Abstract

Distributions of wave fields in three-dimensional domains are analyzed, synthesized, and generated experimentally. Fundamental limits are discussed and sampling conditions are derived for their generation, with use of a single diffractive element. A general design procedure, based on optimization algorithms, is developed and implemented. Experimental results show that special three-dimensional light distributions can be achieved with low-information-content elements in on-axis configurations.

© 1996 Optical Society of America

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References

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  1. D. Gabor, “A new microscopic principle,” Nature (London)161, 777–778 (1948).
  2. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  3. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  4. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
    [CrossRef]
  5. W. J. Dallas, “Computer generated holograms,” in Topics in Applied Physics, B. R. Frieden, ed. (Springer, Berlin, 1980), Vol. 41, pp. 291–366.
    [CrossRef]
  6. A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik (Stuttgart)51, 105–117 (1978).
  7. R. P. Porter, “Generalized holography with applications to inverse scattering and inverse source problems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 317–413.
    [CrossRef]
  8. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  9. W. L. Stutzman, Antenna Theory and Design (Wiley, New York, 1981).
  10. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  11. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett.19, 771–773 (1994).
  12. R. Piestun, B. Spektor, J. Shamir, “Three-dimensional distribution of light generated by a diffractive element,” in Vol. 139 of Institute of Physics Conference Series, Part II (Institute of Physics, Philadelphia, Pa., 1995), pp. 275–278.
  13. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,” IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
    [CrossRef]
  14. L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
    [CrossRef]
  15. R. Aharoni, Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,” Linear Algebra Appl. 120, 165–175 (1989).
    [CrossRef]
  16. R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distributions,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–326 (1995).
    [CrossRef]
  17. H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
    [CrossRef]
  18. B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
    [CrossRef]
  19. C. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
    [CrossRef]
  20. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A 70, 150–151 (1980).
    [CrossRef]
  21. A. VanderLugt, “Optimum sampling of Fresnel transforms,” Appl. Opt. 29, 3352–3361 (1990).
    [CrossRef] [PubMed]
  22. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,” J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  24. D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–152.
    [CrossRef]
  25. J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
  26. T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).
  27. J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 845–847 (1994).
    [CrossRef]
  28. B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping using computer-generated holograms,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
    [CrossRef]

1995 (1)

1994 (2)

J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 845–847 (1994).
[CrossRef]

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett.19, 771–773 (1994).

1990 (1)

1989 (1)

R. Aharoni, Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,” Linear Algebra Appl. 120, 165–175 (1989).
[CrossRef]

1987 (1)

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,” IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

1980 (2)

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A 70, 150–151 (1980).
[CrossRef]

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

1979 (1)

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

1978 (1)

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik (Stuttgart)51, 105–117 (1978).

1967 (3)

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[CrossRef] [PubMed]

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

1964 (1)

1949 (1)

C. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature (London)161, 777–778 (1948).

Aharoni, R.

R. Aharoni, Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,” Linear Algebra Appl. 120, 165–175 (1989).
[CrossRef]

Allebach, J. P.

Arsenault, H.

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

Boivin, A.

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

Censor, Y.

R. Aharoni, Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,” Linear Algebra Appl. 120, 165–175 (1989).
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).

Cohen, N.

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).

Colombeau, B.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer generated holograms,” in Topics in Applied Physics, B. R. Frieden, ed. (Springer, Berlin, 1980), Vol. 41, pp. 291–366.
[CrossRef]

Fineup, J. R.

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

Froehly, C.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature (London)161, 777–778 (1948).

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–152.
[CrossRef]

Goodman, J. W.

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

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Kotzer, T.

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik (Stuttgart)51, 105–117 (1978).

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[CrossRef] [PubMed]

McCutchen, C. W.

Nazarathy, M.

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A 70, 150–151 (1980).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Paris, D. P.

Piestun, R.

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett.19, 771–773 (1994).

R. Piestun, B. Spektor, J. Shamir, “Three-dimensional distribution of light generated by a diffractive element,” in Vol. 139 of Institute of Physics Conference Series, Part II (Institute of Physics, Philadelphia, Pa., 1995), pp. 275–278.

