Abstract

A useful analytic model describing the response of a photon-counting (PC) system has been developed. The model describes the nonlinear count loss and apparent count gain arising from the overlap of photomultiplier tube (PMT) pulses, taking into account the distribution in amplitude of the PMT output pulses and the effect of the pulse-height discrimination threshold. Comparisons between the model and Monte Carlo simulations show excellent agreement. The model has been applied to a PC lidar system with favorable results. Application of the model has permitted us to extend the linear operating range of the PC system and to quantify accurately the response of the system in its nonlinear operating regime, thus increasing the useful dynamic range of the system by 1 order of magnitude.

© 1993 Optical Society of America

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  1. E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
    [CrossRef]
  2. J. D. Ingle, S. R. Crouch, “Pulse overlap effects on linearity and signal-to-noise ratio in photon counting,” Anal. Chem. 44, 777–784 (1972).
    [CrossRef] [PubMed]
  3. K. C. Ash, E. H. Piepmeier, “Double beam photon counting photomotor with dead time compensation,” Anal. Chem. 43, 26–32 (1971).
    [CrossRef]
  4. R. W. Engstrom, Photomultiplier Handbook (RCA Corporation, Lancaster, Pa., 1980), pp. 66–69.
  5. A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
    [CrossRef]
  6. R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York, 1955), pp. 785–787.
  7. J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
    [CrossRef]
  8. E. Funck, “Dead time effects from linear amplifiers and discriminators in single detector systems,” Nucl. Instrum. Methods 245, 519–524 (1985).
  9. K. Omote, “Dead-time effects in photon counting distributions,” Nucl. Instrum. Methods 293, 582–588 (1990).
    [CrossRef]
  10. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 32–37.
  11. J. R. Prescott, “A statistical model for photomultiplier single-electron statistics,” Nucl. Instrum. Methods 39, 173–179 (1966).
    [CrossRef]
  12. L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
    [CrossRef]
  13. A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
    [CrossRef]
  14. C. W. Helstrom, “Output distributions of electrons in a photomultiplier,” J. Appl. Phys. 55, 2786–2792 (1984).
    [CrossRef]
  15. J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
    [CrossRef]
  16. M. Omori, “A simple method to estimate small photoelectron numbers for a photomultiplier using the log-normal function,” Nucl. Instrum. Methods 276, 602–607 (1989).
    [CrossRef]
  17. B. H. Candy, “Photomultiplier characteristics and practice relevant photon counting,” Rev. Sci. Instrum. 56, 183–193 (1985).
    [CrossRef]
  18. D. N. Whiteman, S. H. Melfi, R. A. Ferrare, “Raman lidar system for the measurement of water vapor and aerosols in the Earth's atmosphere,” Appl. Opt. 31, 3068–3082 (1992).
    [CrossRef] [PubMed]

1992

1991

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

1990

K. Omote, “Dead-time effects in photon counting distributions,” Nucl. Instrum. Methods 293, 582–588 (1990).
[CrossRef]

1989

M. Omori, “A simple method to estimate small photoelectron numbers for a photomultiplier using the log-normal function,” Nucl. Instrum. Methods 276, 602–607 (1989).
[CrossRef]

1985

B. H. Candy, “Photomultiplier characteristics and practice relevant photon counting,” Rev. Sci. Instrum. 56, 183–193 (1985).
[CrossRef]

E. Funck, “Dead time effects from linear amplifiers and discriminators in single detector systems,” Nucl. Instrum. Methods 245, 519–524 (1985).

1984

C. W. Helstrom, “Output distributions of electrons in a photomultiplier,” J. Appl. Phys. 55, 2786–2792 (1984).
[CrossRef]

J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
[CrossRef]

1979

E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
[CrossRef]

1975

A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
[CrossRef]

1973

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[CrossRef]

1972

J. D. Ingle, S. R. Crouch, “Pulse overlap effects on linearity and signal-to-noise ratio in photon counting,” Anal. Chem. 44, 777–784 (1972).
[CrossRef] [PubMed]

1971

K. C. Ash, E. H. Piepmeier, “Double beam photon counting photomotor with dead time compensation,” Anal. Chem. 43, 26–32 (1971).
[CrossRef]

1967

L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
[CrossRef]

1966

J. R. Prescott, “A statistical model for photomultiplier single-electron statistics,” Nucl. Instrum. Methods 39, 173–179 (1966).
[CrossRef]

Ash, K. C.

K. C. Ash, E. H. Piepmeier, “Double beam photon counting photomotor with dead time compensation,” Anal. Chem. 43, 26–32 (1971).
[CrossRef]

Cafolla, A. A.

