Abstract

We introduce a profilometry sensor that combines phase shifting with a Talbot self-image of a sinusoidal grating as the illumination part of the sensor. Contrast of the Talbot diffraction pattern produced with a sinusoidal grating in a diverging beam is theoretically discussed and verified experimentally. The mathematical relationship that is used to convert the phase measured with this sensor to the corresponding relief of an object is derived in the Appendix. A ceramic former used in the production of lenses was profiled with this sensor, and measurement results are presented.

© 1994 Optical Society of America

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References

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  1. R. E. Brooks, L. O. Heflinger, “Moiré gauging using optical interference patterns,” Appl. Opt. 8, 935–939 (1969)
    [CrossRef] [PubMed]
  2. D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of surface contours by Moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  3. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
    [CrossRef] [PubMed]
  4. P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
    [CrossRef]
  5. G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. 17, 2930–2933 (1978).
    [CrossRef] [PubMed]
  6. D. T. Moore, B. E. Truax, “Phase-locked Moiré fringe analysis for automated contouring of diffuse surfaces,” Appl. Opt. 18, 91–96 (1979).
    [CrossRef] [PubMed]
  7. G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring Moiré topography,” Opt. Laser Technol. 16(6), 315–319 (December1984).
    [CrossRef]
  8. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef] [PubMed]
  9. B. W. Bell, “Digital heterodyne topography,” in Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 754–768 (1987).
  10. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).
  11. A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection Moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greiven-kamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 120–129 (1992).
  12. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  13. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
    [CrossRef] [PubMed]
  14. V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
    [CrossRef] [PubMed]
  15. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, Chap. 5, pp. 350–393.
    [CrossRef]
  16. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  17. J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  18. X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
    [CrossRef]
  19. B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).
  20. M. Gruber, G. Hausler, “Simple, robust and accurate phase-measuring triangulation,” Optik 89(3), 118–122 (1992).
  21. P. Chavel, T. C. Strand, “Range measurement using Talbot diffraction imaging of gratings,” Appl. Opt. 23, 862–871 (1984).
    [CrossRef] [PubMed]
  22. L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
    [CrossRef]
  23. D. Joyeux, Y. Cohen-Sabban, “High magnification self-imaging,” Appl. Opt. 21, 625–627 (1982).
    [CrossRef] [PubMed]
  24. P. Latimer, R. F. Course, “Talbot effect reinterpreted,” Appl. Opt. 31, 80–89 (1992).
    [CrossRef] [PubMed]
  25. K. Patorski, S. Kozak, “Self-imaging with nonparabolic approximation of spherical wave fronts,” J. Opt. Soc. Am. A 5, 1322–1327 (1988).
    [CrossRef]
  26. K. Patorski, “Self-imaging and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1989), Vol. 27, Chap. 1, pp. 1–108.
    [CrossRef]
  27. A. Lohmann, “Grating diffraction spectra as coherent light sources for two- or three-beam interferometry,” Opt. Acta 9, 1–12 (1962).
    [CrossRef]
  28. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]

1993 (3)

J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

1992 (3)

1988 (1)

1985 (1)

1984 (3)

1982 (2)

1979 (1)

1978 (1)

1975 (2)

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

1974 (1)

1970 (2)

1969 (1)

1962 (1)

A. Lohmann, “Grating diffraction spectra as coherent light sources for two- or three-beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

Allen, J. B.

Bell, B. W.

B. W. Bell, “Digital heterodyne topography,” in Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 754–768 (1987).

Benoit, P.

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Bourimborde, L. V.

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

Brangaccio, D. J.

Brooks, R. E.

Bruning, J. H.

Chavel, P.

Cohen-Sabban, Y.

Colautti, C. M. V.

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

Course, R. F.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, Chap. 5, pp. 350–393.
[CrossRef]

Fagg, A. H.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection Moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greiven-kamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 120–129 (1992).

Fairman, P.

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).

Gallagher, J. E.

Ghaffari, M.

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).

Glatt, I.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

Gruber, M.

M. Gruber, G. Hausler, “Simple, robust and accurate phase-measuring triangulation,” Optik 89(3), 118–122 (1992).

Hales, B.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection Moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greiven-kamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 120–129 (1992).

Halioua, M.

Hausler, G.

M. Gruber, G. Hausler, “Simple, robust and accurate phase-measuring triangulation,” Optik 89(3), 118–122 (1992).

Heflinger, L. O.

Herriott, D. R.

Hormiere, J.

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Huntley, J. M.

Ina, H.

Indebetouw, G.

Johnson, W. O.

Joyeux, D.

Kafri, O.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

Kobayashi, S.

Kozak, S.

Larkin, K. G.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).

Latimer, P.

Liu, H. C.

Lohmann, A.

