Abstract

The real image of a line object, located in water of refractive index n w and recorded on an in-line Fraunhofer hologram, is calculated by use of the Huygens–Fresnel principle. The presence of the water–glass and glass–air interfaces or the change in effective wavelength between recording and replay introduce wave-front aberrations. Spherical aberration dominates for a perfectly aligned finite-aperture hologram, and its effect on the replayed image of a finite-width line object is evaluated. Numerical results are compared with experimental data of a 10-μm wire located in water 50.0 mm from a 10-mm-thick glass window, and good agreement is demonstrated. It is shown that the error on the linewidth is less than 1.5%, and the shift in focal plane from the Gaussian plane is less than 16 μm, for a replay-to-recording wavelength ratio μ in the range 0.98 < μn w < 1.02.

© 1998 Optical Society of America

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References

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  1. C. S. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, UK, 1992).
    [CrossRef]
  2. C. S. Vikram, “Holographic metrology of micro-objects in a dynamic volume,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997).
  3. J. Watson, “Hologrammetry and its applications,” J. Imaging Technol. 15, 38–46 (1984).
  4. J. Watson, “Underwater visual inspection and measurement using optical holography,” Opt. Lasers Eng. 16, 375–390 (1992).
    [CrossRef]
  5. E. Foster, J. Watson, “Holography for underwater inspection and measurement: an overview of current work,” Opt. Laser Technol. 29, 17–23 (1997).
    [CrossRef]
  6. C. Knox, “Holographic microscopy as a technique for recording microscopic subject,” Science 153, 989–990 (1966).
    [CrossRef] [PubMed]
  7. K. L. Carder, “Holographic microvelocimeter for use in studying ocean particle dynamics,” Opt. Eng. 18, 524–525 (1979).
    [CrossRef]
  8. K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
    [CrossRef]
  9. P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
    [CrossRef]
  10. C. Özkul, “Effects of finite aperture on linewidth measurement using in-line Fraunhofer holography,” Opt. Laser Technol. 18, 36–38 (1986).
    [CrossRef]
  11. R. A. Belz, F. M. Shofner, “Characteristics and measurements of an aperture-limited in-line hologram image,” Appl. Opt. 11, 2215–2221 (1972).
    [CrossRef] [PubMed]
  12. P. R. Hobson, A. Raouf, “Finite aperture Fraunhofer hologram of two co-planar discs,” J. Mod. Opt. 39, 807–824 (1992).
    [CrossRef]
  13. J. N. Latta, “Computer-based analysis of hologram imaging and aberrations. I. Hologram types and their monochromatic aberrations,” Appl. Opt. 10, 599–608 (1971).
    [CrossRef] [PubMed]
  14. J. N. Latta, “Fifth-order hologram aberrations,” Appl. Opt. 10, 666–667 (1971).
    [CrossRef] [PubMed]
  15. C. S. Vikram, M. L. Billet, “Aberration limited resolution in Fraunhofer holography with collimated beams,” Opt. Laser Technol. 21, 185–187 (1989).
    [CrossRef]
  16. C. S. Vikram, “Resolution limits due to primary aberrations in Fraunhofer holography,” Optik 99, 29–31 (1995).
  17. C. S. Vikram, “Rayleigh versus Marechal spherical aberration tolerance in in-line Fraunhofer holography,” Opt. Eng. 33, 3715–3717 (1994).
    [CrossRef]
  18. J. Nowak, M. Zajac, “Investigation of the influence of hologram aberrations on the light intensity distribution in the image plane,” Opt. Acta 30, 1749–1767 (1983).
    [CrossRef]
  19. I. Banyasz, “Resolution problems in holography,” in International Colloquium on Diffractive Optical Elements, J. Nowak, M. Zajac, eds., Proc. SPIE1574, 282–293 (1991).
    [CrossRef]
  20. J. N. Latta, “Computer-based analysis of hologram imagery and aberrations. II: Aberrations induced by a wavelength shift,” Appl. Opt. 10, 609–618 (1971).
    [CrossRef] [PubMed]
  21. J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: aberrations in the real image of an underwater object when replayed in air,” J. Phys. D 21, 1701–1705 (1988).
    [CrossRef]
  22. J. Watson, J. M. Kilpatrick, “Optical aberrations in underwater holography and their compensation,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 245–253 (1991).
  23. J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: reduction of aberrations by index compensation,” J. Phys. D 26, 177–182 (1993).
    [CrossRef]
  24. G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
    [CrossRef]
  25. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970), p. 40.
  26. Ref. 25, pp. 211–224.
  27. R. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am 55, 987–992 (1965).
  28. C. S. Vikram, M. L. Billet, “In-line Fraunhofer holography at a few far fields,” Appl. Opt. 23, 3091–3094 (1984).
    [CrossRef] [PubMed]
  29. C. Özkul, “Nonlinearities of an aperture-limited in-line far-field hologram,” Appl. Opt. 25, 3924–3926 (1986).
    [CrossRef] [PubMed]
  30. P. C. Mehta, “Fifth-order aberrations in in-line holograms,” Opt. Acta 21, 1005–1008 (1974).
    [CrossRef]
  31. P. Dunn, B. J. Thompson, “Object shape, fringe visibility, and resolution in far-field holography,” Opt. Eng. 21, 327–332 (1982).
    [CrossRef]

