Abstract

This paper presents new formulas to determine the depth of field (DOF) of optical and digital microscope systems. Unlike the conventional DOF formula, the new methods consider the interplay of geometric and diffraction optics for infinite and finite optical microscopes and for corresponding digital microscope systems. It is shown that in addition to the well understood parameters such as numerical apertures, focal length, and light wavelength, system components such as aperture stops also affect the DOF. For the same objective lens, the DOF is inversely proportional to the size of the aperture stop, and it is proportional to the focal length of the ocular lens. It is also shown that under optimal viewing and operating conditions, the visual accommodation of human observers has no meaningful impact on DOF. The new formulas reported are useful for accurately calculating the DOF of microscopes.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. M. S. Elliot and W. C. K. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Colloid Interface Sci. 92, 133–194 (2001).
    [CrossRef] [PubMed]
  5. S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
    [CrossRef]
  6. C. Maurer, S. Khan, S. Fassl, S. Bernet, and M. Ritsch-Marte, “Depth of field multiplexing in microscopy,” Opt. Express 18, 3023–3034 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [PubMed]
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    [PubMed]
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    [CrossRef]
  14. H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.
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    [CrossRef]
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    [CrossRef]
  18. A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
    [CrossRef]
  19. F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, “Increase in depth of field taking into account deconvolution by optimization of pupil mask,” Opt. Lett. 34, 2970–2972(2009).
    [CrossRef] [PubMed]
  20. B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
    [CrossRef]

2011

2010

C. Maurer, S. Khan, S. Fassl, S. Bernet, and M. Ritsch-Marte, “Depth of field multiplexing in microscopy,” Opt. Express 18, 3023–3034 (2010).
[CrossRef] [PubMed]

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

2009

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, “Increase in depth of field taking into account deconvolution by optimization of pupil mask,” Opt. Lett. 34, 2970–2972(2009).
[CrossRef] [PubMed]

2008

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

2007

2001

M. S. Elliot and W. C. K. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Colloid Interface Sci. 92, 133–194 (2001).
[CrossRef] [PubMed]

1987

P. Gualtieri and L. Barsanti, “Identification of cellular and subcellular features by means of digital microscopy,” Int. J. Biomed. Comput. 20, 79–86 (1987).
[CrossRef] [PubMed]

J. W. Bacus and L. J. Grace, “Optical microscope system for standardized cell measurements and analysis,” Appl. Opt. 26, 3280–3293 (1987).
[CrossRef] [PubMed]

1983

1972

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Al-Akwaa, N.

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

Bacus, J. W.

Barsanti, L.

P. Gualtieri and L. Barsanti, “Identification of cellular and subcellular features by means of digital microscopy,” Int. J. Biomed. Comput. 20, 79–86 (1987).
[CrossRef] [PubMed]

Bernet, S.

Bhatia, A. B.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Blechinger, F.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.

Born, M.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Chebbi, B.

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

Cho, H.

Clemmow, P. C.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Cox, I. J.

Diaz, F.

Durand, F.

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

P. Green, W. Sun, W. Matusik, and F. Durand, “Multi-aperture photography,” ACM Trans. Graph. 26, 68 (2007).
[CrossRef]

Elliot, M. S.

M. S. Elliot and W. C. K. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Colloid Interface Sci. 92, 133–194 (2001).
[CrossRef] [PubMed]

Fassl, S.

Freeman, W. T.

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

Futaba, D. N.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Gabor, D.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Golub, I.

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

Goudail, F.

Grace, L. J.

Green, P.

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

P. Green, W. Sun, W. Matusik, and F. Durand, “Multi-aperture photography,” ACM Trans. Graph. 26, 68 (2007).
[CrossRef]

Gross, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.

Gualtieri, P.

P. Gualtieri and L. Barsanti, “Identification of cellular and subcellular features by means of digital microscopy,” Int. J. Biomed. Comput. 20, 79–86 (1987).
[CrossRef] [PubMed]

Hasinoff, S. W.

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

Hata, K.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Häusler, G.

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Hong, D.

Hua, H.

Huignard, J.-P.

Iijima, S.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Inoue, S.

S. Inoue and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

Khan, S.

Kim, M.

Levin, A.

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

Liu, S.

Loiseaux, B.

Martin, L. C.

L. C. Martin, The Theory of the Microscope (Elsevier, 1966), pp. 192–204.

Matusik, W.

P. Green, W. Sun, W. Matusik, and F. Durand, “Multi-aperture photography,” ACM Trans. Graph. 26, 68 (2007).
[CrossRef]

Maurer, C.

Michael, B.

B. Michael, Handbook of Optics Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).
[PubMed]

Minko, S.

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

Park, K.

Peschka, M.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.

Piller, H.

H. Piller, Microscope Photometry, Vol. 16 (Springer-Verlag, 1977), p. 16.

Poon, W. C. K.

