Abstract

Models for the transmission of light through either rough-surface or bulk diffusers are presented. Since the transmission function contains a dependence on the input angle, it is convenient to use an angular-spectrum representation for the input illumination in the general case. Computer simulations of far-zone speckle patterns and analytical calculations of the ensemble-averaged radiation patterns are made with and without the angle dependence in the transmission function. We find that this angular dependence is relatively unimportant with input illumination near normal incidence. However, the computer simulation demonstrates that the angle dependence can make a significant difference in the detail of the speckle pattern for a complex input spectrum or even for a single plane wave at large (≥30°) angles of incidence. The essential effect of the angle dependence on the averaged radiation patterns is to increase the effective roughness of the diffuser.

© 1987 Optical Society of America

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References

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  1. Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
    [CrossRef]
  2. L. I. Goldfischer, “Autocorrelation function and power spectral density of laser-produced speckle patterns,”J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  3. S. Lowenthal, D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,”J. Opt. Soc. Am. 61, 847–851 (1971).
    [CrossRef]
  4. E. Jakeman, P. N. Pusey, “The statistics of light scatterred by a random phase screen,”J. Phys. A 6, L88–L92 (1973).
    [CrossRef]
  5. E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,”J. Phys. A 8, 369–391 (1975).
    [CrossRef]
  6. N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [CrossRef]
  7. J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, 1975).
  8. A. Zardecki, “Statistical features of phase screens from scattering data,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 155–192.
    [CrossRef]
  9. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.
  10. W. P. Brown, “Propagation in random media—cumulative effect of weak inhomogeneities,”IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).
    [CrossRef]
  11. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  12. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
    [CrossRef]
  13. P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
    [CrossRef]
  14. M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
    [CrossRef]
  15. L. G. Shirley, N. George, “Diffuser transmission functions and far-zone speckle patterns,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 63–69 (1985).
    [CrossRef]
  16. Unless otherwise indicated, the limits of integration on all integrals are from −∞ to ∞.
  17. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  18. W. T. Welford, “Laser speckle and surface roughness,” Contemp. Phys. 21, 401–412 (1980).
    [CrossRef]
  19. H. Fujii, J. Uozumi, T. Asakura, “Computer simulation study of image speckle patterns with relation to object surface profile,”J. Opt. Soc. Am. 66, 1222–1236 (1976).
    [CrossRef]
  20. J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
    [CrossRef]
  21. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
    [CrossRef]
  22. E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
    [CrossRef]
  23. Strictly speaking, only the second class, with even powers of r, satisfies the conditions of smoothness and small slope assumed in the derivations of the plane-wave transmission functions. See, e.g., D. E. Barrick, “Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering,” Radio Sci. 5, 647–654 (1970).
    [CrossRef]

1984 (1)

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

1981 (1)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

1980 (1)

W. T. Welford, “Laser speckle and surface roughness,” Contemp. Phys. 21, 401–412 (1980).
[CrossRef]

1977 (1)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

1976 (1)

1975 (2)

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,”J. Phys. A 8, 369–391 (1975).
[CrossRef]

1974 (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

1973 (1)

E. Jakeman, P. N. Pusey, “The statistics of light scatterred by a random phase screen,”J. Phys. A 6, L88–L92 (1973).
[CrossRef]

1971 (1)

1970 (2)

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Strictly speaking, only the second class, with even powers of r, satisfies the conditions of smoothness and small slope assumed in the derivations of the plane-wave transmission functions. See, e.g., D. E. Barrick, “Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering,” Radio Sci. 5, 647–654 (1970).
[CrossRef]

1967 (1)

W. P. Brown, “Propagation in random media—cumulative effect of weak inhomogeneities,”IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).
[CrossRef]

1965 (1)

1950 (1)

Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Asakura, T.

Barrick, D. E.

Strictly speaking, only the second class, with even powers of r, satisfies the conditions of smoothness and small slope assumed in the derivations of the plane-wave transmission functions. See, e.g., D. E. Barrick, “Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering,” Radio Sci. 5, 647–654 (1970).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
[CrossRef]

Booker, H. G.

Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Brown, W. P.

W. P. Brown, “Propagation in random media—cumulative effect of weak inhomogeneities,”IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).
[CrossRef]

Carter, W. H.