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distributions,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–326 (1995).
[CrossRef]

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Porter, R. P.

R. P. Porter, “Generalized holography with applications to inverse scattering and inverse source problems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 317–413.
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Rosen, J.

B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping using computer-generated holograms,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
[CrossRef]

J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 845–847 (1994).
[CrossRef]

Salik, B.

Seldowitz, M. A.

Shamir, J.

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett.19, 771–773 (1994).

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A 70, 150–151 (1980).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distributions,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–326 (1995).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Three-dimensional distribution of light generated by a diffractive element,” in Vol. 139 of Institute of Physics Conference Series, Part II (Institute of Physics, Philadelphia, Pa., 1995), pp. 275–278.

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).

Shannon, C.

C. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

Spektor, B.

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distributions,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–326 (1995).
[CrossRef]

R. Piestun, B. Spektor, J. Shamir, “Three-dimensional distribution of light generated by a diffractive element,” in Vol. 139 of Institute of Physics Conference Series, Part II (Institute of Physics, Philadelphia, Pa., 1995), pp. 275–278.

Stutzman, W. L.

W. L. Stutzman, Antenna Theory and Design (Wiley, New York, 1981).

Sweeney, D. W.

Vampouille, M.

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

VanderLugt, A.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,” IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

Wyrowski, F.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

Yariv, A.

B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping using computer-generated holograms,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
[CrossRef]

J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 845–847 (1994).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,” IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—Theory,” IEEE Trans. Med. Imaging TMI-1, 81–94 (1982).
[CrossRef]

J. Appl. Phys. (1)

H. Arsenault, A. Boivin, “An axial form of the sampling theorem and its application to optical diffraction,” J. Appl. Phys. 38, 3988–3990 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

B. Salik, J. Rosen, A. Yariv, “One-dimensional beam shaping using computer-generated holograms,” J. Opt. Soc. Am. A 12, 1702–1706 (1995).
[CrossRef]

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. A 70, 150–151 (1980).
[CrossRef]

Linear Algebra Appl. (1)

R. Aharoni, Y. Censor, “Block-iterative projection methods for parallel computation of solutions to convex feasibility problems,” Linear Algebra Appl. 120, 165–175 (1989).
[CrossRef]

Nature (London) (1)

D. Gabor, “A new microscopic principle,” Nature (London)161, 777–778 (1948).

Opt. Commun. (1)

B. Colombeau, C. Froehly, M. Vampouille, “Fourier structure of the axial patterns diffracted from optical pupils, examples and applications,” Opt. Commun. 28, 35–38 (1979).
[CrossRef]

Opt. Eng. (1)

J. R. Fineup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).

Opt. Lett. (2)

J. Rosen, A. Yariv, “Synthesis of an arbitrary axial field profile by computer-generated holograms,” Opt. Lett. 19, 845–847 (1994).
[CrossRef]

R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett.19, 771–773 (1994).

Optik (Stuttgart) (1)

A. W. Lohmann, “Three-dimensional properties of wave-fields,” Optik (Stuttgart)51, 105–117 (1978).

Proc. IRE (1)

C. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10–21 (1949).
[CrossRef]

USSR Comput. Math. Math. Phys. (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point of convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Other (11)

R. Piestun, B. Spektor, J. Shamir, “Diffractive optics for unconventional light distributions,” in Diffractive and Holographic Optics Technology II, I. Cindrich, S. H. Lee, eds., Proc. SPIE2404, 320–326 (1995).
[CrossRef]

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

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 111–152.
[CrossRef]

R. P. Porter, “Generalized holography with applications to inverse scattering and inverse source problems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 317–413.
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

W. L. Stutzman, Antenna Theory and Design (Wiley, New York, 1981).

R. Piestun, B. Spektor, J. Shamir, “Three-dimensional distribution of light generated by a diffractive element,” in Vol. 139 of Institute of Physics Conference Series, Part II (Institute of Physics, Philadelphia, Pa., 1995), pp. 275–278.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

W. J. Dallas, “Computer generated holograms,” in Topics in Applied Physics, B. R. Frieden, ed. (Springer, Berlin, 1980), Vol. 41, pp. 291–366.
[CrossRef]

T. Kotzer, N. Cohen, J. Shamir, Y. Censor, “Summed squared distance error reduction of simultaneous multi-projections and applications,” Publ. 909 (Department of Electrical Engineering, Technion-I.I.T., Haifa, Israel, 1994).

