## Statistical connection of binomial photon counting and photon averaging in high dynamic range beam-scanning microscopy |

Optics Express, Vol. 20, Issue 9, pp. 10406-10415 (2012)

http://dx.doi.org/10.1364/OE.20.010406

Acrobat PDF (948 KB)

### Abstract

Data from photomultiplier tubes are typically analyzed using either counting or averaging techniques, which are most accurate in the dim and bright signal limits, respectively. A statistical means of adjoining these two techniques is presented by recovering the Poisson parameter from averaged data and relating it to the statistics of binomial counting from Kissick *et al.* [Anal. Chem. **82**, 10129 (2010)]. The point at which binomial photon counting and averaging have equal signal to noise ratios is derived. Adjoining these two techniques generates signal to noise ratios at 87% to approaching 100% of theoretical maximum across the full dynamic range of the photomultiplier tube used. The technique is demonstrated in a second harmonic generation microscope.

© 2012 OSA

## 1. Introduction

3. L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. Lond. **72**(6), 1037–1048 (1958). [CrossRef]

6. C. D. Whitmore, D. Essaka, and N. J. Dovichi, “Six orders of magnitude dynamic range in capillary electrophoresis with ultrasensitive laser-induced fluorescence detection,” Talanta **80**(2), 744–748 (2009). [CrossRef] [PubMed]

7. O. O. Dada, D. C. Essaka, O. Hindsgaul, M. M. Palcic, J. Prendergast, R. L. Schnaar, and N. J. Dovichi, “Nine orders of magnitude dynamic range: picomolar to millimolar concentration measurement in capillary electrophoresis with laser induced fluorescence detection employing cascaded avalanche photodiode photon counters,” Anal. Chem. **83**(7), 2748–2753 (2011). [CrossRef] [PubMed]

*et al*. [8

8. J. M. Soukka, A. Virkki, P. E. Hänninen, and J. T. Soini, “Optimization of multi-photon event discrimination levels using Poisson statistics,” Opt. Express **12**(1), 84–89 (2004). [CrossRef] [PubMed]

*n*quickly resulted in overlap between the PHDs of multiple photon events through convolution, such that in practice only a relatively small number of thresholds can be used reliably with this approach and only for detectors with exceptionally low variance in the one-photon PHD.

12. V. J. Nau and T. A. Nieman, “Photometric instrument with automatic switching between photon counting and analog modes,” Anal. Chem. **53**(2), 350–354 (1981). [CrossRef]

*et al.*[13] patented a similar device. The drawbacks of these systems are that the calibration can be defeated by fluctuation in the gain or DC offset of the PMT [14

14. R. E. Santini, “Signal-to-noise characteristics of real photomultiplier and photodiode detection systems. Comments,” Anal. Chem. **44**(9), 1708–1709 (1972). [CrossRef]

*et al*. [9

9. D. J. Kissick, R. D. Muir, and G. J. Simpson, “Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation,” Anal. Chem. **82**(24), 10129–10134 (2010). [CrossRef] [PubMed]

## 2. Adjoining binomial photon counting and photon averaging: theoretical foundation

*n*photons being generated isThe mean and variance of the Poisson distribution are both equal to λ.

*n*photoelectrons generated by the detector, the

*n*-photon probability density function (PDF) is described as the single photon lognormal PDF convolved with itself

*n*-1 times. Although no closed form solution exists for the convolution of two lognormal PDFs, the resulting distribution is often well-approximated by another lognormal PDF [16]. The overall PDF of detected voltage for any point in the sample is a combination of these Poisson and lognormal processes, and can be intuited as the linear combination of each n-photon lognormal PDF, weighted by the Poisson probability of event

*n*. In the binomial counting technique, the probability of observing a count,

*p*, is the probability of a signal event exceeding a user-defined threshold: In Eqs. (3) and (4),

*μ*and

_{n}*σ*

_{n}correspond to the mean and standard deviation, respectively, of the

*n*-photon voltage lognormal PDF describing the peak height distribution, while

*M*and

_{n}*S*are the standard lognormal parameters, corresponding to the mean and standard deviation, respectively, of ln(V).

_{n}*et al.*[9

9. D. J. Kissick, R. D. Muir, and G. J. Simpson, “Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation,” Anal. Chem. **82**(24), 10129–10134 (2010). [CrossRef] [PubMed]

8. J. M. Soukka, A. Virkki, P. E. Hänninen, and J. T. Soini, “Optimization of multi-photon event discrimination levels using Poisson statistics,” Opt. Express **12**(1), 84–89 (2004). [CrossRef] [PubMed]

9. D. J. Kissick, R. D. Muir, and G. J. Simpson, “Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation,” Anal. Chem. **82**(24), 10129–10134 (2010). [CrossRef] [PubMed]

17. M. I. Bell and R. N. Tyte, “Pulsed dye laser system for Raman and luminescence spectroscopy,” Appl. Opt. **13**(7), 1610–1614 (1974). [CrossRef] [PubMed]

**82**(24), 10129–10134 (2010). [CrossRef] [PubMed]

*N*is the number of events capable of producing photons, in this case equal to the number of laser pulses.