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

Chandley, P. J.

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

Dainty, J. C.

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Fujii, H.

Garcia, N.

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

George, N.

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

L. G. Shirley, N. George, “Diffuser transmission functions and far-zone speckle patterns,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 63–69 (1985).
[CrossRef]

Goldfischer, L. I.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

Jain, A.

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Jakeman, E.

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,”J. Phys. A 8, 369–391 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, “The statistics of light scatterred by a random phase screen,”J. Phys. A 6, L88–L92 (1973).
[CrossRef]

Joyeux, D.

Lowenthal, S.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

Pusey, P. N.

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,”J. Phys. A 8, 369–391 (1975).
[CrossRef]

E. Jakeman, P. N. Pusey, “The statistics of light scatterred by a random phase screen,”J. Phys. A 6, L88–L92 (1973).
[CrossRef]

Ratcliffe, J. A.

Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Shinn, D. H.

Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Shirley, L. G.

L. G. Shirley, N. George, “Diffuser transmission functions and far-zone speckle patterns,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 63–69 (1985).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Uozumi, J.

Welford, W. T.

W. T. Welford, “Laser speckle and surface roughness,” Contemp. Phys. 21, 401–412 (1980).
[CrossRef]

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

Wolf, E.

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

Zardecki, A.

A. Zardecki, “Statistical features of phase screens from scattering data,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 155–192.
[CrossRef]

Appl. Phys. (1)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Contemp. Phys. (1)

W. T. Welford, “Laser speckle and surface roughness,” Contemp. Phys. 21, 401–412 (1980).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

W. P. Brown, “Propagation in random media—cumulative effect of weak inhomogeneities,”IEEE Trans. Antennas Propag. AP-15, 81–89 (1967).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A (2)

E. Jakeman, P. N. Pusey, “The statistics of light scatterred by a random phase screen,”J. Phys. A 6, L88–L92 (1973).
[CrossRef]

E. Jakeman, P. N. Pusey, “Non-Gaussian fluctuations in electromagnetic radiation scattered by a random phase screen. I. Theory,”J. Phys. A 8, 369–391 (1975).
[CrossRef]

Opt. Acta (2)

M. Nieto-Vesperinas, N. Garcia, “A detailed study of the scattering of scalar waves from random rough surfaces,” Opt. Acta 28, 1651–1672 (1981).
[CrossRef]

J. C. Dainty, “Some statistical properties of random speckle patterns in coherent and partially coherent illumination,” Opt. Acta 17, 761–772 (1970).
[CrossRef]

Opt. Commun. (1)

E. Wolf, W. H. Carter, “Fields generated by homogeneous and by quasi-homogeneous planar secondary sources,” Opt. Commun. 50, 131–136 (1984).
[CrossRef]

Opt. Quantum Electron. (2)

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

P. J. Chandley, W. T. Welford, “A re-formulation of some results of P. Beckmann for scattering from rough surfaces,” Opt. Quantum Electron. 7, 393–397 (1975).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

Many authors have used a phase-changing screen to model diffraction from the ionosphere. See, for example, H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionospheric problems,” Philos. Trans. R. Soc. London Ser. A 242, 579–607 (1950); J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Rep. Prog. Phys. 19, 188–218 (1956); A. Hewish, “The diffraction of radio waves in passing through a phase-changing ionosphere,” Proc. R. Soc. London Ser. A 209, 81–96 (1951); R. P. Mercier, “Diffraction by a screen causing large random phase fluctuations,” Proc. Cambridge Philos. Soc. 58, 382–400 (1962).
[CrossRef]

Radio Sci. (1)

Strictly speaking, only the second class, with even powers of r, satisfies the conditions of smoothness and small slope assumed in the derivations of the plane-wave transmission functions. See, e.g., D. E. Barrick, “Unacceptable height correlation coefficients and the quasi-specular component in rough surface scattering,” Radio Sci. 5, 647–654 (1970).
[CrossRef]

Other (8)

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics VI, E. Wolf, ed. (North-Holland, Amsterdam, 1967), pp. 53–69.
[CrossRef]

L. G. Shirley, N. George, “Diffuser transmission functions and far-zone speckle patterns,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.556, 63–69 (1985).
[CrossRef]

Unless otherwise indicated, the limits of integration on all integrals are from −∞ to ∞.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

J. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, 1975).