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Figures (10)

Fig. 1
Fig. 1

Propagation of a coherent wave field in a source-free half-space.

Fig. 2
Fig. 2

(a) Generalized aperture and definition of parameters rM and rm relative to a point O at the entrance plane. (b) Illustration of the zero-order circular harmonic of the field at z = 0, its derivative, and the autocorrelation of its derivative.

Fig. 3
Fig. 3

Optimum (nonuniform) axial sampling: (a) representation of the axial spatial frequency cutoff, (b) axial sampling spacing as a function of the distance from the entrance plane for a circular aperture (rM = 6.4 mm, rm = 0, λ = 633 nm), (c) axial sampling spacing as a function of the transverse coordinate for the same aperture.

Fig. 4
Fig. 4

Available 3-D domain when a sampled diffractive element is implemented with rectangular grids. The 3-D distribution of light must be compactly supported within this region in order to avoid overlapping between orders (aliasing).

Fig. 5
Fig. 5

Diffractive optical problem under consideration: Given an arbitrary wave front, design a thin planar diffractive element to obtain a desired light distribution within the 3-D region D.

Fig. 6
Fig. 6

Weighted summed squared distance error (WSSDE) reduction as a function of the iteration number for the POCS algorithm (WSSDEpocs) and for the BIWP algorithm (WSSDEbiwp). For comparison we show the behavior of the regular SSDE in both cases (SSDEpocs and SSDEbiwp).

Fig. 7
Fig. 7

Binary amplitude diffractive element for the generation of a 3-D light distribution (see the text).

Fig. 8
Fig. 8

Axial profile of the field intensity distribution.

Fig. 9
Fig. 9

Experimental results: the central part of the intensity distribution at different distances; z increases from left to right and from top to bottom (from z = 1 m to z = 1.5 m).

Fig. 10
Fig. 10

Detailed experimental results: (a) comparison of the intensity distribution in a peak (z = 142 cm) and in a valley (z = 136 cm), (b) transverse view of the intensity distribution at z = 142 cm.

Equations (46)

