Abstract

We propose a three-dimensional phase contrast digital holographic microscopy. The object to be observed is a low-contrast transparent refractive index distribution sample, such as biological tissue. Low contrast phase objects are converted to high contrast images through the microscopy we propose. In order to gain high three-dimensional resolution, the direction of pump plane wave is scanned, and separate holographic images produced at each angle are acquired and decoded into complex amplitude in Fourier space. The three-dimensional image is reconstructed in a computer from all information acquired through the system. The resolution in the direction of the optical axis is increased by utilizing a 4π configuration of objective lenzes.

© 2007 Optical Society of America

Full Article  |  PDF Article

Errata

Naoki Fukutake and Tom D. Milster, "Proposal of three-dimensional phase contrast holographic microscopy: erratum," Opt. Express 16, 5964-5964 (2008)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-16-8-5964

References

  • View by:
  • |
  • |
  • |

  1. T. Wilson, Confocal Microscopy (Academic Press, 1990).
  2. W. B. Amos, J. G. White, and M. Fordham, "Use of confocal imaging in the study of biological structures," Appl. Opt. 26, 3239 (1987).
    [CrossRef] [PubMed]
  3. G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
    [PubMed]
  4. I. Freund and M. Deutsch, "2nd-harmonic microscopy of biological tissue," Opt. Lett. 11, 94 (1986).
    [CrossRef] [PubMed]
  5. P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
    [CrossRef] [PubMed]
  6. J. Mertz and L. Moreaux, "Second -harmonic generation by focused excitation of inhomogeneously distributed scatterers," Opt. Commun. 196, 325 (2001).
    [CrossRef]
  7. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, "Nonlinear scanning laser microscopy by third-harmonic generation," Appl. Phys. Lett. 70, 922 (1997).
    [CrossRef]
  8. M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
    [CrossRef] [PubMed]
  9. M. D. Duncan, J. Reintjes, and T. J. Manuccia, "Scanning coherent anti-Stokes Raman microscope," Opt. Lett. 7, 350 (1982).
    [CrossRef] [PubMed]
  10. A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).
  11. F. Zernike, "Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung," Z. Tech. Phys. 16, 454 (1935).
  12. F. Zernike, "How I discovered phase contrast," Science 121, 345 (1955).
    [CrossRef] [PubMed]
  13. W. S. Haddad, D. Cullen, J. C. Solem, J. W. Longworth, A. McPherson, K. Boyer, and C. K. Rhodes, "Fourier-transform holographic microscope," Appl. Opt. 31, 4973 (1992).
    [CrossRef] [PubMed]
  14. U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179 (1994).
    [CrossRef] [PubMed]
  15. J. H. Massig, "Digital off-axis holography with a synthetic aperture," Opt. Lett. 27, 2179 (2002).
    [CrossRef]
  16. S. Kostianovski, S. G. Lipson, and E. N. Ribak, "Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution," Appl. Opt. 32,4744 (1993).
    [CrossRef] [PubMed]
  17. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
    [CrossRef] [PubMed]
  18. T. Dresel, G. Hausler, and H. Venzke, "Three-dimensional sensing of rough surfaces by coherence radar," Appl. Opt. 31, 919 (1992).
    [CrossRef] [PubMed]
  19. M. Mansuripur, Classical Optics and its Applications (Cambridge University Press 2002)
  20. M. Born and E. Wolf, Principles of Optics 5th. ed., (Pergamon Press, 1974).
  21. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

2002

2001

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

J. Mertz and L. Moreaux, "Second -harmonic generation by focused excitation of inhomogeneously distributed scatterers," Opt. Commun. 196, 325 (2001).
[CrossRef]

1999

A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).

1998

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

1997

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

1994

1993

1992

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

1988

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

1987

1986

1982

1969

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

1955

F. Zernike, "How I discovered phase contrast," Science 121, 345 (1955).
[CrossRef] [PubMed]

1935

F. Zernike, "Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung," Z. Tech. Phys. 16, 454 (1935).

Amos, W. B.