A. Zardecki, “Statistical features of phase screens from scattering data,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 155–192.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vols. 1 and 2.

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Figures (8)

Fig. 1
Fig. 1

Bulk diffuser consisting of a planar slab of thickness H with a constant index of refraction (Section 2) and an inhomogeneous n(x, y) (Section 3).

Fig. 2
Fig. 2

Phase error Δϕλ/H of Eq. (31) plotted against internal angle θ1 for Δn/ n ¯ in the range (a) −0.3–−0.003 and (b) 0.3–0.003. The dashed lines indicate the maximum internal angle θ1 that can be obtained for a given value of n ¯.

Fig. 3
Fig. 3

Rough-surface diffuser of average thickness H and constant index of refraction n.

Fig. 4
Fig. 4

Computer simulation of far-field speckle patterns calculated with paraxial, dashed line, and wide-angle, solid line, transmission functions for plane-wave incidence (a) at 15°, (b) at 30°, and (c) for the sum of three equal-amplitude plane waves at 0° and ±5°.

Fig. 5
Fig. 5

Numerical evaluation of the series D(w0fr)/w02 for (a) an exponential autocorrelation function and (b) a Gaussian autocorrelation function. Separate curves for S in the range of 0.01–4.0 are plotted.

Fig. 6
Fig. 6

Log–log plot of the width Δ(frw0), curves (1) and (2), and the peak D0, curves (a) and (b), against S. The autocorrelation function is an exponential in curves (1) and (a) and a Gaussian in curves (2) and (b). Width is defined as the half-width at half-maximum in the long-dashed lines and by (2D(0)/D″(0)]1/2 in the solid lines. The straight dashed lines at the ends of curves (a) and (b) are asymptotes.

Fig. 7
Fig. 7

Dependence of the shape of D on S. The curves are normalized, and their widths are scaled to have the same curvature at the peak. The autocorrelation function is an exponential for the upper set of curves and a Gaussian for the. lower set. The dashed curves show the limiting shapes that occur for either S ≪ 1 or S ≫ 1.

Fig. 8
Fig. 8

Ensemble-averaged radiation patterns as calculated with paraxial, dashed line, and wide-angle, solid line, transmission functions for plane-wave incidence at 0° and 45°. The autocorrelation function is an exponential in (a) and a Gaussian in (b). Both plots are for a bulk diffuser with S0 = 2.5, w0 = 25λ0, and n ¯ = 1.3.

Equations (90)

