Abstract

We study the properties of a double-exposure image specklegram of a diffuse object obtained by use of a double-aperture pupil. A phase object is placed in front of one aperture during the first or the second exposure. Also, it is assumed that a uniform displacement of the diffuser between exposures is produced. The recorded specklegram is coherently illuminated and analyzed by Fourier transform operations. The average intensity distribution and the interference fringe visibility in the Fourier plane are investigated. On this basis, an alternative interference technique to detect phase objects is proposed.

© 2003 Optical Society of America

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References

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  1. M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
  2. D. Duffy, “Moiré gauging of in-plane displacement using double aperture imaging,” Appl. Opt. 11, 1778–1781 (1972).
    [CrossRef] [PubMed]
  3. H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
    [CrossRef]
  4. E. Vanidhis, J. Spyridelis, “An image multiplexing method, through internal modulation of speckle patterns,” Optik (Stuttgart) 61, 195–208 (1982).
  5. L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
    [CrossRef]
  6. M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
    [CrossRef]
  7. M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
    [CrossRef]
  8. L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
    [CrossRef]
  9. D. Sharma, R. Sirohi, M. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542–1546 (1984).
    [CrossRef] [PubMed]
  10. R. Mohanty, C. Joenathan, R. Sirohi, “Speckle and speckle-shearing interferometers combined for the simultaneous determination of out-of-plane displacement and slope,” Appl. Opt. 24, 3106–3109 (1985).
    [CrossRef]
  11. L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Phase object analysis by a novel speckle interferometer,” Opt. Lett. 27, 506–508 (2002).
    [CrossRef]
  12. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 11–76.
  13. I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Ámsterdam, 1985), Vol. XXII, pp. 271–339.
  14. D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser-speckle interferometry,” J. Opt. Soc. Am. A 2, 657–666 (1985).
    [CrossRef]
  15. G. H. Kaufmann, “Numerical processing of speckle photography data by Fourier transform,” Appl. Opt. 20, 4277–4280 (1981).
    [CrossRef] [PubMed]
  16. F. Chiang, R. Khetan, “Strain analysis by one-beam laser speckle interferometry. 2: Multiaperture method,” Appl. Opt. 18, 2175–2186 (1979).
    [CrossRef] [PubMed]
  17. N. Halliwell, C. Pickering, “Analysis methods in laser speckle photography and particle image velocimetry,” in Interferogram Analysis. Digital Fringe Pattern Measurement Techniques, D. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 230–261.
  18. J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).
  19. D. Robinson, “Automatic fringe analysis with the computed image-processing system,” Appl. Opt. 22, 2169–2176 (1983).
    [CrossRef]

2002 (1)

2001 (1)

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

2000 (2)

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

1999 (1)

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

1994 (1)

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

1985 (2)

1984 (1)

1983 (1)

1982 (1)

E. Vanidhis, J. Spyridelis, “An image multiplexing method, through internal modulation of speckle patterns,” Optik (Stuttgart) 61, 195–208 (1982).

1981 (1)

1980 (1)

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

1979 (1)

1972 (1)

Angel, L.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Phase object analysis by a novel speckle interferometer,” Opt. Lett. 27, 506–508 (2002).
[CrossRef]

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Arizaga, R.

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

Bolognini, N.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Phase object analysis by a novel speckle interferometer,” Opt. Lett. 27, 506–508 (2002).
[CrossRef]

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

Chen, J. B.

Chiang, F.

Chiang, F. P.

Duffy, D.

Françon, M.

M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).

Garavaglia, M.

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 11–76.

Halliwell, N.

N. Halliwell, C. Pickering, “Analysis methods in laser speckle photography and particle image velocimetry,” in Interferogram Analysis. Digital Fringe Pattern Measurement Techniques, D. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 230–261.

Joenathan, C.

Kaufmann, G. H.

Khetan, R.

Kothiyal, M.

Li, D. W.

Mohanty, R.

Pickering, C.

N. Halliwell, C. Pickering, “Analysis methods in laser speckle photography and particle image velocimetry,” in Interferogram Analysis. Digital Fringe Pattern Measurement Techniques, D. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 230–261.

Pomarico, J.

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

Rabal, H.

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

Robinson, D.

Sharma, D.

Sicre, E.

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

Sirohi, R.

Spyridelis, J.

