Abstract

A model for analyzing the effects of material shrinkage on volume holograms is presented. This model is based on the fringe-plane rotation model used for describing the effects of plane-wave grating holograms that undergo shrinkage. A computer was used to exercise the model for a simple input object typical of those used in digital holographic memory applications and stored as a Fourier-transform hologram. The theoretical formulation of the model is presented as well as the results of the numerical analysis.

© 1994 Optical Society of America

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  1. T. K. Gaylord, “Digital data storage,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 379–413.
  2. A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
    [CrossRef]
  3. O. Changsuk, P. Hankyu, “Real-time Fourier-transformed holographic associative memory with photorefractive material,” in Optical Computing '88,P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 554–559 (1988).
  4. H. Mada, “Architecture for optical computing using holographic associative memories,” Appl. Opt. 24, 2063–2066 (1985).
    [CrossRef] [PubMed]
  5. M. S. Cohen, “Design of a new medium for volume holographic information processing,” Appl. Opt. 25, 2288–2294 (1986).
    [CrossRef] [PubMed]
  6. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
    [CrossRef] [PubMed]
  7. M. R. Feldman, C. C. Guest, “Computer-generated holographic optical elements for optical interconnection of very large scale integrated circuits,” Appl. Opt. 26, 4377–4384 (1987).
    [CrossRef] [PubMed]
  8. T. Drabik, “Optically interconnected parallel processor arrays,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1990).
  9. W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).
  10. W. S. Colburn, K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
    [CrossRef] [PubMed]
  11. B. L. Booth, “Photopolymer material for holography,” Appl. Opt. 14, 593–601 (1975).
    [CrossRef] [PubMed]
  12. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971), pp. 287–289.
  13. D. H. R. Vilkomerson, D. Bostwick, “Some effects of emulsion shrinkage on a hologram's image space,” Appl. Opt. 6, 1270–1272 (1967).
    [CrossRef] [PubMed]
  14. P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, London, 1984), pp. 47–56, 88–115.
  15. N. Chen, “Aberrations of volume holographic grating,” Opt. Lett. 10, 472–474 (1985).
    [CrossRef] [PubMed]
  16. E. A. Chandross, W. J. Tomlinson, G. D. Aumiller, “Latent-imaging photopolymer systems,” Appl. Opt. 17, 566–573 (1978).
    [CrossRef] [PubMed]
  17. E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
    [CrossRef] [PubMed]
  18. L. H. Lin, C. V. LoBianco, “Experimental techniques in making multi-color white light reconstructed holograms,” Appl. Opt. 6, 1255–1258 (1967).
    [CrossRef] [PubMed]
  19. A. A. Friesem, J. L. Walker, “Experimental investigations of some anomalies in photographic plates,” Appl. Opt. 8, 1504–1506(1969).
    [CrossRef] [PubMed]
  20. O. Bryngdahl, “Can detrimental effects in photographic volume holography be compensated for?” Appl. Opt. 11, 195 (1972).
    [CrossRef] [PubMed]
  21. G. D. Mintz, D. K. Morland, W. M. Boerner, “Holographic simulation of parabolic mirrors,” Appl. Opt. 14, 564–565 (1975).
    [CrossRef] [PubMed]
  22. L. Joly, R. Vanhorebeek, “Development effects in white-light reflection holography,” Photogr. Sci. Eng. 24(2), 108–113 (1980).
  23. P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: the influence of halide type and concentration on diffraction efficiency and scattering,” Appl. Opt. 26, 3895–3898 (1987).
    [CrossRef] [PubMed]
  24. P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms. II. Spatial frequency effects,” Appl. Opt. 27, 3852–3854 (1988).
    [CrossRef] [PubMed]
  25. N. J. Phillips, “The role of silver halide materials in the formation of holographic images,” in Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.532, 29–38 (1985).
  26. M. P. Jordan, L. Solymar, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
    [CrossRef]
  27. P. Fiala, J. Ruzek, T. Jerie, “Behavior and properties of real holographic recording materials,” in Practical Holography II, T. H. Jeong, ed., Proc. Soc. Photo-Opt. Instrum. Eng.747, 74–81 (1987).
  28. T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
    [CrossRef]
  29. L. B. Au, J. C. W. Newell, L. Solymar, “Non-uniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
    [CrossRef]
  30. J. T. Gallo, M. L. Jones, C. M. Verber, “Computer modeling of the effects of apertures in the Fourier-transform plane of Fourier-transform imaging systems,” Appl. Opt. 33, 2891–2899 (1994).
    [CrossRef] [PubMed]
  31. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 49–53.
  32. The authors are indebted to a reviewer for helping to clarify this point.
  33. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  34. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  35. A. V. Oppenheim, A. S. Willsky, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983), pp. 184–185.

