Abstract

We demonstrate how real-time holographic interferometry yielding two-dimensional fringes can be recorded and used to determine changes in three-dimensional attitude of a model airplane through digital image processing. A simple bench-top experiment with a model airplane as a test object is conducted to demonstrate interference fringes superposed on the image due to changes in attitudes (pitch, yaw, and roll) as well as distortion. A novel second-generation thermoplastic camera suitable for dynamic multiple reversible registration of thin-phase holograms using thermoplastic and semiconductor film on glass substrate is used for in situ recording and readout during real-time holographic interferometry. Thin-phase holograms also offer the advantage of exact image reconstruction from forward-phase conjugation.

© 2008 Optical Society of America

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  1. N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
    [CrossRef]
  2. N. Noginova, N. Kukhtarev, T. Kukhtareva, M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, D. Parker, and P. P. Banerjee, “Photoinduced electric current in Fe-doped KNbO3,” J. Opt. Soc. Am. B 14, 1390-1395 (1997).
    [CrossRef]
  3. N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
    [CrossRef]
  4. M. R. R. Gesualdi, M. Mori, M. Muramatsu, E. A. Liberti, and E. Munin, “Phase-shifting real-time holographic interferometry applied to load transmission evaluation in dried human skull,” Appl. Opt. 46, 5419-5429 (2007).
    [CrossRef]
  5. J. D. Liou, C. K. Lee, and K. C. Wu, “Photorefractive crystal-based holographic interferometry system for full-field wave propagation metrology,” Opt. Express 15, 5460-5472(2007).
    [CrossRef]
  6. T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.
  7. H. Jinhu and M. Tianxiang, “The application of moiré interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip,” Acta Mech. Sinica 7, 376-382 (1991).
    [CrossRef]
  8. Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
    [CrossRef]
  9. D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).
  10. D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).
  11. P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).
  12. C. M. Vest, Holographic Interferometry (Wiley, 1979).
  13. T. Kreis, Holographic Interferometry: Principles and Methods (Verlag, 1996).
  14. D. Dirksen and G. Von Bally, “Holographic double-exposure interferometry in the near real time with photorefractive crystals,” J. Opt. Soc. Am. B 11, 1858-1863 (1994).
  15. A. Lyalikov, “Real-time holographic interferometry using superposed compensating holograms,” Tech. Phys. 52, 1040-1045 (2007).
    [CrossRef]
  16. L. H. Lin and H. L. Beauchamp, “Write-read-erase in situ optical memory using thermoplastic holograms,” Appl. Opt. 9, 2088-2092 (1970).
  17. G. Pedrini, W. Osten, and M. E. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456-3462 (2006).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  19. D. W. Robinson, “Automatic fringe analysis with a computer image-processing system,” Appl. Opt. 22, 2169-2176 (1983).

2007 (4)

Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
[CrossRef]

A. Lyalikov, “Real-time holographic interferometry using superposed compensating holograms,” Tech. Phys. 52, 1040-1045 (2007).
[CrossRef]

J. D. Liou, C. K. Lee, and K. C. Wu, “Photorefractive crystal-based holographic interferometry system for full-field wave propagation metrology,” Opt. Express 15, 5460-5472(2007).
[CrossRef]

M. R. R. Gesualdi, M. Mori, M. Muramatsu, E. A. Liberti, and E. Munin, “Phase-shifting real-time holographic interferometry applied to load transmission evaluation in dried human skull,” Appl. Opt. 46, 5419-5429 (2007).
[CrossRef]

2006 (1)

2005 (1)

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

2002 (1)

D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).

1997 (1)

1995 (1)

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

1994 (1)

1991 (1)

H. Jinhu and M. Tianxiang, “The application of moiré interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip,” Acta Mech. Sinica 7, 376-382 (1991).
[CrossRef]

1983 (1)

1970 (1)

Baba, M.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Banerjee, P. P.

