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We report a spectrally resolved fluorescence lifetime imaging system based on time gated single photon detection with a fixed gate width of 200 ps and 7 spectral channels. Time gated systems can operate at high count rates but usually have large gate widths and sample only part of the fluorescence decay curve. In the system presented in this work, the fluorescence signal is sampled using a high speed transceiver. An error analysis is carried out to characterize the performance of both lifetime and spectral detection. The effect of gate width and spectral channel width on the accuracy of estimated lifetimes and spectral widths is described. The performance of the whole instrument is evaluated at count rates of up to 12 MHz. Accurate fluorescence lifetimes (error < 2%) are recorded at count rates as high as 5 MHz. This is limited by the PMT performance, not by the electronics. Analysis of the large spectral lifetime image sets is challenging and time-consuming. Here, we demonstrate the use of lifetime and spectral phasors for analyzing images of fibroblast cells with 2 different labeled components. The phasor approach provides a fast and intuitive way of analyzing the results of spectrally resolved fluorescence lifetime imaging experiments.
©2013 Optical Society of America
1. Introduction
Lifetime and spectral imaging are complementary techniques that offer noninvasive solutions for quantitative biomedical microscopy. These techniques can, for instance, be used for extracting information about metabolic processes, identification of biochemical compounds and visualization of molecular interactions. The decay kinetics and the emission spectrum of the excited chromospheres can be spatially mapped using a standard microscope equipped with time resolved detection systems for lifetime imaging or spectrographs for spectral imaging. Emission spectra of the excited molecules can be used in co-localization studies of different molecules via unmixing algorithms [1,2], to assess pH [3] and intermolecular energy transfer [4].
Fluorescence lifetime imaging adds another dimensionality to fluorescence imaging and can be employed to assess environmental parameters like calcium concentration [5,6], pH [7], oxygen saturation [8], discriminating bound and free NADH [9]. It can also be used to identify individual components in tissues and their fractional contribution to the fluorescence signal [10]. Importantly, lifetime imaging has become the technique of choice to study molecular interactions. To this end two molecules, donor and acceptor, are fluorescently labeled with different, matched fluorophores. When the molecules are within 10nm of each other Förster Resonance Energy Transfer (FRET) takes place which results in shortening of the donor fluorescence lifetime. This application turned lifetime imaging into a major tool for biological applications [11,12].
Time resolved and spectral imagings provide complementary and sometimes overlapping information. Simultaneous recording of spectra and lifetimes offers the prospect to obtain more information and achieve higher accuracies than with a single technique. To fully take advantage of this, novel instrumentation that is both robust and affordable is required.
Spectrally resolved lifetime imaging systems have been developed that employ frequency [13] and time domain [14,15] based lifetime detection.
Multispectral lifetime imaging has been employed to study the properties of fluorescent molecules in live cells using a time- and space-correlated single-photon counting detector [16]. Furthermore, the application of multispectral lifetime analysis for FRET imaging has also been reported [17,18]. Here, spectral selection was carried out using a polychromator in front of the detector.
Other techniques are also reported. Beule et al. employ an electron multiplying CCD (EMCCD) equipped with a gated intensifier to record series of time gated images at different delays after excitation [19]. The spectrum is recorded by dispersing the fluorescence signal on an electron multiplying CCD (EMCCD) camera resulting in 32 spectral channels with 27nm spectral channel width. Another method uses two beam splitters to guide the light into three fibers with different lengths. This separates the decays of different spectral channels are in time. A single PMT is used as a detector to collect the entire signal [20]. These systems have been used to improve the accuracy of FRET analyses by including information from both donor and acceptor channels [21,22]. Another application is the unmixing of fluorescence components without a priori knowledge of the sample [23].
To date, all the time domain multispectral FLIM systems are based on time correlated single photons counting (TCSPC) [24]. FLIM systems based on continuous sampling, including some time-gating implementations, offer advantages over TCSPC systems. Due to the low dead time of the electronics these systems can operate at high count rates (~10 MHz) without noticeable pile-up effects due to detector electronics, which makes them well suited for rapid lifetime imaging. The maximum count rate of the whole system is usually limited by the detector. In Time Gating the gates are usually comparatively long and only a limited number of gates are employed. As a result part of the decay curve is not recorded and the curve is often sampled with comparatively long gate widths. The effects of truncation and sampling of the decay curve are described in [10,25].
