Abstract

This paper analyzes the dynamics of laser speckles and fringes, formed in an imaging-speckle-pattern interferometer with the purpose of sensing linear three-dimensional motion and out-of-plane components of rotation in real time, using optical spatial-filtering-velocimetry techniques. The ensemble-average definition of the cross-correlation function is applied to the intensity distributions, obtained in the observation plane at two positions of the object. The theoretical analysis provides a description for the dynamics of both the speckles and the fringes. The analysis reveals that both the magnitude and direction of all three linear displacement components of the object movement can be determined. Simultaneously, out-of-plane rotation of the object including the corresponding directions can be determined from the spatial gradient of the in-plane fringe motion throughout the observation plane. The theory is confirmed by experimental measurements.

© 2011 Optical Society of America

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References

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  1. A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomenon, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 203–253.
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    [CrossRef]
  3. M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
    [CrossRef]
  4. I. Yamaguchi, “Fringe formation in deformation and vibration and measurements using laser light,” in Vol.  22 of Progress in Optics, E.Wolf, ed. (Elsevier, 1985), pp. 272–340.
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).
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    [CrossRef]
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    [CrossRef]
  14. A. E. Siegman, Lasers (University Science, 1986).
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    [CrossRef]
  16. U. Schnell, J. Piot, and R. Dändliker, “Detection of movement with laser speckle patterns: statistical properties,” J. Opt. Soc. Am. A 15, 207–216 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
    [CrossRef]
  20. H. T. Yura, B. Rose, and S. G. Hanson, “Speckle dynamics from in-plane rotating diffuse objects in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1167–1173 (1998).
    [CrossRef]
  21. N. Takai, T. Iwai, and T. Asakura, “Real time velocity measurements for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
    [CrossRef]
  22. R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. 20, 83–87 (1980).
    [CrossRef]
  23. M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
    [CrossRef]
  24. M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
    [CrossRef]

2011 (1)

2010 (1)

M. L. Jakobsen and S. G. Hanson, “Miniaturised optical sensors for industrial applications,” Proc. SPIE 7726, 77260P(2010).
[CrossRef]

2009 (1)

2008 (1)

M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
[CrossRef]

2005 (1)

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
[CrossRef]

2003 (1)

1998 (4)

1997 (1)

1996 (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

1987 (1)

1983 (1)

1980 (2)

N. Takai, T. Iwai, and T. Asakura, “Real time velocity measurements for a diffuse object using zero-crossing of laser speckle,” J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. 20, 83–87 (1980).
[CrossRef]

1970 (1)

1968 (1)

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).
[CrossRef]

1966 (1)

Aizu, Y.

Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).

Arsenault, H.

Asakura, T.

Ator, J. T.

Barakat, R.

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. 20, 83–87 (1980).
[CrossRef]

Burch, J. M.

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).
[CrossRef]

Chang, Y. H.

Dändliker, R.

Ennos, A. E.

A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomenon, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 203–253.

Fricke-Begemann, T.

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Hanson, S. G.

T. F. Q. Iversen, M. L. Jakobsen, and S. G. Hanson, “Speckle-based three-dimensional velocity measurement using spatial filtering velocimetry,” Appl. Opt. 50, 1523–1533(2011).
[CrossRef] [PubMed]

M. L. Jakobsen and S. G. Hanson, “Miniaturised optical sensors for industrial applications,” Proc. SPIE 7726, 77260P(2010).
[CrossRef]

M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
[CrossRef]

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
[CrossRef]

H. T. Yura, B. Rose, and S. G. Hanson, “Speckle dynamics from in-plane rotating diffuse objects in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1167–1173 (1998).
[CrossRef]

H. T. Yura, B. Rose, and S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
[CrossRef]

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931–1948 (1987).
[CrossRef]

Iversen, T. F. Q.

Iwai, T.

Jakobsen, M. L.

