Abstract

The notion of the coefficient of correlation between two interfering wave fronts is applied to the study of fringe formation and visibility in speckle interferometry. The parameters affecting the visibility of the in-plane displacement fringes are analyzed in the case of an arbitrary displacement and deformation of the object surface. The influence of strain as well as defocusing on the visibility is explicated. Physical significance of the results is discussed. The work is partly backed by experimental investigation.

© 1981 Optical Society of America

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  1. J. C. Charmet, F. Montel, Rev. Fr. Phys. Appl. 12, 603 (1977).
    [CrossRef]
  2. J. Ebbeni, J. C. Charmet, Appl. Opt. 16, 2543 (1977).
    [CrossRef] [PubMed]
  3. J. C. Charmet, D.Sc Thesis, U. Pierre et Marie Curie, Paris (1977).
  4. Y. Y. Hung, Opt. Commun. 11, 732 (1974).
    [CrossRef]
  5. J. N. Butters, J. A. Leendertz, J. Sci. Instrum. 5, 1107 (1973).
  6. Y. Y. Hung, R. E. Rowlands, I. M. Daniel, Appl. Opt. 14, 618 (1975).
    [CrossRef] [PubMed]
  7. Y. Y. Hung, A. J. Durelli, Strain Anal. 14, 81 (1979).
    [CrossRef]
  8. J. A. Leendertz, J. Phys. E: 3, 214 (1970).
    [CrossRef]
  9. E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
    [CrossRef]
  10. R. Jones, Opt. Laser Technol. 8, 215 (1976).
    [CrossRef]
  11. R. Jones, C. Wykes, Opt. Acta 24, 533 (1977).
    [CrossRef]
  12. J. N. Butters, R. Jones, C. Wykes, Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), Chap. 6.
  13. J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, (Springer, 1975), Vol. 9, pp. 9–75.
    [CrossRef]
  14. I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1962).
    [CrossRef]
  15. P. M. Boone, Opt. Technol. 2, 94 (1970).
    [CrossRef]
  16. C. A. Sciammarella, J. A. Gilbert, Exp. Mech. 16, 215 (1976).
    [CrossRef]
  17. J. A. Gilbert, G. A. Exner, Exp. Mech. 18, 382 (1978).
    [CrossRef]
  18. W. J. Beranek, A. J. A. Bruinsma, at SESA Spring Meeting, San Francisco (May, 1979).
  19. P. K. Rastogi, Ph.D. thesis, Besançon (1979).
  20. M. Schlüter, A. Nowatzyk, Opt. Acta 27, 799 (1980).
    [CrossRef]
  21. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  22. P. K. Rastogi, M. Spajer, J. Monneret, Opt. Lasers Eng. accepted for publication in Vol. 2, No. 2, 1981.
    [CrossRef]
  23. E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).
  24. M. Françon, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), vol. 24.
  25. K. J. Ebeling, Opt. Acta 26, 1505 (1979).
    [CrossRef]

1980 (1)

M. Schlüter, A. Nowatzyk, Opt. Acta 27, 799 (1980).
[CrossRef]

1979 (2)

Y. Y. Hung, A. J. Durelli, Strain Anal. 14, 81 (1979).
[CrossRef]

K. J. Ebeling, Opt. Acta 26, 1505 (1979).
[CrossRef]

1978 (1)

J. A. Gilbert, G. A. Exner, Exp. Mech. 18, 382 (1978).
[CrossRef]

1977 (4)

E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).

R. Jones, C. Wykes, Opt. Acta 24, 533 (1977).
[CrossRef]

J. C. Charmet, F. Montel, Rev. Fr. Phys. Appl. 12, 603 (1977).
[CrossRef]

J. Ebbeni, J. C. Charmet, Appl. Opt. 16, 2543 (1977).
[CrossRef] [PubMed]

1976 (2)

R. Jones, Opt. Laser Technol. 8, 215 (1976).
[CrossRef]

C. A. Sciammarella, J. A. Gilbert, Exp. Mech. 16, 215 (1976).
[CrossRef]

1975 (1)

1974 (1)

Y. Y. Hung, Opt. Commun. 11, 732 (1974).
[CrossRef]

1973 (1)

J. N. Butters, J. A. Leendertz, J. Sci. Instrum. 5, 1107 (1973).

1972 (1)

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

1970 (2)

J. A. Leendertz, J. Phys. E: 3, 214 (1970).
[CrossRef]

P. M. Boone, Opt. Technol. 2, 94 (1970).
[CrossRef]

1962 (1)

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

Archbold, E.