E. Vanidhis, J. Spyridelis, “An image multiplexing method, through internal modulation of speckle patterns,” Optik (Stuttgart) 61, 195–208 (1982).

Tebaldi, M.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Phase object analysis by a novel speckle interferometer,” Opt. Lett. 27, 506–508 (2002).
[CrossRef]

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Torroba, R.

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

Trivi, M.

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Phase object analysis by a novel speckle interferometer,” Opt. Lett. 27, 506–508 (2002).
[CrossRef]

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

L. Angel, M. Tebaldi, N. Bolognini, M. Trivi, “Speckle photography with different pupils in a multiple-exposure scheme,” J. Opt. Soc. Am. A 17, 107–119 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

Vanidhis, E.

E. Vanidhis, J. Spyridelis, “An image multiplexing method, through internal modulation of speckle patterns,” Optik (Stuttgart) 61, 195–208 (1982).

Yamaguchi, I.

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Ámsterdam, 1985), Vol. XXII, pp. 271–339.

Appl. Opt. (6)

J. Mod. Opt. (1)

L. Angel, M. Tebaldi, M. Trivi, N. Bolognini, “Fringe visibility analysis with different scale apertures in speckle photography,” J. Mod. Opt. 43, 1749–1765 (2001).
[CrossRef]

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

Opt. Commun. (3)

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “Optical operations based on speckle modulation by using a photorefractive crystal,” Opt. Commun. 168, 55–64 (1999).
[CrossRef]

M. Tebaldi, L. Angel, M. Trivi, N. Bolognini, “New multiple aperture arrangements for speckle photography,” Opt. Commun. 182, 95–105 (2000).
[CrossRef]

H. Rabal, N. Bolognini, E. Sicre, M. Garavaglia, “Optical image subtraction through speckle modulated by Young fringes,” Opt. Commun. 34, 7–10 (1980).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (2)

E. Vanidhis, J. Spyridelis, “An image multiplexing method, through internal modulation of speckle patterns,” Optik (Stuttgart) 61, 195–208 (1982).

J. Pomarico, R. Arizaga, R. Torroba, H. Rabal, “Algorithm to compute spacing of digital speckle correlation fringes,” Optik (Stuttgart) 95, 125–127 (1994).

Other (4)

M. Françon, “Information processing using speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).

N. Halliwell, C. Pickering, “Analysis methods in laser speckle photography and particle image velocimetry,” in Interferogram Analysis. Digital Fringe Pattern Measurement Techniques, D. Robinson, G. T. Reid, eds. (Institute of Physics, London, 1993), pp. 230–261.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 11–76.

I. Yamaguchi, “Fringe formations in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Ámsterdam, 1985), Vol. XXII, pp. 271–339.

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Figures (13)

Fig. 1
Fig. 1

Experimental setup: D, diffuser; S, specklegram image plane; F, Fourier plane; L1 and L2, lenses; P, pupil aperture; W, phase object.

Fig. 2
Fig. 2

(a) Zero-order and (b) lateral-order visibility behavior as a function of t2 when different parallel plates are introduced.

Fig. 3
Fig. 3

(a) Zero-order and (b) lateral-order visibility for t1=1, t2=0.8, and different τ12 values.

Fig. 4
Fig. 4

Φ12(U) behavior applied to (1) τ12=0 and t1t2=1, (2) τ12=798 m-1 and t1t2=1, (3) τ12=1653 m-1 and t1t2=1, (4) τ12=798 m-1 and t1t2=0.8, and (5) τ12=1653 m-1 and t1t2=0.8.

Fig. 5
Fig. 5

Zero-order average intensity theoretical simulations and their corresponding profiles along the U axis. The thin-edge wedge and the diffuser displacement (ΔX12=200 µm, ΔY12=0) are perpendicular. The parameters are t1=1 and t2=0.8 throughout and τ1=τ2=τ12=0 in (a), τ12=1653 m-1 in (b), and τ12=-1653 m-1 in (c). The set of parameters λR=633 nm, λW=514 nm, D=3.8 mm, d=10 mm, ZC=485 mm, Z0=135 mm, and f=100 mm is also used in Figs. 6 and 813 below.

