Abstract

In a 4Pi confocal fluorescence microscope two opposing microscope objective lenses were used to illuminate a fluorescent object from both sides and to collect the fluorescence emissions on both sides. Constructive interference of either the illumination wave fronts in the common focus or the detection wave fronts in the common detector pinhole resulted in an axial resolution approximately four times higher than that in a confocal fluorescence microscope. A precise 4Pi confocal fluorescence microscope that uses simultaneous illumination was built. The full width at half-maximum of the depth discrimination was determined experimentally to be approximately 110 nm when lenses with a numerical aperture of 1.4, an excitation of 633 nm, and detection of approximately 725 nm were used.

© 1992 Optical Society of America

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References

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  1. T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990).
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  3. M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).
  4. D. Gerlach, Das Lichtmikroskop (Georg Thieme Verlag, Stuttgart, 1976).
  5. S. Hell, German patent applicationP40 40 441.2 (filed December18, 1990; published 1992).
  6. S. Hell, “The physical basis of confocal fluorescence microscopy,” presented as part of the Scandinavian Course in Modern Image Analysis Technology, Jyvaskyla, Finland;Solubiologi 3, 183–185 (1991).
  7. S. Hell, E. H. K. Stelzer, “A 4Pi confocal microscope has an improved axial resolution,” presented at the 4th International Conference on Confocal Microscopy, Amsterdam, The Netherlands, 1992.
  8. C. J. R. Sheppard, Y. Gong, “Improvement in axial resolution by interference confocal microscopy,” Optik 87, 129–132 (1991).
  9. K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlöf, N. Åslund, “Three-dimensional microscopy using a confocal laser scanning microscope,” Opt. Lett. 10, 53–55 (1985).
    [CrossRef] [PubMed]
  10. R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
    [CrossRef]
  11. G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
    [CrossRef]
  12. C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy,T. Wilson, ed. (Academic, London, 1990).
    [CrossRef]
  13. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  14. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  15. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field.” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  16. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–368 (1959).
    [CrossRef]
  17. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  18. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, England, 1986), p. 468.
  19. H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in high-aperture fluorescence confocal microscopy: a numerical analysis,” J. Microsc. (Oxford) Pt.1 158, 43–54, (1990).
    [CrossRef]
  20. S. Hell, E. Lehtonen, E. H. K. Stelzer, “Confocal fluorescence microscopy: wave optics considerations and applications to cell biology,” in New Dimensions of Visualization in Biomedical Microscopies,A. Kriete, ed.(Verlag Chemie, Weinheim, Germany, 1992).
  21. T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. (Oxford) 154, 143–156 (1989).
    [CrossRef]
  22. E. H. K. Stelzer, R. W. Wijnaendts-van-Resandt, “Optical cell slicing with the confocal fluorescence microscope: microtomoscopy,” in Confocal Microscopy,T. Wilson, ed., (Academic, London, 1990).
  23. S. Hell, “Abbildung transparenter Mikrostrukturen im konfokalen Mikroskop,” Ph.D. dissertation (Universitat Heidelberg, Heidelberg, Germany, 1990).
  24. W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
    [CrossRef] [PubMed]
  25. S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal microscope using two-photon excitation,” Opt. Commun. (to be published).

1991

C. J. R. Sheppard, Y. Gong, “Improvement in axial resolution by interference confocal microscopy,” Optik 87, 129–132 (1991).

1990

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

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in high-aperture fluorescence confocal microscopy: a numerical analysis,” J. Microsc. (Oxford) Pt.1 158, 43–54, (1990).
[CrossRef]

1989

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. (Oxford) 154, 143–156 (1989).
[CrossRef]

1987

1986

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

1985

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlöf, N. Åslund, “Three-dimensional microscopy using a confocal laser scanning microscope,” Opt. Lett. 10, 53–55 (1985).
[CrossRef] [PubMed]

1967

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field.” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–368 (1959).
[CrossRef]

Åslund, N.

Boivin, A.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Brakenhoff, G. J.