E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Beranek, W. J.

W. J. Beranek, A. J. A. Bruinsma, at SESA Spring Meeting, San Francisco (May, 1979).

Boone, P. M.

P. M. Boone, Opt. Technol. 2, 94 (1970).
[CrossRef]

Bruinsma, A. J. A.

W. J. Beranek, A. J. A. Bruinsma, at SESA Spring Meeting, San Francisco (May, 1979).

Burch, J. M.

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

Butters, J. N.

J. N. Butters, J. A. Leendertz, J. Sci. Instrum. 5, 1107 (1973).

J. N. Butters, R. Jones, C. Wykes, Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), Chap. 6.

Charmet, J. C.

J. C. Charmet, F. Montel, Rev. Fr. Phys. Appl. 12, 603 (1977).
[CrossRef]

J. Ebbeni, J. C. Charmet, Appl. Opt. 16, 2543 (1977).
[CrossRef] [PubMed]

J. C. Charmet, D.Sc Thesis, U. Pierre et Marie Curie, Paris (1977).

Daniel, I. M.

Durelli, A. J.

Y. Y. Hung, A. J. Durelli, Strain Anal. 14, 81 (1979).
[CrossRef]

Ebbeni, J.

Ebeling, K. J.

K. J. Ebeling, Opt. Acta 26, 1505 (1979).
[CrossRef]

Ennos, A. E.

E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

Exner, G. A.

J. A. Gilbert, G. A. Exner, Exp. Mech. 18, 382 (1978).
[CrossRef]

Françon, M.

M. Françon, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), vol. 24.

Gilbert, J. A.

J. A. Gilbert, G. A. Exner, Exp. Mech. 18, 382 (1978).
[CrossRef]

C. A. Sciammarella, J. A. Gilbert, Exp. Mech. 16, 215 (1976).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, (Springer, 1975), Vol. 9, pp. 9–75.
[CrossRef]

Hung, Y. Y.

Y. Y. Hung, A. J. Durelli, Strain Anal. 14, 81 (1979).
[CrossRef]

Y. Y. Hung, R. E. Rowlands, I. M. Daniel, Appl. Opt. 14, 618 (1975).
[CrossRef] [PubMed]

Y. Y. Hung, Opt. Commun. 11, 732 (1974).
[CrossRef]

Jones, R.

R. Jones, C. Wykes, Opt. Acta 24, 533 (1977).
[CrossRef]

R. Jones, Opt. Laser Technol. 8, 215 (1976).
[CrossRef]

J. N. Butters, R. Jones, C. Wykes, Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), Chap. 6.

Leendertz, J. A.

J. N. Butters, J. A. Leendertz, J. Sci. Instrum. 5, 1107 (1973).

J. A. Leendertz, J. Phys. E: 3, 214 (1970).
[CrossRef]

Monneret, J.

P. K. Rastogi, M. Spajer, J. Monneret, Opt. Lasers Eng. accepted for publication in Vol. 2, No. 2, 1981.
[CrossRef]

Montel, F.

J. C. Charmet, F. Montel, Rev. Fr. Phys. Appl. 12, 603 (1977).
[CrossRef]

Nowatzyk, A.

M. Schlüter, A. Nowatzyk, Opt. Acta 27, 799 (1980).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, M. Spajer, J. Monneret, Opt. Lasers Eng. accepted for publication in Vol. 2, No. 2, 1981.
[CrossRef]

P. K. Rastogi, Ph.D. thesis, Besançon (1979).

Reed, I. S.

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

Rowlands, R. E.

Schlüter, M.

M. Schlüter, A. Nowatzyk, Opt. Acta 27, 799 (1980).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella, J. A. Gilbert, Exp. Mech. 16, 215 (1976).
[CrossRef]

Spajer, M.

P. K. Rastogi, M. Spajer, J. Monneret, Opt. Lasers Eng. accepted for publication in Vol. 2, No. 2, 1981.
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Virdee, A.

E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).

Wykes, C.