Barad, Y.

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

Boyer, K.

Brakenhoff, G. J.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

Campagnola, P. J.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Clark, H. A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

Cullen, D.

Deutsch, M.

Dresel, T.

Duncan, M. D.

Eisenberg, H.

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

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Fordham, M.

Freund, I.

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Haddad, W. S.

Hausler, G.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Holtom, G. R.

A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).

Horowitz, M.

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

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Jüptner, W.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

Kostianovski, S.

Lewis, A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Lipson, S. G.

Loew, L. M.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

Longworth, J. W.

Manuccia, T. J.

Massig, J. H.

McPherson, A.

Mertz, J.

J. Mertz and L. Moreaux, "Second -harmonic generation by focused excitation of inhomogeneously distributed scatterers," Opt. Commun. 196, 325 (2001).
[CrossRef]

Mohler, W. A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

Moreaux, L.

J. Mertz and L. Moreaux, "Second -harmonic generation by focused excitation of inhomogeneously distributed scatterers," Opt. Commun. 196, 325 (2001).
[CrossRef]

Muller, M.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

Nanninga, N.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Reintjes, J.

Rhodes, C. K.

Ribak, E. N.

Schnars, U.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[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 (1997).
[CrossRef]

Solem, J. C.

Squier, J.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

van der Voort, H. T. M.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

van Spronsen, E. A.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

Venzke, H.

White, J. G.

Wilson, K. R.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

Xie, X. S.

A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).

Zernike, F.

F. Zernike, "How I discovered phase contrast," Science 121, 345 (1955).
[CrossRef] [PubMed]

F. Zernike, "Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung," Z. Tech. Phys. 16, 454 (1935).

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).

Appl. Opt.

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 (1997).
[CrossRef]

Bell Syst. Tech. J

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

J. Biomed. Opt.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, "Second -harmonic imaging microscopy of living cells," J. Biomed. Opt. 6, 277 (2001).
[CrossRef] [PubMed]

J. Microsc.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, "3D microscopy of transparent objects using third-harmonic generation," J. Microsc. 191, 266 (1998).
[CrossRef] [PubMed]

Opt. Commun.

J. Mertz and L. Moreaux, "Second -harmonic generation by focused excitation of inhomogeneously distributed scatterers," Opt. Commun. 196, 325 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

A. Zumbusch, G. R. Holtom, and X. S. Xie, "Vibrational microscopy using coherent anti-Stokes Raman scattering," Phys. Rev. Lett. 82, 4014 (1999).

Scanning Microsc.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronsen, and N. Nanninga, "3-Dimensional imaging of biological structures by high resolution confocal scanning laser microscopy," Scanning Microsc. 2, 33 (1988).
[PubMed]

Science

F. Zernike, "How I discovered phase contrast," Science 121, 345 (1955).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science  254, 1178 (1991).
[CrossRef] [PubMed]

Z. Tech. Phys.

F. Zernike, "Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung," Z. Tech. Phys. 16, 454 (1935).

Other

T. Wilson, Confocal Microscopy (Academic Press, 1990).

M. Mansuripur, Classical Optics and its Applications (Cambridge University Press 2002)

M. Born and E. Wolf, Principles of Optics 5th. ed., (Pergamon Press, 1974).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Schematic of the setup of the 3-D phase contrast holographic microscopy. BS: Beam splitter. λ/2 : Half wave plate. λ/4 : Quarter wave plate. PBS: Polarized beam splitter.

Fig. 2.
Fig. 2.

Description of the position of the 0-order light and the circular area of the scattered wave on the observation plane. The circular area shifts during the scanning of the pump direction. On the other hand, the arrival position of the 0-order light is fixed at the center.

Fig. 3.
Fig. 3.

Schematic of the sample holder and the mechanical aperture between the two Fourier lenses facing each other. The Fourier lens satisfies the condition of h = F sin θ.