*p*,

*λ*can also be extracted from signal averaged analyses. The mean voltage of the pooled samples

*μ*is found in signal averaging, which analytically corresponds to the expectation value of the underlying distribution. From the standard definition, the expectation value

_{sample}*µ*is given in Eq. (8):Note that within this equation, the integral of

_{sample}*V*multiplied by the lognormal distribution of

*V*will lead to the expectation value of that lognormal distribution for all

*n*.

*n*and summing over all values results in yet another expectation value. The mean of the Poisson distribution is λ, so Eq. (9) reduces to the result we could have expected from the outset:Thus, dividing the measured mean voltage by the mean of the one photon lognormal PDF will recover the mean value of the underlying Poisson distribution, effectively providing a “calibration”to relate the averaged signal directly back to the average number of photons. This result is consistent with the expectation value for a sum of a random amount of random numbers [18], where a Poisson random number of lognormal random variables are summed.

*n-*photon lognormal distribution,

*n*in the first term will combine with the infinite sum and reduce to λ. The

*n*

^{2}in the second term will combine with the infinite sum and reduce to

*SNR*for signal averaging is obtained by dividing the sample mean from Eq. (10) for

*N*laser pulses by the standard deviation from Eq. (14):The maximum SNR theoretically achievable is defined by

*SNR*of the underlying Poisson distribution,

_{Poisson}*SNR*goes toTherefore, the performance of signal averaging in comparison to the theoretical maximum is defined entirely by the performance of the detector. This result is in agreement with simulation, as shown in Fig. 1 .

_{ave}/SNR_{Poisson}*p*approaches one. For the dim signal regime (λ< 0.5), the binomial counting algorithm offers a considerable SNR advantage. For such a preferential analysis, there is some intermediate signal value where the SNR for both binomial counting and signal averaging are equal, and that value of λ will be the preferential crossover point for the analysis. This crossover point is pictorially described in Fig. 1. It can be derived by comparing the SNR for photon averaging to the SNR for binomial photon counting.

_{1}/𝜎

_{1}< 10. Therefore, if either counting or averaging yields a value of λ that is less than that obtained through Eq. (19), binomial photon counting will produce a larger SNR. Conversely, for signals greater than this threshold value, photon averaging offers a SNR advantage. In the limit of negligible Johnson noise, the preferential crossover point is defined entirely by the mean and variance per photon in the response of the PMT.

## 3. Experimental

## 4. Adjoining binomial photon counting and photon averaging: application in SHG microscopy

## 5. Summary

## Acknowledgments

## References and links

1. | J. D. Ingle, Jr, and S. R. Crouch, |

2. | J. B. Pawley, |

3. | L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. Lond. |

4. | B. P. Roe, |

5. | W. Becker, |

6. | C. D. Whitmore, D. Essaka, and N. J. Dovichi, “Six orders of magnitude dynamic range in capillary electrophoresis with ultrasensitive laser-induced fluorescence detection,” Talanta |

7. | O. O. Dada, D. C. Essaka, O. Hindsgaul, M. M. Palcic, J. Prendergast, R. L. Schnaar, and N. J. Dovichi, “Nine orders of magnitude dynamic range: picomolar to millimolar concentration measurement in capillary electrophoresis with laser induced fluorescence detection employing cascaded avalanche photodiode photon counters,” Anal. Chem. |

8. | J. M. Soukka, A. Virkki, P. E. Hänninen, and J. T. Soini, “Optimization of multi-photon event discrimination levels using Poisson statistics,” Opt. Express |

9. | D. J. Kissick, R. D. Muir, and G. J. Simpson, “Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation,” Anal. Chem. |

10. | J. M. Harris and F. E. Lytle, “Measurement of subnanosecond fluorescence decays by sampled single-photon detection,” Rev. Sci. Instrum. |

11. | T. L. Gustafson, F. E. Lytle, and R. S. Tobias, “Sampled photon counting with multilevel discrimination,” Rev. Sci. Instrum. |

12. | V. J. Nau and T. A. Nieman, “Photometric instrument with automatic switching between photon counting and analog modes,” Anal. Chem. |

13. | K. L. Staton, A. N. Dorsel, and A. Schleifer, “Large dynamic range light detection,” U.S. Patent 6,355,921 B1 (March 12, 2002). |

14. | R. E. Santini, “Signal-to-noise characteristics of real photomultiplier and photodiode detection systems. Comments,” Anal. Chem. |