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

U ( 0 , 0 , z ) = exp ( j k z ) j λ z 0 2 π 0 r f ( r , θ ) exp ( j k 2 z r 2 ) d r d θ = 2 π exp ( j k z ) j λ z 0 r f ¯ ( r ) exp ( j k 2 z r 2 ) d r ,
f ¯ ( r ) = 1 2 π 0 2 π f ( r , θ ) d θ
v = 1 2 λ z , q = r 2 ,
U ( 0 , 0 , z ) = U ¯ ( v ) = - j 2 π v exp ( j π λ 2 v ) F - 1 { f ^ ( q ) } = exp ( j k z ) F - 1 { d f ^ ( q ) d q } ,
f ^ ( q ) = { f ¯ ( r ) if q 0 ( r 0 ) 0 if q < 0 .
S [ d f ^ ( q ) d q ] S [ f ^ ( q ) ] = [ r m 2 , r M 2 ] ,
Δ v field = 1 2 λ z - 1 2 λ ( z + Δ z f i e l d ) = 1 r M 2 - r m 2 .
Δ z field = 4 π z 2 k ( r M 2 - r m 2 ) - 4 π z ,
U ¯ ( v ) 2 = | F - 1 { d f ^ ( q ) d q } | 2
F { U ¯ ( v ) 2 } = d f ^ ( q ) d q d f ^ ( q ) d q ,
S [ F { U ¯ ( v ) 2 ] = S [ d f ^ ( q ) d q d f ^ ( q ) d q ] = [ - ( r M 2 - r m 2 ) , r M 2 - r m 2 ] .
BW U ¯ ( v ) 2 r M 2 - r m 2 .
Δ z I = 2 π z 2 k ( r M 2 - r m 2 ) - 2 π z .
U ( x , y , z ) = FrT [ f ( x , y ) ] = exp ( j k z ) j λ z Q [ 1 / z ] V [ 1 λ z ] F Q [ 1 / z ] f ( x , y ) ,
Δ x field = λ z Δ r x ,
Δ x I = Δ x field 2 .
Δ z ¯ I = 2 π z 2 k R 2 - 2 π z λ z 2 R 2 .
U ( x , y , z ) = - F ( p , q ) × exp [ j 2 π ( p x + q y + m z ) ] d p d q ,
λ p , λ q , and λ m = 1 - ( λ p ) 2 - ( λ q ) 2
F ( p , q ) = - f ( x , y ) exp [ - j 2 π ( p x + q y ) ] d x d y
u ( p , q , m ) = - U ( x , y , z ) × exp [ - j 2 π ( p x + q y + m z ) ] d x d y d z .
u ( p , q , m ) = F ( p , q ) δ [ m - 1 λ 1 - ( λ p ) 2 - ( λ q ) 2 ] .
p 2 + q 2 < 1 / λ 2 .
δ z ( z ) = 4 π z 2 k ( r M 2 - r m 2 ) - 4 π z 2 λ z 2 r M 2 - r m 2 .
δ x = λ z Δ r x .
f ( x , y ) = [ g ( x , y ) - - δ ( x - m Δ x , y - n Δ y ) ] * s ( x , y ) ,
U ( x , y , z ) = exp ( j k z ) j λ z V [ 1 λ z ] Q [ λ 2 z ] { 1 Δ x Δ y × - - δ ( x - m Δ x , y - n Δ y ) * F Q [ 1 / z ] g ( x , y ) } .
F ~ W Δ Ω λ 2 ,
N = v 1 - v 2 Δ v field = r M 2 - r m 2 λ ( 1 z 1 - 1 z 2 ) ,
F N ~ π Δ Ω λ ( 1 / z 1 - 1 / z 2 ) π Δ Ω z 1 λ ,
F p = W δ x δ y .
F p N ~ π λ δ x δ y ( 1 / z 1 - 1 / z 2 ) π λ z 1 δ x δ y .
f - g = inf v C i f - v ,
P w f = i = 1 n w ( i ) P i f .
f k + 1 = P w k f k ,
f C = i = 1 n C i .
g = P z i f = FrT - 1 ( P ¯ z i FrT f )
P ¯ d 1 + ( 8 k + l ) Δ z 1 [ F ( x , y , d 1 + ( 8 k + l ) Δ z ) ] = min { F [ x , y , d 1 + ( 8 k + l ) Δ z ] , c ( x , y ) } × exp { i Φ [ x , y , d 1 + ( 8 k + l ) Δ z ] } ,
P ¯ d 1 + ( 8 k + l ) Δ z 2 { F [ x , y , d 1 + ( 8 k + l ) Δ z ] } = { A exp [ i Φ ( 0 , 0 ) ] if ( x , y ) = ( 0 , 0 ) F [ x , y , d 1 + ( 8 k + l ) Δ z ] otherwise ,
A = { M if l = 1 , 2 c ( 0 , 0 ) if l = 3 , , 8 .
P 0 { f ( x , y ) } = { Re ( f ) if Re ( f ) 0 0 if Re ( f ) < 0 .
f 8 k + l = P w ( 8 k + l ) f 8 k + l - 1 = 0.01 P d 1 + ( 8 k + l ) Δ z 1 f 8 k + l - 1 + 0.99 P d 1 + ( 8 k + l ) Δ z 2 f 8 k + l - 1 ,
f 33 = P 0 f 32 ,
P w ( k ) = P w ( k + 33 ) .
WSSDE ( f ) = i = 1 n w ( i ) d 2 [ P i f , f ] = i = 1 n w ( i ) p , q P , Q ( P i f p , q - f p , q ) 2 = i = 1 n w ( i ) p , q P , Q ( P ¯ z i FrT f p , q - FrT f p , q ) 2 .
m = v 1 - v 2 Δ _ v I = ( 1 z 1 - 1 z 2 ) R 2 λ R 2 λ z 1 .

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