A. Lohmann, “Grating diffraction spectra as coherent light sources for two- or three-beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

Mathieu, E.

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Meadows, D. M.

Messer, H. I.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring Moiré topography,” Opt. Laser Technol. 16(6), 315–319 (December1984).
[CrossRef]

Moore, D. T.

Oreb, B. F.

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).

Patorski, K.

K. Patorski, S. Kozak, “Self-imaging with nonparabolic approximation of spherical wave fronts,” J. Opt. Soc. Am. A 5, 1322–1327 (1988).
[CrossRef]

K. Patorski, “Self-imaging and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1989), Vol. 27, Chap. 1, pp. 1–108.
[CrossRef]

Reid, G. T.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring Moiré topography,” Opt. Laser Technol. 16(6), 315–319 (December1984).
[CrossRef]

Rixon, R. C.

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring Moiré topography,” Opt. Laser Technol. 16(6), 315–319 (December1984).
[CrossRef]

Rosenfeld, D. P.

Saldner, H.

Sicre, E. E.

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

Srinivasan, V.

Stahl, H. P.

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection Moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greiven-kamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 120–129 (1992).

Strand, T. C.

Su, X. Y.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Takasaki, H.

Takeda, M.

Thomas, A.

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Tonso, A. O.

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

Truax, B. E.

von Bally, G.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Vukicevic, D.

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

White, A. D.

Wyant, J. C.

Appl. Opt. (13)

R. E. Brooks, L. O. Heflinger, “Moiré gauging using optical interference patterns,” Appl. Opt. 8, 935–939 (1969)
[CrossRef] [PubMed]

D. M. Meadows, W. O. Johnson, J. B. Allen, “Generation of surface contours by Moiré patterns,” Appl. Opt. 9, 942–947 (1970).
[CrossRef] [PubMed]

H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef] [PubMed]

J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, D. R. Herriott, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14, 2622–2626 (1975).
[CrossRef] [PubMed]

V. Srinivasan, H. C. Liu, M. Halioua, “Automated phase-measuring profilometry: a phase mapping approach,” Appl. Opt. 24, 185–188 (1985).
[CrossRef] [PubMed]

G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. 17, 2930–2933 (1978).
[CrossRef] [PubMed]

D. T. Moore, B. E. Truax, “Phase-locked Moiré fringe analysis for automated contouring of diffuse surfaces,” Appl. Opt. 18, 91–96 (1979).
[CrossRef] [PubMed]

J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

P. Chavel, T. C. Strand, “Range measurement using Talbot diffraction imaging of gratings,” Appl. Opt. 23, 862–871 (1984).
[CrossRef] [PubMed]

D. Joyeux, Y. Cohen-Sabban, “High magnification self-imaging,” Appl. Opt. 21, 625–627 (1982).
[CrossRef] [PubMed]

P. Latimer, R. F. Course, “Talbot effect reinterpreted,” Appl. Opt. 31, 80–89 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

Nouv. Rev. Opt. (1)

P. Benoit, E. Mathieu, J. Hormiere, A. Thomas, “Characterization and control of three dimensional objects using fringe projection techniques,” Nouv. Rev. Opt. 6, 67–86 (1975).
[CrossRef]

Opt. Acta (1)

A. Lohmann, “Grating diffraction spectra as coherent light sources for two- or three-beam interferometry,” Opt. Acta 9, 1–12 (1962).
[CrossRef]

Opt. Commun. (2)

L. V. Bourimborde, A. O. Tonso, C. M. V. Colautti, E. E. Sicre, “Real-time measurement of the meniscus shape using the Talbot effect,” Opt. Commun. 102, 397–400 (1993).
[CrossRef]

X. Y. Su, G. von Bally, D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141–150 (1993).
[CrossRef]

Opt. Laser Technol. (1)

G. T. Reid, R. C. Rixon, H. I. Messer, “Absolute and comparative measurements of three-dimensional shape by phase measuring Moiré topography,” Opt. Laser Technol. 16(6), 315–319 (December1984).
[CrossRef]

Optik (1)

M. Gruber, G. Hausler, “Simple, robust and accurate phase-measuring triangulation,” Optik 89(3), 118–122 (1992).

Other (6)

K. Patorski, “Self-imaging and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1989), Vol. 27, Chap. 1, pp. 1–108.
[CrossRef]

B. F. Oreb, K. G. Larkin, P. Fairman, M. Ghaffari, “Moiré based optical surface profiler for the minting industry,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greivenkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 48–57 (1992).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Pergamon, New York, 1988), Vol. 26, Chap. 5, pp. 350–393.
[CrossRef]

B. W. Bell, “Digital heterodyne topography,” in Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.814, 754–768 (1987).

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley Interscience, New York, 1990).

