Direct Measurement of Fast Transients by Using Boot-strapped Waveform Averaging
DDirect measurement of fast transients by using boot-strapped waveform averaging
Mattias Olsson, Fredrik Edman, and Khadga Jung Karki a)1) Department of Electrical and Information Technology, Lund University,Ole R¨omers v¨ag 3, 22363, Lund, Sweden Chemical Physics, Lund University, Naturevetarv¨agen 16, 22362, Lund,Sweden
An approximation to coherent sampling, also known as boot-strapped waveform av-eraging, is presented. The method uses digital cavities to determine the conditionfor coherent sampling. It can be used to increase the effective sampling rate of arepetitive signal and the signal to noise ratio simultaneously. The method is demon-strated by using it to directly measure the fluorescence lifetime from rhodamine 6Gby digitizing the signal from a fast avalanche photodiode. The obtained lifetime of4.4 ± . a) Electronic mail: [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] S e p . INTRODUCTION The analog to digital converters (ADCs) that are used in mordern signal digitizers havefixed sampling rates, that can range up to few giga samples per second for digitizers with 8bit ADCs. In order to further increase the sampling rate, one uses some form of interleavingmethods. The interleaving techniques may be classified into two groups, one based onhardware and other based on signal processing.The hardware based interleaving uses multiple ADCs, which digitize the signal in parallel.In this technique, the signal is simultaneously channeled to the different ADCs. The ADCsare triggered with slight phase difference such that they sample different sections of thesignal. Finally, the data stream from all the ADCs are merged based on the trigger sequence.In this technique, the effective sampling rate is the multiple of the sampling rate of one ADCand the number of the ADCs used in parallel.Random interleaved sampling (RIS) is another interleaving method based on the signalprocessing. It is specialized feature used in some of the digital sampling oscilloscopes (DSO) .In RIS, the signal is sampled multiple times by changing the trigger position using a time-to-digital converter (TDC). The value from the TDC is used to arrange and interleave thedifferent waveforms. If n waveforms are interleaved then the end result is a single waveformsampled at an effective sampling rate that is n times the digitization speed of the ADC. Oneof the drawbacks of this technique is that it requires TDC for controlling the trigger, whichis not available in general ADC boards.Coherent sampling is yet another method that can be used to increase the effectivesampling rate of repetitive signal. In this method, the sampling condition that fulfills thecondition in equation (1) f IN f s = MN , (1)where f s is the sampling frequency, f IN is the signal frequency, M is the integer number ofcycles in the data record and N is the integer number of samples in the record. Coherentsampling is primarily used in sinewave testing of ADCs. Nevertheless, the M cycles of thewaveform obtained by coherent sampling can be unwarped to increase the effective samplingrate. However, finding the right condition for coherent sampling is complex. Here, we implement an approximation of coherent sampling based on the digital cavities,which we call boot-strapped waveform averaging. The technique is simple, does not require2xternal trigger from the signal source during digitization, can be used to analyze any repet-itive signals and the effective sampling rate can be varied. We demonstrate the technique byusing it to measure the lifetime of fluorescence from a dye molecule, Rhodamine 6G, with atime precision of 10 ps using a digitizer with the normal sampling rate of 1 GSa/s.
II. ALGORITHMS OF BOOT-STRAPPED WAVE-FORM AVERAGING
The boot-strapped wave-form averaging is an approximation to the coherent sampling.In order to describe the algorithm, we assume that the signal is digitized at regular intervals, τ . In the method, one first approximates the window that holds complete M cycles by usingusing the digital cavities. The data stream is folded m times into the cavities of varyinglength. When the cavity length approximately matches the coherent sampling condition,the signal within the cavity enhances. The folding of the data in the cavity also reduces therandom noise by √ m . If the length of the cavity for coherent build up of the signal is N ,then the approximate time period of the waveform is given by T = N τM . (2)Using T , one can “boot-strap” the different cycles of the waveform in the digital cavity togenerate one waveform that is effectively sampled at the regular intervals of τ /M . III. MEASUREMENT OF FLUORESCENCE LIFETIME USINGBOOT-STRAPPED WAVEFORM AVERAGING
In order to demonstrate an implementation of the algorithm, we have used it to measurethe lifetime of fluorescence from Rhodamine 6G. The schematics of the experimental setupis shown in Figure 1. The optical setup has been described elsewhere.