A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
[CrossRef]

Candy, B. H.

B. H. Candy, “Photomultiplier characteristics and practice relevant photon counting,” Rev. Sci. Instrum. 56, 183–193 (1985).
[CrossRef]

Carswell, A. I.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Carter, J. N.

A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
[CrossRef]

Crouch, S. R.

J. D. Ingle, S. R. Crouch, “Pulse overlap effects on linearity and signal-to-noise ratio in photon counting,” Anal. Chem. 44, 777–784 (1972).
[CrossRef] [PubMed]

Darland, E. J.

E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
[CrossRef]

Delaney, C. F. G.

A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
[CrossRef]

Dietz, L. A.

L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
[CrossRef]

Engstrom, R. W.

R. W. Engstrom, Photomultiplier Handbook (RCA Corporation, Lancaster, Pa., 1980), pp. 66–69.

Enke, C. G.

E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
[CrossRef]

Evans, R. D.

R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York, 1955), pp. 785–787.

Ferrare, R. A.

Funck, E.

E. Funck, “Dead time effects from linear amplifiers and discriminators in single detector systems,” Nucl. Instrum. Methods 245, 519–524 (1985).

Hance, A. B.

L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
[CrossRef]

Hanrahan, L. R.

L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, “Output distributions of electrons in a photomultiplier,” J. Appl. Phys. 55, 2786–2792 (1984).
[CrossRef]

Hyman, L. G.

J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
[CrossRef]

Ingle, J. D.

J. D. Ingle, S. R. Crouch, “Pulse overlap effects on linearity and signal-to-noise ratio in photon counting,” Anal. Chem. 44, 777–784 (1972).
[CrossRef] [PubMed]

Leroi, G. E.

E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
[CrossRef]

Melfi, S. H.

Müller, J. W.

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[CrossRef]

O'Callaghan, J. P.

J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
[CrossRef]

Omori, M.

M. Omori, “A simple method to estimate small photoelectron numbers for a photomultiplier using the log-normal function,” Nucl. Instrum. Methods 276, 602–607 (1989).
[CrossRef]

Omote, K.

K. Omote, “Dead-time effects in photon counting distributions,” Nucl. Instrum. Methods 293, 582–588 (1990).
[CrossRef]

Pal, S. R.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Piepmeier, E. H.

K. C. Ash, E. H. Piepmeier, “Double beam photon counting photomotor with dead time compensation,” Anal. Chem. 43, 26–32 (1971).
[CrossRef]

Prescott, J. R.

J. R. Prescott, “A statistical model for photomultiplier single-electron statistics,” Nucl. Instrum. Methods 39, 173–179 (1966).
[CrossRef]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 32–37.

Stanek, R.

J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
[CrossRef]

Steinbrecht, W.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Ulitsky, A.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Wang, T-Y

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Whiteman, D. N.

Whiteway, J. A.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

Anal. Chem.

E. J. Darland, G. E. Leroi, C. G. Enke, “Pulse (photon) counting: determination of optimum measurement system parameters,” Anal. Chem. 51, 240–245 (1979).
[CrossRef]

J. D. Ingle, S. R. Crouch, “Pulse overlap effects on linearity and signal-to-noise ratio in photon counting,” Anal. Chem. 44, 777–784 (1972).
[CrossRef] [PubMed]

K. C. Ash, E. H. Piepmeier, “Double beam photon counting photomotor with dead time compensation,” Anal. Chem. 43, 26–32 (1971).
[CrossRef]

Appl. Opt.

Can. J. Phys.

A. I. Carswell, S. R. Pal, W. Steinbrecht, J. A. Whiteway, A. Ulitsky, T-Y Wang, “Lidar measurements of the middle atmosphere,” Can. J. Phys. 69, 1076–1086 (1991).
[CrossRef]

J. Appl. Phys.

C. W. Helstrom, “Output distributions of electrons in a photomultiplier,” J. Appl. Phys. 55, 2786–2792 (1984).
[CrossRef]

Nucl. Instrum. Methods

J. P. O'Callaghan, R. Stanek, L. G. Hyman, “On estimating the photoelectron yield and the resultant inefficiency of a photomultiplier-based detector,” Nucl. Instrum. Methods 225, 153–163 (1984).
[CrossRef]

M. Omori, “A simple method to estimate small photoelectron numbers for a photomultiplier using the log-normal function,” Nucl. Instrum. Methods 276, 602–607 (1989).
[CrossRef]

J. R. Prescott, “A statistical model for photomultiplier single-electron statistics,” Nucl. Instrum. Methods 39, 173–179 (1966).
[CrossRef]

J. W. Müller, “Dead-time problems,” Nucl. Instrum. Methods 112, 47–57 (1973).
[CrossRef]

E. Funck, “Dead time effects from linear amplifiers and discriminators in single detector systems,” Nucl. Instrum. Methods 245, 519–524 (1985).

