Abstract

Standard deviation measurements of intensity profiles of stationary single fluorescent molecules are useful for studying axial localization, molecular orientation, and a fluorescence imaging system’s spatial resolution. Here we report on the analysis of the precision of standard deviation measurements of intensity profiles of single fluorescent molecules imaged using an EMCCD camera. We have developed an analytical expression for the standard deviation measurement error of a single image which is a function of the total number of detected photons, the background photon noise, and the camera pixel size. The theoretical results agree well with the experimental, simulation, and numerical integration results. Using this expression, we show that single-molecule standard deviation measurements offer nanometer precision for a large range of experimental parameters.

© 2010 Optical Society of America

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2009

2008

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008).
[CrossRef] [PubMed]

2007

M. H. Ulbrich and E. Y. Isacoff, "Subunit counting in membrane-bound proteins," Nature Methods 4, 319-321 (2007).
[PubMed]

2006

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006).
[CrossRef] [PubMed]

2005

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

2004

A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004).
[CrossRef]

2003

J. Hynecek and T. Nishiwaki, "Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs," IEEE Trans. on Electron Devices 50, 239-245 (2003).
[CrossRef]

M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

2002

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

2000

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

1998

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

1986

N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152-1157 (1986).
[CrossRef]

Adachi, K.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Aguet, F.

Austin, R. H.

Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006).
[CrossRef] [PubMed]

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Balci, H.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

Bates, M.

B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Bobroff, N.

N. Bobroff, "Position measurement with a resolution and noise-limited instrument," Rev. Sci. Instrum. 57, 1152-1157 (1986).
[CrossRef]

Brakenhoff, G. J.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

Buranachai, C.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

Cox, E. C.

Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006).
[CrossRef] [PubMed]

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Enderlein, J.

Florin, E.-L.

M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003).
[CrossRef] [PubMed]

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Geissbühler, S.

Golding, I.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Goldman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Guan, X.-J.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Guo, L.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Ha, T.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Harada, Y.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Huang, B.

B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Hynecek, J.

J. Hynecek and T. Nishiwaki, "Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs," IEEE Trans. on Electron Devices 50, 239-245 (2003).
[CrossRef]

Isacoff, E. Y.

M. H. Ulbrich and E. Y. Isacoff, "Subunit counting in membrane-bound proteins," Nature Methods 4, 319-321 (2007).
[PubMed]

Ishitsuka, Y.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

Jonas, A.

M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003).
[CrossRef] [PubMed]

Joo, C.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

Kinosita, K.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Köhler, J.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Lasser, T.

Märki, I.

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Müller, M.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

Nishiwaki, T.

J. Hynecek and T. Nishiwaki, "Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs," IEEE Trans. on Electron Devices 50, 239-245 (2003).
[CrossRef]

Noji, H.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Reisner, W.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Riehn, R.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Schmidt, J.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

Selvin, P. R.

J. Enderlein, E. Toprak, and P. R. Selvin, "Polarization effect on position accuracy of fluorophore localization," Opt. Express 14, 8111-8120 (2006).
[CrossRef] [PubMed]

A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004).
[CrossRef]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Speidel, M.

M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003).
[CrossRef] [PubMed]

Sturm, J.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Tegenfeldt, J.

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Tomishige, M.

A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004).
[CrossRef]

Toprak, E.

Ulbrich, M. H.

M. H. Ulbrich and E. Y. Isacoff, "Subunit counting in membrane-bound proteins," Nature Methods 4, 319-321 (2007).
[PubMed]

Unser, M.

Vale, R. D.

A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004).
[CrossRef]

van Oijen, A. M.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

Wang, W.

B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Wang, Y. M.

Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006).
[CrossRef] [PubMed]

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Yasuda, R.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Yildiz, A.

A. Yildiz, M. Tomishige, R. D. Vale, and P. R. Selvin, "Kinesin walks hand-over-hand," Science 303, 676-678 (2004).
[CrossRef]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, "Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization," Science 300, 2061-2065 (2003).
[CrossRef] [PubMed]

Yoshida, M.