T. F. Q. Iversen, M. L. Jakobsen, and S. G. Hanson, “Speckle-based three-dimensional velocity measurement using spatial filtering velocimetry,” Appl. Opt. 50, 1523–1533(2011).
[CrossRef] [PubMed]

M. L. Jakobsen and S. G. Hanson, “Miniaturised optical sensors for industrial applications,” Proc. SPIE 7726, 77260P(2010).
[CrossRef]

M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
[CrossRef]

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
[CrossRef]

Larsen, H. E.

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
[CrossRef]

Lehmann, M.

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Lin, S. T.

Lowenthal, S.

Mertz, L.

Pedersen, F.

M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
[CrossRef]

Piot, J.

Rose, B.

Saldner, H. O.

Schnell, U.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sjödahl, M.

Takai, N.

Tokarski, J. M. J.

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, “Fringe formation in deformation and vibration and measurements using laser light,” in Vol.  22 of Progress in Optics, E.Wolf, ed. (Elsevier, 1985), pp. 272–340.
[CrossRef]

Yeh, S. L.

Yura, H. T.

Appl. Opt. (5)

Inf. Sci. (1)

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. 20, 83–87 (1980).
[CrossRef]

J. Opt. A (1)

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for submicron, in-plane vibration measurements,” J. Opt. A 7, S303–S307(2005).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

J. M. Burch and J. M. J. Tokarski, “Production of multiple beam fringes from photographic scatters,” Opt. Acta 15, 101–111 (1968).
[CrossRef]

Opt. Commun. (1)

M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. 128, 325–340 (1996).
[CrossRef]

Opt. Lasers Eng. (1)

M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. L. Jakobsen, F. Pedersen, and S. G. Hanson, “Zero-crossing detection algorithm for arrays of optical spatial filtering velocimetry sensors,” Proc. SPIE 7003, 70030T (2008).
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Miniaturised optical sensors for industrial applications,” Proc. SPIE 7726, 77260P(2010).
[CrossRef]

Other (5)

A. E. Ennos, “Speckle Interferometry,” in Laser Speckle and Related Phenomenon, J.C.Dainty, ed. (Springer-Verlag, 1984), pp. 203–253.

I. Yamaguchi, “Fringe formation in deformation and vibration and measurements using laser light,” in Vol.  22 of Progress in Optics, E.Wolf, ed. (Elsevier, 1985), pp. 272–340.
[CrossRef]

Y. Aizu and T. Asakura, Spatial Filtering Velocitmetry: Fundamentals and Applications (Springer-Verlag, 2006).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (12)

Fig. 1
Fig. 1

Setup for describing the imaging-speckle-pattern interferometer.

Fig. 2
Fig. 2

Ensemble-averaged correlation function plotted versus q x in units of M w i .

Fig. 3
Fig. 3

Four detectors define a single unit cell from a spatial bandpass filter with N unit cells. The four detectors provide two differential signals with a phase lag of π / 2 .

Fig. 4
Fig. 4

Spatial power spectrum [Eq. (28)] of a spatial filter designed to respond to the presence of speckles. The spatial filter is designed with a moderate selectivity, defined by N = 9 unit cells in the array. The unit of the spatial frequency is 1 / Λ x / y . The fundamental frequency of the spatial filter is designed to coincide with the spectral content of the speckles (dashed line). The logarithmic scales for the two power spectra are equivalent. However, the absolute magnitudes are arbitrary.

Fig. 5
Fig. 5

Spatial power spectrum [Eq. (28)] of a spatial filter designed to respond to the presence of fringes. The spatial filter is designed with a high selectivity, defined by N = 65 unit cells in the array. The unit of the spatial frequency is 1 / Λ z . The fundamental frequency of the spatial filter is designed to coincide with the spectral content of the fringe patterns (dashed line). The logarithmic scales for the two power spectra are equivalent. However, their absolute magnitudes are arbitrary.