A Ti:Sapphireoscillator (Synergy, Femtolasers) is used as the optical source. The center wavelength ofthe output is at 790 nm and the pulse duration is about 10 fs. The oscillator generatesabout 70.17 million pulses every second. An inverted microscope (Nikon Ti-S) is usedto focus the laser beam onto the sample in a flow-cell. The microscope setup has beendescribed elsewhere. A millimolar solution of rhodamine 6G in water is used as the sample.The fluorescence from the sample excited by two-photon absorption is directed to a fast3valanche photodiode (APD) (APD210, MenloSystems GmbH). The APD has a bandwidthof 1.6 GHz. The signal from the APD is digitized by a fast digitizer (ATS9870, Alazartech)at the rate of 1 GSa/s (giga samples per second). μ m
400 nm
FIG. 1. Experimental setup. Laser source is a mode-locked oscillator. The center wavelength is790 nm, repetition rate is about 70.17 MHz and the duration of the laser pulses is about 10 fs.The schematic of the microscope is shown in the right. Two-photon fluorescence from the sampleis detected by a fast avalanche photodiode. The signal from the photodiode is digitized by a fastdigitizer and saved for further signal processing.
A sample of the raw digitized signal is shown in Figure 2 ( a. ). Figure 2( b. ) shows the first57 data points. As shown in the figure, the data has a poor temporal resolution of 1 ns and israther noisy because of which one does not observe the clear decay profile of the fluorescencesignal. Previously, we have shown that digital cavities can be used to improve the signal tonoise ratio in sinusoidal signals. Here, although the signal is not sinusoidal, it is repetitive.Nevertheless, a repetitive signal can be decomposed into a fundamental sinusoidal signaland its harmonics, one can still use the algorithm of digital cavity to improve the signal tonoise ratio.A digital cavity is defined by its length, i.e. the number of bins in which the data pointsare accumulated. The application of the cavity refers to the repeated folding of the data4
IG. 2. Raw data. (a) shows 2000 samples and (b) shows first 57 samples. stream in the cavity. If the period of the repetitive signal matches the cavity length it getsenhanced by the number of times the data is folded, otherwise the signal gets averaged out.In order to find the period after which the signal repeats, we vary the cavity length. Figure 3shows the amplitude of the signal after averaging 80 times in the digital cavities of varyinglength. We observe maximum amplitudes when the length of the cavity is 57 ns or 1411 ns.The signal approximately fulfills the condition of coherent sampling at these lengths of thecavity.
FIG. 3. Amplitude of the signal filtered by digital cavities of various lengths.
Figure 4( a. ) shows the average signal obtained from the digital cavity of length 57ns. Compared to the raw data in Figure 2( b. ), the signal to noise ratio has improvedconsiderably such that the fast decay of fluorescence is clearly visible. There are four cyclesof the fluorescence transients, which are not identical. The difference in the four transientsobserved in figure 4( a. ) is due to the mismatch in the repetition rate of the laser and the5ate of digitization of the signal. As the repetition rate of the laser is ∼ b. ) shows the merged transient labeledas “boot-strapped data”. The data has been shifted such that the maximum of the signalis at time zero. The effective sampling rate of the data is 4 GSa/s and the correspondingtemporal resolution is 250 ps. The dashed line is the exponential decay fit of the data, whichhas a decay constant of 4.5 ± FIG. 4. (a) and (c) are Coherently averaged waveforms using cavities of lengths 57 and 1411ns, respectively. (b) and (d) show bood-strapped data of (a) and (c), respectively, together withexponential fits and detector response.
The effective sampling rate can be increased further by using the digital cavity of longer6ength. Figure 4( c. ) shows the average signal obtained from the digital cavity of length 1141ns. The appropriate length of the cavity is chosen based on the second maxima amplitudein figure 2. The averaged signal in the cavity has 99 cycles, from which we obtain therepetition period of ∼ d. )has the effective sampling rate of 99 GSa/s and the corresponding temporal resolution ofabout 10 ps. The time constant of the exponential decay (also known as the lifetime of thefluorescence) is 4.4 ± . b. and d. ) agree withthe known lifetime of rhodamine 6G in water (about 4.08 ns). Thus, in this case the cavityof only 57 bins, which gives effective sampling rate of 4 GSa/s, is good enough to evaluatethe fluorescence lifetime. However, for faster transients one would require longer cavity andcorrespondingly higher effective sampling rate.There are many methods, such as time correlated single photon counting (TCSPC) ,phase modulation , gated fluorescence detection and direct recording of full transientsusing fast digitizers , that can used to measure the lifetime of fluorescence. TCSPC isthe most commonly used technique, which has comparatively advantageous features suchas low systematic errors and best signal-to-noise ratio. However, the measurement time ofa single transient is rather long, which has been a disadvantage in applications thatrequire fast measurements. More recently, direct recording of the fluorescence transients using fast detectors anddata acquisition systems have been used as an alternative lifetime measurement techniquein clinical applications.