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

v 2 ( x , y ) = v 1 ( x , y ) t ( x , y ) ,
t ( x , y ) = exp [ - i ϕ ( x , y ) ]
Δ v 2 ( x , y ; f x 0 , f y 0 ) = exp [ i 2 π ( f x 0 x + f y 0 y ) ] t ( x , y ; f x 0 , f y 0 ) .
E ( x , y , z ; ν ) = E y r ( x , y , z ; t ) exp ( - i 2 π ν t ) d t
E y r ( x , y , z ; t ) = E ( x , y , z ; ν ) exp ( i 2 π ν t ) d ν .
v ( x , y , z ; ν ) = [ 1 + sgn ( ν ) ] E ( x , y , z ; ν ) ,
sgn ( ν ) = { 1 , ν > 0 - 1 , ν < 0 .
E y ( x , y , z ; t ) = v ( x , y , z ; ν ) exp ( i 2 π ν t ) d ν .
v ( x , y , z ; ν ) = v ( x , y , 0 ; ν ) S n d x d y ,
S n = exp ( - i k R 1 ) 2 π R 1 ( i k + 1 R 1 ) z R 1 .
R 1 = [ ( x - x ) 2 + ( y - y ) 2 + z 2 ] 1 / 2 .
V ( f x , f y ; z ; ν ) = v ( x , y , z ; ν ) exp [ - i 2 π ( f x x + f y y ) ] d x d y ,
v ( x , y , z ; ν ) = V ( f x , f y ; z ; ν ) exp [ i 2 π ( f x x + f y y ) ] d f x d f y .
V ( f x , f y ; z ; ν ) = V ( f x , f y ; 0 ; ν ) exp { - i 2 π z [ ( k / 2 π ) 2 - f x 2 - f y 2 ] 1 / 2 } .
E y r ( x , y , z ; t ) = cos [ 2 π ν 0 t - k sin θ 0 ( x cos ϕ 0 + y sin ϕ 0 - k z cos θ 0 - Φ ) ] ,
v ( x , y ; z ; ν ) = δ ( ν - ν 0 ) exp { - i [ k sin θ 0 ( x cos ϕ 0 + y sin ϕ 0 ) + k z cos θ 0 + Φ ] }
V ( f x , f y ; z ; ν ) = δ ( ν - ν 0 ) δ ( f x + sin θ 0 cos ϕ 0 / λ ) δ ( f y + sin θ 0 × cos ϕ 0 / λ ) exp [ - i ( k z cos θ 0 + Φ ) ] .
f x 0 = - sin θ 0 cos ϕ 0 λ = - x λ R 0
f y 0 = - sin θ 0 sin ϕ 0 λ = - y λ R 0 ,
R 0 = ( x 2 + y 2 + z 2 ) 1 / 2 .
cos θ 0 = [ 1 - λ 2 ( f x 0 2 + f y 0 2 ) ] 1 / 2 .
v 1 ( x , y ) = exp [ i 2 π ( f x x + f y y ) ]
sin θ 1 = sin θ 0 n ( x , y ) ,             ϕ 1 = ϕ 0 .
v 2 ( x , y ) = exp [ - i k 0 n ( x , y ) sin θ 1 ( x cos ϕ 1 + y sin ϕ 1 ) ] × exp [ - i k 0 n ( x , y ) H cos θ 1 ] ,
t ( x , y ; f x , f y ) = exp [ - i k 0 n ( x , y ) H cos θ 1 ] = exp ( - i 2 π H { [ n ( x , y ) λ 0 ] 2 - f x 2 - f y 2 } 1 / 2 ) .
Δ Φ = k 0 n H ( 1 - cos θ 1 ) .
n ( x , y ) = n ¯ + Δ n ( x , y ) ,
Δ n n ¯ 2 cos 2 θ ¯
cos θ ¯ = [ 1 - ( λ 0 / n ¯ ) 2 ( f x 2 + f y 2 ) ] 1 / 2 ,
t ( x , y ; f x , f y ) = exp ( - i k 0 n ¯ H cos θ ¯ ) exp [ - i k 0 Δ n ( x , y ) H / cos θ ¯ ] .
Δ Φ = k 0 H [ n ¯ cos θ ¯ - ( n ¯ + Δ n ) cos θ 1 + Δ n / cos θ ¯ ] .
Φ 1 = - k 0 n [ H + h ( x , y ) ] cos θ 1 .
Φ 2 = k 0 h ( x , y ) cos θ 2 .
t ( x , y ; f x , f y ) = exp { - i k 0 [ H n cos θ 1 + h ( x , y ) × ( n cos θ 1 - cos θ 2 ) ] } .
cos θ 2 = ( { ( 1 + h x 2 + h y 2 ) ( 1 - n 2 ) + [ sin θ 0 ( h x cos ϕ 0 + h y sin ϕ 0 ) - n cos θ 1 ] 2 } 1 / 2 + sin θ 0 ( h x cos ϕ 0 + h y sin ϕ 0 ) + ( h x 2 + h y 2 ) n cos θ 1 ) / ( 1 + h x 2 + h y 2 ) ,