Fig. 6
Fig. 6

Spatial power spectrum [Eq. (28)] of a spatial filter designed to respond to the presence of fringes. The spatial filter is designed with a very low selectivity, defined by N = 3 unit cells in the array. The unit of the spatial frequency is 1 / Λ x / y . The fundamental frequency of the spatial filter is designed to coincide with the spectral content of the fringe patterns (dashed line); see Eq. (30). The logarithmic scales for the two power spectra are equivalent. However, their absolute magnitudes are arbitrary.

Fig. 7
Fig. 7

Schematic of the setup for obtaining the experimental data. HeNe, laser; BS, nonpolarizing beam splitters; f 1 and f 2 , lenses of the imaging system; σ, circular aperture; CMOS, CMOS camera; R, radius of the object. The object rotates with an angular velocity of ω = Δ θ x / Δ t .

Fig. 8
Fig. 8

A small fraction of the transmittance mask illustrates a unit cell, being a part of a spatial filter, producing the photocurrent i ϕ ( s ) when processing a sequence of images. A second mask, which produces the photocurrent i ϕ + π / 2 ( s ) in phase quadrature with i ϕ ( s ) , is equivalent to the first mask but shifted one column.

Fig. 9
Fig. 9

The two time records illustrate the two constructed differential photocurrents, i 1 and i 2 , as quasi-sinusoidal functions of time. The two photocurrents have a mutual phase lag of approximately π / 2 .

Fig. 10
Fig. 10

Fringes and speckles overlaid with a vector map that illustrates the displacement rate ( q x ) of the fringes as the object rotates about the y axis with an angular displacement of Δ θ x 1 .

Fig. 11
Fig. 11

The signal frequencies are plotted as the local fringe displacement rates versus the horizontal position of the spatial filter in the observation plane. The three calibrated angular velocities are addressed.

Fig. 12
Fig. 12

The signal frequencies errors are plotted versus the horizontal position of the spatial filter in the observation plane. Three different object radii are addressed; R = 3.0 mm , R = 4.0 mm , and R = 5.0 mm .

Tables (2)

Tables Icon

Table 1 Fitted Parameters of a Linear Regression Applied to Angular Velocities Estimated from the Measured Signal Frequencies Plotted in Fig. 11, Specified as f s , x measured = c 0 + c 1 p x a

Tables Icon

Table 2 Fitted Parameters of a Linear Regression Applied to the Measured Velocity Data Plotted Specified as f s , x measured f s , x theoretical = c 0 + c 1 p x a

Equations (35)

