Abstract

This paper proposes an elegant technique for the simultaneous measurement of in-plane and out-of-plane displacements of a deformed object in digital holographic interferometry. The measurement relies on simultaneously illuminating the object from multiple directions and using a single reference beam to interfere with the scattered object beams on the CCD plane. Numerical reconstruction provides the complex object wave-fields or complex amplitudes corresponding to prior and postdeformation states of the object. These complex amplitudes are used to generate the complex reconstructed interference field whose real part constitutes a moiré interference fringe pattern. Moiré fringes encode information about multiple phases which are extracted by introducing a spatial carrier in one of the object beams and subsequently using a Fourier transform operation. The information about the in-plane and out-of-plane displacements is then ascertained from the estimated multiple phases using sensitivity vectors of the optical configuration.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Schnars and W. P. O. Juptner, “Digital recording and reconstruction of holograms in hologram interferometry and shearography,” Appl. Opt. 33, 4373–4377 (1994).
    [CrossRef] [PubMed]
  2. G. Pedrini and H. J. Tiziani, “Digital double-pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
    [CrossRef]
  3. G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Technol. 29, 249–256 (1997).
    [CrossRef]
  4. G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
    [CrossRef] [PubMed]
  5. S. Schedin, G. Pedrini, H. J. Tiziani, and F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
    [CrossRef]
  6. P. Picart, E. Moisson, and D. Mounier, “Twin-sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947–1957 (2003).
    [CrossRef] [PubMed]
  7. C. Kohler, M. R. Viotti, and J. G. Armando Albertazzi, “Measurement of three-dimensional deformations using digital holography with radial sensitivity,” Appl. Opt. 49, 4004–4009 (2010).
    [CrossRef] [PubMed]
  8. E. Kolenovic, W. Osten, R. Klattenhoff, S. Lai, C. Von Kopylow, and W. Juptner, “Miniaturized digital holography sensor for distal three-dimensional endoscopy,” Appl. Opt. 42, 5167–5172 (2003).
    [CrossRef] [PubMed]
  9. S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
    [CrossRef]
  10. Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
    [CrossRef]
  11. C. A. Sciammarella, “Holographic moiré, an optical tool for the determination of displacements, strains, contours, and slopes of surfaces” Opt. Eng. 21, 447–457 (1982).
  12. C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
    [CrossRef]
  13. E. S. Simova and K. N. Stoev, “Automated Fourier transform fringe-pattern analysis in holographic moiré,” Opt. Eng. 32, 2286–2294 (1993).
    [CrossRef]
  14. A. Patil and P. Rastogi, “Phase determination in holographic moiré in presence of nonsinusoidal waveforms and random noise,” Opt. Commun. 257, 120–132 (2006).
    [CrossRef]
  15. A. Patil, R. Langoju, and P. Rastogi, “Constraints in dual phase shifting interferometry,” Opt. Express 14, 88–102(2006).
    [CrossRef] [PubMed]
  16. A. Patil, P. Rastogi, and B. Raphael, “A stochastic method for generalized data reduction in holographic moiré,” Opt. Commun. 248, 395–405 (2005).
    [CrossRef]
  17. A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. 30, 2227–2229 (2005).
    [CrossRef] [PubMed]
  18. A. Patil, R. Langoju, and P. Rastogi, “Model-based processing of a holographic moiré,” Opt. Lett. 30, 2870–2872 (2005).
    [CrossRef] [PubMed]
  19. A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13, 4070–4084(2005).
    [CrossRef] [PubMed]
  20. A. Patil and P. Rastogi, “Rotational invariance approach for the evaluation of multiple phases in interferometry in the presence of nonsinusoidal waveforms and noise,” J. Opt. Soc. Am. A 22, 1918–1928 (2005).
    [CrossRef]
  21. R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. 30, 3326–3328 (2005).
    [CrossRef]
  22. R. Langoju, A. Patil, and P. Rastogi, “Estimation of multiple phases in interferometry in the presence of nonlinear arbitrary phase steps,” Opt. Express 14, 7686–7691 (2006).
    [CrossRef] [PubMed]
  23. A. Patil, R. Langoju, P. Rastogi, and S. Ramani, “Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 794–813 (2007).
    [CrossRef]
  24. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  25. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).
  26. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]

2010

2008

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

2007

2006

2005

2003

2002

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1999

1997

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

1995

G. Pedrini and H. J. Tiziani, “Digital double-pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

1994

1993

E. S. Simova and K. N. Stoev, “Automated Fourier transform fringe-pattern analysis in holographic moiré,” Opt. Eng. 32, 2286–2294 (1993).
[CrossRef]

1982

C. A. Sciammarella, “Holographic moiré, an optical tool for the determination of displacements, strains, contours, and slopes of surfaces” Opt. Eng. 21, 447–457 (1982).