In this paper we introduce a novel multi-channel detection method for spectrally resolved fluorescence lifetime imaging. This “Lambda-Tau” detector employs an FPGA (Field-programmable gate array) equipped with 8 high speed transceivers running at 5 GHz to record whole decay curves at high count rates and high time resolution (200ps). The multi-channel feature of the FPGA enables simultaneous recording of decay curves for each spectral channel and offers a powerful, scalable solution for spectrally resolved fluorescence lifetime imaging.
2. Materials and methods
2.1 Instrument details
The setup consists of a confocal microscope (Nikon C1, Japan) equipped with a 473 nm solid state diode laser (Becker & Hickl, BDL-473-SMC) with a pulse repetition rate of 20-80 MHz and pulse-width of 40 ps. Imaging of solutions of Fluorescein is performed using a 10x objective (NA = 0.3 Nikon, Japan) and cells are imaged with a 60x water immersion objective (NA = 1.2, Nikon, Japan). In all cell images the field of view is 60x60 um2 and the image size is 160x160 pixels. The confocal microscope is equipped with a fibre coupled output port and a 900 µm diameter fibre is used to couple the microscope to a prism based spectrograph. The latter consists of a 100 mm achromatic lens, to make the fluorescence beam parallel, an anti-reflection coated SF10 prism and another lens with the same focal length to focus the dispersed spectrum onto a 32 channel multi-anode linear PMT (Hamamatsu, H7260). The spectral range from 510 nm to 690 nm covers the 7 anodes of the PMT used in our experiments. The spectrograph is wavelength calibrated using a white light super continuum source (NKT photonics, SuperK) equipped with an Acousto-Optic Tuneable Filters (AOTF). Single photon output pulses of the PMT channels are fed into the Lambda-Tau electronics module. The total measurement time window in our experiments depends on the laser frequency and varies from 12.5 ns to 50 ns. The dead time of the electronics itself is negligible, but at present only one photon can be detected per laser pulse per detection channel. A schematic overview of the detection system is shown in Fig. 1 . The output signals from the multi-anode PMT are first amplified by a fast 8 channel amplifier (bandwidth 1.5GHz, Phillips scientific 774-s-50). Next the pulses are thresholded using a high-speed comparator (MAX9600). Signals above the threshold are fed into a multiplexer which is used to select between the thresholded signal and an internally generated signal used for calibration purposes. The multiplexed signal is fed into a programmable configurable fine delay with a resolution of 1ps. The delayed signal is fed into the input of a high speed configurable transceiver of the FPGA. The FPGA, an Altera Stratix II GX family FPGA, is equipped with eight high speed (5 Gbit) transceivers.
The transceivers act as a deserializer and convert the serial input signal into 40-bit words and buffer the 40bit words using a FIFO. Another programmable delay with coarse resolution of 200 ps is employed. Both programmable delays are employed to compensate the inter channel time delays, optical delays, PMT colour effects [26], signal path length differences and component delays. Here, the delays of the whole system are calibrated by recording the single exponential fluorescence decay from an aqueous solution of Fluorescein (1 mM at pH 10). Delays are found by fitting an exponential decay convoluted with the experimentally recorded instrument response function and using the shift as a fit parameter.
The laser trigger signal is treated in a similar way. One of the transceivers is reserved for recording the laser trigger signal and the remaining channels are employed for recording the signal from the single fluorescence photon events. The laser trigger signal serves as timing reference for the event time measurement of all channels. The deserializer output is analysed for the presence of the laser trigger and event signals and used to determine the time difference between the rising edges of the laser trigger and event signals with a time resolution of 200 ps.
The performance of the system is tested using a fast pulse generator. A sustained count rate of 25 MHz can be detected; this is limited by the communication speed of the USB2 port. The event times of all channels are stored in the 512 MByte buffer memory via the FPGA memory controller before transfer to the host PC through the USB 2.0 port. A total count rate of 125 MHz can be recording for 4s before filling up the entire buffer memory.
2.2 Data analysis
Spectral lifetime imaging results in very large data sets and requires fast and reliable methods for data analyses. The most commonly used method to analyse fluorescence lifetime data is iterative convolution based on non-linear least square minimization [27]. This is a standard method that is time consuming and not capable of providing fast (real-time) analyses of the acquired data. Although techniques have been developed to improve analyses speed [27–29], this approach remains computationally intensive and comparatively slow. Lambda-Tau images are typically one order of magnitude larger than standard lifetime images. Therefore, implementation of nonlinear least square fitting on a pixel by pixel basis is not suitable for fast analyses of Lambda-Tau images.
