Abstract

My purpose here is to outline a method for calculating the fundamental behavior of speckle patterns in imaging systems. The theory of speckle displacement and decorrelation to include imaging at a general oblique angle is extended to more imaging systems, and explicit formulas are given for the image-point–object-point relationship that is important when defocused speckle is used. The intermediate results can be reused for optical systems other than those presented here. The image-speckle displacement analyzed in the three systems is expressed equivalently. The speckle decorrelation is in general larger in a single-lens system than in a two-lens system and can be minimized by proper design of the system.

© 1995 Optical Society of America

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References

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  1. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  2. M. Sjödahl, “Systematic and random errors in electronic speckle photography,” Appl. Opt. 33, 7461–7471 (1994).
    [CrossRef] [PubMed]
  3. J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
    [CrossRef]
  4. O. Løkberg, “Electronic speckle pattern interferometry,” Phys. Technol. 11, 16–22 (1980).
    [CrossRef]
  5. H. J. Tiziani, “A study of the use of laser speckle to measure small tilts,” Opt. Commun. 5, 271–276 (1972).
    [CrossRef]
  6. D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Lasers Technol. 8, 201–213 (1976).
    [CrossRef]
  7. K. A. Stetson, “Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions,” J. Opt. Soc. Am. 66, 1267–1271 (1976).
    [CrossRef]
  8. P. Jacquot, P. K. Rastogi, “Speckle motions induced by rigid-body movements in free space geometry: an explicit investigation and extension to new cases,” Appl. Opt. 18, 2022–2032 (1979).
    [CrossRef] [PubMed]
  9. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  10. I. Yamaguchi, “Theory and applications of speckle displacement and decorrelation,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 1–39.
  11. 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]
  12. F. P. Chiang, D. W. Li, “Random (speckle) patterns for displacement and strain measurement: some recent advances,” Opt. Eng. 24, 936–943 (1985).
  13. D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
    [CrossRef]
  14. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  16. Y. C. Fung, Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, N.J., 1965).
  17. M. G. Pedretti, F. P. Chiang, “Effect of magnification change in laser speckle photography,” J. Opt. Soc. Am. 68, 1742–1748 (1978).
    [CrossRef]
  18. I. Yamaguchi, “Fringe formation in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1985), Vol. 22, pp. 272–340.
    [CrossRef]
  19. K. A. O’Donnell, “Correlations of time-varying speckle near the focal plane,” J. Opt. Soc. Am. 72, 191–197 (1982).
    [CrossRef]
  20. K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 556, 337–346 (1985).
  21. M. Sjödahl, “Electronic speckle photography: measurement of in-plane strain fields through the use of defocused-laser speckle,” Appl. Opt. 34, 5799–5808 (1995).
    [CrossRef] [PubMed]
  22. I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
    [CrossRef]

1995

1994

1993

I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
[CrossRef]

1986

1985

F. P. Chiang, D. W. Li, “Random (speckle) patterns for displacement and strain measurement: some recent advances,” Opt. Eng. 24, 936–943 (1985).

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]

1982

1981

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1980

O. Løkberg, “Electronic speckle pattern interferometry,” Phys. Technol. 11, 16–22 (1980).
[CrossRef]

1979

1978

1976

K. A. Stetson, “Problem of defocusing in speckle photography, its connection to hologram interferometry, and its solutions,” J. Opt. Soc. Am. 66, 1267–1271 (1976).
[CrossRef]

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Lasers Technol. 8, 201–213 (1976).
[CrossRef]

1972

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

1970

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Chen, J. B.

Chiang, F. P.

Creath, K.

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 556, 337–346 (1985).

Ennos, A. E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Fung, Y. C.

Y. C. Fung, Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, N.J., 1965).

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. 9–75.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gregory, D. A.

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Lasers Technol. 8, 201–213 (1976).
[CrossRef]

Jacquot, P.

Kobayashi, K.

I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
[CrossRef]

Leendertz, J. A.

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Li, D. W.

Løkberg, O.

O. Løkberg, “Electronic speckle pattern interferometry,” Phys. Technol. 11, 16–22 (1980).
[CrossRef]

O’Donnell, K. A.

Pedretti, M. G.

Rastogi, P. K.

Sjödahl, M.

Stetson, K. A.

Takemori, T.

I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
[CrossRef]

Tiziani, H. J.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Yamaguchi, I.