K. Omote, “Dead-time effects in photon counting distributions,” Nucl. Instrum. Methods 293, 582–588 (1990).
[CrossRef]

A. A. Cafolla, J. N. Carter, C. F. G. Delaney, “A computation on secondary electron emission statistics and its application to single electron spectra in photo- and electron multipliers,” Nucl. Instrum. Methods 128, 157–161 (1975).
[CrossRef]

Rev. Sci. Instrum.

L. A. Dietz, L. R. Hanrahan, A. B. Hance, “Single-electron response of a porous KC1 transmission dynode and application of Polya statistics to particle counting in an electron multiplier,” Rev. Sci. Instrum. 38, 176–183 (1967).
[CrossRef]

B. H. Candy, “Photomultiplier characteristics and practice relevant photon counting,” Rev. Sci. Instrum. 56, 183–193 (1985).
[CrossRef]

Other

R. D. Evans, The Atomic Nucleus (McGraw-Hill, New York, 1955), pp. 785–787.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 32–37.

R. W. Engstrom, Photomultiplier Handbook (RCA Corporation, Lancaster, Pa., 1980), pp. 66–69.

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Figures (21)

Fig. 1
Fig. 1

Schematic representation of a photon-counting (PC) system.

Fig. 2
Fig. 2

Pulse pileup effects: A, count loss resulting from pulse pileup; B, count gain resulting from pulse pileup.

Fig. 3
Fig. 3

Pulse pileup of idealized pulses. The dashed lines represent individual pulses, and the solid lines show the sum voltage.

Fig. 4
Fig. 4

Results of Monte Carlo simulations compared with Eq. (16) and Nl, showing the effect of changing the pulse width. Both simulations represent averages over 500 trials with T = 0.5.

Fig. 5
Fig. 5

Results of Monte Carlo simulations compared with Eq. (16) and Nl, showing the effect of changing the discriminator level. Both simulations represent averages over 500 trials with τd = 0.05 μs.

Fig. 6
Fig. 6

N/Nl as a function of count rate and discriminator level as predicted by Eq. (16). Here T ranges from T = 0.9 to T = 0.1 in steps of 0.05.

Fig. 7
Fig. 7

Sample standard deviation of the data shown in Fig. 4 compared with that expected for a Poisson process ( N ). The points correspond to the simulation data, and the dashed curves show Eq. (18) for τd/t = 0.075 and τd/t = 0.05. Here T = 0.5.

Fig. 8
Fig. 8

Sample standard deviation of the data shown in Fig. 5 compared with that expected for a Poisson process ( N ). The points correspond to the simulation data for two discriminator levels, while the dashed curve shows Eq. (18) for τd/t = 0.05.

Fig. 9
Fig. 9

Example of two computed single-photoelectron pulse-height distributions as functions of the pulse height relative to the mean. Distribution B is quasi-exponential, while C is quasi-Gaussian. For comparison, the uniform distribution A is also shown.

Fig. 10
Fig. 10

Results of Monte Carlo simulations using the pulse-height distributions shown in Fig. 9. The solid curves labeled B and C are from Eq. (3) with the coefficients calculated directly from the corresponding distributions shown in Fig. 9. Here τd = 0.05 μs. The solid curve corresponding to A is from Eq. (16).

Fig. 11
Fig. 11

Relative error as a function of observed count rate obtained by a fit of Eq. (19) to the exact curve for distribution C at T = 0.8 shown in Fig. 10.

Fig. 12
Fig. 12

Sample standard deviation of the A, B, and C data shown in Fig. 10 compared with that expected for a Poisson process ( N ) and for a zero discriminator level [Eq. (18)]. The dashed curve shows Eq. (18) for τd/t = 0.05.

Fig. 13
Fig. 13

Example of N and Nl obtained by measurement of the return from the middle atmosphere. Nl was obtained with a ND filter with Tf ≈ 0.01. The intense near-field signal was blocked with a mechanical chopper.

Fig. 14
Fig. 14

Observed N/Nl as a function of count rate and discriminator level for PMT 1.

Fig. 15
Fig. 15

Example of fit of Eq. (21) to data taken with PMT 1 with a discriminator threshold of −42 mV.