Alternatively, recently introduced phasor approaches can be employed to analyse lifetime [10,30] or spectral images [2]. These are graphical approaches that are comparatively simple and fast, but need prior calibration on well characterized samples to obtain quantitative results [31]. Briefly, the real and imaginary parts of the Fourier transform of the fluorescence decay curve or fluorescence emission spectrum are used as coordinates in the phasor plot. The phasor plot and the image are correlated; every point in the phasor plot can be traced back to pixels with the same property in the image. Moreover, every decay or spectrum is mapped onto a unique position in the phasor plot and the position of the phasor determines the lifetime or the spectrum. A region of interest in the phasor plot can be back projected to the pixels correlated with the selected phasor points. This results in fast and convenient image segmentation.
The application of the phasor approach to time domain data with different time gate settings and acquisition periods has been shown before [10]. This theoretical frame work can be applied to different time gate configurations. The general phasor semi-circle is expressed by:
where K is the number of time gates, T is the total acquisition period and τ is the lifetime. R is a complex number and the reference semicircle is generated by drawing the imaginary part of R versus real part. Figure 2(a) shows the modified reference semicircle for different gate numbers and the same acquisition period for a lifetime range from to . As the lifetime increases the phasor moves on the semicircle from right to left. When , R converges to the standard phasor curve [10]:The average lifetime can be estimated by the following equation:where G is the Imaginary and S the real part of the phasor. For a binary system with invariant lifetimes and a spread in the relative contributions of the components, a line can be fitted through the phasor points and the intersection of this line with the phasor semicircle yields the lifetimes of the two components:where v and u are the slope and the intercept of the fitted line.
Fig. 2 a) Plot of the modified semi-circle for different gate numbers. b) The phasor of a bi-exponential decay curve with lifetimes , and fractional intensity (open circle).
Similar to lifetime images, the spectral images recorded with the Lambda-Tau detector can be analyzed using the spectral phasor approach; at present the number of spectral channels of the hardware is limited to 7 but this can be easily extended. The spectral phasor approach is versatile and can be employed at low and high numbers of spectral channels see [2]. Briefly, based on the shape and the maximum emission wavelength, the Fourier transform of the spectra are mapped to a unique position in the phasor plot. The peak position (center of gravity) defines the phase in the phasor plot. When the emission moves from red to blue, the phasor rotates from small angles to large angles. The width of the spectrum is related to the modulus of the phasor and as the width gets broader the phasor moves toward the center of the phasor plot. The phasors of artificial Gaussian spectrums with variable peak position and widths are shown in Fig. 3 for different numbers of spectral detection channels (K = 8, K = 16 and K = 128). The phasor curves for zero width (w = 0, corresponding to a delta function) show the maximum extent of the phasors. None zero width spectra fall inside these curves. In Fig. 3 curves are shown for widths w = 0, w = K/8 and w = K/4. The reduction in extent of the semicircle for larger widths is due to the truncation of the Gaussian spectra at the edges of the spectrograph; part of the spectra is not recorded. The phasors for peak positions k0 = K/4, k0 = K/2 and k0 = 3K/4 are also shown in Fig. 3. For the zero width spectra, these phasors have phase angles of π/2, π and 3π/2 radians respectively. By increasing the spectral widths the length of the phasor curves shortens; a deviation for k0 = K/4 and k0 = 3K/4 phasors is observed which is due to the non-symmetrical recording of the spectra. The spectra are truncated at the edges of the spectrograph at one end. This effect is not observed for the spectra which are positioned at the center of spectral range of the spectrograph.
Fig. 3 Spectral phasor plots for Gaussian spectra recorded with K spectral channels and with widths 0, K/8 and K/4. k0 indicates the peak position (wavelength) of the spectrum.
2.3 Sample details
The Lambda-Tau detector is tested using different specimens. 1 mM aqueous solutions are prepared from Fluorescein (Sigma-Aldrich F6377). FluoCells slide #6 from Invitrogen (F-35925) are used in the imaging experiments. They contain fixed, permeabilized, and labeled muntjac skin fibroblast cells. Mitochondria are labeled with mouse anti–OxPhos Complex V inhibitor protein antibody and visualized using orange-fluorescent Alexa Fluor 555 goat anti–mouse IgG antibody. F-actin is labeled with green-fluorescent Alexa Fluor 488 phalloidin, and the nucleus is stained with TO-PRO-3 iodide.