K. Adachi, R. Yasuda, H. Noji, Y. Harada, M. Yoshida, and K. Kinosita, "Stepping rotation of F1-ATPase visualized through angle-resolved single-fluorophore imaging," Proc. Natl. Acad. Sci. USA 97, 7243-7247 (2000).
[CrossRef] [PubMed]

Zhuang, X.

B. Huang, W. Wang, M. Bates, and X. Zhuang, "Three-dimensional super-resolution imaging by stochastic optical reconstruction microscopy," Science 319, 810-813 (2008).
[CrossRef] [PubMed]

Annu. Rev. Biochem.

C. Joo, H. Balci, Y. Ishitsuka, C. Buranachai, and T. Ha, "Advances in single-molecule fluorescence methods for molecular biology," Annu. Rev. Biochem. 77, 51-76 (2008).
[CrossRef] [PubMed]

Biophys. J.

R. E. Thompson, D. R. Larson, and W. W. Webb, "Precise nanometer localization analysis for individual fluorescent probes," Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Chem. Phys. Lett.

A. M. van Oijen, J. Köhler, J. Schmidt, M. Müller, and G. J. Brakenhoff, "3-Dimensional super-resolution by spectrally selective imaging," Chem. Phys. Lett. 292, 183-187 (1998).
[CrossRef]

IEEE Trans. on Electron Devices

J. Hynecek and T. Nishiwaki, "Excess noise and other important characteristics of low light level imaging using charge multiplying CCDs," IEEE Trans. on Electron Devices 50, 239-245 (2003).
[CrossRef]

Nature Methods

M. H. Ulbrich and E. Y. Isacoff, "Subunit counting in membrane-bound proteins," Nature Methods 4, 319-321 (2007).
[PubMed]

Opt. Express

Opt. Lett.

M. Speidel, A. Jonas, and E.-L. Florin, "Three-dimensional tracking of fluorescent nanoparticles with subnanometer precision by use of off-focus imaging," Opt. Lett. 28 (2003).
[CrossRef] [PubMed]

Phys. Rev. Lett.

Y. M. Wang, R. H. Austin, and E. C. Cox, "Single molecule measurements of repressor protein 1D diffusion on DNA," Phys. Rev. Lett. 97, 048302 (2006).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA

Y. M. Wang, J. Tegenfeldt, W. Reisner, R. Riehn, X.-J. Guan, L. Guo, I. Golding, E. C. Cox, J. Sturm, and R. H. Austin, "Single-molecule studies of repressor-DNA interactions show long-range interactions," Proc. Natl. Acad. Sci. USA 102, 9796-9801 (2005).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1.
Fig. 1.

(A) Representative images with increasing N of 151, 393, and 1891 photons of single streptavidin-Cy3 molecules. (B) 1D intensity profiles (circles) of the molecules in (A) and their Gaussian fits (lines). The respective 1D SD values are 195.4 nm, 140.5 nm, and 110.9 nm. The scale bar is 500 nm.

Fig. 2.
Fig. 2.

Comparing ∆sx,rms vs N obtained by using four different methods: experimental measurements (solid squares), simulations (circles), numerical integrations (crosses), and analytical calculations (dashed line). Each experimental ∆sx,rms data point is the SD from the Gaussian fit to the sx distribution of a single streptavidin-Cy3 monomer. For each data point, its experimental N and background distributions were used for simulation, and its experimental 〈N〉, 〈sx,y 〉, σb , and 〈b〉 values were used for the numerical integrations and analytical calculations. The experimental data are on average 57% higher than the analytical calculation data.

Fig. 3.
Fig. 3.

sx,rms vs a/s 0 studied by simulations. In these simulations, there were no fluctuations in N or s 0. The vertical dashed line at a/s 0 = 1.18 is where the theoretical ∆sx,rms minimum occurs, determined by differentiating Eq. (15) with respect to a.

