Abstract

The optical transfer function (OTF) is widely used to compare the performance of different optical systems. Conventionally, the OTF is normalized to unity for zero spatial frequency, but in some cases it is better to consider the unnormalized OTF, which gives the absolute value of the image signal. Examples are in confocal microscopy and image scanning microscopy, where the signal level increases with pinhole or array size. Comparison of the respective unnormalized OTFs gives useful insight into their relative performance. The significance of other properties of the general OTF is discussed.

© 2016 Optical Society of America

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References

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  1. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).
    [Crossref]
  2. M. Gu and C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9(1), 151–153 (1992).
    [Crossref]
  3. C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).
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    [Crossref]
  5. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
    [Crossref]
  6. C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).
  7. C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
    [Crossref] [PubMed]
  8. A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
    [Crossref] [PubMed]
  9. C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
    [Crossref] [PubMed]
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    [Crossref]
  12. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1(8), 801–808 (1984).
    [Crossref]
  13. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3(9), 1428–1432 (1986).
    [Crossref]
  14. R. N. Bracewell, The Fourier Transform and its Applications (McGraw Hill, 1978).
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    [Crossref]
  16. X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15(4), 187–192 (1993).
    [Crossref]
  17. T. Wilson and D. K. Hamilton, “Difference confocal scanning microscopy,” Opt. Acta (Lond.) 31(4), 452–465 (1984).
    [Crossref]
  18. V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).
  19. R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
    [Crossref] [PubMed]
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    [Crossref]
  23. S. Roth, C. J. R. Sheppard, and R. Heintzmann, “Superconcentration of light: circumventing the classical limit to achievable irradiance,” Opt. Lett. 41(9), 2109–2112 (2016).
    [Crossref] [PubMed]

2016 (1)

2014 (1)

2013 (4)

2012 (1)

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

2010 (1)

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

2003 (1)

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

1998 (1)

R. Gauderon and C. J. R. Sheppard, “Improvement in imaging in confocal fluorescent microscopes using detector arrays,” Bioimaging 6(3), 126–129 (1998).
[Crossref]

1997 (2)

V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

1993 (1)

X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15(4), 187–192 (1993).
[Crossref]

1992 (2)

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165(3), 377–390 (1992).
[Crossref]

M. Gu and C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9(1), 151–153 (1992).
[Crossref]

1988 (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).

1986 (1)

1984 (2)

1972 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).
[Crossref]

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[Crossref]

Azar, L. N.

Boyer, G.

V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).

Brandt, R. A. J.

Breedijk, R. M. P.

Chitnis, A. B.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Combs, C. A.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

de Jong, B. E.

De Koninck, Y.

De Luca, G. M. R.

Dehez, H.

Enderlein, J.

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Fischer, R. S.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Gan, X. S.

X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15(4), 187–192 (1993).
[Crossref]

Gauderon, R.

R. Gauderon and C. J. R. Sheppard, “Improvement in imaging in confocal fluorescent microscopes using detector arrays,” Bioimaging 6(3), 126–129 (1998).
[Crossref]

Goodman, J. W.

Gu, M.

M. Gu and C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9(1), 151–153 (1992).
[Crossref]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165(3), 377–390 (1992).
[Crossref]

Hamilton, D. K.

T. Wilson and D. K. Hamilton, “Difference confocal scanning microscopy,” Opt. Acta (Lond.) 31(4), 452–465 (1984).
[Crossref]

Hanley, Q. S.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Heintzmann, R.

S. Roth, C. J. R. Sheppard, and R. Heintzmann, “Superconcentration of light: circumventing the classical limit to achievable irradiance,” Opt. Lett. 41(9), 2109–2112 (2016).
[Crossref] [PubMed]

C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
[Crossref]

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Hoebe, R. A.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).
[Crossref]

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[Crossref]

Johnson, C.

V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).

Jovin, T. M.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Kuang, C.

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Li, Y.

Liu, X.

Manders, E. M.

Mehta, S. B.

Mione, M.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Müller, C. B.

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Munroe, P.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Nailon, J.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Nogare, D. D.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Parekh, S. H.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Piché, M.

Rong, Z.

Roth, S.

S. Roth, C. J. R. Sheppard, and R. Heintzmann, “Superconcentration of light: circumventing the classical limit to achievable irradiance,” Opt. Lett. 41(9), 2109–2112 (2016).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
[Crossref]

Sarafis, V.

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).

Sheppard, C. J. R.

S. Roth, C. J. R. Sheppard, and R. Heintzmann, “Superconcentration of light: circumventing the classical limit to achievable irradiance,” Opt. Lett. 41(9), 2109–2112 (2016).
[Crossref] [PubMed]

C. J. R. Sheppard, S. B. Mehta, and R. Heintzmann, “Superresolution by image scanning microscopy using pixel reassignment,” Opt. Lett. 38(15), 2889–2892 (2013).
[Crossref] [PubMed]

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
[Crossref]

R. Gauderon and C. J. R. Sheppard, “Improvement in imaging in confocal fluorescent microscopes using detector arrays,” Bioimaging 6(3), 126–129 (1998).
[Crossref]

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15(4), 187–192 (1993).
[Crossref]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165(3), 377–390 (1992).
[Crossref]

M. Gu and C. J. R. Sheppard, “Confocal fluorescent microscopy with a finite-sized circular detector,” J. Opt. Soc. Am. A 9(1), 151–153 (1992).
[Crossref]

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).

