Abstract

Live biological specimens exhibit time-varying behavior on the microscale in all three dimensions. Although scanning confocal and two-photon microscopes are able to record three-dimensional image stacks through these specimens, they do so at relatively low speeds which limits the time resolution of the biological processes that can be observed. One way to improve the data acquisition rate is to image only the regions of a specimen that are of interest and so researchers have recently begun to acquire two-dimensional images of inclined planes or surfaces extending significantly into the z-direction. As the resolution is not uniform in x, y and z, the images possess non-isotropic resolution. We explore this theoretically and show that images of an oblique plane may contain spectral content that could not have been generated by specimen features lying wholly within the plane but must instead arise from a spatial variation in another direction. In some cases we find that the image contains frequencies three times higher than the resolution limit for in-plane features. We confirm this finding through numerical simulations and experiments on a novel, oblique-plane imaging system and suggest that care be taken in the interpretation of such images.

© 2011 Optical Society of America

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References

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  1. T. Wilson, and C. J. R. Sheppard, Theory and practice of Scanning Optical Microscopy (Academic Press, 1984).
  2. W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
    [CrossRef] [PubMed]
  3. I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986).
    [CrossRef] [PubMed]
  4. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
    [CrossRef]
  5. W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007).
    [CrossRef] [PubMed]
  6. N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
    [CrossRef] [PubMed]
  7. P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
    [CrossRef]
  8. M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
    [CrossRef]
  9. S. Vembu, “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. 12, 165–168 (1961).
    [CrossRef]
  10. E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
    [CrossRef]
  11. E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
    [CrossRef]
  12. O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
    [CrossRef]
  13. D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
    [CrossRef] [PubMed]
  14. G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
    [CrossRef]

2010

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

2009

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

2008

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

2007

W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007).
[CrossRef] [PubMed]

2003

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

1997

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

1990

W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1986

I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986).
[CrossRef] [PubMed]

1961

S. Vembu, “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. 12, 165–168 (1961).
[CrossRef]

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Beaurepaire, E.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Bollensdorff, C.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Booth, M.

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

Botcherby, E.

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

Bourgine, P.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Bub, G.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Burton, R.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Camelliti, P.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Carbone, G.

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

Clarke, K.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Débarre, D.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

Denk, W.

W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Deutsch, M.

I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986).
[CrossRef] [PubMed]

Duloquin, L.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Eisenberg, H.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Elston, S.

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

Faure, E.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Freund, I.

I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986).
[CrossRef] [PubMed]

Göbel, W.

W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007).
[CrossRef] [PubMed]

Haeberlé, O.

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

Helmchen, F.

W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007).
[CrossRef] [PubMed]

Horowitz, M.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Juškaitis, R.

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

Kohl, P.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Luengo-Oroz, M.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Olivier, N.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Peyríeras, N.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Picton, G.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Raynes, E.

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

Salter, P.

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

Santos, A.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Savy, T.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Smith, C.

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

Solinas, X.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Srinivas, S.

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

Strickler, J.

W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Stuckey, D.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Veilleux, I.

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

Vembu, S.

S. Vembu, “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. 12, 165–168 (1961).
[CrossRef]

Watanabe, T.

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

Webb, W.

W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Wilson, T.

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

Am. J. Physiol. Heart Circ. Physiol.

G. Bub, P. Camelliti, C. Bollensdorff, D. Stuckey, G. Picton, R. Burton, K. Clarke, and P. Kohl, “Measurement and analysis of sarcomere length in rat cardiomyocytes in situ and in vitro,” Am. J. Physiol. Heart Circ. Physiol. 298, 1616–1625 (2010).
[CrossRef]

Appl. Phys. Lett.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

J. Neurophysiol.

W. Göbel, and F. Helmchen, “New angles on neuronal dendrites in vivo,” J. Neurophysiol. 98, 3770–3779 (2007).
[CrossRef] [PubMed]

Opt. Commun.

E. Botcherby, R. Juškaitis, M. Booth, and T. Wilson, “An optical technique for refocusing in microscopy,” Opt. Commun. 281, 880–887 (2008).
[CrossRef]

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

Opt. Lett.

D. Débarre, E. Botcherby, T. Watanabe, S. Srinivas, M. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef] [PubMed]

I. Freund, and M. Deutsch, “Second-harmonic microscopy of biological tissue,” Opt. Lett. 11, 94–96 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett.

