Abstract

Three-dimensional (3D) imaging by holographic tomography can be performed for a fixed detector through rotation of either the object or the illumination beam. We have previously presented a paraxial treatment to distinguish between these two approaches using transfer function analysis. In particular, the cutoff of the transfer function when rotating the illumination about one axis was calculated analyt ically using one-dimensional Fourier integration of the defocused transfer function. However, high numerical aperture objectives are usually used in experimental arrangements, and the previous paraxial model is not accurate in this case. Hence, in this analysis, we utilize 3D analytical geometry to derive the imaging behavior for holographic tomography under high-aperture conditions. As expected, the cutoff of the new transfer function leads to a similar peanut shape, but we found that there was no line singularity as was previously observed in the paraxial case. We also present the theory of coherent transfer function for holographic tomography under object rotation while the detector is kept stationary. The derived coherent transfer functions offer quantitative insights into the image formation of a diffractive tomography system.

© 2009 Optical Society of America

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References

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  1. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial Mathematics, 2001).
    [CrossRef]
  2. H. Stark, Image Recovery: Theory and Application (Academic, 1987).
  3. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
    [CrossRef] [PubMed]
  4. J. R. Price, P. R. Bingham, and C. E. Thomas, “Improving resolution in microscopic holography by computationally fusing multiple, obliquely illuminated object waves in the Fourier domain,” Appl. Opt. 46, 827-833 (2007).
    [CrossRef] [PubMed]
  5. G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
    [CrossRef] [PubMed]
  6. S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15, 13640-13648 (2007).
    [CrossRef] [PubMed]
  7. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. (Oxford) 205, 165-176 (2002).
    [CrossRef]
  8. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
    [CrossRef] [PubMed]
  9. F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178-180 (2006).
    [CrossRef] [PubMed]
  10. S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
    [CrossRef]
  11. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  12. R. W. James, Optical Principles of the Diffraction of X-Rays, 1st ed. (G. Bell, 1982).
  13. D. W. Sweeney and C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649-2664 (1973).
    [CrossRef] [PubMed]
  14. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  15. R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323-328 (1970).
    [CrossRef]
  16. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Jena) 72, 131-133 (1986).
  17. S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography,” Opt. Lett. 33, 2362-2364(2008).
    [CrossRef] [PubMed]
  18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  19. C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377-390 (1991),
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).
  21. A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC Press, 2006).
  22. A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111-112 (1982).
    [CrossRef] [PubMed]
  23. A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374-376 (1981).
    [CrossRef] [PubMed]
  24. C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
    [CrossRef] [PubMed]
  25. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).
  26. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266-277 (2009).
    [CrossRef] [PubMed]

2009 (3)

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266-277 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (3)

2006 (2)

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

F. Charrière, A. Marian, F. Montfort, J. Kühn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178-180 (2006).
[CrossRef] [PubMed]

2002 (1)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. (Oxford) 205, 165-176 (2002).
[CrossRef]

1993 (1)

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
[CrossRef] [PubMed]

1991 (1)

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377-390 (1991),
[CrossRef]

1986 (1)

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Jena) 72, 131-133 (1986).

1982 (2)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

A. J. Devaney, “Inversion formula for inverse scattering within the Born approximation,” Opt. Lett. 7, 111-112 (1982).
[CrossRef] [PubMed]

1981 (1)

1973 (1)

1970 (1)

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323-328 (1970).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Abbena, E.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC Press, 2006).

Alexandrov, S. A.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

Badizadegan, K.

Belkebir, K.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Bingham, P. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

Charrière, F.

Chaumet, P. C.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Chernov, L. A.

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

Choi, W.

Colomb, T.

Connolly, T. J.

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
[CrossRef] [PubMed]

Cuche, E.

Dändliker, R.

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323-328 (1970).
[CrossRef]

Dasari, R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Dasari, R. R.

Delaunay, J.-J.

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

Depeursinge, C.

Devaney, A.

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Devaney, A. J.

Drsek, F.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Fang-Yen, C.

Feld, M.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Feld, M. S.

Giovannini, H.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Girard, J.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gray, A.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC Press, 2006).

Gu, M.

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
[CrossRef] [PubMed]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377-390 (1991),
[CrossRef]

Gutzler, T.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

Haeberlé, O.

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

Hillman, T. R.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

James, R. W.