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C ( p 1 , p 2 ; Δ r , Δ z ) = C ( p , p + q ; Δ r , Δ z ) = I ( p , t 1 ) I * ( p q , t 2 ) ,
C ( p , p + q ; Δ r , Δ z ) = ( U o ( p , t 1 ) + U r ( p ) ) ( U o ( p , t 1 ) + U r ( p ) ) * ( U o ( p q , t 2 ) + U r ( p q ) ) ( U o ( p q , t 2 ) + U r ( p q ) ) *
U o ( p , t i ) = S d 2 r U s ( r , t i ) G ( r , p , z ( t i ) ) ,
G ( r , p ; z i ) = i k 2 π B exp ( i k ( L + z i ) ) exp ( i k 2 B ( A r 2 2 r · p + D p 2 ) )
U s ( r , t ) = U i ( r ) Ψ ( r , t ) ,
B ψ ( r 1 , r 2 ) = Ψ ( r 1 , t 1 ) Ψ * ( r 2 , t 2 ) = 4 π k 2 ( 2 π r c 2 exp [ 2 | r 2 r 1 | 2 r c 2 ] ) ,
B Ψ ( r 1 , r 2 ) = ( 4 π k 2 ) δ ( r 2 r 1 ) ,
C ( p , p + q ; Δ r , Δ z ) = C o o o o ( p , p + q ; Δ r , Δ z ) + C r r r r ( p , p + q ; Δ r , Δ z ) + C o o r r ( p , p + q ; Δ r , Δ z ) + C o r o r ( p , p + q ; Δ r , Δ z ) + C o o o r ( p , p + q ; Δ r , Δ z ) + C o r r r ( p , p + q ; Δ r , Δ z ) ,
C o o o o ( p , p + q ; Δ r , Δ z ) = | U o ( p , t 1 ) U o * ( p q , t 2 ) | 2 + I o ( p , t 1 ) I o ( p q , t 2 ) ,
C o o o o ( p , p + q ; Δ r , Δ z ) = | S r d 2 r 1 d 2 r 2 [ Ψ ( r 1 , t 1 ) Ψ * ( r 2 , t 2 ) × U i ( r 1 ) G ( r 1 , p , z 1 ) U i * ( r 2 ) G * ( r 2 , p q , z 2 ) ] | 2 + S r d 2 r 1 d 2 r 1 [ Ψ ( r 1 , t 1 ) Ψ * ( r 1 , t 1 ) × U i ( r 1 ) G ( r 1 , p , z 1 ) U i * ( r 1 ) G * ( r 1 , p , z 1 ) ] × S r d 2 r 2 d 2 r 2 [ Ψ ( r 2 , t 2 ) Ψ * ( r 2 , t 2 ) × U i ( r 2 ) G ( r 2 , p q , z 2 ) U i * ( r 2 ) G * ( r 2 , p q , z 2 ) ] .
C o o o o ( p , p + q ; Δ r , Δ z ) = ( 4 π k 2 ) 2 | d 2 r U i ( r ) G ( r , p , z 1 ) U i * ( r + Δ r ) G * ( r + Δ r , p q , z 1 + Δ z ) | 2 + d r I i ( r ) G ( r , p , z 1 ) G * ( r , p , z 1 ) × d r I i ( r + Δ r ) G ( r + Δ r , p q , z 2 ) G * ( r + Δ r , p q , z 2 ) .
C r r r r ( p , p + q ; Δ r , Δ z ) = I r ( p ) I r ( p q ) ,
C o r o r ( p , p + q ; Δ r , Δ z ) = U r * ( p ) U r * ( p q ) U o ( p , t 1 ) U o ( p q , t 2 ) + c c + U r * ( p ) U r ( p q ) U o ( p , t 1 ) U o * ( p q , t 2 ) + c c ,
C o o r r ( p , p + q ; Δ r , Δ z ) = I r ( p q ) I o ( p , t 1 ) + I r ( p ) I o ( p q , t 2 ) .
C o r o r ( p , p + q ; Δ r , Δ z ) = ( 4 π k 2 ) U r * ( p ) U r ( p q ) d 2 r U i ( r ) G ( r , p , z 1 ) U i * ( r + Δ r ) G * ( r + Δ r , p q , z 1 + Δ z ) + c c .