N. Noginova, N. Kukhtarev, T. Kukhtareva, M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, D. Parker, and P. P. Banerjee, “Photoinduced electric current in Fe-doped KNbO3,” J. Opt. Soc. Am. B 14, 1390-1395 (1997).
[CrossRef]

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Beauchamp, H. L.

Bütefisch, K. A.

D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).

Caulfield, H. J.

N. Noginova, N. Kukhtarev, T. Kukhtareva, M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, D. Parker, and P. P. Banerjee, “Photoinduced electric current in Fe-doped KNbO3,” J. Opt. Soc. Am. B 14, 1390-1395 (1997).
[CrossRef]

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Dirksen, D.

Edwards, M. E.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

Fujigaki, M.

Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
[CrossRef]

Garcia, D.

D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).

Gesualdi, M. R. R.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gusev, M. E.

Hariharan, P.

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

Hesselink, L.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Jinhu, H.

H. Jinhu and M. Tianxiang, “The application of moiré interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip,” Acta Mech. Sinica 7, 376-382 (1991).
[CrossRef]

Jones, J.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

Kato, N.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Kojima, A.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Kreis, T.

T. Kreis, Holographic Interferometry: Principles and Methods (Verlag, 1996).

Kukhtarev, N.

Kukhtarev, N. V.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Kukhtareva, T.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

N. Noginova, N. Kukhtarev, T. Kukhtareva, M. A. Noginov, H. J. Caulfield, P. Venkateswarlu, D. Parker, and P. P. Banerjee, “Photoinduced electric current in Fe-doped KNbO3,” J. Opt. Soc. Am. B 14, 1390-1395 (1997).
[CrossRef]

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Lee, C. K.

Liberti, E. A.

Lin, L. H.

Liou, J. D.

Lyalikov, A.

A. Lyalikov, “Real-time holographic interferometry using superposed compensating holograms,” Tech. Phys. 52, 1040-1045 (2007).
[CrossRef]

Lyuksyutov, S. F.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

Matsumoto, T.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Mori, M.

Morimoto, Y.

Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
[CrossRef]

Munin, E.

Muramatsu, M.

Noginov, M. A.

Noginova, N.

Orteu, J. J.

D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).

Osten, W.

Pallek, D.

D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).

Parker, D.

Pedrini, G.

Penazzi, L.

D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).

Quest, J.

D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).

Reagan, M. A.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

Robinson, D. W.

Strudthoff, W.

D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).

Tamiwa, M.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Tianxiang, M.

H. Jinhu and M. Tianxiang, “The application of moiré interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip,” Acta Mech. Sinica 7, 376-382 (1991).
[CrossRef]

Venkateswarlu, P.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, 1979).

Von Bally, G.

Wang, J.

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

Watanabe, T.

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

Wu, K. C.

Yamamoto, Y.

Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
[CrossRef]

Yu, H.-L.

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Acta Mech. Sinica (1)

H. Jinhu and M. Tianxiang, “The application of moiré interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip,” Acta Mech. Sinica 7, 376-382 (1991).
[CrossRef]

Appl. Opt. (4)

J. Appl. Phys. (1)

N. V. Kukhtarev, T. Kukhtareva, M. E. Edwards, J. Jones, J. Wang, S. F. Lyuksyutov, and M. A. Reagan, “Smart photogalvanic running-grating interferometer,” J. Appl. Phys. 97, 054301 (2005).
[CrossRef]

J. Mater. Process. Technol. (1)

D. Garcia, J. J. Orteu, and L. Penazzi, “A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming,” J. Mater. Process. Technol. 125-126, 736-742(2002).