I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

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

I. Yamaguchi, “Theory and applications of speckle displacement and decorrelation,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 1–39.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. E

J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E 3, 214–218 (1970).
[CrossRef]

Opt. Acta

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Commun.

H. J. Tiziani, “A study of the use of laser speckle to measure small tilts,” Opt. Commun. 5, 271–276 (1972).
[CrossRef]

Opt. Eng.

F. P. Chiang, D. W. Li, “Random (speckle) patterns for displacement and strain measurement: some recent advances,” Opt. Eng. 24, 936–943 (1985).

I. Yamaguchi, T. Takemori, K. Kobayashi, “Stabilized and accelerated speckle strain gauge,” Opt. Eng. 32, 618–625 (1993).
[CrossRef]

Opt. Lasers Technol.

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Lasers Technol. 8, 201–213 (1976).
[CrossRef]

Phys. Technol.

O. Løkberg, “Electronic speckle pattern interferometry,” Phys. Technol. 11, 16–22 (1980).
[CrossRef]

Other

I. Yamaguchi, “Theory and applications of speckle displacement and decorrelation,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 1–39.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Y. C. Fung, Foundations of Solid Mechanics (Prentice-Hall, Englewood Cliffs, N.J., 1965).

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

K. Creath, “Phase-shifting speckle interferometry,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 556, 337–346 (1985).

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Figures (4)

Fig. 1
Fig. 1

Basic principle of the optical setup and definitions used to calculate the mutual intensity in the object plane.

Fig. 2
Fig. 2

Definitions used to analyze the speckle displacement and decorrelation in a single-lens imaging system.

Fig. 3
Fig. 3

Definitions used to analyze speckle displacement and decorrelation in a telecentric imaging system.

Fig. 4
Fig. 4

Definitions used to analyze speckle displacement and decorrelation in an afocal imaging system.

Equations (66)