1994 (1)

1988 (1)

1987 (4)

1986 (2)

A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
[CrossRef]

M. S. Cohen, “Design of a new medium for volume holographic information processing,” Appl. Opt. 25, 2288–2294 (1986).
[CrossRef] [PubMed]

1985 (2)

1983 (1)

1980 (1)

L. Joly, R. Vanhorebeek, “Development effects in white-light reflection holography,” Photogr. Sci. Eng. 24(2), 108–113 (1980).

1979 (1)

T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
[CrossRef]

1978 (2)

1975 (2)

1972 (1)

1971 (1)

1969 (2)

A. A. Friesem, J. L. Walker, “Experimental investigations of some anomalies in photographic plates,” Appl. Opt. 8, 1504–1506(1969).
[CrossRef] [PubMed]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967 (2)

1966 (1)

Au, L. B.

L. B. Au, J. C. W. Newell, L. Solymar, “Non-uniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

Aumiller, G. D.

Boerner, W. M.

Booth, B. L.

Bostwick, D.

Bryngdahl, O.

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971), pp. 287–289.

Chandross, E. A.

Changsuk, O.

O. Changsuk, P. Hankyu, “Real-time Fourier-transformed holographic associative memory with photorefractive material,” in Optical Computing '88,P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 554–559 (1988).

Chen, N.

Chidley, C. M.

Cohen, M. S.

Colburn, W. S.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971), pp. 287–289.

Drabik, T.

T. Drabik, “Optically interconnected parallel processor arrays,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1990).

Feldman, M. R.

Fiala, P.

P. Fiala, J. Ruzek, T. Jerie, “Behavior and properties of real holographic recording materials,” in Practical Holography II, T. H. Jeong, ed., Proc. Soc. Photo-Opt. Instrum. Eng.747, 74–81 (1987).

Friesem, A. A.

Gallo, J. T.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

T. K. Gaylord, “Digital data storage,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 379–413.

Goodman, J. W.

Guest, C. C.

Haines, K. A.

Hankyu, P.

O. Changsuk, P. Hankyu, “Real-time Fourier-transformed holographic associative memory with photorefractive material,” in Optical Computing '88,P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 554–559 (1988).

Hariharan, P.

Hesselink, L.

Jerie, T.

P. Fiala, J. Ruzek, T. Jerie, “Behavior and properties of real holographic recording materials,” in Practical Holography II, T. H. Jeong, ed., Proc. Soc. Photo-Opt. Instrum. Eng.747, 74–81 (1987).

Joly, L.

L. Joly, R. Vanhorebeek, “Development effects in white-light reflection holography,” Photogr. Sci. Eng. 24(2), 108–113 (1980).

Jones, M. L.

Jordan, M. P.

M. P. Jordan, L. Solymar, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kostuk, R. K.

Kozma, A.

Kubota, T.

T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
[CrossRef]

Kwong, S.-K.

A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
[CrossRef]

Kyuma, K.

A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
[CrossRef]

Leith, E. N.

Lin, L. H.

L. H. Lin, C. V. LoBianco, “Experimental techniques in making multi-color white light reconstructed holograms,” Appl. Opt. 6, 1255–1258 (1967).
[CrossRef] [PubMed]

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971), pp. 287–289.

LoBianco, C. V.

Mada, H.

Marks, J.

Massey, N.

Mickish, D. J.

W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).

Mintz, G. D.

Moharam, M. G.

Morland, D. K.

Newell, J. C. W.