Fig. 6
Fig. 6

Zero-order average intensity theoretical simulations. The thin-edge wedge and the diffuser displacement (ΔX12=0, ΔY12=200 µm) are parallel. The parameters are t1=1 and t2=0.8 throughout and τ12=-992 m-1 in (a), τ12=992 m-1 in (b), and τ12=1653 m-1 in (c).

Fig. 7
Fig. 7

Scheme utilized to discuss the diffracted-order visibility behavior.

Fig. 8
Fig. 8

Lateral-order average intensity theoretical simulations and their corresponding profile along the U axis. The thin-edge wedge and the diffuser displacement (ΔX12=200 µm, ΔY12=0) are perpendicular. The parameters are t1=1 and t2=0.8 throughout and τ12=0 in (a), τ12=992 m-1 in (b), and τ12=-992 m-1 in (c).

Fig. 9
Fig. 9

Three-dimensional plot of the lateral-order average intensity theoretical simulations. The thin-edge wedge and the diffuser displacement are perpendicular. The parameters are ΔX12=200 µm, ΔY12=0, t1=1, and t2=0.8 throughout and τ12=0 in (a), τ12=1653 m-1 in (b), and τ12=-1653 m-1 in (c).

Fig. 10
Fig. 10

Lateral-order average intensity theoretical simulations. The thin-edge wedge and the diffuser displacement are parallel. The parameters are ΔX12=0, ΔY12=200 µm, t1=1, and t2=0.8 throughout and τ12=798 m-1 in (a), τ12=1653 m-1 in (b), and τ12=2363 m-1 in (c).

Fig. 11
Fig. 11

Experimental results (a) without and (b) with a parallel plate in front of one aperture in the second exposure and (c) and (d) the corresponding theoretical simulations. The central images in (c) and (d) represent the zero order, whereas the lateral images represent the (-1) and (+1) orders, respectively. The parameters are ΔX12=0, ΔY12=142×10-3 mm, t2=0.8, and γ12=π.

Fig. 12
Fig. 12

Experimental results (a) without and (b) with a wedge in front of one aperture in the second exposure and (c) and (d) the corresponding theoretical simulations. The thin-edge wedge and the diffuser displacement are parallel. The central images in (c) and (d) represent the zero order, whereas the lateral images represent the (-1) and (+1) orders, respectively. The parameters are ΔX12=0, ΔY12=94×10-3 mm, t2=0.8, γ12=0.7, and τ12=850 m-1.

Fig. 13
Fig. 13

Experimental results (a) without and (b) with a wedge in front of one aperture in the second exposure and (c) and (d) the corresponding theoretical simulations. The thin-edge wedge and the diffuser displacement are perpendicular. The central images in (c) and (d) represent the zero order, whereas the lateral images represent the (-1) and (+1) orders, respectively. The parameters are ΔX12=94×10-3 mm, ΔY12=0, t2=0.8, γ12=0.7, and τ12=850 m-1.

Equations (28)