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

Meas. Sci. Technol. (1)

Y. Yamamoto, Y. Morimoto, and M. Fujigaki, “Two-directional phase-shifting moiré interferometry and its application to thermal deformation measurement of an electronic device,” Meas. Sci. Technol. 18, 561-566 (2007).
[CrossRef]

Opt. Eng. (1)

N. V. Kukhtarev, T. Kukhtareva, H. J. Caulfield, P. P. Banerjee, H.-L. Yu, and L. Hesselink, “Broadband dynamic, holographically selfrecorded, and static hexagonal scattering patterns in photorefractive potassium niobate,” Opt. Eng. 34, 2261-2265 (1995).
[CrossRef]

Opt. Express (1)

Tech. Phys. (1)

A. Lyalikov, “Real-time holographic interferometry using superposed compensating holograms,” Tech. Phys. 52, 1040-1045 (2007).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

P. Hariharan, Optical Interferometry, 2nd ed. (Academic, 2003).

C. M. Vest, Holographic Interferometry (Wiley, 1979).

T. Kreis, Holographic Interferometry: Principles and Methods (Verlag, 1996).

D. Pallek, K. A. Bütefisch, J. Quest, and W. Strudthoff, “Model deformation measurement in ETW using the moiré technique,” presented at 20th International Congress on Instrumentation in Aerospace Simulation Facilities, Göttingen, Germany, 25-29 August 2003).

T. Matsumoto, A. Kojima, N. Kato, T. Watanabe, M. Tamiwa, and M. Baba, “Deformation analysis of the human femur by holographic interferometry,” in Proceedings of 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (IEEE, 2007), pp. 4699-4702.

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Figures (11)

Fig. 1
Fig. 1

Experimental setup for holographic interferometry of diffuse object (attitude and deformation determination).

Fig. 2
Fig. 2

Recording and reconstruction of holograms from diffuse objects.

Fig. 3
Fig. 3

Coordinate system with airplane, showing pitch, yaw, and roll axes.

Fig. 4
Fig. 4

(a)–(c) Interferograms superposed on image for different amounts of pitch and roll, respectively. (d) Interferogram superposed on image for pitch and yaw.

Fig. 5
Fig. 5

(a) Interferogram upon initial deformation of the airplane. (b) Interferogram after relaxation of the airplane for 5 min .

Fig. 6
Fig. 6

Typical plots of fringe periods in x and y, and fringe angle, for the case of pitch, yaw, and roll with Δ ϕ = 0.01 , Δ ψ = 0.01 , and Δ θ = 0.01 .

Fig. 7
Fig. 7

(a)–(e) Figures derived from Fig. 4 with dots indicating the locations where the attitude changes have been calculated.

Fig. 8
Fig. 8

Water bubble reconstruction using the same setup used for the plane model system. The non-Bragg order on the left constitutes the phase conjugate, which is the reconstructed water bubble. The fringes around it are due to the vibration of the slide that holds the water bubble. The middle beam is the reference beam and the right beam is the signal beam. The phase doubled non-Bragg order, which appears to the right of the signal beam, is not shown.

Fig. 9
Fig. 9

(a), (b) Localized sections from Figs. 7d, 7e, respectively, where the fringe periods and angles are calculated at the locations of the crosses (which correspond to the location of the dots in Figs. 7d, 7e.

Fig. 10
Fig. 10

Plot of localized intensity averaging around the cross in Fig. 9b along the vertical direction versus the angle by which Fig. 9b is rotated. The rotation angle at which maximum averaged intensity is achieved yields the angle of the fringes.

Fig. 11
Fig. 11

Plot of intensity values versus horizontal position in the localized image, which has been rotated by the angle derived from the results of Fig. 10. This rotated image now contains vertical fringes. The local fringe period is calculated by digital processing of the intensity variation to first enhance the peaks and, thereafter, by determining the average distance between successive peaks.