R. Jones, C. Wykes, Opt. Acta 24, 533 (1977).
[CrossRef]

J. N. Butters, R. Jones, C. Wykes, Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), Chap. 6.

Appl. Opt. (2)

Exp. Mech. (2)

C. A. Sciammarella, J. A. Gilbert, Exp. Mech. 16, 215 (1976).
[CrossRef]

J. A. Gilbert, G. A. Exner, Exp. Mech. 18, 382 (1978).
[CrossRef]

IRE Trans. Inf. Theory (1)

I. S. Reed, IRE Trans. Inf. Theory IT-8, 194 (1962).
[CrossRef]

J. Phys. E: (1)

J. A. Leendertz, J. Phys. E: 3, 214 (1970).
[CrossRef]

J. Sci. Instrum. (1)

J. N. Butters, J. A. Leendertz, J. Sci. Instrum. 5, 1107 (1973).

Opt. Acta (4)

K. J. Ebeling, Opt. Acta 26, 1505 (1979).
[CrossRef]

M. Schlüter, A. Nowatzyk, Opt. Acta 27, 799 (1980).
[CrossRef]

E. Archbold, J. M. Burch, A. E. Ennos, Opt. Acta 19, 253 (1972).
[CrossRef]

R. Jones, C. Wykes, Opt. Acta 24, 533 (1977).
[CrossRef]

Opt. Commun. (1)

Y. Y. Hung, Opt. Commun. 11, 732 (1974).
[CrossRef]

Opt. Laser Technol. (1)

R. Jones, Opt. Laser Technol. 8, 215 (1976).
[CrossRef]

Opt. Technol. (1)

P. M. Boone, Opt. Technol. 2, 94 (1970).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

E. Archbold, A. E. Ennos, A. Virdee, Proc. Soc. Photo-Opt. Instrum. Eng. 136, 258 (1977).

Rev. Fr. Phys. Appl. (1)

J. C. Charmet, F. Montel, Rev. Fr. Phys. Appl. 12, 603 (1977).
[CrossRef]

Strain Anal. (1)

Y. Y. Hung, A. J. Durelli, Strain Anal. 14, 81 (1979).
[CrossRef]

Other (8)

J. C. Charmet, D.Sc Thesis, U. Pierre et Marie Curie, Paris (1977).

J. N. Butters, R. Jones, C. Wykes, Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), Chap. 6.

J. W. Goodman, Laser Speckle and Related Phenomena, J. C. Dainty, (Springer, 1975), Vol. 9, pp. 9–75.
[CrossRef]

W. J. Beranek, A. J. A. Bruinsma, at SESA Spring Meeting, San Francisco (May, 1979).

P. K. Rastogi, Ph.D. thesis, Besançon (1979).

M. Françon, in Handbuch der Physik, S. Flugge, Ed. (Springer, Berlin, 1956), vol. 24.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

P. K. Rastogi, M. Spajer, J. Monneret, Opt. Lasers Eng. accepted for publication in Vol. 2, No. 2, 1981.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Optical setup used for the visualization of in-plane displacements by speckle interferometry.

Fig. 2
Fig. 2

Shift of the two interfering speckle fields in the pupil plane due to out-of-plane rotation of the object. Shaded region is the effective pupil.

Fig. 3
Fig. 3

Pupil plane shift due to in-plane rotation of the object.

Fig. 4
Fig. 4

Influence of strain on the speckle shift in the pupil plane.

Fig. 5
Fig. 5

Characteristics of 8E75 emulsion: (a) normal development process (negative transparency); and (b) development process resulting in the inversion of the emulsion (positive transparency).

Fig. 6
Fig. 6

Normalized profile of correlation fringes obtained in mask technique using high contrast emulsion. Dash line is obtained with Is/〈I〉 = log2, continuous line with Is/〈I〉 = 1. Dashes belong to curve |sinΦ|.

Fig. 7
Fig. 7

Microdensitometric trace of correlation fringes obtained with sandwich technique using 10E75 photographic plates.

Fig. 8
Fig. 8

Optical system for observing correlation fringes obtained by sandwich technique.

Fig. 9
Fig. 9

Variation of the visibility of correlation fringes in function of the lateral object translation with (a) unpolarized speckle (mat diffusing point) and (b) polarized speckle.

Fig. 10
Fig. 10

(a) Correlation fringes obtained in the case of a disk under diametral compression, where additional system of displacement fringes has been added to facilitate (b) the formation of moiré pattern corresponding to strain in optically shearing the two identical images of the correlation fringe pattern.

