMT (Right column). The average detected photon rate was 0.7 Mcps in the experiment. For a 4-µs pixel, 2.8 photons were dedicated in average. In all the plots, the amplitudes were normalized by the average values so that the amplitudes could be directly interpreted as the inverse of the SNR.

A number of high amplitude noise pulses were clearly observed for the analog-mode PMT as seen in Fig. 8, while the results of the analog SPC did not show this kind of *impulse*
*noises*. The histogram of the analog SPC output just obeyed the characteristic Poisson distribution of the optical shot noise. Figure 8(c) also shows that the impulse noises in the analog-mode PMT made the height distribution deviate from the Poisson distribution at high amplitudes, and these must have originated from other noise sources rather than the shot noise. Even though the occurrence frequency of these exceptionally high-amplitude impulse noises is relatively low, their effect on the image quality is significant. This must be a more serious problem in fluorescence microscopy in which the desired signals often appear as a set of small dots, especially in the case of single-molecule imaging techniques [12, 13]. Note that the images of Fig. 8(b) should be considered as a part of the full-size images because of their small effective numbers of pixels. Typical sizes of microscope images exceed 500×500 (250,000) pixels, which are ~100 times larger than the images of Fig. 8(b) in terms of area. So, the number of those noise impulses is thought to be ~100 times higher for a full-size image.

These high-amplitude noisy impulses were also observed for the single-photon response as shown in the inset graph of Fig. 5(a). Even for a low count rate with negligible multiple-photon collisions, erroneous pulses with significantly high amplitudes can be produced by a PMT. Based on the extraordinarily high amplitudes of these noise spikes, the origin of this noise is thought to be the ionization current of residual gases in a PMT or high-energy particles. Even though high-energy particles like cosmic rays of muons can generate such results, their occurring frequency is expected to be extremely low, and they are not likely to be the major source. Ionization current of residual gases is believed to be the most probable source of this effect for the occurrence frequency and high amplitude.

As illustrated in Fig. 2, a PMT is composed of a photocathode and multiple dynodes. In the middle of the electron trajectories, collisions of high-energy electrons and residual gases may occur inside the tube and, as a consequence, result in ionization of the gases. Then, the positively charged massive ions return to the photocathode and generate multiple free electrons. This effect is also known as *ion*
*feedback* and *afterpulse* because it produces a secondary pulse that follows the original photo-electronic pulse after a delay. The afterpulse count rate is proportional to the photoelectron count rate and increases by the increase in the signal intensity. All the kinds of PMTs based on the vacuum-tube technologies can suffer this problem. Helium, the major residual gas species can penetrate from the outside into the glass tubes, especially when a PMT is used in a helium-rich environment for a long time. Therefore, it is deduced that the alternative PMTs such as HPDs and CPMs also generate this kind of impulse noises.

It is obvious that the proposed method of analog SPC regulates these high-amplitude impulses. Even though these noise pulses are also counted, the output amplitudes are not bigger than those of single photons. Because of their low appearance probabilities, they rarely make a significant effect on the visual quality as long as they are regulated by the SPC scheme. In order to quantify the strength of the impulse noises, we introduce a new measure of signal-noise ratio i.e. the *signal-to-noise-peak-ratio (SNPR)*. SNPR^{m}, *m*-th order SNPR is defined as the ratio of the signal to the noise peak amplitude of 10^{-m} integrated probability of appearance. For an area-normalized PDF of height distribution for a CW optical signal that can be measured by the vertical histogram, *P*(*ν*), the PDF of noise height distribution, *P*
_{n}(*ν*
_{n}) is given by *P*
_{n}(*ν*
_{n})=*P*(*ν*
_{n}+*S*), where *S*≡∫^{∞}
_{0}
*ν*
*P*(*ν*)*d*
*ν*is the mean amplitude as the desired signal. The noise peak of *m*-th order, *Ω*
^{m} is determined as

And the SNPR^{m} is defined by *SNPR*
^{m}≡*S*/*Ω*
^{m}. For example, SNPR^{4} can be interpreted as the ratio of the signal amplitude to the maximum amplitude of the noise that can be observed statistically in 100×100 data points at least. On the other hand, SNR is defined as the ratio of the signal amplitude to the standard deviation of the noise:

SNR and SNPR were evaluated as functions of the optical irradiation power by acquiring the vertical histograms of the analog SPC output and the analog-mode PMT output, respectively. Figure 9 shows the SNR and SNPR^{4} curves for the SPC, analog mode and the theoretical Poisson distribution, respectively. The improvement of our analog SPC in SNR value was not considerably high, just ~10% better to the analog-mode PMT. Note that the SNR performance of the analog-mode PMT is already near the theoretical limit. On the contrary, the improvement in SNPR^{4} was found to be very significant, and was more obvious for low-intensity irradiations. While the analog-mode PMT suffered from impulse noises, our analog SPC exhibited a good SNPR performance similar to that of the theoretical shot-noise limited photodetection. This observation well explains the results of Fig. 8. In terms of SNPR^{4}, improvement of factor 3 was achieved by the analog SPC for the case of Fig. 8, and it clearly distinguishes the image quality obtained by the analog SPC from that of the analog mode. After all, this result demonstrates that our analog SPC scheme provides a significantly better noise characteristic.

## 5. Conclusion

We have introduced a new version of SPC based on simple analog electronics circuits, mainly aiming at better dynamic ranges which have been limited by the low maximum count rates in the conventional SPC technique. We have verified that the basic photon counting characteristics of the developed analog SPC are sufficiently good in discrimination precision, stability and noise suppression. The operation stability was evaluated to be more than 3 times better than the analog-mode counterpart, which means the photodetection performance is more immune to the various intrinsic and extrinsic changes of the PMT gain. This feature must be useful in many applications in which signals of multiple frames are compared quantitatively. The effective maximum count rate was significantly improved due to the simplicity of the structure of our analog SPC method.

For the effective noise analysis of a photon detection scheme, we have introduced a new noise characterizing parameter: *m-th order signal-to-noise-peak-ratio* (SNPR^{m}). Conventional measures of noise property such as SNR only have focused on standard deviations of the noise amplitudes and cannot efficiently show the effects of noises with high amplitudes but low occurrence probabilities. We have observed that the afterpulse problem is a typical problem for a PMT, and this may degrade the image quality significantly for the analog-mode operation. By measuring SNPR^{m} for our analog SPC scheme and comparing it with that of analog-mode PMT output and, we have demonstrated that this problem can be effectively solved by our analog SPC detection scheme.

The overall performance of the analog SPC is between that of an analog-mode PMT and that of a digital single-photon counting method. In this context, this method can be called as *regulated analog-mode operation*, emphasizing its analog features as well as analog single-photon counting, stressed on the SPC-like properties. This method provides a hybrid approach which mixes those two conventional approaches. We believe that various features of our analog SPC method are very attractive in the high-speed scanning microscopy application. Its high-speed operation capability, wider bandwidth (higher measurement repetition frequency) and higher maximum count rate enables faster data collection with sufficiently high signal amplitudes along with its noise suppression capabilities. Furthermore, its analog output interface provides with direct compatibility to the conventional microscopy instruments that usually use analog-mode PMTs. This technique must be very suitable for nonlinear-optic microscopy techniques such as MPE or CARS microscopy in which signals are relatively weak and ultimate performances are required for the photodetectors.

## Acknowledgments

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

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