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I 1 ( X 1 , Y 1 ) I 2 ( X 2 , Y 2 ) = u 1 ( X 1 , Y 2 ) u 1 * ( X 1 , Y 1 ) u 2 ( X 2 , Y 2 ) u 2 * ( X 2 , Y 2 ) ,
I 1 ( X 1 , Y 1 ) I 2 ( X 2 , Y 2 ) = I 1 ( X 1 , Y 1 ) I 2 ( X 2 , Y 2 ) + J ( X 1 , Y 1 ; X 2 , Y 2 ) 2 ,
J ( X 1 , Y 1 ; X 2 , Y 2 ) = u 1 ( X 1 , Y 1 ) u 2 * ( X 2 , Y 2 )
Δ I 1 ( X 1 , Y 1 ) Δ I 2 ( X 2 , Y 2 ) = J ( X 1 , Y 1 ; X 2 , Y 2 ) 2
N 1 = cos ( q 1 ) b 1 + sin ( q 1 ) sin ( q 2 ) b 2 + sin ( q 1 } cos ( q 2 ) b 3 ,
N 2 = cos ( q 2 ) b 2 - sin ( q 2 ) b 3 ,
N 3 = - sin ( q 1 ) b 1 + cos ( q 1 ) sin ( q 2 ) b 2 + cos ( q 1 ) cos ( q 2 ) b 3 .
J ( r , r ' ) = I exp { i k [ L A ( r ) - L A ( r ' ) ] } δ [ r + a ( r ) - r ' ] ,
J ( s , s ) = A Σ exp ( i k { L A ( r ) - L A [ r + a ( r ) ] } ) × exp ( i k [ L B ( r , s ) - L B [ r + a ( r ) , s ] } ) d 2 r ,
a ( r ) L A ( r ) , L B ( r , s ) ,             which gives L A [ r + a ( r ) ] = L A ( r ) - l ^ A ( r ) . a ( r ) ,
L B [ r + a ( r ) , s ' ] = L B ( r , s ' ) - l ^ B ( r , s ' ) . a ( r ) ,
L B ( r , s ) = ξ 1 b 1 + ψ 1 b 2 + ( Δ L + L 0 ) b 3 - x N 1 - y N 2 Δ L + L 0 - sin ( q 1 ) cos ( q 2 ) x + sin ( q 2 ) y + [ cos ( q 1 ) x ] 2 + [ cos ( q 2 ) y + sin ( q 1 ) sin ( q 2 ) x ] 2 2 ( Δ L + L 0 ) + ξ 1 2 + ψ 1 2 2 ( Δ L + L 0 ) - ξ 1 cos ( q 1 ) x + ψ 1 [ cos ( q 2 ) y + sin ( q 1 ) sin ( q 2 ) x ] Δ L + L 0 ,
L B ( r , s ' ) = ξ 2 b 1 + ψ 2 b 2 + ( Δ L + L 0 ) b 3 - x N 1 - y N 2 Δ L + L 0 - sin ( q 1 ) cos ( q 2 ) x + sin ( q 2 ) y + [ cos ( q 1 ) x ] 2 + [ cos ( q 2 ) y + sin ( q 1 ) sin ( q 2 ) x ] 2 2 ( Δ L + L 0 ) + ξ 2 2 + ψ 2 2 2 ( Δ L + L 0 ) - ξ 2 cos ( q 1 ) x + ψ 2 [ cos ( q 2 ) y + sin ( q 1 ) sin ( q 2 ) x ] Δ L + L 0 ,
m ( r , s ) · a ( r ) = l ^ A ( r R ) · a ( r R ) + L 0 b 3 + s - r R L 0 · a ( r R ) + { [ m ( r , s ) · a ( r ) ] } r R · ( r - r R ) ,
J ( ξ 1 , ψ 1 ; ξ 2 , ψ 2 ) = A exp ( i k θ 1 ) exp { i k [ ξ 2 b 1 · a ( r R ) + ψ 2 b 2 · a ( r R ) L 0 + ξ 1 2 - ξ 2 2 + ψ 1 2 - ψ 2 2 2 ( Δ L + L 0 ) ] } Σ exp { i k x × [ cos ( q 1 ) ( ξ 2 - ξ 1 ) + sin ( q 1 ) sin ( q 2 ) ( ψ 2 - ψ 1 ) Δ L + L 0 + { [ m ( r , s ' ) · a ( r ) ] } r R · N 1 ] } × exp { i k y [ cos ( q 2 ) ( ψ 2 - ψ 1 ) Δ L + L 0 + { [ m ( r , s ) · a ( r ) ] } r R · N 2 ] } d x d y ,
J ( ξ 1 , ψ 1 ; ξ 2 , ψ 2 ) = P ( ξ 1 , ψ 1 ) P * ( ξ 2 , ψ 2 ) J ( ξ 1 , ψ 1 ; ξ 2 , ψ 2 ) × exp ( - i k ξ 1 2 - ξ 2 2 + ψ 1 2 - ψ 2 2 2 f ) × δ [ ξ 2 - ξ 1 - ( Δ L + L 0 ) Δ ξ ; ψ 2 - ψ 1 - ( Δ L + L 0 ) Δ ψ ] ,