L. B. Au, J. C. W. Newell, L. Solymar, “Non-uniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983), pp. 184–185.

Phillips, N. J.

N. J. Phillips, “The role of silver halide materials in the formation of holographic images,” in Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.532, 29–38 (1985).

Ruzek, J.

P. Fiala, J. Ruzek, T. Jerie, “Behavior and properties of real holographic recording materials,” in Practical Holography II, T. H. Jeong, ed., Proc. Soc. Photo-Opt. Instrum. Eng.747, 74–81 (1987).

Smothers, W. K.

W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).

Solymar, L.

L. B. Au, J. C. W. Newell, L. Solymar, “Non-uniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

M. P. Jordan, L. Solymar, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

Tomlinson, W. J.

Trout, T. J.

W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).

Upatnieks, J.

Vanhorebeek, R.

L. Joly, R. Vanhorebeek, “Development effects in white-light reflection holography,” Photogr. Sci. Eng. 24(2), 108–113 (1980).

Verber, C. M.

Vilkomerson, D. H. R.

Walker, J. L.

Weber, A. M.

W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983), pp. 184–185.

Yariv, A.

A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
[CrossRef]

Appl. Opt. (16)

E. N. Leith, A. Kozma, J. Upatnieks, J. Marks, N. Massey, “Holographic data storage in three-dimensional media,” Appl. Opt. 5, 1303–1311 (1966).
[CrossRef] [PubMed]

L. H. Lin, C. V. LoBianco, “Experimental techniques in making multi-color white light reconstructed holograms,” Appl. Opt. 6, 1255–1258 (1967).
[CrossRef] [PubMed]

O. Bryngdahl, “Can detrimental effects in photographic volume holography be compensated for?” Appl. Opt. 11, 195 (1972).
[CrossRef] [PubMed]

B. L. Booth, “Photopolymer material for holography,” Appl. Opt. 14, 593–601 (1975).
[CrossRef] [PubMed]

E. A. Chandross, W. J. Tomlinson, G. D. Aumiller, “Latent-imaging photopolymer systems,” Appl. Opt. 17, 566–573 (1978).
[CrossRef] [PubMed]

H. Mada, “Architecture for optical computing using holographic associative memories,” Appl. Opt. 24, 2063–2066 (1985).
[CrossRef] [PubMed]

M. S. Cohen, “Design of a new medium for volume holographic information processing,” Appl. Opt. 25, 2288–2294 (1986).
[CrossRef] [PubMed]

P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms: the influence of halide type and concentration on diffraction efficiency and scattering,” Appl. Opt. 26, 3895–3898 (1987).
[CrossRef] [PubMed]

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Computer-generated holographic optical elements for optical interconnection of very large scale integrated circuits,” Appl. Opt. 26, 4377–4384 (1987).
[CrossRef] [PubMed]

P. Hariharan, C. M. Chidley, “Rehalogenating bleaches for photographic phase holograms. II. Spatial frequency effects,” Appl. Opt. 27, 3852–3854 (1988).
[CrossRef] [PubMed]

J. T. Gallo, M. L. Jones, C. M. Verber, “Computer modeling of the effects of apertures in the Fourier-transform plane of Fourier-transform imaging systems,” Appl. Opt. 33, 2891–2899 (1994).
[CrossRef] [PubMed]

W. S. Colburn, K. A. Haines, “Volume hologram formation in photopolymer materials,” Appl. Opt. 10, 1636–1641 (1971).
[CrossRef] [PubMed]

A. A. Friesem, J. L. Walker, “Experimental investigations of some anomalies in photographic plates,” Appl. Opt. 8, 1504–1506(1969).
[CrossRef] [PubMed]

D. H. R. Vilkomerson, D. Bostwick, “Some effects of emulsion shrinkage on a hologram's image space,” Appl. Opt. 6, 1270–1272 (1967).
[CrossRef] [PubMed]

G. D. Mintz, D. K. Morland, W. M. Boerner, “Holographic simulation of parabolic mirrors,” Appl. Opt. 14, 564–565 (1975).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