1997 (2)

E. Foster, J. Watson, “Holography for underwater inspection and measurement: an overview of current work,” Opt. Laser Technol. 29, 17–23 (1997).
[CrossRef]

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

1995 (1)

C. S. Vikram, “Resolution limits due to primary aberrations in Fraunhofer holography,” Optik 99, 29–31 (1995).

1994 (1)

C. S. Vikram, “Rayleigh versus Marechal spherical aberration tolerance in in-line Fraunhofer holography,” Opt. Eng. 33, 3715–3717 (1994).
[CrossRef]

1993 (1)

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: reduction of aberrations by index compensation,” J. Phys. D 26, 177–182 (1993).
[CrossRef]

1992 (2)

P. R. Hobson, A. Raouf, “Finite aperture Fraunhofer hologram of two co-planar discs,” J. Mod. Opt. 39, 807–824 (1992).
[CrossRef]

J. Watson, “Underwater visual inspection and measurement using optical holography,” Opt. Lasers Eng. 16, 375–390 (1992).
[CrossRef]

1989 (1)

C. S. Vikram, M. L. Billet, “Aberration limited resolution in Fraunhofer holography with collimated beams,” Opt. Laser Technol. 21, 185–187 (1989).
[CrossRef]

1988 (1)

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: aberrations in the real image of an underwater object when replayed in air,” J. Phys. D 21, 1701–1705 (1988).
[CrossRef]

1986 (2)

C. Özkul, “Effects of finite aperture on linewidth measurement using in-line Fraunhofer holography,” Opt. Laser Technol. 18, 36–38 (1986).
[CrossRef]

C. Özkul, “Nonlinearities of an aperture-limited in-line far-field hologram,” Appl. Opt. 25, 3924–3926 (1986).
[CrossRef] [PubMed]

1984 (2)

C. S. Vikram, M. L. Billet, “In-line Fraunhofer holography at a few far fields,” Appl. Opt. 23, 3091–3094 (1984).
[CrossRef] [PubMed]

J. Watson, “Hologrammetry and its applications,” J. Imaging Technol. 15, 38–46 (1984).

1983 (1)

J. Nowak, M. Zajac, “Investigation of the influence of hologram aberrations on the light intensity distribution in the image plane,” Opt. Acta 30, 1749–1767 (1983).
[CrossRef]

1982 (2)

K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
[CrossRef]

P. Dunn, B. J. Thompson, “Object shape, fringe visibility, and resolution in far-field holography,” Opt. Eng. 21, 327–332 (1982).
[CrossRef]

1979 (1)

K. L. Carder, “Holographic microvelocimeter for use in studying ocean particle dynamics,” Opt. Eng. 18, 524–525 (1979).
[CrossRef]

1976 (1)

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

1974 (1)

P. C. Mehta, “Fifth-order aberrations in in-line holograms,” Opt. Acta 21, 1005–1008 (1974).
[CrossRef]

1972 (1)

1971 (3)

1966 (1)

C. Knox, “Holographic microscopy as a technique for recording microscopic subject,” Science 153, 989–990 (1966).
[CrossRef] [PubMed]

1965 (1)

R. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am 55, 987–992 (1965).

Banyasz, I.

I. Banyasz, “Resolution problems in holography,” in International Colloquium on Diffractive Optical Elements, J. Nowak, M. Zajac, eds., Proc. SPIE1574, 282–293 (1991).
[CrossRef]

Belz, R. A.

Betzer, P. R.

K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
[CrossRef]

Billet, M. L.