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in high-aperture fluorescence confocal microscopy: a numerical analysis,” J. Microsc. (Oxford) Pt.1 158, 43–54, (1990).
[CrossRef]

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

Carlsson, K.

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy,T. Wilson, ed. (Academic, London, 1990).
[CrossRef]

Danielsson, P. E.

Davoust, J.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Denk, W.

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

Dow, J.

Furtak, T. E.

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

Gerlach, D.

D. Gerlach, Das Lichtmikroskop (Georg Thieme Verlag, Stuttgart, 1976).

Gong, Y.

C. J. R. Sheppard, Y. Gong, “Improvement in axial resolution by interference confocal microscopy,” Optik 87, 129–132 (1991).

Hell, S.

S. Hell, “Abbildung transparenter Mikrostrukturen im konfokalen Mikroskop,” Ph.D. dissertation (Universitat Heidelberg, Heidelberg, Germany, 1990).

S. Hell, “The physical basis of confocal fluorescence microscopy,” presented as part of the Scandinavian Course in Modern Image Analysis Technology, Jyvaskyla, Finland;Solubiologi 3, 183–185 (1991).

S. Hell, E. Lehtonen, E. H. K. Stelzer, “Confocal fluorescence microscopy: wave optics considerations and applications to cell biology,” in New Dimensions of Visualization in Biomedical Microscopies,A. Kriete, ed.(Verlag Chemie, Weinheim, Germany, 1992).

S. Hell, German patent applicationP40 40 441.2 (filed December18, 1990; published 1992).

S. Hell, E. H. K. Stelzer, “A 4Pi confocal microscope has an improved axial resolution,” presented at the 4th International Conference on Confocal Microscopy, Amsterdam, The Netherlands, 1992.

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal microscope using two-photon excitation,” Opt. Commun. (to be published).

Klein, M. V.

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

Lehtonen, E.

S. Hell, E. Lehtonen, E. H. K. Stelzer, “Confocal fluorescence microscopy: wave optics considerations and applications to cell biology,” in New Dimensions of Visualization in Biomedical Microscopies,A. Kriete, ed.(Verlag Chemie, Weinheim, Germany, 1992).

Lenz, R.

Liljeborg, A.

Majlöf, L.

Marsman, H. J. B.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Matthews, H. J.

Nanninga, H.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field.” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–368 (1959).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, Y. Gong, “Improvement in axial resolution by interference confocal microscopy,” Optik 87, 129–132 (1991).

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy,T. Wilson, ed. (Academic, London, 1990).
[CrossRef]

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

Spronsen, E. A.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, England, 1986), p. 468.

Stelzer, E. H. K.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

S. Hell, E. Lehtonen, E. H. K. Stelzer, “Confocal fluorescence microscopy: wave optics considerations and applications to cell biology,” in New Dimensions of Visualization in Biomedical Microscopies,A. Kriete, ed.(Verlag Chemie, Weinheim, Germany, 1992).

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal microscope using two-photon excitation,” Opt. Commun. (to be published).

S. Hell, E. H. K. Stelzer, “A 4Pi confocal microscope has an improved axial resolution,” presented at the 4th International Conference on Confocal Microscopy, Amsterdam, The Netherlands, 1992.

E. H. K. Stelzer, R. W. Wijnaendts-van-Resandt, “Optical cell slicing with the confocal fluorescence microscope: microtomoscopy,” in Confocal Microscopy,T. Wilson, ed., (Academic, London, 1990).

Strickler, J. H.

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

Strieker, R.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

van der Voort, H. T. M.

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in high-aperture fluorescence confocal microscopy: a numerical analysis,” J. Microsc. (Oxford) Pt.1 158, 43–54, (1990).
[CrossRef]

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

Webb, W. W.

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

Wijnaendts-van-Resandt, R. W.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

E. H. K. Stelzer, R. W. Wijnaendts-van-Resandt, “Optical cell slicing with the confocal fluorescence microscope: microtomoscopy,” in Confocal Microscopy,T. Wilson, ed., (Academic, London, 1990).

