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.

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2011

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]

2010

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]

2009

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]

2004

1981

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]

1978

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]

1977

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]

1974

1972

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

1958

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

Abràmoff, M. D.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int.11, 36–42 (2004).

Bell, M. I.

Dada, O. O.

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]

Dovichi, N. J.

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]

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]

Essaka, D.

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]

Essaka, D. C.

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]

Gustafson, T. L.

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]

Hänninen, P. E.

Harris, J. M.

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]

Hindsgaul, O.

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]

Kissick, D. J.

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]

Lytle, F. E.

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]

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]

Magalhães, P. J.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int.11, 36–42 (2004).

Mandel, L.

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

Muir, R. D.

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]

Nau, V. J.

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]

Nieman, T. A.

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]

Palcic, M. M.

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]

Prendergast, J.

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]

Ram, S. J.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int.11, 36–42 (2004).

Santini, R. E.

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

Schnaar, R. L.

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]

Simpson, G. J.

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]

Soini, J. T.

Soukka, J. M.

Tobias, R. S.

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]

Tyte, R. N.

Virkki, A.

Whitmore, C. D.

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]

Anal. Chem.

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]

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]

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

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]

Appl. Opt.

Biophotics Int.

M. D. Abràmoff, P. J. Magalhães, and S. J. Ram, “Image processing with ImageJ,” Biophotics Int.11, 36–42 (2004).

Opt. Express

Proc. Phys. Soc. Lond.

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

Rev. Sci. Instrum.

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]

Talanta

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]

Other

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

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

D. Blumenfeld, Operations Research Calculations Handbook (CRC, 2009).

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.

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.

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Figures (3)

Fig. 1
Fig. 1

(Left) SNR of photon averaging and binary counting. (Right) SNR of photon averaging and binary counting as a ratio of the theoretical maximum SNR (the SNR of the underlying Poisson distribution). SNR at the crossover point is at ~87% of the theoretical limit. μ1 = 7.2 mV, σ1 = 4 mV, σJ = 0.3 mV. Data were simulated per sample by summing a Poisson distributed random number of lognormal random numbers, with an additional normally distributed random number also added to represent the Johnson noise (between 2 × 106 values at low λ to 5000 at high λ). Binary counting offers higher SNR at low λ, and photon counting offers higher SNR at high λ.

Fig. 2
Fig. 2

Approximation of crossover point from a power function, where 0.75 < ��1/��1 < 10

Fig. 3
Fig. 3

SHG images of crystalline urea. (Left Column) Full contrast image, (Right Column) Contrast adjusted to λmax = 0.02. (A,B) Analysis with photon averaging only. The majority of the image is silhouetted in comparison to the brightest pixels in (A), up to λ = 74. The dimmest pixels are evident in (B), but are as prominent as the horizontal streaks and noise in the image. (C,D) Analysis with binomial counting only. The largest recoverable value was λ = 6.23; pixels brighter than this were clipped to this value in (C). The dimmest pixels are easily identifiable in (D), and the instrument noise is not evident. (E,F) Preferential crossover analysis incorporating photon averaging and binomial counting. Pixels brighter than λ = 0.48 were analyzed by photon averaging, so the full upper range of detection is preserved in (E). Pixels dimmer than λ = 0.48 were analyzed by binomial counting, so the lower range of detection is preserved in (F). Selection of a crossover point defined by Eq. (18) maximizes SNR across the entire range of detection.

Equations (4)

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f P (n)= λ n n! exp(λ)
p= n=1 ( λ n exp( λ ) n! )( threshold 1 (2π) S n V exp( (ln( V ) M n ) 2 2 S n 2 )dV )
σ V 2  = n=0 [ λ n exp( λ ) n! ] ×[ 0 V 2 1 (2π) S n V exp( ( ln( V ) M n ) 2 2 S n 2 )dV ] λ 2 μ 1 2
SN R ave SN R Poisson = μ 1 2 μ 1 2 + σ 1 2

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