C o o r r ( p , p + q ; Δ r , Δ z ) = ( 4 π k 2 ) I r ( p q ) d 2 r I i ( r ) G ( r , p , z 1 ) G * ( r , p , z 1 ) + ( 4 π k 2 ) I r ( p ) d 2 r I i ( r + Δ r ) × G ( r + Δ r , p q , z 2 ) G * ( r + Δ r , p q , z 2 ) .
U i ( r ) = E i exp ( | r | 2 w i 2 ) ,
U r ( p ) = E r exp ( | p | 2 w r 2 + i k φ · p ) .
C o o o o ( p , p + q ; Δ r , Δ z ) = 16 E i 4 k 4 ρ 4 ( 1 + f 1 2 f 2 2 ρ 2 2 w i 2 ) 2 × exp ( 2 | | p | 2 + | p + q | 2 | ρ 2 2 + f 2 2 f 1 2 w i 2 ) × ( 1 + exp ( 2 | q + ( f 2 f 1 + 4 f 1 f 2 k 2 σ 2 w i 2 ) Δ r | 2 ρ 2 ( 1 + 4 f 1 2 k 2 σ 2 w i 2 ) ) ) ,
C r r r r ( p , p + q ; Δ r , Δ z ) = E r 4 exp ( 2 ( | p | 2 + | p + q | 2 ) w r 2 ) .
C o o r r ( p , p + q ; Δ r , Δ z ) = 4 E i 2 E r 2 k 2 ρ 2 ( 1 + f 1 2 f 2 2 ρ 2 2 w i 2 ) 2 × exp ( ( 2 w r 2 + 2 ρ 2 2 + f 2 2 f 1 2 w i 2 ) | p | 2 ) × ( exp ( 2 w r 2 ( 2 p · q + | q | 2 ) ) + exp ( 2 ( 2 p · q + | q | 2 ) ρ 2 2 + f 2 2 f 1 2 w i 2 ) ) ,
C o r o r ( p , p + q ; Δ r , Δ z ) = 4 E i 2 E r 2 k 2 ρ 2 ( 1 + f 1 2 f 2 2 ρ 2 2 w i 2 ) × cos ( k ( 2 Δ z + φ · q ) ) × exp ( | q + ( f 2 2 f 1 2 + 4 f 1 f 2 k 2 σ 2 w i 2 ) Δ r | 2 ρ 2 ( 1 + 4 f 1 2 k 2 σ 2 w i 2 ) ) × exp ( ( 1 w r 2 + 1 ρ 2 2 + f 2 2 f 1 2 w i 2 ) ( | p | 2 + | p + q | 2 ) ) .
C o o o o ( p , p + q ; Δ θ , Δ z ) = 16 E i 4 k 4 ρ 4 ( 1 + f 1 2 f 2 2 ρ 2 2 w i 2 ) 2 × exp ( 2 | | p | 2 + | p + q | 2 | ρ 2 2 + f 2 2 f 1 2 w i 2 ) × ( 1 + exp ( 2 | Δ θ | 2 4 k 2 w i 2 + σ 2 f 1 2 ) × exp ( 2 | q + ( f 2 f 1 + 4 f 1 f 2 k 2 σ 2 w i 2 ) R Δ θ | 2 ρ 2 ( 1 + 4 f 1 2 k 2 σ 2 w i 2 ) ) ) .
C o r o r ( p , p + q ; Δ θ , Δ z ) = 4 E i 2 E r 2 k 2 ρ 2 ( 1 + f 1 2 f 2 2 ρ 2 2 w i 2 ) cos ( k ( 2 Δ z + φ · q ) + k ( 2 p + q ) · Δ θ f 2 f 1 + ρ 2 2 w i 2 ) × exp ( | q + ( f 2 2 f 1 2 + 4 f 1 f 2 k 2 σ 2 w i 2 ) R Δ θ | 2 ρ 2 ( 1 + 4 f 1 2 k 2 σ 2 w i 2 ) ) × exp ( ( 1 w r 2 1 ρ 2 2 + f 2 2 f 1 2 w i 2 ) ( | p | 2 + | p + q | 2 ) 2 | Δ θ | 2 4 k 2 w i 2 + σ 2 f 1 2 ) .
q x = 2 φ x ( Δ z + Δ θ x p x M ) ,
Δ θ x = φ x M 2 d p ( q x 2 q x 1 ) ,
q x = 2 Δ θ y φ x M p y .
Δ θ y = φ x M 2 d p ( q y 2 q y 1 ) .
Δ ϕ ( Δ θ x ) = 2 k ( Δ z + Δ θ x p x M ) .
f s = 2 λ ( v z + Δ θ x Δ t p x M ) = 1 Λ fr Δ q x Δ t .
i θ ( s ) = d 2 p I ( p s ) h θ ( p ) .
H ( ξ ) = ( 2 ξ 0 N Λ i w ) 2 { n = δ ( ξ ( 2 n + 1 ) ξ 0 ) × sinc 2 ( w ξ ) } sinc 2 ( N Λ i ξ ) ,
Λ x / y = 3 2 λ f 2 σ .
Λ z = Λ fr = λ | φ | .
i ϕ ( s ) = c = 1 N u r = 1 N L α ϕ , r , c i s , r , c .

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