Equations (81)

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2 Φ = Φ 2 - Φ 1 = 2 π λ · 2 S y sin ϑ ,
μ = A A * ( A 2 A 2 ) 1 / 2 ,
C = I I - I I [ ( I - I ) 2 ( I - I ) 2 ] 1 / 2 = μ 2 .
μ 1 = A 1 A 1 * ( A 1 2 A 1 2 ) 1 / 2 = μ 1 exp ( - j Φ 1 ) , μ 2 = A 2 A 2 * [ A 2 2 A 2 2 ] 1 / 2 = μ 2 exp ( - j Φ 2 ) .
μ = ½ ( μ 1 + μ 2 ) , μ 2 = ¼ [ μ 1 2 + μ 2 2 + 2 μ 1 μ 2 cos ( Φ 2 - Φ 1 ) ] .
μ 1 = exp ( - j Φ 1 ) , μ 2 = exp ( - j Φ 2 ) , and μ = cos Φ exp ( - j Ψ ) ,
Ψ = Φ 1 + Φ 2 2 = 2 π λ ( 1 + cos ϑ ) S z ,
Φ = Φ 2 - Φ 1 2 = 2 π λ · sin ϑ · S x .
p ( I ) = 1 I exp ( - I I ) .
p ( I ) = 4 I I 2 exp ( - 2 I I ) .
I = I V + I H , I = I V + I H , I V I H - I V I H = 0 , I V I H - I V I H = 0.
I ( M ) = 4 I 1 ( 1 + cos Φ cos Ψ ) .
ψ A = 2 π λ cos ϑ R · Δ ϑ R · x .
E = 0 I T N ( I ) p ( I , I ) d I d I ,
p ( I , I ) = exp [ - I + I I ( 1 - C ) ] I 2 ( 1 - C ) · I 0 [ 2 I I C I ( 1 - C ) ] ,
I 0 ( x ) = 1 2 π - π + π exp ( x cos ϑ ) d ϑ .
T = 0 T p ( I ) T N ( I ) p ( I , I ) d I d I ,
T min = 0 T p ( I ) T N ( I ) p ( I ) d I = T p ( I ) T N ( I ) ,
T max = 0 T p ( I ) p ( I ) d I 0 T N ( I ) p ( I ) d I = T p ( I ) T N ( I ) ,
V = T p T N - T p T N T p T N + T p T N .
T R ( x ) = 1 i 0 i τ 1 ( x ) τ 2 ( x - x ) d x
T max = 1 i 0 i τ 1 ( x ) τ 2 ( x ) d x ,
T min = 1 i 0 i τ 1 ( x ) τ 2 ( x - i 2 ) d x ;
V = T max - T min T max + T min .
V 1 V 2 2 ( V 1 = V 2 = 1 V = ½ ) ;
U ( x , y ) = exp { i k [ x sin ϑ - ( 1 + cos ϑ ) z ( x , y ) ] } .
X = X 0 + ( x - x 0 ) x - ( y - y 0 ) ω z , Y = Y 0 + ( y - y 0 ) y + ( x - x 0 ) ω z , Z = Z 0 - ( x - x 0 ) ω y + ( y - y 0 ) ω x .
u = ξ 2 λ f , v = η 2 λ f .
u 0 = 1 + cos ϑ λ ω x - sin ϑ λ x , v 0 = sin ϑ λ ω z - 1 + cos ϑ λ ω y .
P i ( u , v ) = exp [ j k W i ( u , v ) ] inside the pupil 0 elsewhere ,
μ 1 = - + P 1 ( u , v ) P 2 * ( u - u 0 , v - v 0 ) exp [ - j 2 π ( u X 0 + v Y 0 ) ] d u d v - + P 1 ( u , v ) 2 d u d v · exp ( j Φ 1 )
P 1 ( u , v ) + P 1 [ u ( 1 - x ) - v ω z , v ( 1 - y ) + u ω z ] ,
X 0 = X 0 ( 1 - x ) + ( x - x 0 ) x + ( y - y 0 - Y 0 ) ω z Y 0 = Y 0 ( 1 - y ) + ( y - y 0 ) y + ( x - x 0 - X 0 ) ω z ,
Φ 1 = 2 π λ { sin ϑ [ X 0 + ( x - x 0 ) x - ( y - y 0 ) ω z ] + ( 1 + cos ϑ ) [ Z 0 - ( x - x 0 ) ω y + ( y - y 0 ) ω x ] } .