Wilson, T.

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. (Oxford) 154, 143–156 (1989).
[CrossRef]

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

T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990).

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–368 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field.” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Ann. N. Y. Acad. Sci.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. Spronsen, H. Nanninga, “Three-dimensional imaging by confocal scanning microscopy,” Ann. N. Y. Acad. Sci. 483, 405–415 (1986).
[CrossRef]

J. Microsc.

R. W. Wijnaendts-van-Resandt, H. J. B. Marsman, J. Davoust, E. H. K. Stelzer, R. Strieker, “Optical Fluorescence microscopy in three dimensions,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

J. Microsc. (Oxford)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. (Oxford) 154, 143–156 (1989).
[CrossRef]

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in high-aperture fluorescence confocal microscopy: a numerical analysis,” J. Microsc. (Oxford) Pt.1 158, 43–54, (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Optik

C. J. R. Sheppard, Y. Gong, “Improvement in axial resolution by interference confocal microscopy,” Optik 87, 129–132 (1991).

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, I. An integral representation of the image field.” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems, II. Structure of the image field in aplanatic systems,” Proc. R. Soc. London Ser. A 253, 358–368 (1959).
[CrossRef]

Science

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

Other

S. Hell, E. H. K. Stelzer, “Fundamental improvement of resolution with a 4Pi-confocal microscope using two-photon excitation,” Opt. Commun. (to be published).

E. H. K. Stelzer, R. W. Wijnaendts-van-Resandt, “Optical cell slicing with the confocal fluorescence microscope: microtomoscopy,” in Confocal Microscopy,T. Wilson, ed., (Academic, London, 1990).

S. Hell, “Abbildung transparenter Mikrostrukturen im konfokalen Mikroskop,” Ph.D. dissertation (Universitat Heidelberg, Heidelberg, Germany, 1990).

C. J. R. Sheppard, C. J. Cogswell, “Three-dimensional imaging in confocal microscopy,” in Confocal Microscopy,T. Wilson, ed. (Academic, London, 1990).
[CrossRef]

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

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, England, 1986), p. 468.

S. Hell, E. Lehtonen, E. H. K. Stelzer, “Confocal fluorescence microscopy: wave optics considerations and applications to cell biology,” in New Dimensions of Visualization in Biomedical Microscopies,A. Kriete, ed.(Verlag Chemie, Weinheim, Germany, 1992).

T. Wilson, “Confocal microscopy,” in Confocal Microscopy, T. Wilson, ed. (Academic, London, 1990).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

M. V. Klein, T. E. Furtak, Optics, 2nd ed. (Wiley, New York, 1986).

D. Gerlach, Das Lichtmikroskop (Georg Thieme Verlag, Stuttgart, 1976).

S. Hell, German patent applicationP40 40 441.2 (filed December18, 1990; published 1992).

S. Hell, “The physical basis of confocal fluorescence microscopy,” presented as part of the Scandinavian Course in Modern Image Analysis Technology, Jyvaskyla, Finland;Solubiologi 3, 183–185 (1991).

S. Hell, E. H. K. Stelzer, “A 4Pi confocal microscope has an improved axial resolution,” presented at the 4th International Conference on Confocal Microscopy, Amsterdam, The Netherlands, 1992.

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Figures (13)

Fig. 1
Fig. 1

Aperture angle in microscopy. In the usual epifluorescence arrangement, always less than half of 4π is used to illuminate the point of interest and to detect the emitted light.

Fig. 2
Fig. 2

Contour plot of the point-spread function for linearly polarized illumination. The horizontal axis lies in the focal plane and is perpendicular to the direction of vibration of the incident electric field. The numerical aperture of the oil (n = 1.518) immersion lens is 1.4; the wavelength is 633 nm. The point-spread function is normalized to unity. The contour lines drop as follows: 0.9, 0.7, 0.5, 0.2, 0.07, 0.03, 0.015, and 0.005. They are indicated by decreasing line thicknesses. This applies to all contour plots shown in this paper.