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Albertazzi, J. G.

Fujigaki, M.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
[CrossRef]

Ina, H.

Jacquot, P.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Juptner, W.

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, “Digital recording and reconstruction of holograms in hologram interferometry and shearography,” Appl. Opt. 33, 4373–4377 (1994).
[CrossRef] [PubMed]

Klattenhoff, R.

Kobayashi, S.

Kohler, C.

Kolenovic, E.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

Lai, S.

Langoju, R.

Matsui, A.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

Matui, T.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
[CrossRef]

Moisson, E.

Morimoto, Y.

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
[CrossRef]

Mounier, D.

Narayanan, R.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Okazawa, S.

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
[CrossRef]

Osten, W.

Patil, A.

A. Patil, R. Langoju, P. Rastogi, and S. Ramani, “Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 794–813 (2007).
[CrossRef]

A. Patil and P. Rastogi, “Phase determination in holographic moiré in presence of nonsinusoidal waveforms and random noise,” Opt. Commun. 257, 120–132 (2006).
[CrossRef]

A. Patil, R. Langoju, and P. Rastogi, “Constraints in dual phase shifting interferometry,” Opt. Express 14, 88–102(2006).
[CrossRef] [PubMed]

R. Langoju, A. Patil, and P. Rastogi, “Estimation of multiple phases in interferometry in the presence of nonlinear arbitrary phase steps,” Opt. Express 14, 7686–7691 (2006).
[CrossRef] [PubMed]

R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. 30, 3326–3328 (2005).
[CrossRef]

A. Patil and P. Rastogi, “Rotational invariance approach for the evaluation of multiple phases in interferometry in the presence of nonsinusoidal waveforms and noise,” J. Opt. Soc. Am. A 22, 1918–1928 (2005).
[CrossRef]

A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13, 4070–4084(2005).
[CrossRef] [PubMed]

A. Patil, P. Rastogi, and B. Raphael, “A stochastic method for generalized data reduction in holographic moiré,” Opt. Commun. 248, 395–405 (2005).
[CrossRef]

A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. 30, 2227–2229 (2005).
[CrossRef] [PubMed]

A. Patil, R. Langoju, and P. Rastogi, “Model-based processing of a holographic moiré,” Opt. Lett. 30, 2870–2872 (2005).
[CrossRef] [PubMed]

Pedrini, G.

S. Schedin, G. Pedrini, H. J. Tiziani, and F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
[CrossRef]

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini and H. J. Tiziani, “Digital double-pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Picart, P.

Ramani, S.

Raphael, B.

A. Patil, P. Rastogi, and B. Raphael, “A stochastic method for generalized data reduction in holographic moiré,” Opt. Commun. 248, 395–405 (2005).
[CrossRef]

Rastogi, P.

A. Patil, R. Langoju, P. Rastogi, and S. Ramani, “Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 794–813 (2007).
[CrossRef]

R. Langoju, A. Patil, and P. Rastogi, “Estimation of multiple phases in interferometry in the presence of nonlinear arbitrary phase steps,” Opt. Express 14, 7686–7691 (2006).
[CrossRef] [PubMed]

A. Patil, R. Langoju, and P. Rastogi, “Constraints in dual phase shifting interferometry,” Opt. Express 14, 88–102(2006).
[CrossRef] [PubMed]

A. Patil and P. Rastogi, “Phase determination in holographic moiré in presence of nonsinusoidal waveforms and random noise,” Opt. Commun. 257, 120–132 (2006).
[CrossRef]

A. Patil, P. Rastogi, and B. Raphael, “A stochastic method for generalized data reduction in holographic moiré,” Opt. Commun. 248, 395–405 (2005).
[CrossRef]

A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. 30, 2227–2229 (2005).
[CrossRef] [PubMed]

A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13, 4070–4084(2005).
[CrossRef] [PubMed]

A. Patil, R. Langoju, and P. Rastogi, “Model-based processing of a holographic moiré,” Opt. Lett. 30, 2870–2872 (2005).
[CrossRef] [PubMed]

A. Patil and P. Rastogi, “Rotational invariance approach for the evaluation of multiple phases in interferometry in the presence of nonsinusoidal waveforms and noise,” J. Opt. Soc. Am. A 22, 1918–1928 (2005).
[CrossRef]

R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. 30, 3326–3328 (2005).
[CrossRef]

Rastogi, P. K.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Santoyo, F. M.

Schedin, S.

Schnars, U.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, “Digital recording and reconstruction of holograms in hologram interferometry and shearography,” Appl. Opt. 33, 4373–4377 (1994).
[CrossRef] [PubMed]

Sciammarella, C. A.