Equations (37)

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χ 2 ( s ) = i ( y i N i ) 2 σ i , photon 2 ,
σ i , photon 2 = N i + σ b 2 .
( Δ s ) 2 = 1 i ( N i 2 / σ i , photon 2 ) .
f ( n * ) = 1 M exp ( n * / M ) ,
F 2 = 1 M 2 σ out , camera 2 σ in , photon 2 2
σ i * 2 = 2 N i M 2 + ( σ b 2 + b ) M 2 .
σ i 2 = σ i * 2 / M 2 = 2 N i + σ b 2 + b .
( Δ s ) 2 = 1 i ( N i 2 / σ i 2 ) .
N i = Na 2 πs exp ( ( ia ) 2 / 2 s 2 ) ,
( Δ s ) 2 = s 0 2 N ,
( Δ s ) 2 = 8 π s 0 3 ( σ b 2 + b ) 3 a N 2 .
( Δ s ) 2 = s 0 2 N + 8 π s 0 3 ( σ b 2 + b ) 3 a N 2 .
s 0 †2 = s 0 2 + a 2 12 .
( Δ s ) 2 = s 0 2 + a 2 12 N + 8 π ( s b 2 + a 2 12 ) 3 / 2 ( σ b 2 + b ) 3 a N 2 .
( Δ s x ) 2 = s 0 x 2 + a 2 12 N + 16 π ( s 0 x 2 + a 2 12 ) 3 / 2 ( s 0 y 2 + a 2 12 ) 1 / 2 ( σ b 2 b ) 3 a 2 N 2 .
f ( x , y ) = f 0 exp ( ( x x 0 ) 2 2 s x 2 ( y y 0 ) 2 2 s y 2 ) + b ,
M = ( σ i * 2 σ b * 2 ) / 2 ( N i * b * ) ,
( Δ x ) 2 = 2 ( s 0 x 2 + a 2 12 ) N + 8 π ( s 0 x 2 + a 2 12 ) 3 / 2 ( s 0 y 2 + a 2 12 ) 1 / 2 ( σ b 2 + b ) a 2 N 2 .
s ( z ) = s 0 ( 1 + z 2 D 2 ) 1 / 2 ,
Δ z = D s 0 ( 1 s 0 2 s ( z ) 2 ) 1 / 2 Δ s rms ( z ) .
f y i = 1 2 π σ i exp ( Δ y i 2 2 σ i 2 ) ,
Δ y i = 0 ,
( Δ y i ) 2 = σ i 2 .
d χ 2 ( s ) ds = i d ds ( y N i ) 2 σ i 2 = i 2 ( y i N i ) ( y i N i ) σ i 2 ( y i N i ) 2 · 2 σ i σ i σ i 4 .
i 2 ( y i N i ) ( y i N i ) σ i 2 = i ( y i N i ) 2 . 2 σ i σ i σ i 4 .
y i N i ( s ) = y i ( N i ( s 0 ) + N i Δ s ) = Δ y i N i Δ s ,
( y i N i ) = N i ,
σ i 2 = 2 N i ( s ) + 2 σ b 2 = 2 ( N i ( s 0 ) + N i Δ s ) + 2 σ b 2 ,
2 σ i σ i = 2 N i .
i 2 ( Δ y i + N i Δ s ) ( N i ) σ i 2 = i ( Δ y i + N i Δ s ) 2 · 2 N i σ i 4
i ( Δ y i 2 + 2 Δ y i N i Δ s ) · 2 N i σ i 4 .
Δ s i ( N i 2 σ i 2 2 Δ y i N i 2 σ i 4 ) = i ( Δ y i 2 N i σ i 4 Δ y i N i σ i 2 )
Δ s = i Δ y i N i ' σ i 2 ( 1 Δ y i σ i 2 ) i N i 2 σ i 2 ( 1 2 Δ y i σ i 2 ) .
Δ s i Δ y i N i ' σ i 2 i N i 2 σ i 2 .
( Δ s ) 2 = i Δ y i N i σ i 2 j Δ y i N j σ j 2 ( i N i 2 σ i 2 ) 2 = i , j Δ y i Δ y j N i N j σ i 2 σ j 2 ( i N i 2 σ i 2 ) 2 .
N i , j = N a 2 2 π s x s y exp ( ( ia ) 2 2 s x 2 ( ja ) 2 2 s y 2 ) ,
( Δ s x 2 ) = 1 i ( d ( N i ) d s x ) 2 σ i 2 .

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