R. W. James, Optical Principles of the Diffraction of X-Rays, 1st ed. (G. Bell, 1982).

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial Mathematics, 2001).
[CrossRef]

Konan, D.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Kou, S. S.

Kühn, J.

Lauer, V.

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. (Oxford) 205, 165-176 (2002).
[CrossRef]

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Maire, G.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Marian, A.

Marquet, P.

Montfort, F.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Price, J. R.

Salamon, S.

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC Press, 2006).

Sampson, D. D.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

Sentenac, A.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Sheppard, C. J. R.

S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography,” Opt. Lett. 33, 2362-2364(2008).
[CrossRef] [PubMed]

S. S. Kou and C. J. R. Sheppard, “Imaging in digital holographic microscopy,” Opt. Express 15, 13640-13648 (2007).
[CrossRef] [PubMed]

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
[CrossRef] [PubMed]

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377-390 (1991),
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Jena) 72, 131-133 (1986).

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial Mathematics, 2001).
[CrossRef]

Stark, H.

H. Stark, Image Recovery: Theory and Application (Academic, 1987).

Sung, Y.

Sweeney, D. W.

Talneau, A.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Thomas, C. E.

Vertu, S.

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

Vest, C. M.

Weiss, K.

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323-328 (1970).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

Yamada, I.

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

Appl. Opt. (2)

Central Eur. J. Phys. (1)

S. Vertu, J.-J. Delaunay, I. Yamada, and O. Haeberlé, “Diffraction microtomography with sample rotation: influence of a missing apple core in the recorded frequency space,” Central Eur. J. Phys. 7, 22-31 (2009).
[CrossRef]

J. Microsc. (1)

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377-390 (1991),
[CrossRef]

J. Microsc. (Oxford) (1)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. (Oxford) 205, 165-176 (2002).
[CrossRef]

Nat. Methods (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717-719 (2007).
[CrossRef] [PubMed]

Opt. Commun. (2)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323-328 (1970).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Optik (Jena) (1)

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Jena) 72, 131-133 (1986).

Phys. Rev. Lett. (3)

C. J. R. Sheppard, T. J. Connolly, and M. Gu, “Scattering by a one-dimensional rough surface and surface reconstruction by confocal imaging,” Phys. Rev. Lett. 70, 1409-1412(1993).
[CrossRef] [PubMed]

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture Fourier holographic optical microscopy,” Phys. Rev. Lett. 97, 168102-168104 (2006).
[CrossRef] [PubMed]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe's limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905-213904 (2009).
[CrossRef] [PubMed]

Ultrason. Imaging (1)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (7)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial Mathematics, 2001).
[CrossRef]

H. Stark, Image Recovery: Theory and Application (Academic, 1987).

R. W. James, Optical Principles of the Diffraction of X-Rays, 1st ed. (G. Bell, 1982).

L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

A. Gray, E. Abbena, and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed. (Chapman & Hall/CRC Press, 2006).

Supplementary Material (2)

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Figures (7)

Fig. 1
Fig. 1

Analytical geometry for deriving the high-aperture object rotation case in transmission: (a). 3D view and (b). projective 2D view.

Fig. 2
Fig. 2

Three-dimensional PSF for holographic transmission tomography in a 4 π configuration.

Fig. 3
Fig. 3

Three-dimensional transfer function cutoff for object rotation in transmission. (a) Spindle torus for 4 π configuration, (b) cross-sectional view of the 3D CTF with its weighting density, (c) torus for object rotation with aperture semiangle equal to π / 3 , (d) cross-sectional view of the CTF with aplanatic weighting density.

Fig. 4
Fig. 4

Analytical geometry for deriving the high-aperture transmission illumination rotation case at a normalized n value of 0.6.

Fig. 5
Fig. 5

Cross-sectional views of the transmission CTF with aplanatic weighting density at normalized n of (a) 0, (b) 0.6, (c) 0.8.

Fig. 6
Fig. 6

Three-dimensional CTF spatial cutoff for illumination rotation in high-aperture transmission imaging with an aperture semiangle equal to π / 3 : left: top view; right, flipped bottom view.

Fig. 7
Fig. 7

Three-dimensional CTF spatial cutoff for illumination rotation in high-aperture reflection imaging with an aperture semiangle equal to π / 3 : left, top view; right, flipped bottom view.