Direct recording of the transients is particularly advantageousin samples that are highly fluorescent. In this case, it is even possible to record a transientin real time. Previously, sampling oscilloscopes have been used to record the data at thesampling rate of about 5 GSa/s (giga samples per second), which gives a temporal resolu-tion in data acquisition of about 200 ps.
Traditional interleaved sampling methodshave been used to improve the effective temporal resolution. However, such methods aretime consuming, which offsets the benefits of fast measurement. The technique we havepresented in this note provides a new method to rapidly carry out wave-form averaging andinterleaving without slowing the measurements. In our measurements, the data used tocarry out waveform averaging with the cavities of lengths 57 and 1141 have data acquisitiontimes of about 4.6 µ s and 113 µ s, respectively.7lthough, we have shown the use of boot-strapped waveform averaging for the measure-ment of fluorescence lifetimes, the method is general and can be used to increase the effectivesampling rate in any repetitive signal. IV. CONCLUSION
We have presented an approximate method of coherent sampling and waveform averagingto increase the signal to noise ratio in a repetitive signal and at the same time increase theeffective sampling rate. We have demonstrated its use by measuring the fluorescence lifetimeof rhodamine 6G. The method is general and can be used to rapidly analyze any repetitivesignal.
Acknowledgments
Financial support from Lund University Innovation System and VINNOVA is gratefullyacknowledged.
REFERENCES “Interleaving ADCs for Higher Sample Rates,” , accessed: 2017-09-24. “Random Interleaved Sampling (RIS),” http://cdn.teledynelecroy.com/files/whitepapers/wp_ris_102203.pdf , accessed: 2017-09-24. “A Tutorial in Coherent and Windowed Sampling with A/D Converters,” , accessed: 2017-09-24. K. J. Karki, L. Kringle, A. H. Marcus, and T. Pullerits, J. Opt. , 015504 (2016). V. A. Osipov, X. Shang, T. Hansen, T. Pullerits, and K. J. Karki, Phys. Rev. A , 1(2016). K. J. Karki, M. Abdellah, W. Zhang, and T. Pullerits, Appl. Phys. Lett. Photonics , 1(2016). K. J. Karki, M. Torbj¨ornsson, J. R. Widom, A. H. Marcus, and T. Pullerits, J. Instrum. , 1 (2013). 8 S. Fu, A. Sakurai, L. Liu, F. Edman, T. Pullerits, V. ¨Owall, and K. J. Karki, Rev. Sci.Instrum. (2013). “Lifetime Data of Selected Fluorophores,” , accessed: 2017-09-24. D. V. OConnor and D. Phillips,
Time-correlated single photon counting (Academic Press,London, 1984). I. Salmeen and L. Rimai, Biophys. J. , 335 (1977). E. Gratton and M. Limkeman, Biophys. J. , 315 (1983). S. S. Brody, Rev. Sci. Instrum. , 1021 (1957). J. G. C. Brown, Rev. Sci. Instrum. , 414 (1963). J. D. Pitts and M.-A. Mycek, Rev. Sci. Instrum. , 3061 (2001). T. J. Pfefer, D. Y. Paithankar, J. M. Poneros, K. T. Schomacker, and N. S. Nishioka,Lasers Surg. Med. , 10 (2003). Q. Fang, T. Papaioannou, J. A. Jo, R. Vaitha, K. Shastry, and L. Marcu, Rev. Sci.Instrum. , 151 (2004). P. A. A. D. Beule, C. Dunsby, N. P. Galletly, G. W. Stamp, A. C. Chu, U. Anand,P. Anand, C. D. Benham, A. Naylor, and P. M. W. French, Rev. Sci. Instrum. , 1(2007). A. J. Thompson, S. Coda, M. B. S. rensen, G. Kennedy, R. Patalay, U. Waitong-Br¨amming,P. A. A. D. Beule, M. A. A. Neil, S. Andersson-Engels, N. B. e, P. M. W. French, K. Svan-berg, and C. Dunsby, J. Biophotonics , 1 (2012). S. Coda, A. J. Thompson, G. T. Kennedy, K. L. Roche, L. Ayaru, D. S. Bansi, G. W.Stamp, A. V. Thillainayagam, P. M. W. French, and C. Dunsby, Biomed. Opt. Express , 515 (2014). L. Marcu, Ann. Biomed. Eng. , 304 (2012). V. G. Ivchenko, A. N. Kalashnikov, R. E. Challis, and B. R. Hayes-Gill, IEEE Trans.Instrum. Meas.56