h x = h ( x , y ) x ,             h y = h ( x , y ) y ,
sin θ 1 = sin θ 0 n .
cos θ 2 cos θ 0 - tan θ 0 ( h x cos ϕ 0 + h y sin ϕ 0 ) ( n cos θ 1 - cos θ 0 ) .
t ( x , y ; f x , f y ) = exp ( - i k 0 { H n cos θ 1 + h ( x , y ) ( n cos θ 1 - cos θ 0 ) [ 1 + tan θ 0 ( h x cos ϕ 0 + h y sin ϕ 0 ) ] } ) .
tan θ 0 ( h x cos ϕ 0 + h y sin ϕ 0 ) h / λ 0 1
t ( x , y ; f x , f y ) = exp ( - i k 0 H n cos θ 1 ) × exp [ - i k 0 h ( x , y ) ( n cos θ 1 - cos θ 0 ) ] .
d v 2 ( x , y ) = V 1 ( f x , f y ) exp [ i 2 π ( f x x + f y y ) ] t ( x , y ; f x , f y ) d f x d f y .
v 2 ( x , y ) = V 1 ( f x , f y ) t ( x , y ; f x , f y ) exp [ i 2 π ( f x x + f y y ) ] d f x d f y .
v 2 ( x , y ) = v 1 ( x , y ) g ( x , y ; x , y ) d x d y ,
g ( x , y ; x , y ) = t ( x , y ; f x , f y ) exp { i 2 π [ f x ( x - x ) + f y ( y - y ) ] } d f x d f y .
t ( x , y ; f x , f y ) = exp [ - i 2 π ( f x x + f y y ) ] g ( x , y ; x , y ) × exp [ i 2 π ( f x x + f y y ) ] d x d y .
v ( x , y , z ) = i exp ( - i k R 0 ) R 0 λ cos θ v ( x , y , 0 ) × exp [ - i 2 π ( f x x + f y y ) ] d x d y ,
v ( x , y , z ) = i exp ( - i k R 0 ) R 0 λ cos θ V ( f x , f y ; 0 ) ,
V 2 a ( f x , f y ) = d x d y a ( x , y ) exp [ - i 2 π ( f x x + f y y ) ] × V 1 ( f x , f y ) t ( x , y ; f x , f y ) × exp [ i 2 π ( f x x + f y y ) ] d f x d f y ,
V 2 a ( f x , f y ) = d x d y exp [ - i 2 π ( f x x + f y y ) ] × V 1 a ( f x , f y ) × t ( x , y ' f x , f y ) exp [ i 2 π ( f x x + f y y ) ] d f x d f y ,
V 1 a ( f x , f y ) = V 1 ( f x , f y ) * A ( f x , f y ) .
U ( f x , f y ) = V 2 a ( f x , f y ) 2 ,
v ( x , y , z ) 2 = ( cos θ R 0 λ ) 2 U ( f x , f y ) .
R Δ n ( x 2 - x 1 ) = Δ n ( x 1 ) Δ n ( x 2 ) / Δ n 2 = exp [ - ( x 2 - x 1 ) 2 / w 0 2 ] .
U ( f x , f y ) = V 2 a ( f x , f y ) 2 .
v ( x , y , z ) 2 = ( cos θ R 0 λ ) 2 U ( f x , f y ) .
R t ( x 2 - x 1 , y 2 - y 1 ; f x , f y ) = t * ( x 1 , y 1 ; f x , f y ) t ( x 2 , y 2 ; f x , f y ) .
U ( f x , f y ) = d f x d f y V 1 a ( f x , f y ) 2 R t ( x , y ; f x , f y ) × exp { - i 2 π [ ( f x - f x ) x + ( f y - f y ) y ] } d x d y .
U ( f x , f y ) = d f ξ 1 d f η 1 d f ξ 2 d f η 2 d f x d f y V 1 * ( f ξ 1 , f η 1 ) V 1 ( f ξ 2 , f η 2 ) × A * ( f x , f y ) A ( f x + f ξ 1 - f ξ 2 , f y + f η 1 - f η 2 ) × t * ( 0 , 0 ; f ξ 1 , f η 1 ) t ( x , y ; f ξ 2 , f η 2 ) × exp { - i 2 π [ ( f x - f x - f ξ 1 ) x + ( f y - f y - f η 1 ) y ] } d x d y .