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aj(u, v)=a(u-uj, v-vj)=1insidethejthaperture,j=1,20otherwise,
Pk(u, v)=a(u-u1, v-v1)+Wka(u-u2, v-v2),k=1,2,
Ak(Xa, Ya)[Al(Xb, Yb)]*=KF{Pkl(u, v)}×Xa-XbλWZC-ΔxklλWZ0, Ya-YbλWZC-ΔyklλWZ0fork, l=1, 2,
Ik(Xa, Ya)Il(Xb, Yb)=IkIl+|Ak(Xa, Ya)×[Al(Xb, Yb)]*|2fork, l=1, 2.
Ik=Ak(X, Y)[Ak(X, Y)]*F{|Pk(u, v)|2}(0, 0)[1+(tk)2]F{a(u, v)}(0, 0)fork=1, 2,
Pkl(u, v)=Pk(u, v)[Pl(u, v)]*=a(u-u1, v-v1)+tktl×exp[-i(γkl+τklu)]×a(u-u2, v-v2),
Ak(Xa, Ya)[Al(Xb, Yb)]*exp[-i2π(u1ξ+v1η)]F{a(u, v)}(ξ, η)+tktl exp[-i2π(u2ξ+v2η)]exp[-i(γkl+u2τkl)]×F{a(u, v)}ξ+τkl2π, η,
(ξ, η)Xa-XbλWZC-ΔxklλWZ0, Ya-YbλWZC-ΔyklλWZ0.
|Ak(Xa, Ya)[Al(Xb, Yb)]*|2[F{a(u, v)}(ξ, η)]2+(tktl)2×F{a(u, v)}ξ+τkl2π, η2+2tktlF{a(u, v)}(ξ, η)F{a(u, v)}ξ+τkl2π, η×cos[2π(u12ξ+v12η)+γkl+u2τkl],
If(U, V)k,l=12 Ik(Xa, Ya)Il(Xb, Yb)×exp-i 2πλRf [U(Xa-Xb)+V(Ya-Yb)]×dXadYadXbdYb.
If(U, V)δ(U, V)+k,l=12 F{Pkl(u, v)}×Xa-XbλWZC-ΔxklλWZ0, Ya-YbλWZC-ΔyklλWZ02×exp-i 2πλRf [U(Xa-Xb)+V(Ya-Yb)]×dXadYadXbdYb.
If(U, V)k,l=12 exp-i 2πλRf (UΔXkl+VΔYkl)×F{|F{Pkl(u, v)}(X, Y)|2}(ϑU, ϑV),
F{|F{Pkl(u, v)}(X, Y)|2}(ϑU, ϑV)=[1+(tktl)2 exp(iτklϑU)]a(ϑU, ϑV)a(-ϑU, -ϑV)+tktl exp[i(γkl+τklu2)]×a(-ϑU+u12, -ϑV+v12){exp[iτkl(ϑU-u12)]a(ϑU-u12, ϑV-v12)}+tktl exp[-i(γkl+τklu2)]×a(ϑU+u12, ϑV+v12){exp[iτkl(ϑU+u12)]a(-ϑU-u12, -ϑV-v12)},
If(U, V)=If0(U, V)+If+1(U, V)+If-1(U, V),
If0(U, V)2+(t1)4+(t2)4+2[1+(t1t2)4+2(t1t2)2 cos(ϑτ12U)]1/2×cos2πλRf (UΔX12+VΔY12)-Φ12(U)×a(ϑU, ϑV)a(-ϑU, -ϑV),
If±1(U, V)[(t1)2+(t2)2]a(ϑU+u12, ϑV+v12)a(±ϑU-u12, ±ϑV-v12)+t1t2×2 Reexp-i2πλRf (UΔX12+VΔY12)γ12τ12u2a(ϑU+u12, ϑV+v12){exp[iτ12(ϑUu12)]a(±ϑU-u12, ±ϑV-v12)}.
Φ12(U)=tan-1(t1t2)2 sin(ϑτ12U)1+(t1t2)2 cos(ϑτ12U)
If0(U, V)D2 trianϑUDtrianϑVD2+(t1)4+(t2)4+2[1+(t1t2)4+2(t1t2)2 cos(ϑτ12U)]1/2×cos2πλRf (UΔX12+VΔY12)-Φ12(U),
If±1(U, V)D2 trianϑUu12DtrianϑVv12D×(t1)2+(t2)2+2t1t2×sinc τ12D2π 1-ϑUu12D×sign(ϑUu12)cos2πλRf UΔX12-λWZC4π τ12+VΔY12γ12u1+u22 τ12,
V0(U)=If0(U, V)max-If0(U, V)minIf0(U, V)max+If0(U, V)min=22+(t1)4+(t2)4 [1+(t1t2)4+2(t1t2)2 cos(ϑτ12U)]1/2,
0[1-(t1t2)2]21+(t1t2)4+2(t1t2)2 cos(ϑτ12U)[1+(t1t2)2]22+(t1)4+(t2)422.
V±1(U)=2t1t2(t1)2+(t2)2×sincDτ122π 1-ϑUu12D×sign(ϑUu12).
If0(U, V)D2 trianϑUDtrianϑVD×2+(t1)4+(t2)4+2[1+(t1t2)2]×cos2πλRf (UΔX12+VΔY12),
If±1(U, V)D2 trianϑUu12DtrianϑVv12D×(t1)2+(t2)2+2t1t2×cos2πλRf (UΔX12+VΔY12)γ12.
V0=2+2(t1t2)22+(t1)4+(t2)4,
V±1=2t1t2(t1)2+(t2)2.
2t1t2 sincτ12D2π {1-[(ϑUu12)/D]sign(ϑUu12)}
V±1(d)=2t1t2(t1)2+(t2)2 |sinc(Dτ12/2π)|

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