Abstract

Since the advantages of noncontact, strong antidisturbing capability and wide measurement range, moiré tomography has been considered a powerful diagnostic tool for flow fields. In this paper, the volume computerized tomography is introduced to obtain the real three-dimensional reconstruction based on moiré deflectometry. In order to realize volume moiré tomography (VMT), double cross gratings are applied in the moiré deflected system to gain the shearing phase distribution of the moiré deflected projection in two mutually perpendicular directions simultaneously. Thus, the scalar diffraction theory is used for analyzing the imaging process of VMT based on double cross gratings to achieve the explicit form of shearing phase. Finally, the real temperature distribution of a propane flame is reconstructed, which can confirm the VMT method.

© 2012 Optical Society of America

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References

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  1. R. Snyder and L. Hesselink, “Optical tomography for flow visualization of the density field around a revolving helicopter rotor blade,” Appl. Opt. 23, 3650–3656 (1984).
    [CrossRef]
  2. Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.
  3. T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
    [CrossRef]
  4. W. Lv, H.-C. Zhou, and J.-R. Zhu, “Implementation of tridirectional large lateral shearing displacement interferometry in temperature measurement of a diffused ethylene flame,” Appl. Opt. 50, 3924–3936 (2011).
    [CrossRef]
  5. C. Tian, Y. Yang, Y. Zhuo, T. Wei, and T. Ling, “Tomographic reconstruction of three-dimensional refractive index fields by use of a regularized phase-tracking technique and a polynomial approximation method,” Appl. Opt. 50, 6495–6504(2011).
    [CrossRef]
  6. H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010–5019 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. Y.-Y. Chen, Y. Song, A.-Z. He, and Z.-H. Li, “Applicability of moiré deflection tomography for diagnosing arc plasmas,” Appl. Opt. 48, 489–496 (2009).
    [CrossRef]
  15. E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, “Temperature mapping in flames by moire deflectometry,” Appl. Opt. 22, 698–705 (1983).
    [CrossRef]
  16. A. J. Senol and G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” J. Spacecr. Rockets 23, 39–46 (1986).
    [CrossRef]
  17. Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101 (2006).
    [CrossRef]
  18. Y. Song, Y. Y. Chen, A. He, and Z. Zhao, “Theoretical analysis for moiré deflectometry from diffraction theory,” J. Opt. Soc. Am. A 26, 882–889 (2009).
    [CrossRef]
  19. K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
    [CrossRef]
  20. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express 16, 7806–7817 (2008).
    [CrossRef]
  21. J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
    [CrossRef]
  22. Y.-S. Cheng, “Fringe formation with a cross-grating interferometer,” Appl. Opt. 25, 4185–4191 (1986).
    [CrossRef]
  23. Y.-S. Cheng, “Analysis of the interference pattern in a cross-grating interferometer,” Appl. Opt. 27, 3025–3034 (1988).
    [CrossRef]
  24. Y.-S. Cheng, “Interference patterns in cross-grating interferometers: further analysis,” Appl. Opt. 28, 556–564 (1989).
    [CrossRef]
  25. S. Orlov, “Theory of three dimensional reconstruction: I. Conditions for a complete set of projections,” Sov. Phys. Crystallogr. 20, 312–314 (1975).
  26. Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).
  27. M. D. Pritt, “Unweighted least squares phase unwrapping by means of multigrid techniques,” in Synthetic Aperture Radar and Passive Microwave Sensing, G. Franceschetti, C. J. Oliver, J. C. Shiue, and S. Tajbakhsh, eds. (SPIE, 1995).
  28. Y.-Y. Chen, Z.-H. Li, Y. Song, and A.-Z. He, “Extension of the Gladstone–Dale equation for flame flow field diagnosis by optical computerized tomography,” Appl. Opt. 48, 2485–2490 (2009).
    [CrossRef]

2011 (3)

2009 (3)

2008 (1)

2007 (1)