C. S. Vikram, M. L. Billet, “Aberration limited resolution in Fraunhofer holography with collimated beams,” Opt. Laser Technol. 21, 185–187 (1989).
[CrossRef]

C. S. Vikram, M. L. Billet, “In-line Fraunhofer holography at a few far fields,” Appl. Opt. 23, 3091–3094 (1984).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970), p. 40.

Carder, K. L.

K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
[CrossRef]

K. L. Carder, “Holographic microvelocimeter for use in studying ocean particle dynamics,” Opt. Eng. 18, 524–525 (1979).
[CrossRef]

Dunn, P.

P. Dunn, B. J. Thompson, “Object shape, fringe visibility, and resolution in far-field holography,” Opt. Eng. 21, 327–332 (1982).
[CrossRef]

Foster, E.

E. Foster, J. Watson, “Holography for underwater inspection and measurement: an overview of current work,” Opt. Laser Technol. 29, 17–23 (1997).
[CrossRef]

Hobson, P. R.

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

P. R. Hobson, A. Raouf, “Finite aperture Fraunhofer hologram of two co-planar discs,” J. Mod. Opt. 39, 807–824 (1992).
[CrossRef]

Kilpatrick, J. M.

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: reduction of aberrations by index compensation,” J. Phys. D 26, 177–182 (1993).
[CrossRef]

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: aberrations in the real image of an underwater object when replayed in air,” J. Phys. D 21, 1701–1705 (1988).
[CrossRef]

J. Watson, J. M. Kilpatrick, “Optical aberrations in underwater holography and their compensation,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 245–253 (1991).

Knox, C.

C. Knox, “Holographic microscopy as a technique for recording microscopic subject,” Science 153, 989–990 (1966).
[CrossRef] [PubMed]

Krantz, E. P.

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

Lampitt, R. S.

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

Latta, J. N.

Mehta, P. C.

P. C. Mehta, “Fifth-order aberrations in in-line holograms,” Opt. Acta 21, 1005–1008 (1974).
[CrossRef]

Meier, R.

R. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am 55, 987–992 (1965).

Nowak, J.

J. Nowak, M. Zajac, “Investigation of the influence of hologram aberrations on the light intensity distribution in the image plane,” Opt. Acta 30, 1749–1767 (1983).
[CrossRef]

Özkul, C.

C. Özkul, “Effects of finite aperture on linewidth measurement using in-line Fraunhofer holography,” Opt. Laser Technol. 18, 36–38 (1986).
[CrossRef]

C. Özkul, “Nonlinearities of an aperture-limited in-line far-field hologram,” Appl. Opt. 25, 3924–3926 (1986).
[CrossRef] [PubMed]

Raouf, A.

P. R. Hobson, A. Raouf, “Finite aperture Fraunhofer hologram of two co-planar discs,” J. Mod. Opt. 39, 807–824 (1992).
[CrossRef]

Rogerson, A.

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

Shofner, F. M.

Steward, R. G.

K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
[CrossRef]

Thompson, B. J.

P. Dunn, B. J. Thompson, “Object shape, fringe visibility, and resolution in far-field holography,” Opt. Eng. 21, 327–332 (1982).
[CrossRef]

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

Vikram, C. S.

C. S. Vikram, “Resolution limits due to primary aberrations in Fraunhofer holography,” Optik 99, 29–31 (1995).

C. S. Vikram, “Rayleigh versus Marechal spherical aberration tolerance in in-line Fraunhofer holography,” Opt. Eng. 33, 3715–3717 (1994).
[CrossRef]

C. S. Vikram, M. L. Billet, “Aberration limited resolution in Fraunhofer holography with collimated beams,” Opt. Laser Technol. 21, 185–187 (1989).
[CrossRef]

C. S. Vikram, M. L. Billet, “In-line Fraunhofer holography at a few far fields,” Appl. Opt. 23, 3091–3094 (1984).
[CrossRef] [PubMed]

C. S. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, UK, 1992).
[CrossRef]

C. S. Vikram, “Holographic metrology of micro-objects in a dynamic volume,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997).

Watson, J.

E. Foster, J. Watson, “Holography for underwater inspection and measurement: an overview of current work,” Opt. Laser Technol. 29, 17–23 (1997).
[CrossRef]

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: reduction of aberrations by index compensation,” J. Phys. D 26, 177–182 (1993).
[CrossRef]

J. Watson, “Underwater visual inspection and measurement using optical holography,” Opt. Lasers Eng. 16, 375–390 (1992).
[CrossRef]

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: aberrations in the real image of an underwater object when replayed in air,” J. Phys. D 21, 1701–1705 (1988).
[CrossRef]

J. Watson, “Hologrammetry and its applications,” J. Imaging Technol. 15, 38–46 (1984).

J. Watson, J. M. Kilpatrick, “Optical aberrations in underwater holography and their compensation,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 245–253 (1991).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970), p. 40.