C. A. Sciammarella, “Holographic moiré, an optical tool for the determination of displacements, strains, contours, and slopes of surfaces” Opt. Eng. 21, 447–457 (1982).

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

Simova, E. S.

E. S. Simova and K. N. Stoev, “Automated Fourier transform fringe-pattern analysis in holographic moiré,” Opt. Eng. 32, 2286–2294 (1993).
[CrossRef]

Stoev, K. N.

E. S. Simova and K. N. Stoev, “Automated Fourier transform fringe-pattern analysis in holographic moiré,” Opt. Eng. 32, 2286–2294 (1993).
[CrossRef]

Takeda, M.

Tiziani, H. J.

S. Schedin, G. Pedrini, H. J. Tiziani, and F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38, 7056–7062 (1999).
[CrossRef]

G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional spatial carrier,” Appl. Opt. 36, 786–792 (1997).
[CrossRef] [PubMed]

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

G. Pedrini and H. J. Tiziani, “Digital double-pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Viotti, M. R.

Von Kopylow, C.

Zou, Y. L.

Appl. Mech. Mater.

S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005).
[CrossRef]

Appl. Opt.

Exp. Mech.

C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Narayanan, “Holographic moiré in real time,” Exp. Mech. 22, 52–63 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Measurement

G. Pedrini and H. J. Tiziani, “Digital double-pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Opt. Commun.

A. Patil and P. Rastogi, “Phase determination in holographic moiré in presence of nonsinusoidal waveforms and random noise,” Opt. Commun. 257, 120–132 (2006).
[CrossRef]

A. Patil, P. Rastogi, and B. Raphael, “A stochastic method for generalized data reduction in holographic moiré,” Opt. Commun. 248, 395–405 (2005).
[CrossRef]

Opt. Eng.

C. A. Sciammarella, “Holographic moiré, an optical tool for the determination of displacements, strains, contours, and slopes of surfaces” Opt. Eng. 21, 447–457 (1982).

E. S. Simova and K. N. Stoev, “Automated Fourier transform fringe-pattern analysis in holographic moiré,” Opt. Eng. 32, 2286–2294 (1993).
[CrossRef]

Opt. Express

Opt. Laser Technol.

G. Pedrini and H. J. Tiziani, “Quantitative evaluation of two-dimensional dynamic deformations using digital holography,” Opt. Laser Technol. 29, 249–256 (1997).
[CrossRef]

Opt. Lett.

Strain

Y. Morimoto, T. Matui, M. Fujigaki, and A. Matsui, “Three-dimensional displacement analysis by windowed phase-shifting digital holographic interferometry,” Strain 44, 49–56(2008).
[CrossRef]

Other

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Multibeam illumination of the object.

Fig. 2
Fig. 2

(a) First phase Δ ϕ 1 ( x , y ) in radians. (b) Second phase Δ ϕ 2 ( x , y ) in radians. (c) Moiré fringe pattern corresponding to Γ ( x , y ) . (d) Moiré fringe pattern corresponding to Γ c ( x , y ) . (e) Fourier spectrum of Γ ( x , y ) . (f) Fourier spectrum of Γ c ( x , y ) .

Fig. 3
Fig. 3

(a) Wrapped estimate of phase Δ ϕ 1 ( x , y ) . (b) Wrapped estimate of phase Δ ϕ 2 ( x , y ) . (c) Unwrapped estimate of phase Δ ϕ 1 ( x , y ) in radians. (d) Unwrapped estimate of phase Δ ϕ 2 ( x , y ) in radians. (e) Sum of estimated phases in radians. (f) Difference of estimated phases in radians.

Fig. 4
Fig. 4

DHM schematic: BS1–BS2, beam splitters; BE1–BE3, beam expanders; M1–M5, mirrors; OBJ, diffuse object.

Fig. 5
Fig. 5

(a) Intensity of the numerically reconstructed hologram using discrete Fresnel transform. (b) DHM fringe pattern. (c) 2D FT of Γ ( x , y ) . (d) Carrier fringes. (e) Wrapped estimate of first phase Δ ϕ 1 ( x , y ) . (f) Wrapped estimate of second phase Δ ϕ 2 ( x , y ) .

Fig. 6
Fig. 6

(a) Unwrapped phase Δ ϕ 1 ( x , y ) in radians. (b) Unwrapped phase Δ ϕ 2 ( x , y ) in radians. (c) Sum of estimated phases in radians. (d) Difference of estimated phases in radians. (e) Wrapped form of sum phase. (f) Wrapped form of difference phase.