2006 (2)

2005 (1)

1999 (2)

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

1989 (1)

1988 (1)

1986 (2)

Y.-S. Cheng, “Fringe formation with a cross-grating interferometer,” Appl. Opt. 25, 4185–4191 (1986).
[CrossRef]

A. J. Senol and G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” J. Spacecr. Rockets 23, 39–46 (1986).
[CrossRef]

1985 (1)

O. Kafri and I. Glatt, “Moire deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 246944 (1985).
[CrossRef]

1984 (1)

1983 (1)

1980 (1)

1975 (1)

S. Orlov, “Theory of three dimensional reconstruction: I. Conditions for a complete set of projections,” Sov. Phys. Crystallogr. 20, 312–314 (1975).

1969 (1)

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Anastasio, M. A.

Anzhi, H.

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

Bar-Ziv, E.

Bernabeu, E.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Buzug, T. M.

T. M. Buzug, Computed Tomography (Springer, 2008).

Chen, Y. Y.

Chen, Y.-Y.

Cheng, Y.-S.

Crespo, D.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Donglou, W.

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

Glatt, I.

O. Kafri and I. Glatt, “Moire deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 246944 (1985).
[CrossRef]

He, A.

He, A.-Z.

Hesselink, L.

Hudgins, D.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

Ishino, Y.

Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.

Iwata, K.

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Kafri, O.

Kanhirodan, R.

Kechribaris, C. N.

Keren, E.

Li, Z.-H.

Ling, T.

Lv, W.

Maniatis, T. A.

Meneses-Fabian, C.

Nagae, S.

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Nagata, R.

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).

Nikita, K. S.

Ohiwa, N.

Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.

Okuno, G.

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Orlov, S.

S. Orlov, “Theory of three dimensional reconstruction: I. Conditions for a complete set of projections,” Sov. Phys. Crystallogr. 20, 312–314 (1975).

Pan, X.

Pritt, M. D.

M. D. Pritt, “Unweighted least squares phase unwrapping by means of multigrid techniques,” in Synthetic Aperture Radar and Passive Microwave Sensing, G. Franceschetti, C. J. Oliver, J. C. Shiue, and S. Tajbakhsh, eds. (SPIE, 1995).

Quiroga, J. A.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Robledo-Sánchez, C.

Rodriguez-Zurita, G.

Romine, G. L.

A. J. Senol and G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” J. Spacecr. Rockets 23, 39–46 (1986).
[CrossRef]

Senol, A. J.

A. J. Senol and G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” J. Spacecr. Rockets 23, 39–46 (1986).
[CrossRef]

Sgulim, S.

Shiga, S.

Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.

Sidky, E. Y.

Snyder, R.

Song, Y.

Takeuchi, K.

Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.

Thayyullathil, H.

Tian, C.

Toto-Arellano, N.-I.

Upton, T.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

Uzunoglu, N. K.

Vasu, R. M.

Vázquez-Castillo, J. F.

Verhoeven, D.

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

Vouldis, A. T.

Wei, T.

Wei, Y.

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

Wubbeling, F.

F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).

Yang, Y.

Zhang, B.

Zhao, Z.

Zhendong, W.

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

Zhou, H.-C.

Zhu, J.-R.

Zhuo, Y.

Zou, Y.

Acta Opt. Sin. (1)

Y. Wei, W. Donglou, W. Zhendong, and H. Anzhi, “Wave front reconstruction from opaque object containing interferogram using modified multigrid algorithm,” Acta Opt. Sin. 19, 171–180 (1999).