Fig. 3
Fig. 3

Contour plot of the detection point-spread function. The numerical aperture of the lens is 1.4, the detection wavelength is 725 nm, and the index of refraction is n = 1.518.

Fig. 4
Fig. 4

Contour plot of the confocal point-spread function. It is the product of the functions shown in Figs. 2 and 3.

Fig. 5
Fig. 5

Contour plot of the illumination point-spread function in a type-A 4Pi confocal fluorescence microscope. The two illumination wave fronts interfere constructively in the common focus. Both lenses have a numerical aperture of 1.4; the wavelength is 633 nm. The wave fronts are linearly polarized, and the x axis is perpendicular to the plane of vibration of the incident electric field.

Fig. 6
Fig. 6

Contour plot of the confocal point-spread function in a type-A 4Pi confocal fluorescence microscope. This point-spread function is calculated by multiplying the point-spread function for the normal detection (Fig. 3) and the point-spread function shown in Fig. 5.

Fig. 7
Fig. 7

Contour plot of the detection-intensity point-spread function in a type-B or type-C 4Pi confocal,fluorescence microscope.

Fig. 8
Fig. 8

Contour plot of the confocal point-spread function in a type-C 4Pi confocal fluorescence microscope. This point-spread function is calculated by multiplying the point-spread functions shown in Figs. 5 and 7.

Fig. 9
Fig. 9

Theoretical (z)responses to an infinitely thin fluorescent layer: (a) the response of a confocal fluorescence microscope, (b) the response of a type-A 4Pi confocal fluorescence microscope, (c) the response of a type-C 4Pi confocal fluorescence microscope. In all three cases, objective lenses with an N.A. of 1.4, an excitation wavelength of 633 nm, and an emission wavelength of 725 nm were assumed.

Fig. 10
Fig. 10

Theoretical response to an edge along the optical axis: (a) the response of a confocal fluorescence microscope, (b) the response of a type-A 4Pi confocal fluorescence microscope, (c) the response of a type-C 4Pi confocal fluorescence microscope.

Fig. 11
Fig. 11

Schematic drawing of the 4Pi confocal fluorescence microscope at the European Molecular Biology Laboratory.

Fig. 12
Fig. 12

Experimental response of a type-A 4Pi confocal fluorescence microscope to a 590-nm-thick fluorescent layer. The intensity is normalized to 100. The distance along the optical axis (z) is given in micrometers.

Fig. 13
Fig. 13

First derivative of the experimental responses to a fluorescent layer shown in Fig. 12. (a) The z responses of the type-A 4Pi confocal fluorescence microscope and the z responses of the confocal fluorescence microscope (envelope), (b) the z response shown in the same scale as the theoretical predictions for this experiment.

Tables (1)

Tables Icon

Table 1 Summary of Theoretical Predictiona

Equations (10)

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

E = ( e x , e y , e z ) ,
e x ( r ) = i ( I 0 + I 2 cos 2 ϕ ) , e y ( r ) = i I 2 sin 2 ϕ , e x ( r ) = 2 I 1 cos ϕ .
I = h ill = | E | 2 = | I 0 | 2 + 4 | I 1 | 2 cos 2 ϕ + | I 2 | 2 + 2 cos 2 ϕ Re ( I 0 I 2 * ) .
h det = | I 0 | 2 + 2 | I 1 | 2 + | I 2 | 2 .
H = h ill h det .
H 4 Pi , ill = | E 1 , ill + E 2 , ill | 2 | E det | 2 .
H 4 Pi , ill = | E ill | 2 | E 1 , det + E 2 , det | 2 .
H 4 Pi , 4 Pi = | E 1 , ill + E 2 , ill | 2 | E 1 , det + E 2 , det | 2 .
I layer ( z ) = h ill ( x , y , z ) h det ( x , y , z ) d x d y
I edge ( z ) = z I layer ( z ) d z

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