Fig. 4.
Fig. 4.

Description of the digital hologram and the generation process of a single “twin partial sphere” in 3-D matrix. The circular parts of the complex amplitude generated by the digital hologram are projected onto the partial spheres in the frequency space.

Fig. 5.
Fig. 5.

Positional relation among the partial spheres corresponding to each pump direction.

Fig. 6.
Fig. 6.

3-D entrance pupil function composed of the two pupil functions on the transmission and reflection sides. The pupil function is considered to be in the frequency space.

Fig. 7.
Fig. 7.

Optical transfer function in the case where the primary objective with NA =1.2 in water is used. The OTF has rotational symmetry in the fz direction.

Fig. 8.
Fig. 8.

Error in object reconstruction in the case of the test sample (a cube of 3.0 μ m in size) where the wavelength of 850nm is used.

Equations (80)

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

NA · F = μN 4 β ,
E x t = E ( i ) x t + rot x rot x α ( x ) N ( x ) E ( x , t x x n 0 c ) x x d 3 x ,
α ( x ) N ( x ) = 3 4 π { n ( x ) n 0 } 2 1 { n ( x ) n 0 } 2 + 2 ε 2 π n 0 g ( x ) .
E ( x , t x x n 0 c ) = E 0 ( x ) exp [ i n 0 k 0 x x ] e iωt ,
E r E θ E φ = 0 α ( x ) N ( x ) sin θ n 0 2 k 0 2 E 0 ( x ) exp [ i k 0 x x ] x x 0 e iωt ,
E s x x t = α ( x ) N ( x ) n 0 2 k 0 2 E 0 ( x ) exp [ i k 0 x x ] x x e iωt .
E 0 ( x ) = E 0 ( i ) ( x ) + ε k 0 2 n 0 2 π g ( x ) E 0 ( x ) exp [ i n 0 k 0 x x ] x x d 3 x .
E 0 ( x ) = E 0 ( i ) ( x ) + ε k 0 2 n 0 2 π g ( x ) E 0 ( i ) ( x ) exp [ i n 0 k 0 x x ] x x d 3 x
= exp [ i n 0 k 0 · x ] + ε k 0 2 n 0 2 π g ( x ) exp [ i n 0 k 0 · x ] exp [ i n 0 k 0 x x ] x x d 3 x .
E 0 ( x ) = exp [ i n 0 k 0 · x ] + F 0 ( x ) ,
F 0 ( x ) = n 0 4 π ε k 0 2 exp [ i κ · x ] exp [ i n 0 k 0 · x ] exp [ i n 0 k 0 x x ] x x d 3 x
= n 0 4 π ε k 0 2 exp [ i { ( n 0 k x + κ x ) x + ( n 0 k y + κ y ) y + ( n 0 k z + κ z ) z } ] exp [ i n 0 k 0 R ] R d 3 x ,
x 2 + y 2 + z 2 x′ 2 + y′ 2 + z′ 2
x x + y y + z z x 2 + y 2 + z 2 ,
R x 2 + y 2 + z 2 x x + y y + z z x 2 + y 2 + z 2 + x 2 + y 2 + z 2 2 x 2 + y 2 + z 2 .
F 0 ( x ) = n 0 4 π ε k 0 2 exp [ i n 0 k 0 r ] r′ exp [ i ( n 0 k x + κ x n 0 k x ) x ] exp [ i n 0 k 0 2 r x 2 ] d x