3. Results
3.1 Error analysis
The Lambda-Tau makes use of continuous sampling of the laser trigger signal and the fluorescence signal. Therefore, it can operate at almost any laser frequency and the total sampling period of the decay equals the time between two laser pulses. The width of the time gates, however, is determined by the clock frequency of the high speed transceivers and fixed at 200 ps. Two major factors determine the accuracy of lifetime acquisition systems: the length of the measurement window and the gate widths. The latter affects the estimation of short lifetimes due to under sampling and the former the estimation of long lifetimes by truncation effects. In order to investigate the sensitivity of the system and compare different settings, a figure of merit F [32] was calculated. F does not depend on the number of detected photons and its magnitude is always larger than 1. Moreover, lower values of F correspond to higher sensitivities for the lifetime measurements. Fτ for lifetime measurements is defined as:
whereis the lifetime and is the standard deviation of the estimated lifetime. It is assumed that the andare governed by Poisson statistics. Therefore we can rewrite F as:The effect of the gate width and the length of the measurement window can be incorporated by employing the results of Köllner et.al [33]. The analytical solution of the variance in the lifetime estimation can be calculated using Fischer information theory and is given by:where K is the number of time gates, T is the total length of the measurement window, is the lifetime and Using Eq. (7), Fτ can be written as:whereFigure 4(a) . shows the figure of merit Fτ as a function of lifetime for a total measurement time windows of 12.5ns, 20ns and 50 ns, which corresponds to laser frequencies of 80MHz, 50MHz and 20 MHz respectively. The figure shows that for short lifetimes the Fτ values for different total time windows converge to the same curve, determined by the fixed gate width of 200ps. For long lifetimes Fτ increases due to truncation of the decay curve. Part of the signal is not included in the analyses and the sensitivity of the measurement goes down.Fig. 4 a) Figure of merit for total time windows of 12.5ns, 20ns and 50 ns and a gate width of 200 ps and b) for a fixed total time window of 50ns and K = 1000, 500 and 250 corresponding to gate widths of 50ps, 100ps and 200 ps respectively.
In order to demonstrate the effect of the gate width on Fτ, Fig. 4(b) shows Fτ as a function of lifetime for gate widths of 50ps, 100ps and 200 ps and a total measurement window of 50ns. This corresponds to K = 1000, 500 and 250 respectively. As expected, now the figure of merit for longer lifetimes takes on the same values for different gate widths, but for short lifetimes, Fτ increases for large gate widths.
The required number of photons to realize certain accuracy can be calculated using:
For instance, to analyze a mono exponential decay with a lifetime of 200 ps using gate widths of 200 ps, the required number of photons for realizing a 10% accuracy amounts to 108. This is, only 8% higher than the number of photons for Fτ = 1.In order to investigate the sensitivity of the spectral imaging system a figure of merit Fλ was defined and calculated for different configurations. Assuming a Gaussian spectrum with spectral width σ, k spectral channels and a total spectral detection range L, the variance in the spectral width can be calculated using Fischer information theory:
Where erf is the error function and N the number of detected photons. The Figure of merit for the spectral width estimation can now be calculated using Eq. (6). Spectral figure of merit curves are shown in Fig. 5 for a fixed spectral window of 300 nm for 4, 8, 16 and 32 spectral channels. The maximum emission wavelength is assumed to be fixed and at the centre of the spectral range. For large spectral widths the figure of merit curves exhibits the same behavior for all numbers of spectral channels and the figure of merit gradually goes up for increasing wavelengths. This is due to the truncation of the spectra at the edges of the spectrograph. For small spectral widths the figures of merit rapidly increases; as expected this effect starts sooner for spectrographs with a low number of channels. For typical organic fluorescent dye molecules the spectral widths are in the order of 40nm. Increasing the number of spectral channels from 8 to 32 only slightly improves the performance of the spectrograph at this spectral width. Fλ changes from 0.77 to 0.72, corresponding to a decrease from 58 photons to 51photons for realizing a 10% accuracy. We note that the current configuration of Lamda-Tau has 7 spectral channels. The Figure of merit curves for 7 and 8 spectral channels almost completely overlap.3.2 Lifetime accuracy and count rate