15. | W. A. Kester, |

16. | J. X. Wu, N. B. Mehta, and J. Zhang, “Flexible lognormal sum approximation method,” in |

17. | M. I. Bell and R. N. Tyte, “Pulsed dye laser system for Raman and luminescence spectroscopy,” Appl. Opt. |

18. | D. Blumenfeld, |

19. | M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int. |

**OCIS Codes**

(030.4280) Coherence and statistical optics : Noise in imaging systems

(030.5260) Coherence and statistical optics : Photon counting

(030.5290) Coherence and statistical optics : Photon statistics

(030.6600) Coherence and statistical optics : Statistical optics

(040.5250) Detectors : Photomultipliers

(190.2620) Nonlinear optics : Harmonic generation and mixing

(180.4315) Microscopy : Nonlinear microscopy

(320.7085) Ultrafast optics : Ultrafast information processing

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: January 18, 2012

Manuscript Accepted: February 20, 2012

Published: April 20, 2012

**Virtual Issues**

Vol. 7, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Ryan D. Muir, David J. Kissick, and Garth J. Simpson, "Statistical connection of binomial photon counting and photon averaging in high dynamic range beam-scanning microscopy," Opt. Express **20**, 10406-10415 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-9-10406

Sort: Year | Journal | Reset

### References

- J. D. Ingle, Jr, and S. R. Crouch, Spectrochemical Analysis (Prentice Hall, 1988).
- J. B. Pawley, Handbook of Biological Confocal Microscopy, 3rd ed. (Springer, 2006), p. xxviii.
- L. Mandel, “Fluctuations of photon beams and their correlations,” Proc. Phys. Soc. Lond.72(6), 1037–1048 (1958). [CrossRef]
- B. P. Roe, Probability and Statistics in Experimental Physics (Springer Verlag, 2001).
- W. Becker, Advanced Time-Correlated Single Photon Counting Techniques, Springer Series in Chemical Physics (Springer, Berlin; 2005), p. xix.
- C. D. Whitmore, D. Essaka, and N. J. Dovichi, “Six orders of magnitude dynamic range in capillary electrophoresis with ultrasensitive laser-induced fluorescence detection,” Talanta80(2), 744–748 (2009). [CrossRef] [PubMed]
- O. O. Dada, D. C. Essaka, O. Hindsgaul, M. M. Palcic, J. Prendergast, R. L. Schnaar, and N. J. Dovichi, “Nine orders of magnitude dynamic range: picomolar to millimolar concentration measurement in capillary electrophoresis with laser induced fluorescence detection employing cascaded avalanche photodiode photon counters,” Anal. Chem.83(7), 2748–2753 (2011). [CrossRef] [PubMed]
- J. M. Soukka, A. Virkki, P. E. Hänninen, and J. T. Soini, “Optimization of multi-photon event discrimination levels using Poisson statistics,” Opt. Express12(1), 84–89 (2004). [CrossRef] [PubMed]
- D. J. Kissick, R. D. Muir, and G. J. Simpson, “Statistical treatment of photon/electron counting: extending the linear dynamic range from the dark count rate to saturation,” Anal. Chem.82(24), 10129–10134 (2010). [CrossRef] [PubMed]
- J. M. Harris and F. E. Lytle, “Measurement of subnanosecond fluorescence decays by sampled single-photon detection,” Rev. Sci. Instrum.48(11), 1469–1476 (1977). [CrossRef]
- T. L. Gustafson, F. E. Lytle, and R. S. Tobias, “Sampled photon counting with multilevel discrimination,” Rev. Sci. Instrum.49(11), 1549–1550 (1978). [CrossRef] [PubMed]
- V. J. Nau and T. A. Nieman, “Photometric instrument with automatic switching between photon counting and analog modes,” Anal. Chem.53(2), 350–354 (1981). [CrossRef]
- K. L. Staton, A. N. Dorsel, and A. Schleifer, “Large dynamic range light detection,” U.S. Patent 6,355,921 B1 (March 12, 2002).
- R. E. Santini, “Signal-to-noise characteristics of real photomultiplier and photodiode detection systems. Comments,” Anal. Chem.44(9), 1708–1709 (1972). [CrossRef]
- W. A. Kester, Data Conversion Handbook (Newnes, 2005).
- J. X. Wu, N. B. Mehta, and J. Zhang, “Flexible lognormal sum approximation method,” in IEEE Global Telecommunications Conference, 2005. GLOBECOM '05 (IEEE, 2005), pp. 3413–3417
- M. I. Bell and R. N. Tyte, “Pulsed dye laser system for Raman and luminescence spectroscopy,” Appl. Opt.13(7), 1610–1614 (1974). [CrossRef] [PubMed]
- D. Blumenfeld, Operations Research Calculations Handbook (CRC, 2009).
- M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int.11, 36–42 (2004).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.