A. Yariv, S.-K. Kwong, K. Kyuma, “Demonstration of an all-optical associative holographic memory,” Appl. Phys. Lett. 48, 1114–1116(1986).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Electron. Lett. (1)

M. P. Jordan, L. Solymar, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

J. Mod. Opt. (1)

L. B. Au, J. C. W. Newell, L. Solymar, “Non-uniformities in thick dichromated gelatin transmission gratings,” J. Mod. Opt. 34, 1211–1225 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

T. Kubota, “The bending of interference fringes inside a hologram,” Opt. Acta 26, 731–743 (1979).
[CrossRef]

Opt. Lett. (1)

Photogr. Sci. Eng. (1)

L. Joly, R. Vanhorebeek, “Development effects in white-light reflection holography,” Photogr. Sci. Eng. 24(2), 108–113 (1980).

Other (11)

N. J. Phillips, “The role of silver halide materials in the formation of holographic images,” in Holography, L. Huff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.532, 29–38 (1985).

P. Fiala, J. Ruzek, T. Jerie, “Behavior and properties of real holographic recording materials,” in Practical Holography II, T. H. Jeong, ed., Proc. Soc. Photo-Opt. Instrum. Eng.747, 74–81 (1987).

O. Changsuk, P. Hankyu, “Real-time Fourier-transformed holographic associative memory with photorefractive material,” in Optical Computing '88,P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 554–559 (1988).

T. Drabik, “Optically interconnected parallel processor arrays,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1990).

W. K. Smothers, T. J. Trout, A. M. Weber, D. J. Mickish, “Hologram recording in Dupont's new photopolymer materials,” in Practical Holography TV, S. A. Benton, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1212, 30–39 (1990).

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, Orlando, Fla., 1971), pp. 287–289.

P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, London, 1984), pp. 47–56, 88–115.

A. V. Oppenheim, A. S. Willsky, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983), pp. 184–185.

T. K. Gaylord, “Digital data storage,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), pp. 379–413.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 49–53.

The authors are indebted to a reviewer for helping to clarify this point.

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Figures (9)

Fig. 1
Fig. 1

Fringe-plane rotation model for a plane-wave grating is shown. The reference (R) and signal (S) beams intersect the material at angles of β1 and β2 with respect to the surface normal. The sign of β1 is positive, and the sign of β2 is negative. Before shrinkage, the material has thickness L and the planar parallel fringes are defined by the grating vector K, the fringe spacing Λ, and the slant angle ϕ. The primed values indicate the new parameter values after shrinkage.

Fig. 2
Fig. 2

Fourier-transform hologram writing setup for interference of a reference plane wave with the angular spectrum. The reference beam forms an angle of ϕ z off with the normal to the hologram's surface. The optic axis through the object plane and the Fourier-transform lens intersects the hologram's axis (the z axis) at angles ϕ x 0, ϕ y 0 and ϕ z 0. The angles ϕ x and ϕ y (not shown) represent the angular components in Eq. (6).

Fig. 3
Fig. 3

Results of several shrinkage model analyses are shown. The solid curve represents the input-object function. The other curves are the computed intensity profiles for 0% (σ = 0.00), 1% (σ = 0.01), 2% (σ = 0.02), and 4% (σ = 0.04) shrinkage. The amplitudes of these curves were all normalized to the peak of the 0% shrinkage curve. The system parameters are λ = 0.5145 μm, f = 100 mm, L = 1000 μm, n 1 = 0.001, ϕ x 0 = ϕ y 0 = π/2, ϕ z off = 32°, n = 1.5, and r max = 5 mm.

Fig. 4
Fig. 4

Same as Fig. 3 except that the reference-beam angle is reduced to ϕ z off = 28°.

Fig. 5
Fig. 5

Diffraction efficiency η′, shown as a function of the amount of shrinkage and the grating-fringe angle ϕ zg . Notice that, for any given mean fringe angle, the efficiency is not necessarily maximized when the shrinkage is 0%. The system parameters used in this figure are the same as those in Fig. 3.

Fig. 6
Fig. 6

Diffraction efficiency as a function of the amount of shrinkage for several values of ϕ zg taken from Fig. 5. The decaying sinusoidal dependence on the shrinkage is evident. Note that the peak efficiency does not always correspond to the case when σ = 0. The decay at ϕ zg = 0° is caused by the dependence on L′.