To demonstrate the performance of the system at different count rates, a 0.1mM aqueous solution of Fluorescein with pH 10 is imaged using different neutral density filters in the emission path. In this way effects on the fluorescence emission related to variations in the excitation power, e.g. saturation effects, are avoided. The recorded images are sorted according to their count rates and analysed with iterative convolution technique. The laser is operated at a 20 MHz repetition frequency corresponding to a 50ns time window and 250 time gates. The instrument response is measured by recording scattered laser light from the surface of a cover glass. Figure 6(a) . shows a typical measured decay curve recorded at a count rate of 0.1 MHz and a single exponential fit. The latter is based on iterative convolution that includes the instrument response function; the indicates a good quality fit. The background signal is estimated in a separate fit. Figure 6(b) shows the estimated lifetime as a function of count rate. The fitted lifetime decreases with count rate. This is explained by the dead-time of the PMT; by increasing the count rate the probability of two successive photons arriving at the PMT within the dead-time of the PMT goes up. This pile-up is more probable at the beginning of the decay curve which results in a shortening of the measured lifetime. In addition, the detection efficiency goes down. Below a count rate of 1MHz the shortening in the life time due to the dead-time of the PMT is less than 2% (dotted line). In order to test the Lambda-Tau at higher count rates, the outputs of four PMT are combined using high bandwidth power combiners and connected to a single channel of the Lambda-Tau electronics. Now, the error in the lifetime stays below 2% up to a count rate of about 5MHz, see Fig. 6(c). By increasing the count rate above 5 MHz the error in the estimated lifetime increases and it shows a similar trend as in the case of a single PMT. This demonstrates that the individual channels of the Lambda Tau can operate at high count rates without significant pile-up effects.
Fig. 6 a) Measured and fitted fluorescence decay curves. b) Fitted lifetimes as a function of detection count rate using a single detection channel. c) As in b) using the signal from 4 combined channels. The dashed line indicates an accuracy of 98%.
Importantly, in most spectral imaging experiments the fluorescence signal is spread out over multiple spectral channels; in our 7 channel system count rates as high as 7 MHz can be realized without significant pulse pile-up effects. Basically, the performance is limited by the detector performance and when al 32 PMT channels would be used count rates as high as 32 MHz can be dealt with without significant pile-up effects.
3.2 Analysis of recoded image of cells
Spectral lifetime images (60x60 µm2) of FluoCells test slide #6 are recorded with the Lambda Tau detector and analyzed using the Spectral Phasor and Time Gated Phasor methods. Homemade ImageJ plugins are used for the analyses (plugins are available from http://www.Spechron.com/ Fig. 7 . shows a series of lifetime images recorded at different spectral channels and their corresponding phasor plots. The 7th channel with central wavelength of 754 nm is not shown here because of very low counts collected at this channel. Lifetimes are calculated employing Eq. (3) using K = 62 and T = 200 ps (80 MHz laser frequency). The central wavelength of each spectral channel is indicated at the bottom left in the lifetime images. By moving from the green to the red emission channels, the lifetime phasor gradually moves from the phasor of Alexa 488 with average lifetime of 2.23 ns to the phasor of Alexa 555 with lifetime of about 1.55 ns. In the fourth channel both Alexa 488 and Alexa 555 are present resulting in an elongated phasor. In the lifetime images the same trend is visible from the color coding of the lifetimes; when moving from shorter to longer emission wavelengths the Alexa 555 lifetime becomes visible. In the second image, central wavelength 536nm, most of the pixels have count rates of ~0.5 MHz and some pixels have count rates close to 1MHz. The minimum image acquisition time is determined by the time required to collect enough photons in each pixel. According to Eq. (10) and using K = 62, T = 200 ps and 160x160 pixels, it takes 140 photons to measure a lifetime of 3ns with 10% accuracy. This takes about 4 seconds at 1MHz count rate. Considering 7 spectral channels the total count rate can reach 7 MHz and binning of all channels results in acquisition times of less than 1 second. For the image of Fig. 7 the recording time is almost 30 seconds. This is due to the comparatively low count rate at the longest wavelength channel.