Equations (24)

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

I = | R + O 1 + O 2 | 2 = I 0 + R ( O 1 * + O 2 * ) + R * ( O 1 + O 2 ) ,
Γ 0 ( x , y ) = j λ d exp [ j 2 π d λ ] R ( x , y ) I ( x , y ) exp [ j π λ d ( ( x x ) 2 + ( y y ) 2 ) ] d x d y .
Γ 0 ( x , y ) = j λ d exp [ j 2 π d λ ] [ R I 0 + R 2 ( O 1 * + O 2 * ) + | R | 2 ( O 1 + O 2 ) ] exp [ j π λ d ( ( x x ) 2 + ( y y ) 2 ) ] d x d y .
Γ 1 ( x , y ) = a 1 ( x , y ) exp [ j ϕ 1 ( x , y ) ] + a 2 ( x , y ) exp [ j ϕ 2 ( x , y ) ] .
Γ 2 ( x , y ) = a 1 ( x , y ) exp [ j ( ϕ 1 ( x , y ) + Δ ϕ 1 ( x , y ) ) ] + a 2 ( x , y ) exp [ j ( ϕ 2 ( x , y ) + Δ ϕ 2 ( x , y ) ) ] ,
Γ ( x , y ) = Γ 2 ( x , y ) Γ 1 * ( x , y ) = a 1 2 ( x , y ) exp [ j Δ ϕ 1 ( x , y ) ] + a 2 2 ( x , y ) exp [ j Δ ϕ 2 ( x , y ) ] + a 1 ( x , y ) a 2 ( x , y ) exp [ j ( Δ ϕ 1 ( x , y ) + ϕ 1 ( x , y ) ϕ 2 ( x , y ) ) ] + a 1 ( x , y ) a 2 ( x , y ) exp [ j ( Δ ϕ 2 ( x , y ) + ϕ 2 ( x , y ) ϕ 1 ( x , y ) ) ] .
Γ ( x , y ) = a ( x , y ) exp [ j Δ ϕ 1 ( x , y ) ] + b ( x , y ) exp [ j Δ ϕ 2 ( x , y ) ] + η ( x , y ) ,
s ^ 1 = sin ( θ ) x ^ cos ( θ ) z ^ ,
s ^ 2 = sin ( θ ) x ^ cos ( θ ) z ^ ,
u ^ = z ^ .
Δ ϕ 1 = 2 π λ d · ( u ^ s ^ 1 ) = 2 π λ [ d z ( 1 + cos ( θ ) ) + d x sin ( θ ) ]
Δ ϕ 2 = 2 π λ d · ( u ^ s ^ 2 ) = 2 π λ [ d z ( 1 + cos ( θ ) ) d x sin ( θ ) ] .
Δ ϕ 1 + Δ ϕ 2 = 4 π λ d z ( 1 + cos ( θ ) ) ,
Δ ϕ 1 Δ ϕ 2 = 4 π λ d x sin ( θ ) .
Γ 2 ( x , y ) = a 1 ( x , y ) exp [ j ( ω 1 x + ω 2 y + ϕ 1 ( x , y ) + Δ ϕ 1 ( x , y ) ) ] + a 2 ( x , y ) exp [ j ( ϕ 2 ( x , y ) + Δ ϕ 2 ( x , y ) ) ] ,
Γ ( x , y ) = a ( x , y ) exp [ j ( ω 1 x + ω 2 y + Δ ϕ 1 ( x , y ) ) ] + b ( x , y ) exp [ j Δ ϕ 2 ( x , y ) ] + η ( x , y ) .
G ( ω x , ω y ) = FT { Γ ( x , y ) } = G 1 ( ω x ω 1 , ω y ω 2 ) + G 2 ( ω x , ω y ) + N ( ω x , ω y ) ,
G 1 ( ω x , ω y ) = FT { a ( x , y ) exp [ j ( Δ ϕ 1 ( x , y ) ) ] } ,
G 2 ( ω x , ω y ) = FT { b ( x , y ) exp [ j Δ ϕ 2 ( x , y ) ] } ,
N ( ω x , ω y ) = FT { η ( x , y ) } .
Δ ϕ 1 ( x , y ) = angle { g 1 ( x , y ) } = arctan [ Im { g 1 ( x , y ) } Re { g 1 ( x , y ) } ]
Δ ϕ 2 ( x , y ) = angle { g 2 ( x , y ) } = arctan [ Im { g 2 ( x , y ) } Re { g 2 ( x , y ) } ] .
Γ ( x , y ) = g 1 ( x , y ) + g 2 ( x , y ) = exp [ j Δ ϕ 1 ( x , y ) ] + exp [ j Δ ϕ 2 ( x , y ) ] ,
Γ c ( x , y ) = g 1 ( x , y ) exp [ j ω 1 x ] + g 2 ( x , y ) = exp [ j ( ω 1 x + Δ ϕ 1 ( x , y ) ) ] + exp [ j Δ ϕ 2 ( x , y ) ] .

Metrics