P. Salter, G. Carbone, E. Botcherby, T. Wilson, S. Elston, and E. Raynes, “Liquid crystal director dynamics imaged using two-photon fluorescence microscopy with remote focusing,” Phys. Rev. Lett. 103, 257803 (2009).
[CrossRef]

Proc. SPIE

E. Botcherby, C. Smith, M. Booth, R. Juškaitis, and T. Wilson, “Arbitrary-scan imaging for two-photon microscopy,” Proc. SPIE 7569, 756917 (2010).
[CrossRef]

Q. J. Math.

S. Vembu, “Fourier transformation of the n-dimensional radial delta function,” Q. J. Math. 12, 165–168 (1961).
[CrossRef]

Science

N. Olivier, M. Luengo-Oroz, L. Duloquin, E. Faure, T. Savy, I. Veilleux, X. Solinas, D. Débarre, P. Bourgine, A. Santos, N. Peyríeras, and E. Beaurepaire, “Cell lineage reconstruction of early zebrafish embryos using labelfree nonlinear microscopy,” Science 329, 967–971 (2010).
[CrossRef] [PubMed]

W. Denk, J. Strickler, and W. Webb, “Two-photon fluorescence scanning microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Other

T. Wilson, and C. J. R. Sheppard, Theory and practice of Scanning Optical Microscopy (Academic Press, 1984).

M. Gu, Principles of Three-Dimensional Imaging in Confocal Microscopes (World Scientific, 1996).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) Region of support for the 3D OTF of the microscope. (b) Imaging oblique plane OP with angle θ. (c) When imaging an oblique plane, all k-vectors shown in red have the same projection along the r′-axis so will appear in the image with the same spatial frequency. (d) When imaging an oblique plane, spatial frequency, k1, will appear in the final image whereas a spatial frequency, k2, will not.

Fig. 2
Fig. 2

Limiting values of spatial frequencies along the x′-direction for a 1.4 NA oil immersion objective and a 0.8 NA water immersion objective. Solid lines show the absolute spatial frequency cutoff which includes projections of out-of-plane spatial frequencies whereas dashed lines show the largest spatial frequencies that can be imaged of features lying wholly in the plane.

Fig. 3
Fig. 3

Numerical simulations of images from a 45° oblique plane with a 1.15 NA water immersion objective and associated Fourier transforms. (a) White noise with a projection of the 3D OTF region of support overlain and (b) a spherical shell of diameter 12μm with the cut through the 3D OTF overlain.

Fig. 4
Fig. 4

Oblique plane scanning multi-photon microscope. The lateral scan unit (LSU) scans the focal spot laterally in the specimen and the axial scan unit (ASU) scans the spot axially. Oblique planes are scanned in three dimensions by using a combination of the LSU and ASU.

Fig. 5
Fig. 5

Experimental images from a 45° oblique plane with a 1.15 NA water immersion objective and their associated Fourier transforms. (a) A fluorescent pollen grain with a projection of the OTF overlain and (b) a spherical shell with diameter 12μm with the cut through the OTF overlain.

Equations (7)

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I ( x , y , z ) = C ( m , n , r ) T ( m , n , r ) e i 2 π ( m x + n y + r z ) d m d n d r ,
| r | = { 1 if 0 l 2 , l l 2 4 if 2 < l 4 ,
I ( x , y ) z 0 = C ( m , n , r ) T ( m , n , r ) e i 2 π ( m x + n y + r z 0 ) d m d n d r ,
G ( m , n ) = I ( x , y ) z 0 e i 2 π ( m x + n y ) d x d y
G ( m , n ) = C ( m , n , r ) T ( m , n , r ) e i 2 π r z 0 d r .
m 1 = { 4 n sin ( α ) λ cos θ , 0 tan θ 1 2 cot ( α 2 ) , n cos 2 ( α / 2 ) λ [ 2 tan ( α 2 ) + cot θ ] 2 sin θ , 1 2 cot ( α 2 ) < tan θ tan ( π 2 ) .
m 2 = { 4 n cos 2 ( α / 2 ) λ ( 2 tan ( α 2 ) tan θ ) sec θ , 0 θ α 2 , 4 n sin 2 ( α / 2 ) λ cosec θ , α 2 < θ π 2 .

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