Equations (26)

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

n 2 + s 2 2 s cos θ = 0
c ( l , n ) = π / 2 π / 2 2 δ ( l 2 + n 2 2 l cos θ ) d θ ,
c ( l , n ) = 1 | l sin θ | = 2 [ 4 l 2 ( l 2 + n 2 ) 2 ] 1 / 2 , 2 l > ( l 2 + n 2 ) .
U ( ρ , y ) = sin [ k ( ρ 2 + y 2 ) 1 / 2 ] k ( ρ 2 + y 2 ) 1 / 2 J 0 ( k ρ ) ,
cos β = 1 ( l 2 + n 2 ) / 2 ,
c a obj T ( l , n ) = c ( l , n ) cos 1 / 2 β ,
c a obj T ( l , n ) = 2 [ 2 ( l 2 + n 2 ) ] 1 / 2 [ 4 l 2 ( l 2 + n 2 ) 2 ] 1 / 2 , 4 sin 2 α 2 > 2 l > l 2 + n 2 .
s = cos θ cos γ , ( | m | sin θ ) 2 + n 2 = sin 2 γ ,
sin θ = | m | ( sin 2 γ n 2 ) 1 / 2 , cos γ = s ( n 2 l 2 ) + | m | [ 4 l 2 ( l 2 + n 2 ) 2 ] 1 / 2 2 l 2 ,
s R max = [ cos 2 α m 2 + n 2 + 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α , | m | < sin α + ( sin 2 α n 2 ) 1 / 2 .
S s = cos α ( cos 2 α m 2 n 2 + 2 | m | sin α ) 1 / 2 , sin α ( sin 2 α n 2 ) 1 / 2 < | m | < sin α + ( sin 2 α n 2 ) 1 / 2 .
sin θ = | m | + ( sin 2 α n 2 ) 1 / 2 ,
s R min = [ cos 2 α m 2 + n 2 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α , | m | < sin α ( sin 2 α n 2 ) 1 / 2 .
m 2 + n 2 + s 2 2 | m | sin θ 2 s cos θ = 0 ,
tan θ = | m | s
sin θ = | m | 1 ( 1 n 2 ) 1 / 2 ,
l = ( m 2 + s T 2 ) 1 / 2 = 1 ( 1 n 2 ) 1 / 2 , s T = [ ( 1 ( 1 n 2 ) 1 / 2 ) m 2 ] 1 / 2 .
1 ( 1 n 2 ) 1 / 2 < s < ( cos 2 α + n 2 ) 1 / 2 cos α .
c a illum T ( l , n ) = c ( l , n ) cos 1 / 2 γ ,
[ cos 2 α m 2 + n 2 + 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α > s > cos α [ cos 2 α m 2 n 2 + 2 | m | sin α ] 1 / 2 , sin α + ( sin 2 α n 2 ) 1 / 2 > | m | > sin α ( sin 2 α n 2 ) 1 / 2 ;
[ cos 2 α m 2 + n 2 + 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α > s > [ cos 2 α m 2 + n 2 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α , sin α ( sin 2 α n 2 ) 1 / 2 > | m | > ( sin 2 α n 2 ) 1 / 2 [ 1 ( 1 n 2 ) 1 / 2 ] ( 1 n 2 ) 1 / 2 ;
[ cos 2 α m 2 + n 2 + 2 | m | ( sin 2 α n 2 ) 1 / 2 ] 1 / 2 cos α > s > { [ 1 ( 1 n 2 ) 1 / 2 ] 2 m 2 } 1 / 2 , ( sin 2 α n 2 ) 1 / 2 [ 1 ( 1 n 2 ) 1 / 2 ] ( 1 n 2 ) 1 / 2 > | m | > 0.
c a obj R ( l , n ) = 2 [ ( l 2 + n 2 ) 2 ] 1 / 2 [ 4 l 2 ( l 2 + n 2 ) 2 ] 1 / 2 , 2 l > l 2 + n 2 > 4 sin 2 α 2 .
s T = ( 1 + 1 n 2 ) m 2 .
s R max = cos 2 α m 2 + n 2 + 2 | m | sin 2 α n 2 + cos α , s R min = cos 2 α m 2 + n 2 2 | m | sin 2 α n 2 + cos α .
S s = cos α + cos 2 α m 2 n 2 + 2 | m | sin α .

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