M. S. Elliot and W. C. K. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Colloid Interface Sci. 92, 133–194 (2001).
[CrossRef] [PubMed]

Ritsch-Marte, M.

Sheppard, J. R.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).

Spring, K. R.

S. Inoue and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

Stokes, A. R.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Sun, W.

P. Green, W. Sun, W. Matusik, and F. Durand, “Multi-aperture photography,” ACM Trans. Graph. 26, 68 (2007).
[CrossRef]

Taylor, A. M.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Wayman, P. A.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Wilcock, W. L.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Wolf, E.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

Yasuda, S.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Yumura, M.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Zügge, H.

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.

ACM Trans. Graph.

P. Green, W. Sun, W. Matusik, and F. Durand, “Multi-aperture photography,” ACM Trans. Graph. 26, 68 (2007).
[CrossRef]

A. Levin, S. W. Hasinoff, P. Green, F. Durand, and W. T. Freeman, “4D frequency analysis of computational cameras for depth of field extension,” ACM Trans. Graph. 28, 97 (2009).
[CrossRef]

Adv. Colloid Interface Sci.

M. S. Elliot and W. C. K. Poon, “Conventional optical microscopy of colloidal suspensions,” Adv. Colloid Interface Sci. 92, 133–194 (2001).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

S. Yasuda, D. N. Futaba, M. Yumura, S. Iijima, and K. Hata, “Diagnostics and growth control of single-walled carbon nanotube for using a telecentric optical system for in situ height monitoring,” Appl. Phys. Lett. 93, 143115 (2008).
[CrossRef]

Int. J. Biomed. Comput.

P. Gualtieri and L. Barsanti, “Identification of cellular and subcellular features by means of digital microscopy,” Int. J. Biomed. Comput. 20, 79–86 (1987).
[CrossRef] [PubMed]

Opt. Commun.

B. Chebbi, S. Minko, N. Al-Akwaa, and I. Golub, “Remote control of extended depth of field focusing,” Opt. Commun. 283, 1678–1683 (2010).
[CrossRef]

G. Häusler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Opt. Express

Opt. Lett.

Other

H. Piller, Microscope Photometry, Vol. 16 (Springer-Verlag, 1977), p. 16.

L. C. Martin, The Theory of the Microscope (Elsevier, 1966), pp. 192–204.

M. Born and E. Wolf, with contributions by A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).
[PubMed]

B. Michael, Handbook of Optics Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010).
[PubMed]

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008).

S. Inoue and K. R. Spring, Video Microscopy: the Fundamentals (Plenum, 1997).
[CrossRef]

H. Gross, H. Zügge, M. Peschka, and F. Blechinger, Handbook of Optical Systems, Vol.  3 (Wiley-VCH, 2007), p. 125.

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Figures (11)

Fig. 1
Fig. 1

The objective lens projects an Airy disk of the object point into the image plane.

Fig. 2
Fig. 2

The object point is away from the object plane. It is at the threshold position.

Fig. 3
Fig. 3

Imaging schematics of finite optical visual microscope.

Fig. 4
Fig. 4

Relation between the aperture stop and the exit pupil in the finite optical system based visual microscope.

Fig. 5
Fig. 5

DOF calculation of an infinite optical system based visual microscope.

Fig. 6
Fig. 6

Relation between the aperture stop and the exit pupil for an infinite optical visual microscope.

Fig. 7
Fig. 7

DOF calculation of a finite optical system based digital microscope.

Fig. 8
Fig. 8

Relation between the aperture stop and the exit pupil in the finite optical system based digital microscope.

Fig. 9
Fig. 9

Relation between the exit pupil and the CCD image plane in a digital microscope system.

Fig. 10
Fig. 10

DOF calculation of an infinite optical system based digital microscope.

Fig. 11
Fig. 11

Relation between the aperture stop and the exit pupil in the infinite optical system based digital microscope.

Tables (5)

Tables Icon

Table 1 DOF Results of a Finite Optical System Based Visual Microscope Using the Traditional Calculating Method

Tables Icon

Table 2 DOF Results of a Finite Optical System Based Visual Microscope Using the Improved Calculating Method