A. H. Fagg, B. Hales, H. P. Stahl, “Systematic errors of a projection Moiré contouring system,” in Interferometry: Surface Characterization and Testing, K. Creath, J. E. Greiven-kamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1776, 120–129 (1992).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of a profiling sensor that uses a Talbot self-imaged grating.

Fig. 2
Fig. 2

Visibility of the self-imaged sinusoidal grating in divergent illumination as a function of distance from the physical grating.

Fig. 3
Fig. 3

Profiling sensor that was used in our measurements.

Fig. 4
Fig. 4

Visibility of a Talbot self-imaged grating in the vicinity of the object location shown in Fig. 3.

Fig. 5
Fig. 5

Detected self-imaged irradiance pattern on the surface of the ceramic former.

Fig. 6
Fig. 6

Wrapped phase map of the ceramic former with a cross-sectional profile of the phase.

Fig. 7
Fig. 7

Pseudo-gray-scale relief map of the measured ceramic former.

Fig. 8
Fig. 8

Three-dimensional relief plot of the measured ceramic former.

Fig. 9
Fig. 9

Schematic diagram of the profiling sensor that uses divergent illumination and a Talbot self-imaged grating. TVC, television camera.

Equations (39)

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E c = E 0 ( x , y ) [ 1 + γ ( x , y ) cos ( 2 π O Q / p ) ] ,
ϕ ( x ) = 2 π O Q / p
M ( z ) = 1 + ( z / d ) ,
1 z T + 1 d = λ n p 2 ,
V ( z ) = E ( x , z ) max E ( x , z ) min E ( x , z ) max + E ( x , z ) min ,
V ( z ) coll = E ( 0 , z ) E ( p / 2 , z ) E ( 0 , z ) + E ( p / 2 , z ) .
V ( z ) div = E ( 0 , z ) E [ p / 2 M ( z ) , z ] E ( 0 , z ) + E [ p / 2 M ( z ) , z ] .
E ( x , z ) = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos [ π λ z d p 2 ( z + d ) ] cos [ 2 π x p ( 1 + z / d ) ] + 2 A 1 2 cos [ 4 π x p ( 1 + z / d ) ] ,
T ( x g , y g ) = A 0 + 2 A 1 cos ( 2 π x g / p ) .
V ( z ) div = 4 A 0 A 1 cos [ π λ z d p 2 ( z + d ) ] A 0 2 + 4 A 1 2 .
V ( z ) div = 4 A 0 A 1 cos ( π λ z p 2 ) 1 M ( z ) A 0 2 + 4 A 1 2 .
V ( z ) coll = 4 A 0 A 1 cos ( π λ z p 2 ) A 0 2 + 4 A 1 2 .
λ z d p 2 ( z + d ) = n
x = ± p ( 1 + z d ) z + d 3 λ ,
y = ± p ( 1 + z d ) z + d λ .
E ( x , y ) = E 0 ( x , y ) { 1 + γ ( x , y ) cos [ 2 π x p M ( x ) + Ω ] } ,
E ( x , y ) = E 0 ( x , y ) { 1 + γ ( x , y ) cos [ 2 π x p M ( x ) + Ω ] } .
η = x x .
tan Ψ = x / r = x / ( r + h ) ;
x = x + x h / r .
x z + d = x h tan θ z + d + h sec θ .
x = ( z + d ) cos θ h + ( z + d ) cos θ [ x ( 1 + h r ) h tan θ ] .
η = h ( z + d ) cos θ h + ( z + d ) cos θ { x [ 1 ( z + d ) cos θ 1 r ] + tan θ } .
δ ϕ = 2 π p [ x M ( x ) x M ( x ) ] .
tan β = x cos θ ( z + d + x sin θ ) = x g d .
M ( x ) = x / x g .
M ( x ) = z + d + x sin θ d cos θ .
δ ϕ = 2 π d cos θ ( x x ) p ( z + d ) ( 1 + x k ) ( 1 + x k ) ,
k = sin θ z + d .
δ ϕ = η 2 π d cos θ p ( z + d ) ( 1 + k x ) 2 ( 1 k η 1 + k x ) .
1 k η / ( 1 + k x ) ;
k η / ( 1 + k x ) < 0.028 .
δ ϕ = η 2 π d cos θ p ( z + d ) ( 1 + k x ) 2 .
h = η σ [ 1 η σ ( z + d ) cos θ ] ,
σ = x [ 1 ( z + d ) cos θ 1 r ] + tan θ .
1 η σ ( z + d ) cos θ ;
h η / σ .
h δ ϕ p ( z + d ) ( 1 + k x ) 2 2 π d σ cos θ ,
h = δ ϕ p ( z + d ) ( 1 + k x ) 2 2 π d σ cos θ { 1 + δ ϕ p ( 1 + k x ) 2 π d cos θ [ sin θ ( 1 + k x ) σ cos θ ] } .

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