P u exp { j 2 π λ [ W 1 ( u , v ) - W 2 ( u - u 0 , v - v 0 ) ] } · exp [ j 2 π ( u X 0 + v Y 0 ) ] d u d v .
μ 1 = R u ( X 0 , Y 0 ) R ( 0 , 0 ) exp ( j Φ 1 ) .
μ 1 = exp ( j Φ 1 ) sin π Δ u ( 1 - u 0 Δ u ) π Δ u · sin π Δ v ( 1 - v 0 Δ v ) π Δ v
v 0 = sin ϑ λ ω z .
μ 1 = exp ( j Φ 1 ) exp [ j π λ ( u 0 2 + v 0 2 ) Z 0 ] · p u exp ( j 2 π λ u 2 + v 2 2 Z 0 ) R ( 0 , 0 ) · exp { j 2 π [ u ( X 0 - u 0 Z 0 ) ] + v ( Y 0 - v 0 Z 0 ) } d u d v .
μ 1 ( s t , Z 0 ) = exp ( j Φ 1 ) 0 ρ 0 exp ( j 2 π λ ρ 2 2 Z 0 ) exp ( j 2 π ρ s t ) 2 π ρ d ρ R ( 0 , 0 ) ,
μ 1 = exp ( j π 2 λ Δ u 2 Z 0 ) sinc ( π 2 λ Δ u 2 Z 0 ) .
Z 0 = λ 2 ( 2 f r ) 2 ,
μ 1 = exp ( j Φ 1 ) P u exp [ j 2 π λ u 2 ( 1 - x ) 2 + v 2 ( 1 - y ) 2 2 Δ Z ] R ( 0 , 0 ) · exp [ - j 2 π λ ( u - u 0 ) 2 + ( v + v 0 ) 2 2 ] exp [ j 2 π ( u X 0 + v Y 0 ) ] d u d v .
exp [ j π λ Δ Z ( u 0 2 + v 0 2 ) ] P u exp [ - j 2 π λ Δ Z ( x u 2 + y v 2 ) ] · exp { j 2 π [ u ( X 0 + λ u 0 Δ Z ) + v ( Y 0 + λ v 0 Δ Z ) ] } d u d v .
exp [ - j 2 π λ Δ Z ( x u 2 + y v 2 ) ] ,
2 π λ Δ Z ( 1 + cos ϑ ) ω y sin ϑ x
u 0 = sin ϑ λ x or v 0 = sin ϑ λ ω z
T N = 1 for E < I s , T N = T m for E > I s ,
D = const - log [ T ( Φ ) + 10 - Δ D 2 - 10 - Δ D ( 1 - 10 - Δ D ) 2 ] .
T min = T p T N = 0.
T max = T p T N = exp ( - I s / I ) [ 1 - exp ( - I s / I ) ] ,
σ T 2 = T 2 - T 2 = T ( 1 - T ) .
S N = T max σ T = ( T max 1 - T max ) 1 / 2 .
I s = 0.69 I SNR = 0.58 , I s = I SNR = 0.55 , I s = 0.5 I SNR = 0.56.
T = 0 for I - I < I s , T = 1 for I - I > I s ,
μ 1 = A 1 A 1 * [ A 1 2 A 1 2 ] 1 / 2 .
A 1 ( x , y ) = exp [ j k 2 f ( x 2 + y 2 ) ] j λ ( 2 f ) 2 · - + R 1 ( x - x , y - y ) U ( x , y ) d x d y ,
A 1 ( x , y ) = exp [ j k 2 f ( x 2 + y 2 ) ] j λ ( 2 f ) 2 · - + R 2 ( x - x , y - y ) U ( x , y ) d x d y ,
A 1 A 1 * = 1 λ 2 ( 2 f ) 4 - + R 1 ( x - x 1 , y - y 1 ) · R 2 * ( x - x 2 , y - y 2 ) · U ( x 1 , y 1 ) U * ( x 2 , y 2 ) d x 1 d y 1 d x 2 d y 2 .
U ( x , y ) = exp j k [ x sin ϑ - ( 1 + cos ϑ ) z ( x , y ) ] .
U ( x , y ) = exp j k { ( x + X ) sin ϑ - ( 1 + cos ϑ ) [ z ( x , y ) - Z ] } ,
U ( x 1 , y 1 ) U * ( x 2 , y 2 ) = exp j k { sin ϑ [ x 1 - x 2 - X ( x 2 , y 2 ) ] - ( 1 + cos ϑ ) Z ( x 2 , y 2 ) } · α ( x 1 , y 1 ) α * [ x 2 + X ( x 2 , y 2 ) , y 2 + Y ( x 2 , y 2 ) ] ,