C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3(9), 1428–1432 (1986).
[Crossref]

Shroff, H.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Stallinga, S.

Temprine, K.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Tichenor, D. A.

Timmermans, W.

Wicker, K.

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
[Crossref]

Wilson, T.

T. Wilson and D. K. Hamilton, “Difference confocal scanning microscopy,” Opt. Acta (Lond.) 31(4), 452–465 (1984).
[Crossref]

Wolf, E.

York, A. G.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

You, S.

Zeelenberg, C. H. C.

Bioimaging (1)

R. Gauderon and C. J. R. Sheppard, “Improvement in imaging in confocal fluorescent microscopes using detector arrays,” Bioimaging 6(3), 126–129 (1998).
[Crossref]

Biomed. Opt. Express (1)

Cell Vis. (1)

V. Sarafis, C. Johnson, and G. Boyer, “Confocal microscopy with pinhole super-resolution,” Cell Vis. 4, 264 (1997).

J. Microsc. (1)

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165(3), 377–390 (1992).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Micron (1)

R. Heintzmann, V. Sarafis, P. Munroe, J. Nailon, Q. S. Hanley, and T. M. Jovin, “Resolution enhancement by subtraction of confocal signals taken at different pinhole sizes,” Micron 34(6-7), 293–300 (2003).
[Crossref] [PubMed]

Nat. Methods (1)

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9(7), 749–754 (2012).
[Crossref] [PubMed]

Opt. Acta (Lond.) (1)

T. Wilson and D. K. Hamilton, “Difference confocal scanning microscopy,” Opt. Acta (Lond.) 31(4), 452–465 (1984).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Opt. Nanoscopy (1)

S. Roth, C. J. R. Sheppard, K. Wicker, and R. Heintzmann, “Optical photon reassignment microscopy (OPRA),” Opt. Nanoscopy 2(1), 5 (2013).
[Crossref]

Optik (Stuttg.) (2)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik (Stuttg.) 80, 53–54 (1988).

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik (Stuttg.) 107, 79–87 (1997).

Phys. Rev. Lett. (1)

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 198101 (2010).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. A Math. Phys. Sci. 217(1130), 408–432 (1953).
[Crossref]

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).
[Crossref]

Scanning (1)

X. S. Gan and C. J. R. Sheppard, “Detectability: A new criterion for evaluation of the confocal microscope,” Scanning 15(4), 187–192 (1993).
[Crossref]

Other (1)

R. N. Bracewell, The Fourier Transform and its Applications (McGraw Hill, 1978).

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Figures (5)

Fig. 1
Fig. 1 The normalized OTF for a confocal microscope with different pinhole sizes in Airy units. The OTF for ISM with a large array is also shown.
Fig. 2
Fig. 2 The unnormalized OTF for (a) confocal microscopes and (b) ISM with different pinhole/array sizes.
Fig. 3
Fig. 3 A close up of the unnormalized OTF at high spatial frequency for a confocal microscope with different pinhole sizes (dashed lines), and for ISM with different array sizes (solid lines). The behavior of ISM with a large array is also shown (black line) for comparison. Note that the confocal case results in negative values of OTF, but ISM does not.
Fig. 4
Fig. 4 The normalized spatial frequency for the first zero of the OTF, for confocal imaging with pinhole size in Airy units. The cut-off frequency is 4 for pinhole sizes less than 0.5 AU, and tends to 2 for large pinhole sizes.
Fig. 5
Fig. 5 A logarithmic plot of the unnormalized OTFs for a confocal microscope (dashed lines) and ISM (solid lines) with different pinhole/array sizes. The first positive lobe only of the confocal OTF is shown. The behavior for subtracting images from two pinhole sizes ( I 0.5AU 1 4 I 1AU ) , or using a matched filter with two ring detectors is also shown.

Equations (4)

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

C(l)= v d A C 1 C 2 J 1 ( l 2 v d ) l 2 l d l dϕ,
C 1,2 =Re{ 2 π [ arccos( l 1,2 2 ) l 1,2 2 1 ( l 1,2 2 ) 2 ] }, l 1,2 = l 2 + 1 4 l 2 l l cosϕ ,
C(l)= 1 2π 0 0 v d 0 π/2 h 1 h 2 J 0 (lv)v v dθd v dv,
h 1,2 = 4 J 1 2 ( v 2 + 1 4 v 2 ±v v cosθ ) v 2 + 1 4 v 2 ±v v cosθ .

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