Tables Icon

Table 3 DOF Results of an Infinite Optical System Based Visual Microscope

Tables Icon

Table 4 DOF Results of a Finite Optical System Based Digital Microscope

Tables Icon

Table 5 DOF Results of an Infinite Optical System Based Digital Microscope

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

T g = 250 n ε NA · Γ ,
T p = n λ NA 2 ,
T a = 437.5 n Γ 2 .
T m = T g + T p + T a .
tan α = a r Δ + δ x = r / 2 δ x ,
tan γ = a r Δ δ x = 5 r / 2 δ x .
δ x = r Δ 2 a 3 r ,
δ x = 5 r Δ 2 a + 3 r .
δ x 1 = n β 2 ( r Δ 2 a 3 r ) ,
δ x 2 = n β 2 ( 5 r Δ 2 a + 3 r ) .
δ x = δ x 1 + δ x 2 = n β 2 ( r Δ 2 a 3 r + 5 r Δ 2 a + 3 r ) = 12 n r Δ ( a r ) β 2 ( 4 a 2 9 r 2 ) .
δ = 0.61 λ / NA .
r = 0.61 β λ / NA .
a = ( 250 mm ) · NA Γ .
a a = f e Δ + f e ,
a = 250 NA ( Δ + f e ) f e Γ .
δ x 1 = 0.05368 n 2 NA 2 ( 160 + f e ) ( 1.0065 × 10 3 ) β 2 ,
δ x 2 = 0.2684 n 2 NA 2 ( 160 + f e ) + ( 1.0065 × 10 3 ) β 2 ,
δ x = δ x 1 + δ x 2 = n [ 0.64416 NA 2 ( 160 + f e ) ( 2.16116 × 10 4 ) β 2 ] 4 NA 4 ( 160 + f e ) 2 ( 1.01304 × 10 6 ) β 4 .
tan α = a r f t + δ x = r / 2 δ x ,
tan γ = a r f t δ x = 5 r / 2 δ x .
δ x = r f t 2 a 3 r ,
δ x = 5 r f t 2 a + 3 r ,
δ x 1 = n β 2 ( r f t 2 a 3 r ) ,
δ x 2 = n β 2 ( 5 r f t 2 a + 3 r ) .
δ x = δ x 1 + δ x 2 = n β 2 ( r f t 2 a 3 r + 5 r f t 2 a + 3 r ) = 12 r f t n ( a r ) β 2 ( 4 a 2 9 r 2 ) .
a a = f e f t ,
a = 250 NA · f t f e Γ = NA · f t β .
δ x 1 = 0.0671 n 400 NA 2 ( 1.0065 × 10 3 ) β 2 ,
δ x 2 = 0.3355 n 400 NA 2 + ( 1.0065 × 10 3 ) β 2 ,
δ x = δ x 1 + δ x 2 = n ( 161.04 NA 2 2.70 × 10 4 β 2 ) 1.6 × 10 5 NA 4 1.01304 × 10 6 β 4 .
tan α = a r Δ + δ x = P ccd / β r δ x ,
tan γ = a r Δ δ x = 2 r + P ccd / β r δ x .
δ x = Δ P ccd / β r a r P ccd / β r ,
δ x = 2 r Δ + Δ P ccd / β r a + r + P ccd / β r ,
δ x 1 = n β 2 ( Δ P ccd β r ( a r ) P ccd ) ,
δ x 2 = n β 2 ( 2 r Δ β r + Δ P ccd β r ( a + r ) + P ccd ) ,
δ x = δ x 1 + δ x 2 = n β 2 ( 2 Δ β r ( P ccd + r β r ) ( a r ) β r 2 a 2 ( β r r + P ccd ) 2 ) .
2 a = L ccd 2 + W ccd 2 .
2 a = L ccd 2 + W ccd 2 / β r .
δ x 1 = 1.04 NA · n 3.9905 NA β 2 ( 2.3485 × 10 4 ) β 3 ,
δ x 2 = ( 0.07515 β + 1.04 NA ) · n 3.9905 NA β 2 + ( 2.3485 × 10 4 ) β 3 ,
δ x = δ x 1 + δ x 2 = ( 8.3 NA 2 + 0.3 NA β 1.765 × 10 5 β 2 ) · n 15.924 NA 2 β 2 ( 5.515 × 10 8 ) β 4 .
tan α = a r f t + δ x = P ccd / β r δ x ,
tan γ = a r f t δ x = 2 r + P ccd / β r δ x .
δ x = f t P ccd / β r a r P ccd / β r ,
δ x = 2 r f t + f t P ccd / β r a + r + P ccd / β r ,
δ x 1 = n β 2 ( f t P ccd β r ( a r ) P ccd ) ,
δ x 2 = n β 2 ( 2 r f t β r + f t P ccd β r ( a + r ) + P ccd ) ,
δ x = δ x 1 + δ x 2 = n β 2 ( 2 f t β r ( P ccd + r β r ) ( a r ) β r 2 a 2 ( β r r + P ccd ) 2 ) .
f z = f t / β r .
2 a = L ccd 2 + W ccd 2 ,
2 a = L ccd 2 + W ccd 2 / β r .
δ x 1 = 1.3 NA · n 3.9905 NA β 2 2.3485 × 10 4 β 3 ,
δ x 2 = ( 0.094 β + 1.3 NA ) · n 4.0035 NA β 2 + 2.3485 × 10 4 β 3 ,
δ x = δ x 1 + δ x 2 = ( 10.3922 NA 2 + 0.3751 NA β 2.2076 × 10 5 β 2 ) · n 15.976 NA 2 β 2 3.05 × 10 6 NA β 3 5.5154 × 10 8 β 4 .

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