Zajac, M.

J. Nowak, M. Zajac, “Investigation of the influence of hologram aberrations on the light intensity distribution in the image plane,” Opt. Acta 30, 1749–1767 (1983).
[CrossRef]

Appl. Opt. (6)

J. Geophys. Res. (1)

K. L. Carder, R. G. Steward, P. R. Betzer, “In situ holographic measurements of the sizes and settling of oceanic particulates,” J. Geophys. Res. 87, 5681–5685 (1982).
[CrossRef]

J. Imaging Technol. (1)

J. Watson, “Hologrammetry and its applications,” J. Imaging Technol. 15, 38–46 (1984).

J. Mod. Opt. (1)

P. R. Hobson, A. Raouf, “Finite aperture Fraunhofer hologram of two co-planar discs,” J. Mod. Opt. 39, 807–824 (1992).
[CrossRef]

J. Opt. Soc. Am (1)

R. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am 55, 987–992 (1965).

J. Phys. D (2)

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: aberrations in the real image of an underwater object when replayed in air,” J. Phys. D 21, 1701–1705 (1988).
[CrossRef]

J. M. Kilpatrick, J. Watson, “Underwater hologrammetry: reduction of aberrations by index compensation,” J. Phys. D 26, 177–182 (1993).
[CrossRef]

Opt. Acta (3)

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[CrossRef]

J. Nowak, M. Zajac, “Investigation of the influence of hologram aberrations on the light intensity distribution in the image plane,” Opt. Acta 30, 1749–1767 (1983).
[CrossRef]

P. C. Mehta, “Fifth-order aberrations in in-line holograms,” Opt. Acta 21, 1005–1008 (1974).
[CrossRef]

Opt. Eng. (3)

P. Dunn, B. J. Thompson, “Object shape, fringe visibility, and resolution in far-field holography,” Opt. Eng. 21, 327–332 (1982).
[CrossRef]

K. L. Carder, “Holographic microvelocimeter for use in studying ocean particle dynamics,” Opt. Eng. 18, 524–525 (1979).
[CrossRef]

C. S. Vikram, “Rayleigh versus Marechal spherical aberration tolerance in in-line Fraunhofer holography,” Opt. Eng. 33, 3715–3717 (1994).
[CrossRef]

Opt. Laser Technol. (4)

C. S. Vikram, M. L. Billet, “Aberration limited resolution in Fraunhofer holography with collimated beams,” Opt. Laser Technol. 21, 185–187 (1989).
[CrossRef]

P. R. Hobson, E. P. Krantz, R. S. Lampitt, A. Rogerson, J. Watson, “A preliminary study of the distribution of plankton using hologrammetry,” Opt. Laser Technol. 29, 25–33 (1997).
[CrossRef]

C. Özkul, “Effects of finite aperture on linewidth measurement using in-line Fraunhofer holography,” Opt. Laser Technol. 18, 36–38 (1986).
[CrossRef]

E. Foster, J. Watson, “Holography for underwater inspection and measurement: an overview of current work,” Opt. Laser Technol. 29, 17–23 (1997).
[CrossRef]

Opt. Lasers Eng. (1)

J. Watson, “Underwater visual inspection and measurement using optical holography,” Opt. Lasers Eng. 16, 375–390 (1992).
[CrossRef]

Optik (1)

C. S. Vikram, “Resolution limits due to primary aberrations in Fraunhofer holography,” Optik 99, 29–31 (1995).

Science (1)

C. Knox, “Holographic microscopy as a technique for recording microscopic subject,” Science 153, 989–990 (1966).
[CrossRef] [PubMed]

Other (6)

C. S. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, UK, 1992).
[CrossRef]

C. S. Vikram, “Holographic metrology of micro-objects in a dynamic volume,” in Optical Measurement Techniques and Applications, P. K. Rastogi, ed. (Artech House, Norwood, Mass., 1997).

I. Banyasz, “Resolution problems in holography,” in International Colloquium on Diffractive Optical Elements, J. Nowak, M. Zajac, eds., Proc. SPIE1574, 282–293 (1991).
[CrossRef]

J. Watson, J. M. Kilpatrick, “Optical aberrations in underwater holography and their compensation,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 245–253 (1991).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1970), p. 40.

Ref. 25, pp. 211–224.