U ( f x , f y ; f x 0 , f y 0 ) = d f x d f y A ( f x , f y ) 2 × R t ( x , y ; f x 0 , f y 0 ) × exp { - i 2 π [ ( f x - f x - f x 0 ) x + ( f y - f y - f y 0 ) y ] } d x d y .
U ( f x , f x ; f x 0 , f y 0 ) = A ( f x - f x 0 , f y - f y 0 ) 2 * U D ( f x , f y ; f x 0 , f y 0 ) ,
U D ( f x , f y ; f x 0 , f y 0 ) = R t ( x , y ; f x 0 , f y 0 ) × exp [ - i 2 π ( f x x + f y y ) ] d x d y ,
U ( f x , f y ; f x 0 , f y 0 ) = A ( f x - f x 0 , f y - f y 0 ) 2 .
A ( f x - f x 0 , f y - f y 0 ) 2 A 0 2 δ ( f x - f x 0 , f y - f y 0 ) ,
A 0 2 = A ( f x , f y ) 2 d f x d f y = a ( x , y ) 2 d x d y .
U D ( f x , f y ; f x 0 , f y 0 ) = R t ( , ; f x 0 , f y 0 ) [ δ ( f x ) δ ( f y ) - B ( f x , f y ) ] + b ( x , y ) R t ( x , y ; f x 0 , f y 0 ) × exp [ - i 2 π ( f x x + f y y ) ] d x d y ,
U ( f x , f y ; f x 0 , f y 0 ) = R t ( , ; f x 0 , f y 0 ) [ A ( f x - f x 0 , f y - f y 0 ) 2 - A 0 2 B ( f x - f x 0 , f y - f y 0 ) ] + A 0 2 × b ( x , y ) R t ( x , y ; f x 0 , f y 0 ) × exp { - i 2 π [ ( f x - f x 0 ) x + ( f y - f y 0 ) y ] } × d x d y .
R t ( x , y ; f x , f y ) = exp { i k 0 H [ Δ n ( 0 , 0 ) - Δ n ( x , y ) ] / cos θ ¯ } .
R t ( x , y ; f x , f y ) = C ( η 1 , η 2 ; x , y ) = exp { i [ η 1 Δ n ( 0 , 0 ) + η 2 Δ n ( x , y ) ] } ,
η 1 = - η 2 = k 0 H / cos θ ¯ .
C ( η 1 , η 2 ; x , y ) = exp { - ½ Δ n 2 [ η 1 2 + 2 R Δ n ( x , y ) η 1 η 2 + η 2 2 ] }
R Δ n ( x , y ) = Δ n ( x , y ) Δ n ( x + x , y + y ) Δ n 2 .
R t ( x , y ; f x , f y ) = exp { - S 2 [ 1 - R Δ n ( x , y ) ] } ,
S = Δ n 2 1 / 2 k 0 H / cos θ ¯ .
S = S 0 / cos θ ¯ .
S = ( n cos θ 1 - cos θ 0 ) k 0 h 2 1 / 2 .
S = S 0 ( n cos θ 1 - cos θ 0 ) / ( n - 1 ) .
r = ( x 2 + y 2 ) 1 / 2 .
U ( f x , f y ; f x 0 , f y 0 ) = A ( f x - f x 0 , f y - f y 0 ) 2 * 2 π 0 r J 0 [ 2 π r ( f x 2 + f y 2 ) 1 / 2 ] R t ( r , f x 0 , f y 0 ) d r .
U D ( f r ; f r 0 ) = 2 π 0 r J 0 ( 2 π r f r ) exp { - S 2 [ 1 - R Δ n ( r ) ] } d r ,
f r = ( f x 2 + f y 2 ) 1 / 2 .
R Δ n ( r ) = exp ( - r / w 0 ) ,
R Δ n ( r ) = exp ( - r 2 / w 0 2 ) .
R Δ n ( r ) 1 - r / w 0 ,
R Δ n ( r ) 1 - r 2 / w 0 2 .
U D ( f r ; f r 0 ) = exp ( - S 2 ) { δ ( f x ) δ ( f y ) + 2 π w 0 2 m = 1 S 2 m m 2 m ! × [ 1 + ( 2 π f r w 0 / m ) 2 ] - 3 / 2 } .
U D ( f r ; f r 0 ) = 2 π ( w 0 / S 2 ) 2 [ 1 + ( 2 π f r w 0 / S 2 ) 2 ] - 3 / 2 .
U D ( f r ; f r 0 ) = exp ( - S 2 ) { δ ( f x ) δ ( f y ) + π w 0 2 m = 1 S 2 m m m ! × exp [ - ( π f r w 0 ) 2 / m ] } .
U D ( f r ; f r 0 ) = π ( w 0 / S ) 2 exp [ - ( π f r w 0 / S ) 2 ] .
D ( f r w 0 ) D 0 { 1 - [ f r w 0 / Δ ( f r w 0 ) ] 2 } .

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