Appl. Opt. (11)

Y.-Y. Chen, Z.-H. Li, Y. Song, and A.-Z. He, “Extension of the Gladstone–Dale equation for flame flow field diagnosis by optical computerized tomography,” Appl. Opt. 48, 2485–2490 (2009).
[CrossRef]

Y.-S. Cheng, “Fringe formation with a cross-grating interferometer,” Appl. Opt. 25, 4185–4191 (1986).
[CrossRef]

Y.-S. Cheng, “Analysis of the interference pattern in a cross-grating interferometer,” Appl. Opt. 27, 3025–3034 (1988).
[CrossRef]

Y.-S. Cheng, “Interference patterns in cross-grating interferometers: further analysis,” Appl. Opt. 28, 556–564 (1989).
[CrossRef]

W. Lv, H.-C. Zhou, and J.-R. Zhu, “Implementation of tridirectional large lateral shearing displacement interferometry in temperature measurement of a diffused ethylene flame,” Appl. Opt. 50, 3924–3936 (2011).
[CrossRef]

C. Tian, Y. Yang, Y. Zhuo, T. Wei, and T. Ling, “Tomographic reconstruction of three-dimensional refractive index fields by use of a regularized phase-tracking technique and a polynomial approximation method,” Appl. Opt. 50, 6495–6504(2011).
[CrossRef]

H. Thayyullathil, R. M. Vasu, and R. Kanhirodan, “Quantitative flow visualization in supersonic jets through tomographic inversion of wavefronts estimated through shadow casting,” Appl. Opt. 45, 5010–5019 (2006).
[CrossRef]

R. Snyder and L. Hesselink, “Optical tomography for flow visualization of the density field around a revolving helicopter rotor blade,” Appl. Opt. 23, 3650–3656 (1984).
[CrossRef]

Y.-Y. Chen, Y. Song, A.-Z. He, and Z.-H. Li, “Applicability of moiré deflection tomography for diagnosing arc plasmas,” Appl. Opt. 48, 489–496 (2009).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, and E. Keren, “Temperature mapping in flames by moire deflectometry,” Appl. Opt. 22, 698–705 (1983).
[CrossRef]

Y. Song, B. Zhang, and A. He, “Algebraic iterative algorithm for deflection tomography and its application to density flow fields in a hypersonic wind tunnel,” Appl. Opt. 45, 8092–8101 (2006).
[CrossRef]

Exp. Fluids (1)

T. Upton, D. Verhoeven, and D. Hudgins, “High-resolution computed tomography of a turbulent reacting flow,” Exp. Fluids 50, 125–134 (2011).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Spacecr. Rockets (1)

A. J. Senol and G. L. Romine, “Three dimensional refraction-diffraction of EM waves through rocket exhaust plumes,” J. Spacecr. Rockets 23, 39–46 (1986).
[CrossRef]

Jpn. J. Appl. Phys. (1)

K. Iwata, S. Nagae, R. Nagata, and G. Okuno, “Strain measurement by optical differentiation of cross grating,” Jpn. J. Appl. Phys. 8, 473–477 (1969).
[CrossRef]

Opt. Eng. (2)

O. Kafri and I. Glatt, “Moire deflectometry: a ray deflection approach to optical testing,” Opt. Eng. 24, 246944 (1985).
[CrossRef]

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Sov. Phys. Crystallogr. (1)

S. Orlov, “Theory of three dimensional reconstruction: I. Conditions for a complete set of projections,” Sov. Phys. Crystallogr. 20, 312–314 (1975).

Other (5)

M. D. Pritt, “Unweighted least squares phase unwrapping by means of multigrid techniques,” in Synthetic Aperture Radar and Passive Microwave Sensing, G. Franceschetti, C. J. Oliver, J. C. Shiue, and S. Tajbakhsh, eds. (SPIE, 1995).

Y. Ishino, K. Takeuchi, S. Shiga, and N. Ohiwa, “Measurement of instantaneous 3D-distribution of local burning velocity on a turbulent premixed flame by non-scanning 3D-CT reconstruction,” presented at the 4th European combustion meeting, Vienna, Austria, 14–17 April, 2009.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction (Society for Industrial and Applied Mathematics, 2001).