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Figures (7)

Fig. 1
Fig. 1

Optical arrangement for underwater in-line hologram recording.

Fig. 2
Fig. 2

Geometry of the propagating diffracted wave in water, glass, and air.

Fig. 3
Fig. 3

Fringe intensity distributions at the hologram-recording plane calculated for a 25-μm wire: (a) single path of 80 mm in water and (b) paths of 70 and 10 mm in water with a stage of integration that uses the field at 70 mm.

Fig. 4
Fig. 4

Calculated image intensity distributions from in-line Fraunhofer holograms of an underwater 10-μm wire. The holograms are recorded in air with λ = 633 nm; apertures of 7.1, 14.2, and 20.0 mm; a 10-mm glass window (n = 1.52); and a 10-mm air gap: top row, Gaussian plane and no aberrations arising from the interfaces; middle row, Gaussian plane with aberrations caused by the interfaces; bottom row, best-focus plane with aberrations caused by the interfaces.

Fig. 5
Fig. 5

Replayed real images of an underwater 10-μm wire recorded at 633 nm on a plate in water and a plate in air and replayed at 633, 514, and 488 nm: (a) experimental intensities averaged over 160 pixel rows and (b) calculated intensities.

Fig. 6
Fig. 6

Calculated intensity distributions at the Gaussian image planes replayed with different wavelengths from in-line Fraunhofer holograms of an underwater 10-μm wire. The hologram is recorded at 633 nm with an aperture of 14.2 mm.

Fig. 7
Fig. 7

Calculated intensity distributions at the best-focus planes for the hologram of Fig. 6.

Tables (1)

Tables Icon

Table 1 Calculated Variation with Reconstruction Wavelength of the Aberration-Limited Hologram Aperture, Focal-Plane Shift, and Measured Wire Width for a Hologram of a 10.0-μm Wire in Water (nw = 1.33)a

Equations (19)

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

ψ 1 x 1 = exp ik w z w 1 - 2 a λ w z w 1 / 2 sin α x 1 α x 1 × exp i π x 1 2 λ w z w - π 4 = ψ p x 1 + ψ d x 1 ,
ψ 2 x 2 =   C g D 1 ψ 1 x 1 exp ik g r g d x 1 .
ψ H x H =   C a D 2 ψ 2 x 2 exp ik a r a d x 2 ,
C g - ik g z g 2 π r g 2 = - i   z g λ g r g 2 , C a - ik a z a 2 π r a 2 = - i   z a λ a r a 2 , r g = x 2 - x 1 2 + z g 2 1 / 2 , r a = x H - x 2 2 + z a 2 1 / 2 , D 1 2 1 + n g / n w , D 2 2 1 + 1 / n g ,
ψ H x H =   C g C a D 1 D 2 ψ 1 x 1 × exp i k g r g + k a r a d x 1 d x 2 .
I x H = ψ H ψ H * =   C g C a D 1 D 2 ψ 1 x 1 × exp i k g r g + k a r a d x 1 d x 2 2 .
W 1 x 1 = 2 π n g λ - 1 4   S 1 x 1 4 = - π n g 2 λ   x 1 4 n w 2 z w 3 n w n g 2 - 1 ;
W 2 x 2 = 2 π n a λ - 1 4   S 2 x 2 4 = π n a x 2 4 2 λ n g n g 2 - 1 2 n g n w   z w + z g 3 ;
W H x H = 2 π / λ c - 1 8   ρ 4 S H = - n w π μ 4 λ c z o 3 n w 2 μ 2 - 1 x H 4 ,
μ = λ c / λ ,
W 1 H al + W 2 H al = π 2 ,     W H H al = π 2 .
H al = 1 μ n g 2 λ c n w z w 3 1 - n w n g 2 - n a n g 2 - 1 n g n w   z w + z g 3 1 / 4 ,
H al = 2 λ c z o 3 μ n W n W 2 μ 2 - 1 1 / 4 .
I w = | ψ 1 x 1 | 2 , I a = | ψ H x H | 2 ,
ψ H x H =   C g C a D 1 D 2 ψ 1 x 1 exp i k g r g + k a r a + W 1 + W 2 d x 1 d x 2
ψ w u = - i λ c z c exp ik c z c H   T w x H × exp ik c 2 z c u - x H 2 + iW H d x H ,
ψ a u = - i λ c z c exp ik c z c H   T a x 1 × exp ik c 2 z c u - x 1 2 d x 1 ,
0.98 λ n w λ c 1.02 λ .
Δ z = z c - z o μ n w ,

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