× exp [ i ( n 0 k y + κ y n 0 k y ) y ] exp [ i n 0 k 0 2 r y 2 ] d y
× L 2 L 2 exp [ i ( n 0 k z + κ z n 0 k z ) z ] exp [ i n 0 k 0 2 r z 2 ] d z ,
J x = lim r r r exp [ i ax ] exp [ i b x 2 ] d x ,
J x = lim r β + 0 r′ r′ exp [ i ax ] exp [ ib x 2 ] exp [ β x 2 ] d x
= lim r β + 0 exp [ a 2 4 ( β ib ) ] r′ r′ exp [ ( β ib x ia 2 β ib ) 2 ] d x .
J x = lim r β + 0 exp [ a 2 4 ( β ib ) ] 1 β ib c exp [ z 2 ] d z ,
J x = lim r exp [ a 2 4 ( β ib ) ] 1 β ib r′ r′ exp [ x 2 ] d x
= lim r π β ib exp [ i 4 b a 2 ] exp [ β 4 b 2 a 2 ]
= lim γ i π b exp [ i 4 b a 2 ] exp [ γ a 2 ] .
J x = lim γ i π 2 r n 0 k 0 exp [ i r ( n 0 k x + κ x n 0 k x ) 2 2 n 0 k 0 ] exp [ γ ( n 0 k x + κ x n 0 k x ) 2 ] ,
J y = lim γ i π 2 r n 0 k 0 exp [ i r′ ( n 0 k y + κ y n 0 k y ) 2 2 n 0 k 0 ] exp [ γ ( n 0 k y + κ y n 0 k y ) 2 ] .
J z = L sin c [ L 2 ( n 0 k z + κ z n 0 k z ] ,
F 0 ( r ) = lim γ k 0 2 exp [ i n 0 k 0 r′ ] exp [ i r′ 2 n 0 k 0 { ( n 0 k x + κ x n 0 k x ) 2 + ( n 0 k y + κ y n 0 k y ) 2 } ]
× exp [ γ { ( n 0 k x + κ x n 0 k x ) 2 + ( n 0 k y + κ y n 0 k y ) 2 } ]
× L sin c [ L 2 ( n 0 k z + κ z n 0 k z ) ]
= iπεL λ exp [ i n 0 k 0 r′ ] sin c [ L 2 ( n 0 k z + κ z n 0 k z ) ] , ( as n 0 k x + κ x n 0 k x = 0 , n 0 k y + κ y n 0 k y = 0 ) 0 , ( others ) .
η = ( πεL λ ) 2
D = λN 4 NA = Fλβ μ ,
P T ( f ) = { 1 , ( on shell of transmission side ) 0 , ( others ) .
P R ( f ) = { 1 , ( on shell of reflection side ) 0 , ( others ) .
F ( f , f 0 ) = { O ( x ) e i 2 π f · x d x a δ ( f ) } { P T ( f + f 0 ) + P R ( f + f 0 ) } P T * ( f 0 ) ,
I I ( x′ ) = F ( f , f 0 ) e i 2 π f · x d f 2 d f 0
= [ { O ( x 1 ) e i 2 π f 1 · x 1 d x 1 ( f 1 ) } { P T ( f 1 + f 0 ) + P R ( f 1 + f 0 ) } P T * ( f 0 ) e i 2 π f 1 · x′ d f 1 ]
× [ { O * ( x 2 ) e i 2 π f 2 · x 2 d x 2 ( f 2 ) } { P T * ( f 2 + f 0 ) + P R * ( f 2 + f 0 ) } P T ( f 0 ) e i 2 π f 2 · x′ d f 2 ] d f 0
= [ O ( x 1 ) P T * ( f 0 ) e i 2 π f 0 · ( x x 1 ) { U T ( x x 1 ) + U R ( x x 1 ) } d x 1 a { P T ( f 0 ) + P R ( f 0 ) } P T * ( f 0 ) ]
× [ O * ( x 2 ) P T ( f 0 ) e i 2 π f 0 · ( x x 2 ) { U T * ( x x 2 ) + U R * ( x x 2 ) } d x 2 a { P T * ( f 0 ) + P R * ( f 0 ) } P T ( f 0 ) ] d f 0 ,