Tables (2)

Tables Icon

Table 1 Formulas Relating Changes in Pitch, Yaw, and Roll (and Combinations Thereof) to Positional Changes Δ x , Δ y of a Point a

Tables Icon

Table 2 Derived Values of Pitch, Yaw, and Roll from Figs. 7a, 7b, 7c, 7d, 7e at the Locations Marked on the Figures

Equations (18)

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

exp ( j K x x j K y y ) + [ j k 0 2 π ( z R z 0 ) ] a 0 exp { j k 0 2 ( z R z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } ,
t ( x , y ) e j ϕ ( x , y ) = e j α I ( x , y ) 1 j α I ( x , y ) = 1 j α [ 1 + ( j k 0 2 π ( z R z 0 ) a 0 exp ( j K x x + j K y y ) × exp { j k 0 2 ( z R z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } + c . c . ) ] ,
exp ( j K x x j K y y ) + [ j k 0 2 π ( z R z 0 ) ] a 0 × exp { j k 0 2 ( z R z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } × t ( x , y ) ,
j α [ j k 0 2 π ( z R z 0 ) ] a 0 × exp { j k 0 2 ( z R z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } + ( 1 j α ) [ j k 0 2 π ( z R z 0 ) ] a 0 × exp { j k 0 2 ( z R z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } .
j α [ j k 0 2 π ( z obs z 0 ) ] a 0 × exp { j k 0 2 ( z obs z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } + ( 1 j α ) [ j k 0 2 π ( z obs z 0 ) ] a 0 × exp { j k 0 2 ( z obs z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } .
j α exp { j k 0 2 ( z obs z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } exp { j k 0 2 ( z obs z 0 ) [ ( x x 0 ) 2 + ( y y 0 ) 2 ] } + c . c .
k 0 [ ( x x 0 ) 2 + ( y y 0 ) 2 2 ( z obs z 0 ) ( x x 0 ) 2 + ( y y 0 ) 2 2 ( z obs z 0 ) ] .
z 0 = z 0 + Δ z , x 0 = x 0 + Δ x , y 0 = y 0 + Δ y ,
k 0 2 [ 2 ( x x 0 ) Δ x 2 ( y y 0 ) Δ y ( z obs z 0 ) + ( x x 0 ) 2 Δ z + ( y y 0 ) 2 Δ z ( z obs z 0 ) 2 ] ,
I obs 1 + cos [ k 0 ( ( x x 0 ) Δ x + ( y y 0 ) Δ y ( z obs z 0 ) ( x x 0 ) 2 + ( y y 0 ) 2 2 ( z obs z 0 ) 2 Δ z ) ] .
I obs 1 + cos [ k 0 ( ( x x 0 ) Δ x + ( y y 0 ) Δ y ( z obs z 0 ) ) ] .
s obs = Δ x / Δ y .
Δ x = 2 π ( z obs z 0 ) k 0 Λ x , Δ y = 2 π ( z obs z 0 ) k 0 Λ y .
x 0 Δ x + y 0 Δ y + z 0 Δ z = 0 .
( x 0 y 0 z 0 ) = [ M ] ( x 0 y 0 z 0 ) , [ M ] = [ cos ( φ ) cos ( θ ) cos ( φ ) sin ( θ ) sin ( ψ ) sin ( φ ) cos ( ψ ) sin ( φ ) sin ( ψ ) + cos ( φ ) sin ( θ ) cos ( ψ ) sin ( φ ) cos ( θ ) cos ( φ ) cos ( ψ ) + sin ( φ ) sin ( θ ) sin ( ψ ) sin ( φ ) sin ( θ ) cos ( ψ ) cos ( φ ) sin ( ψ ) sin ( θ ) cos ( θ ) sin ( ψ ) cos ( θ ) cos ( ψ ) ] .
( Δ x Δ y Δ z ) = [ 0 Δ ϕ Δ θ Δ ϕ 0 Δ ψ Δ θ Δ ψ 0 ] ( x 0 y 0 z 0 ) .
Δ ψ = ( y 0 Δ z z 0 Δ y ) / R 2 , Δ θ = ( z 0 Δ x x 0 Δ z ) / R 2 , Δ ϕ = ( x 0 Δ y y 0 Δ x ) / R 2 ) .
| δ Δ x Δ x | = | δ Λ x Λ x | + | δ ( z obs z 0 ) ( z obs z 0 ) | .

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