Fig. 7
Fig. 7

Impact efficiency η I plotted as a function of the amount of shrinkage for reference-beam singles of ϕ z off equal to 32°, 30°, and 28°. The other system parameters are the same as in Fig. 3.

Fig. 8
Fig. 8

Same as Fig. 7 except that η I is plotted for L = 100 and 1000 μm (ϕ z off = 30°).

Fig. 9
Fig. 9

Same as Fig. 8 except that η I is plotted for r max equal to 1. 25, 2.5, and 5 mm (L = 1000 μm).

Equations (28)

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Λ = λ / n 2 sin ( β 1 + β 2 2 ) ,
ϕ = π ( β 1 β 2 ) 2 .
L = ( 1 σ ) L ,
Δ ϕ = Δ θ B = tan 1 [ tan ϕ / ( 1 σ ) ] ϕ ;
ϕ = ϕ + Δ ϕ = tan 1 [ tan ϕ / ( 1 σ ) ] .
A ( ϕ x , ϕ y : z 0 ) = U ( x , y ; z 0 ) × exp [ j 2 π λ ( x cos ϕ x + y cos ϕ y ) ] d x d y
( x 2 + y 2 ) 1 / 2 r max ,
cos ϕ x = x h z 0 ,
cos ϕ y = y h z 0 ,
A 0 ( ϕ x , ϕ y ) = U h ( x h , y h ) x h = z 0 [ cos ( ϕ x ) ] y h = z 0 [ cos ( ϕ y ) ] ,
cos 2 ϕ x 0 + cos 2 ϕ y 0 + cos 2 ϕ z 0 = 1 .
ϕ x g ϕ x 0 + ϕ x + ϕ z off 2 , ϕ y g ϕ y 0 + ϕ y 2 , ϕ z g = cos 1 [ ( 1 cos 2 ϕ x g cos 2 ϕ y g ) 1 / 2 ]
ϕ x g = tan 1 [ ( 1 σ ) tan ϕ x g ] , ϕ y g = tan 1 [ ( 1 σ ) tan ϕ y g ] , ϕ z g = tan 1 ( tan ϕ z g 1 σ ) .
U h ( x h , y h ) = Δ B ( ϕ x , ϕ y ) A 0 ( ϕ x , ϕ y ) cos ( ϕ x ) = x h / z 0 cos ( ϕ y ) = y h / z 0 ,
Δ B ( ϕ x , ϕ y ) = η η ,
η = sin 2 [ ( ν 2 + ξ 2 ) 1 / 2 ] 1 + ξ 2 / ν 2 ,
ν = π n 1 L λ ( c R c S ) 1 / 2 ,
ξ = π L Λ c S [ cos ( ϕ ϕ z off ) λ 2 n Λ ] ,
c R = cos ϕ z off ,
c S = cos ϕ z off λ n Λ cos ϕ ,
ϕ = π / 2 ϕ z g ϕ x g ,
Λ = Λ cos ϕ z g cos ϕ z g ,
U OBJ ( x , y ) = rect ( x δ , y δ ) ,
rect ( x , y ) = { 1 | x | , | y | ½ 0 otherwise ,
U h ( x h , y h ) = A ( ϕ x , ϕ y ) = 1 λ f { U OBJ ( x , y ) } f y = cos ( ϕ y / λ ) f x = cos ( ϕ x / λ ) = δ 2 λ f sinc ( δ cos ϕ x λ , δ cos ϕ y λ ) ,
sinc ( x , y ) = sin ( π x ) π x sin ( π y ) π y .
U IM ( x i , y i ) = 1 λ f m = n = Δ B ( ϕ x , m , ϕ y , n ) A 0 ( ϕ x , m , ϕ y , n ) × exp [ j k ( x im cos ϕ x , m + y im cos ϕ y , n ) ] ,
η I = δ / 2 δ / 2 δ / 2 δ / 2 U i ( x i , y i ) d x i d y i U i ( x i , y i ) d x i d y i ,

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