U T ( x ) = P T ( f ) e i 2 π f · x d f
U R ( x ) = P R ( f ) e i 2 π f · x d f .
U T ( x ) = U R * ( x )
U T ( x ) = U T * ( x )
U R ( x ) = U R * ( x ) .
I I ( x ) = γ ( x 1 x 2 ) O ( x 1 ) O * ( x 2 ) { U T ( x x 1 ) + U R ( x x 1 ) } { U T * ( x x 2 ) + U R * ( x x 2 ) } d x 1 d x 2
a O ( x 1 ) U R ( x x 1 ) { U T ( x x 1 ) + U R ( x x 1 ) } d x 1
a O * ( x 2 ) U T ( x x 2 ) { U T * ( x x 2 ) + U R * ( x x 2 ) } d x 2
+ a 2 P T ( f 0 ) 4 d f 0 ,
γ ( x 1 x 2 ) = P T ( f 0 ) 2 e i 2 π f 0 · ( x 1 x 2 ) d f 0 ,
I I ( x ) = { 1 + ε 0 o ( x ) + ε 0 * o * ( x ) } U T ( x x ) 2 d x
+ ε 0 o ( x ) U R 2 ( x x ) d x + ε 0 * o * ( x ) U T 2 ( x x ) d x
a { 1 + ε 0 o ( x ) } { U T ( x x ) 2 + U R 2 ( x x ) } d x
a { 1 + ε 0 * o * ( x ) } { U T ( x x ) 2 + U T 2 ( x x ) } d x
+ a 2 U T ( x x ) 2 d x ,
I I ( x ) = ( 1 a ) ( 1 a ) U T ( x x ) 2 d x
+ ε 0 o ( x ) { U T ( x x ) 2 + U R 2 ( x x ) } d x
+ ε 0 * o * ( x ) { U T ( x x ) 2 + U T 2 ( x x ) } d x ]
= ( 1 a ) U T ( x ) 2 d x [ ( 1 a )
+ ε 0 o ˜ ( f ) OTF ( f ) e i 2 π f · x d f
+ ε 0 * o ˜ * ( f ) OTF * ( f ) e i 2 π f x d f ] ,
OTF ( f ) = { U T ( x ) 2 + U R 2 ( x ) } e i 2 π f · x d x U T ( x ) 2 d x ,
OTF ( f ) = P ( f ) P T * ( f f ) d f P T ( f ) 2 d f .
I I ( x ) = ( 1 a ) 2 U T ( x x ) 2 d x + ( 1 a ) ε 0 o ( x ) { 2 U T ( x x ) 2 + U R 2 ( x x ) + U T 2 ( x x ) } d x
= ( 1 a ) { ( 1 a ) + 2 ε 0 o ( x ) } PSF ( x x ) / 2 d x ,
PSF ( x x ) = 2 U T ( x x ) 2 + U R 2 ( x x ) + U T 2 ( x x )
= U ( x x ) 2
I I ( x ) = ( 1 a ) { O a ( x ) 2 PSF ( x x ) / 2 d x } .
I II ( x ) = { F f f 0 d f 0 } e i 2 π f · x d f 2
= [ { O ( x ) e i 2 π f · x d x ( f ) } { P T ( f + f 0 ) + P R ( f + f 0 ) } P T * ( f 0 ) d f 0 ] e i 2 π f · x d f 2
= O ( x ) U R ( x x ) { U T ( x x ) + U R ( x x ) } d x a P T ( f 0 ) 2 d f 0 2
= O ( x ) { U T ( x x ) 2 + U R 2 ( x x ) } d x a U T ( x x ) 2 d x 2 ,
I II ( x ) = { 1 + ε 0 o ( x ) } { U T ( x x ) 2 + U R 2 ( x x ) } d x a U T ( x x ) 2 d x 2
= A ( 1 a ) [ ( 1 a ) U T ( x x ) 2 d x
+ ε 0 o ( x ) { U T ( x x ) 2 + U R 2 ( x x ) } d x
+ ε 0 * o * ( x ) { U T ( x x ) 2 + U T 2 ( x x ) } d x ]
= A I I ( x )
RMS ( σ ) = i = 1 N ( I i σ I i 0 ) 2 N ,

Metrics