Δ ξ = - { [ m ( r , s ' ) · a ( r ) ] } r R · N 1 cos ( q 1 ) + sin ( q 1 ) sin ( q 2 ) { [ m ( r , s ' ) · a ( r ) ] } r R · N 2 cos ( q 1 ) cos ( q 2 ) ,
Δ ψ = - { [ m ( r , s ' ) · a ( r ) ] } r R · N 2 cos ( q 2 ) ,
J ( X 1 , Y 1 ; X 2 , Y 2 ) = B exp ( i k θ 2 ) exp ( i k X 1 2 - X 2 2 + Y 1 2 - Y 2 2 2 L 1 ) × exp [ i k ( Δ L + L 0 ) L 1 ( X 2 Δ ξ + Y 2 Δ ψ ) ] × Σ P ( ξ , ψ ) P * [ ξ + ( Δ L + L 0 ) Δ ξ , ψ + ( Δ L + L 0 ) Δ ψ ] × exp { i k ξ [ X 2 - X 1 L 1 + a ( r R ) · b 1 L 0 - ( Δ L + L 0 ) Δ ξ ( 1 ( Δ L + L 0 ) + 1 L 1 - 1 f ) ] } × exp { i k ψ [ Y 2 - Y 1 L 1 + a ( r R ) · b 2 L 0 - ( Δ L + L 0 ) Δ ψ ( 1 ( Δ L + L 0 ) + 1 L 1 - 1 f ) ] } d ξ d ψ ,
X 2 = X 1 + A X ,
Y 2 = Y 1 + A Y ,
A X = - a ( r R ) · b 1 M - Δ L Δ ξ M ,
A Y = - a ( r R ) · b 2 M - Δ L Δ ψ M ,
{ [ m ( r , s ) · a ( r ) ] } r R = [ l ^ A ( r ) x · a ( r ) + l ^ S · a ( r ) x + l ^ B ( r , s ' ) x · a ( r ) + b 3 · a ( r ) x ] r R N 1 + [ l ^ A ( r ) y · a ( r ) + l ^ S · a ( r ) x + l ^ B ( r , s ' ) y · a ( r ) + b 3 · a ( r ) y ] r R N 2 + [ l ^ A ( r ) z · a ( r ) + l ^ S · a ( r ) z + l ^ B ( r , s ' ) z · a ( r ) + b 3 · a ( r ) z ] r R N 3 ,
l ^ A ( r ) x = 1 L s [ ( l S x 2 - 1 ) N 1 + l S x l S y N 2 + l S x l S z N 3 ] r R ,
l ^ A ( r ) y = 1 L s [ l S x l S y N 1 + ( l S y 2 - 1 ) N 2 + l S y l S z N 3 ] r R ,
l ^ B ( r , s ) x = 1 L [ ( l x 2 - 1 ) N 1 + l x l y N 2 + l x l z N 3 ] r R ,
l ^ B ( r , s ) y = 1 L [ l x l y N 1 + ( l y 2 - 1 ) N 2 + l y l z N 3 ] r R ,
a x x = x x ; a x y = x y - ω z , a y x = x y + ω z ; a y y = y y , a z x = x z - ω y ; a z y = y z + ω x .
Δ ξ = - 1 cos ( q 1 ) [ a x ( l S x 2 - 1 L S + l x 2 - 1 L ) + a y ( l S x l S y L S + l x l y L ) + a z ( l S x l S z L S + l x l z L ) ] - x x [ l S x cos ( q 1 ) + sin ( q 1 ) cos ( q 2 ) cos ( q 1 ) ] - ( x y + ω z ) [ l S y cos ( q 1 ) - sin ( q 2 ) cos ( q 1 ) ] - ( x z - ω y ) [ l S z cos ( q 1 ) + cos ( q 2 ) ] + sin ( q 1 ) sin ( q 2 ) cos ( q 1 ) cos ( q 2 ) [ a x ( l S x l S y L S + l x l y L ) + a y ( l S y 2 - 1 L S + l y 2 - 1 L ) + a z ( l S y l S z L S + l y l z L ) ] + ( x y - ω z ) [ l S x sin ( q 1 ) sin ( q 2 ) cos ( q 1 ) cos ( q 2 ) + sin 2 ( q 1 ) sin ( q 2 ) cos ( q 1 ) ] + y y [ l S y sin ( q 1 ) sin ( q 2 ) cos ( q 1 ) cos ( q 2 ) - sin ( q 1 ) sin 2 ( q 2 ) cos ( q 1 ) cos ( q 2 ) ] + ( y z + ω x ) [ l S z sin ( q 1 ) sin ( q 2 ) cos ( q 1 ) cos ( q 2 ) + sin ( q 1 ) sin ( q 2 ) ] ,