α ( x , y ) = exp j k ( 1 + cos ϑ ) z ( x , y ) .
K 0 ( x 1 , y 1 ) 2 δ [ x 1 - ( x 2 + X ) , y 1 - ( y 2 + Y ) ] ,
A 1 A 1 * = E 0 - + R 1 ( x - x 2 - X , y - y 2 - Y ) × R 2 * ( x - x 2 , y - y 2 ) · exp - j k [ X ( x 2 , y 2 ) sin ϑ + ( 1 + cos ϑ ) Z ( x 2 , y 2 ) ] d x 2 d y 2 ,
- + f ( x ) δ ( x - a ) d x = f ( a ) .
X = X 0 + ( x - x 0 ) x - ( y - y 0 ) ω z , Y = Y 0 + ( y - y 0 ) y + ( x - x 0 ) ω z , Z = Z 0 - ( z - z 0 ) ω y + ( y - y 0 ) ω x .
R i ( x , y ) = - + P i ( u , v ) exp [ j 2 π ( u x + u y ) ] d u d v .
R 1 ( x - x - X , y - y - Y ) = - + P 1 ( u , v ) × exp [ - j 2 π ( α 0 u + β 0 v ) ] · exp ( j 2 π { x [ u ( 1 + x ) + v ω z ] + y [ v ( 1 + y ) - u ω z ] } ) d u d v ,
α 0 = x - X 0 + x 0 x - y 0 ω z ;             β 0 = y - Y 0 + y 0 y + x 0 ω z .
u = u ( 1 + x ) + v ω z , v = v ( 1 + y ) - u ω z ,
u ( 1 + y ) - v ω z ( 1 + x ) ( 1 + y ) + ω z 2 u ( 1 - x ) - v ω z , u ω z + v ( 1 + x ) ( 1 + x ) ( 1 + y ) + ω z 2 u ω z + v ( 1 - y ) .
R 1 ( x - x - X , y - y - Y ) = - + P 1 [ u ( 1 - x ) - v ω z , v ( 1 - y ) + u ω z ] · exp ( - j 2 π { α 0 [ u ( 1 - x ) - v ω z ] + β 0 [ v ( 1 - y ) + u ω z ] } ) · exp [ + j 2 π ( u x + v y ) ] d u d v ,
R 2 ( x - x , y - y ) = - + P 2 ( u , v ) exp [ - j 2 π ( u x + v y ) ] · exp [ j 2 π ( u x + v y ) ] d u d v ,
R 1 ( x - x - X , y - y - Y ) = T F - 1 { P 1 ( u , v ) × exp [ - j Θ ( u , v ) ] } ,
R 2 ( x - x , y - y ) = T F - 1 { P 2 ( u , v ) × exp [ - j 2 π ( u x + v y ) ] } ,
A 1 A 1 * = E 0 - + P 1 ( u , v ) exp [ - j Θ ( u , v ) ] · { P 2 ( u , v ) exp [ - j 2 π ( u x + u y ) ] TF - 1 [ exp j 2 π λ [ X sin ϑ + ( 1 + cos ϑ ) Z ] ] * } d u d v ,
A 1 A 1 * = E 0 exp { j 2 π λ [ sin ϑ ( X 0 - x 0 x + y 0 ω z ) + ( 1 + cos ϑ ) ( Z 0 + x 0 ω y - y 0 ω x ) ] } · - + P 1 ( u , v ) exp [ - j Θ ( u , v ) ] P 2 * ( u - u 0 , v - v 0 ) · exp { j 2 π [ ( u - u 0 ) x + ( v - v 0 ) y ] } d u d v ,
u 0 = 1 + cos ϑ λ ω y - sin ϑ λ x , v 0 = sin ϑ λ ω z - 1 + cos ϑ λ ω x ,
P 1 ( u , v ) = P 1 [ u ( 1 - x ) - v ω z , v ( 1 - y ) + u ω z ] ,
Θ ( u , v ) = 2 π u [ ( x - X 0 ) ( 1 - x ) + x 0 x - y 0 ω z + ( y - Y 0 ) ω z ] + 2 π v [ ( y - Y 0 ) ( 1 - y ) + y 0 y + x 0 ω z - ( x - X 0 ) ω z ] .

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