T. M. Buzug, Computed Tomography (Springer, 2008).

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Figures (11)

Fig. 1.
Fig. 1.

Definition of 3-D radon space.

Fig. 2.
Fig. 2.

Optical schematic diagram of VMT.

Fig. 3.
Fig. 3.

Schematic diagram of two cross gratings. The blue lines denote the directions of the grating lines of G1, which are perpendicular. The red lines denote the directions of the grating lines of G2, which are also perpendicular.

Fig. 4.
Fig. 4.

Spectrum distribution of μ2+(x,y).

Fig. 5.
Fig. 5.

Flow chart of 3-D reconstruction method for VMT.

Fig. 6.
Fig. 6.

Spectrum distribution of VMT.

Fig. 7.
Fig. 7.

Grid moiré patterns of order (1,1) (a) without the field and (b) with the field.

Fig. 8.
Fig. 8.

First-order partial derivative of phase of order (1,1) (a) direction of 45° and (b) direction of 135°.

Fig. 9.
Fig. 9.

Process of merging (a) radon transform of Fig. 8(a), (b) radon transform of Fig. 8(b), and (c) radial derivative of the radon transform.

Fig. 10.
Fig. 10.

Filtered projection surface.

Fig. 11.
Fig. 11.

Temperature distribution of the reconstructed propane flame field.

Equations (20)