Δ ψ = - 1 cos ( q 2 ) [ a x ( l S x l S y L S + l x l y L ) + a y ( l S y 2 - 1 L S + l y 2 - 1 L ) + a z ( l S y l S z L S + l y l z L ) ] - ( x y - ω z ) [ l S x cos ( q 2 ) + sin ( q 1 ) ] - y y [ l S y cos ( q 2 ) - sin ( q 2 ) cos ( q 2 ) ] - ( y z + ω x ) [ l S z cos ( q 2 ) + cos ( q 1 ) ] ,
x = - cos ( q 2 ) M X ( L 0 + Δ L ) cos ( q 1 ) cos ( q 2 ) L 0 + cos ( q 1 ) sin ( q 2 ) M Y - sin ( q 1 ) M X ,
y = - [ cos ( q 1 ) M Y - sin ( q 1 ) sin ( q 2 ) M X ] ( L 0 + Δ L ) cos ( q 1 ) cos ( q 2 ) L 0 + cos ( q 1 ) sin ( q 2 ) M Y - sin ( q 1 ) M X .
γ Δ I ( X 1 , Y 1 ; X 2 , Y 2 ) = | Σ P ( ξ , ψ ) P * ( ξ + A ξ , ψ + A ψ ) exp { - i k L 1 [ ξ ( X 1 - X 2 + A x ) + ψ ( Y 1 - Y 2 + A y ) ] } d ξ d ψ Σ P ( ξ , ψ ) 2 d ξ d ψ | 2 1 ,
γ = | Σ P ( ξ , ψ ) P * ( ξ + A ξ , ψ + A ψ ) d ξ d ψ Σ P ( ξ , ψ ) 2 d ξ d ψ | 2 ,
μ 2 = | Σ P ( ξ , ψ ) P * ( ξ + A ξ , ψ + A ψ ) exp [ - i k L 1 [ ξ A x + ψ A Y ) ] ] d ξ d ψ Σ P ( ξ , ψ ) 2 d ξ d ψ | 2 .
J ( α 1 , β 1 ; α 2 , β 2 ) = B exp ( i k θ 2 ) exp ( i k α 1 2 - α 2 2 + β 1 2 - β 2 2 2 f 1 ) exp [ i k ( Δ L + L 0 ) f 1 ( α 2 Δ ξ + β 2 Δ ψ ) ] × Σ exp ( i k { ξ [ α 1 - α 2 f 1 + a ( r R ) · b 1 L 0 - Δ ξ ] + ψ [ β 2 - β 1 f 1 + a ( r R ) · b 2 L 0 - Δ ψ ] } ) d ξ d ψ ,
α 2 = α 1 + A α ,
β 2 = β 1 + A β ,
A α = - f 1 a ( r R ) · b 1 L 0 + f 1 Δ ξ ,
A β = - f 1 a ( r R ) · b 2 L 0 + f 1 Δ ψ ,
J ( α 1 , β 1 ; α 2 , β 2 ) = P ( α 1 , β 1 ) P * ( α 2 , β 2 ) J ( α 1 , β 1 ; α 2 , β 2 ) × exp ( - i k α 1 2 - α 2 2 + β 1 2 - β 2 2 2 f 2 ) × δ [ α 2 - α 1 + f 1 a ( r R ) · b 1 L 0 - f 1 Δ ξ ; β 2 - β 1 + f 1 a ( r R ) · b 2 L 0 - f 1 Δ ψ ] ,
J ( X 1 , Y 1 ; X 2 , Y 2 ) = C exp ( i k θ 3 ) exp ( i k X 1 2 - X 2 2 + Y 1 2 - Y 2 2 2 L 2 ) × exp ( - i k L 2 { X 2 [ f 1 a ( r R ) · b 1 L 0 - f 1 Δ ξ ] + Y 2 [ f 1 a ( r R ) · b 2 L 0 - f 1 Δ ψ ] } ) × Σ P ( α , β ) P * [ α - f 1 a ( r R ) · b 1 L 0 + f 1 Δ ξ , β - f 1 a ( r R ) · b 2 L 0 + f 1 Δ ψ ] × exp ( i k ( 1 f 1 - 1 f 2 + 1 L 2 ) { α [ f 1 a ( r R ) · b 1 L 0 - f 1 Δ ξ ] + β [ f 1 a ( r R ) · b 2 L 0 - f 1 Δ ψ ] } ) × exp { i k [ α ( X 2 - X 1 ) + β ( Y 2 - Y 1 ) L 2 + ( Δ L + L 0 ) ( α Δ ξ + β Δ ψ ) f 1 ] } d α d β ,
X 2 + X 1 + A X ,
Y 2 = Y 1 + A Y ,
A X = - a ( r R ) · b 1 M - Δ L Δ ξ M ,
A Y = - a ( r R ) · b 2 M - Δ L Δ ψ M ,
x = - M X cos ( q 1 ) ,
y = - cos ( q 1 ) M Y - sin ( q 1 ) sin ( q 2 ) M X cos ( q 1 ) cos ( q 2 ) .
J ( α 1 , β 1 ; α 2 , β 2 ) = B exp ( i k θ 2 ) exp [ i k α 1 2 - α 2 2 + β 1 2 - β 2 2 2 ( f 1 + f 2 ) ] × exp [ i k ( Δ L + L 0 ) f 1 + f 2 ( α 2 Δ ξ + β 2 Δ ψ ) ] × Σ exp { i k ξ [ α 2 - α 1 f 1 + f 2 + a ( r R ) · b 1 L 0 - ( Δ L + L 0 ) Δ ξ ( 1 Δ L + L 0 + 1 f 1 + f 2 - 1 f 1 ) ] } × exp { i k ψ [ β 2 - β 1 f 1 + f 2 + a ( r R ) · b 2 L 0 - ( Δ L + L 0 ) Δ ψ ( 1 Δ L + L 0 + 1 f 1 + f 2 - 1 f 1 ) ] } d ξ d ψ ,