Fig. 16
Fig. 16

Corresponding N/Nl curve for data shown in Fig. 11.

Fig. 17
Fig. 17

N/Nl curve for PMT 1 with a discriminator setting of −60 mV.

Fig. 18
Fig. 18

N/Nl curve for PMT 2 with a discriminator setting of −76 mV.

Fig. 19
Fig. 19

Comparison between corrected and uncorrected return signals from a low cloud layer taken in October 1989.

Fig. 20
Fig. 20

Comparison between corrected and uncorrected signals taken a few minutes after the data shown in Fig. 19, but with an additional ND filter in the system that reduced the signal levels.

Fig. 21
Fig. 21

Comparison between depolarization (Dpol) ratios calculated from corrected data shown in Figs. 19 and 20.

Equations (21)

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N = S exp ( S τ d ) ,
N = S P ( a ) exp ( τ d S ) + S P ( 1 b 2 a ) ( τ d S ) exp ( τ d S ) + S P ( 2 b 3 a ) ( τ d S ) 2 2 exp ( τ d S ) + S P ( 3 b 4 a ) ( τ d S ) 3 3 ! × exp ( τ d S ) + ,
N = S exp ( τ d S ) [ P ( a ) + P ( 1 b 2 a ) ( τ d S ) + P ( 2 b 3 a ) ( τ d S ) 2 2 + P ( 3 b 4 a ) ( τ d S ) 3 3 ! + ] ,
P ( 1 b 2 a ) = T P ( a ) , P ( 2 b 3 a ) = T 2 P ( a ) , P ( 3 b 4 a ) = T 3 P ( a ) , .
N = S exp ( τ d S ) P ( a ) { 1 + T ( τ d S ) + T 2 [ ( τ d S ) 2 / 2 ] + } = S exp ( τ d S ) P ( a ) exp ( T τ d S ) = P ( a ) S exp [ P ( a ) S τ d ] .
P n ( x ) d x = + + w ( s 1 ) w ( s 2 ) w ( s n ) d s 1 d s 2 d s n ,
x < i = 1 n s i < x + d x
δ ( x i = 1 n s i ) = 1 2 π + d k exp [ i k ( x i = 1 n s i ) ] .
P n ( x ) = 1 2 π + d k exp ( ikx ) [ + d s exp ( ikx ) w ( s ) ] n .
P 1 ( x ) = θ ( x ) θ ( x 1 ) , P 2 ( x ) = x θ ( x ) 2 ( x 2 ) θ ( x 1 ) + ( x 2 ) θ ( x 2 ) , P 3 ( x ) = x 2 2 θ ( x ) 3 2 ( x 1 ) 2 θ ( x 1 ) + 3 2 ( x 2 ) 2 θ ( x 2 ) ( x 3 ) 2 2 θ ( x 3 ) ,
P 1 ( x T ) = T
P 1 ( x T ) = 1 T .
P ( 1 b 2 a ) = 0 T P 1 ( x ) P 1 ( H T x ) d x ,
P ( 2 b 3 a ) = 0 T P 2 ( x ) P 1 ( H T x ) d x ,
P ( a ) = 1 T , P ( 1 b 2 a ) = T ( T 2 / 2 ! ) , P ( 2 b 3 a ) = ( T 2 / 2 ! ) ( T 3 / 3 ! ) , P ( 3 b 4 a ) = ( T 3 / 3 ! ) ( T 4 / 4 ! ) , .
N = S exp ( τ d S ) [ ( 1 T ) + ( T T 2 2 ! ) τ d S + 1 2 ! ( T 2 2 ! T 3 3 ! ) × ( τ d S ) 2 + 1 3 ! ( T 3 3 ! T 4 4 ! ) ( τ d S ) 3 + ] .
N l = P ( a ) S .
σ 2 = N [ 1 ( 1 τ d / t ) 2 ] N 2 .
N = A S exp ( τ d A S ) [ ( 1 T ) + ( T T 2 2 ! ) τ d A S + 1 2 ! × ( T 2 2 ! T 3 3 ! ) ( τ d A S ) 2 + 1 3 ! ( T 3 3 ! T 4 4 ! ) ( τ d A S ) 3 + ] ,
N f P ( a ) S f = P ( a ) S T f = N l T f ,
N = N l exp [ τ d N l / P ( a ) ] × [ 1 + P ( 1 b 2 a ) P ( a ) ( τ d N l p ( a ) ) + P ( 2 b 3 a ) 2 P ( a ) ( τ d N l P ( a ) ) 2 + ] ,

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