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

f(r)=1(2π)2γ=0πϕ=0π2gγ,ϕ(ξ)ξ2sin(ϕ)dϕdγ,
gγ,ϕ(ξ)=R(φ(x,y)),
gγ,ϕ(ξ)ξ=R(φ(x,y)x)·sinϕ+R(φ(x,y)y)·cosϕ.
u1(x,y)exp{ikφ(x,y)},
g1(x,y)=n1an1exp{in12πd[xcosθysinθ]}×m1am1exp{im12πd[xsinθ+ycosθ]},
u1+(x,y)=u1(x,y)g1(x,y)=n1an1m1am1u1(x,y)exp{i2πxd[n1cosθ+m1sinθ]}×exp{i2πyd[n1sinθ+m1cosθ]}=FT1{n1an1m1am1U1(X1d[n1cosθ+m1sinθ],Y1d[n1sinθ+m1cosθ])},
U1+(X,Y)=FT(u1+(X,Y))=n1an1m1am1U1(X1d[n1cosθ+m1sinθ],Y1d[n1sinθ+m1cosθ]).
U2(X,Y)=exp{ikΔ1λ2((X)2+(Y)2)}×n1an1m1am1U1(X1d[n1cosθ+m1sinθ],Y1d[n1sinθ+m1cosθ]).
g2(x,y)=n2an2exp{in22πd[xcosθ+ysinθ]}×m2am2exp{im22πd[xsinθ+ycosθ]}.
U2+(X,Y)=n1an1m1am1n2an2m2am2exp{ikΔ[1λ22((X)2+(Y)2)]}×exp{ikΔ[λ2(Xd[n2cosθm2sinθ]+Yd[n2sinθ+m2cosθ])]}×exp{ikΔ[λ22d2(n2cosθm2sinθ)2]}×exp{ikΔ[λ22d2(n2sinθ+m2cosθ)2]}×U1(X1d[n1cosθ+m1sinθ+n2cosθm2sinθ],Y1d[n1sinθ+m1cosθ+n2sinθ+m2cosθ]).
u2+(x,y)=n1an1m1am1n2an2m2am2×exp{iπΔλd2{[n2cosθm2sinθ]2+[n2sinθ+m2cosθ]2}}×exp{iπΔλd2{[n2cosθm2sinθ]2[n1cosθ+m1sinθ]2}}×exp{iπΔλd2{[n2sinθ+m2cosθ]2[n1sinθm1cosθ]2}}×exp{i2πd[n1cosθ+m1sinθ+n2cosθm2sinθ]·x}×exp{i2πd[n1sinθ+m1cosθ+n2sinθ+m2cosθ]·y}×u1(xΔλd[n1cosθ+m1sinθ],yΔλd[n1sinθ+m1cosθ]).
Δ=Cd2/λ(Cis an integer),
φ(xΔλd[n1cosθ+m1sinθ],yΔλd[n1sinθ+m1cosθ])=φ(x,y)+φ(x,y)x(Δλd[n1cosθ+m1sinθ])+φ(x,y)y(Δλd[n1sinθ+m1cosθ])+2φ(x,y)x2(Δλd[n1cosθ+m1sinθ])2+2φ(x,y)y2(Δλd[n1sinθ+m1cosθ])2+2φ(x,y)xy(Δλd[n1cosθ+m1sinθ])(Δλd[n1sinθ+m1cosθ])+···.
Δλ/d=Cd.
u2+(x,y)=n1an1m1am1n2an2m2am2exp{iπΔλd2{[n2cosθm2sinθ]2+[n2sinθ+m2cosθ]2}}×exp{iπΔλd2{[n2cosθm2sinθ]2[n1cosθ+m1sinθ]2}}exp{iπΔλd2{[n2sinθ+m2cosθ]2[n1sinθm1cosθ]2}}×exp{i2πd[n1cosθ+m1sinθ+n2cosθm2sinθ]·x}exp{i2πd[n1sinθ+m1cosθ+n2sinθ+m2cosθ]·y}×exp{ikφ(x,y)}exp{i2πΔdφ(x,y)x[n1cosθ+m1sinθ]}exp{i2πΔdφ(x,y)y[n1sinθ+m1cosθ]}.
u3(x,y)=a12a22exp{i2πΔλd2}exp{i2πd[(cosθ+sinθ)x+(cosθsinθ)y]}×exp{ikφ(x,y)}exp{i2πΔdφ(x,y)x(cosθ+sinθ)}exp{i2πΔdφ(x,y)y(cosθsinθ)}+a12a22exp{iπΔλd2}exp{i2πd[(cosθsinθ)x+(cosθsinθ)y]}×exp{ikφ(x,y)}exp{i2πΔdφ(x,y)xcosθ}exp{i2πΔdφ(x,y)ysinθ}+a12a02exp{iπΔλd2}exp{i2πd[(cosθ+sinθ)x+(cosθ+sinθ)y]}exp{ikφ(x,y)}exp{i2πΔdφ(x,y)xsinθ}exp{i2πΔdφ(x,y)ycosθ}+a12a02exp{i2πd[(cosθsinθ)x+(sinθ+cosθ)y]}exp{ikφ(x,y)}.
I=u3·(u3)*=2a14a04{2+2cos(πΔλd2+4πsinθdx2πΔdφ(x,y)xsinθ2πΔdφ(x,y)ycosθ)+2cos(πΔλd24πsinθdy2πΔdφ(x,y)xcosθ+2πΔdφ(x,y)ysinθ)+cos(2πΔλd2+4πsinθd(xy)2πΔdφ(x,y)x(cosθ+sinθ)2πΔdφ(x,y)y(cosθsinθ))+cos(4πsinθd(x+y)2πΔdφ(x,y)x(cosθsinθ)+2πΔdφ(x,y)y(sinθ+cosθ))}.
gγ,ϕ(ξ)=φγ(x,y)δ(xsin(ϕ)+ycos(ϕ)ξ)dxdy,
f(r)=1(2π)2γ=0πϕ=0π2gγ,ϕ(ξ)ξ2sinϕdϕdγ=1(2π)2γ=0πϕ=0πξ[φγ(x,y)xδ(xsinϕ+ycosϕξ)dxdy·sinϕ+φγ(x,y)yδ(xsinϕ+ycosϕξ)dxdy·cosϕ]sinϕdϕdγ.
Ψ(ri)=1(2π)2θ=0π2gγ,θ(ξ)ξ2sin(ϕ)dϕ.

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