α 2 = α 1 + A α ,
β 2 = β 1 + A β ,
A α = - ( f 1 + f 2 ) a ( r R ) · b 1 L 0 + [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ξ ,
A β = - ( f 1 + f 2 ) a ( r R ) · b 2 L 0 + [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ψ ,
J ( α 1 , β 1 ; α 2 , β 2 ) = P ( α 1 , β 1 ) P * ( α 2 , β 2 ) J ( α 1 , β 1 ; α 2 , β 2 ) × exp ( - i k α 1 2 - α 2 2 + β 1 2 - β 2 2 2 f 2 ) × δ ( α 2 - α 1 - A α , β 2 - β 1 - A β ) ,
J ( X 1 , Y 1 ; X 2 , Y 2 ) = C exp ( i k θ 3 ) exp ( i k X 1 2 - X 2 2 + Y 1 2 - Y 2 2 2 L 2 ) × exp ( - i k X 2 L 2 { ( f 1 + f 2 ) a ( r R ) · b 1 L 0 - [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ξ } ) × exp ( - i k Y 2 L 2 { ( f 1 + f 2 ) a ( r R ) · b 2 L 0 - [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ψ } ) × Σ P ( α , β ) P * ( α + A α , β + A β ) × exp [ i k α ( X 2 - X 1 ) + β ( Y 2 - Y 1 ) L 2 ] × exp ( i k α ( 1 f 1 + f 2 - 1 f 2 + 1 L 2 ) { ( f 1 + f 2 ) a ( r R ) · b 1 L 0 - [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ξ } ) × exp ( i k β ( 1 f 1 + f 2 - 1 f 2 + 1 L 2 ) { ( f 1 + f 2 ) a ( r R ) · b 2 L 0 - [ f 1 + f 2 - f 2 ( Δ L + L 0 ) f 1 ] Δ ψ } ) × exp [ i k ( Δ L + L 0 ) ( α Δ ξ + β Δ ψ ) f 1 + f 2 ] d α d β ,
X 2 = X 1 + A X ,
Y 2 = Y 1 + A Y ,
A X = - a ( r R ) · b 1 M - Δ L Δ ξ M ,
A Y = - a ( r R ) · b 2 M - Δ L Δ ψ M ,
x = - cos ( q 2 ) M X [ M ( f 1 + f 2 ) - ( Δ L + L 0 ) ] cos ( q 1 ) cos ( q 2 ) [ M ( f 1 + f 2 ) - L 0 ] + sin ( q 1 ) M X - cos ( q 1 ) sin ( q 2 ) M Y ,
y = - [ cos ( q 1 ) M Y - sin ( q 1 ) sin ( q 2 ) M X ] [ M ( f 1 + f 2 ) - ( Δ L + L 0 ) ] cos ( q 1 ) cos ( q 2 ) [ M ( f 1 + f 2 ) - L 0 ] + sin ( q 1 ) M X - cos ( q 1 ) sin ( q 2 ) M Y .
A x = - a x M + Δ L M [ x x l s x + x y l s y - ω y ( l s z + 1 ) + ω z l s y ] ,
A y = - a y M + Δ L M [ x y l s x + y y l s y + ω x ( l s z + 1 ) - ω z l s x ] ,
A X = - a x cos ( θ 0 ) - a z sin ( θ 0 ) M + Δ L L 1 { - a x [ L 0 cos 2 ( θ s ) L s cos ( θ 0 ) + cos ( θ 0 ) ] + a z [ L 0 sin ( θ s ) cos ( θ s ) L s cos ( θ 0 ) + sin ( θ 0 ) ] } + Δ L M { x x [ sin ( θ s ) cos ( θ 0 ) + tan ( θ 0 ) ] - ω y [ cos ( θ s ) cos ( θ 0 ) + 1 ] } ,
A Y = - a y M - Δ L a y L 1 [ L 0 L s + 1 ] + Δ L M { ( x y - ω z ) [ sin ( θ s ) + sin ( θ 0 ) ] + ω x [ cos ( θ s ) + cos ( θ 0 ) ] } ,

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