Figure 8(a) shows the spectral phasor and the real-color representation of the spectral image [34]. The spectral image is obtained by binning all time channels. As expected for a two component system the phasor is elongated. Two circular regions of interests are selected in the phasor plot and the corresponding images are shown on the right. The region of interest indicated by the upper circle selects the pixels with shorter emission wavelengths and the lower circle selects the pixels with longer emission wavelengths. Segmentation based on back projecting these region into the original image yields images corresponding with the Alexa 488 colored actin fibers and Alexa 555 colored mitochondria. Figure 8(b) shows the time gated phasor for the same image set, now binned over the spectra. Again, two circular regions of interest are selected and their corresponding pixels are shown. The left circle segments the pixels with longer lifetime, corresponding to Alexa 488, and the right circle segments the pixels with shorter lifetime, corresponding to Alexa 555. Good correlation is observed between the segmented images from spectral and time gated phasors. Figure 8(b) also shows the lifetime image at the bottom. Here, the lifetime is color coded and scaled with the intensity (counts). The complete lifetime phasor analyses of 7 spectral channels with 64 time bins each per pixel of an 160x160 pixels image takes about 4 seconds using a single core on a standard PC. This is much faster than standard iterative convolution technique which takes about 1 second per pixel to estimate the lifetime. Phasor analysis facilitates fast analysis of the large spectral lifetime images.
3. Discussion and conclusions
A spectrally resolved fluorescence lifetime imaging system based on FPGA technology is presented. This novel system samples the decay with 200 ps resolution and can record full decay curves. The total measurement window is limited by the laser repetition frequency; it can be extended by lowering the frequency of the laser. An error analysis to investigate the performance of the system for lifetime detection is performed and shows that only 108 photons are required to achieve a 10% accuracy in lifetimes as short as 200 ps. In biological applications the lifetimes vary between ~200 ps to ~8 ns. The current configuration (200 ps gate width, 50 ns time window) is very well suitable for this kind of application. The accuracy for longer lifetimes can be increased by increasing the total measurement window. The system is capable of operating at high count rates which makes it useful for many applications. We tested the performance of the electronics at different count rates and it is found that the system can handle count rates as high as 5 MHz per channel without any significant pulse pile-up effects. The maximum count rate of the electronics is expected to be much higher, but no source of random counts was available to test the electronics at count rates > 5 MHz. In practice, the maximum count rate of the whole system is limited by the PMT performance and it is shown that the multi-anode PMT can operate at count rates as high as 1 MHz per channel without significant errors in the lifetime (<2%). In the multi-spectral approach the donor spectrum is dispersed over several detection channels. Therefore, the total allowable donor count rate can be several MHz. The error in the lifetime is particularly important in FRET experiments. Here, the FRET efficiency is proportional to the lifetime shortening and the error in the lifetime sets a limit to the smallest FRET efficiencies that can be detected. The total spectral window is about 280 nm wide, from 500 nm to 780 nm, and covers the spectra of a large number of fluorophores. However, it can be easily shifted to include more blue spectra.
Compared to single dimensional detectors, like spectral imaging or lifetime imaging systems, this setup provides more information with the same number of detected photons. Combined analyses of spectral and lifetime data can in principle provide more quantitative and more accurate information of biological events. Combined analysis of excitation spectra and lifetimes has been employed already for the unmixing of fluorophores; here fractional intensities and excitation spectra of each component were recovered with minimal a priori information [16].
FRET imaging is a major application for spectral imaging and lifetime imaging. The simultaneous recording of emission spectra and fluorescence lifetimes opens the possibility for a full and simultaneous analyses of the emission of both donor and acceptor molecules. Simultaneous analysis has the potential to provide more accurate energy transfer efficiencies at low signal levels. The analysis of multidimensional images, however, is challenging.
Here, a basic analysis based on a phasor approach is reported. This provides a fast and reliable means to analyses the spectral data and the lifetime data. Currently, the number of spectral channels in our system is limited to 7 but this can easily be extended to meet the requirements of more demanding experiments. Extending the system to handle higher number of spectral channels is under considerations. Multiple (8-channel) units can run in parallel to extend the number of spectral channels. On the other hand photon statistics should be considered as well; extending the number of spectral channels reduces the signal to noise ratio. It is shown that the Lambda-Tau unit can count higher count rates and the limitation is only due to the PMT. Lambda-Tau can be a perfect match for SPAD or hybrid detectors which can handle higher count rates than PMTs [35].
Acknowledgment
We would like to thank Dr. Kees van der Oord for his advice and support during the development of the control software for the Lambda-Tau detector.
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