Abstract

A Fourier optics analysis for a recently developed lateral shearing interferometry—coherent gradient sensing—is presented. The governing equations for the method are explicitly derived. The method of coherent gradient sensing is particularly suitable for investigating the mechanics of fracture of transparent and opaque solids. Several examples demonstrating the applicability of the method to quasistatic and dynamic crack-growth problems is presented.

© 1992 Optical Society of America

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  1. H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
    [CrossRef]
  2. Q. Ru, N. Ohyama, T. Honda, J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
    [CrossRef] [PubMed]
  3. J. Lewandowski, “Lateral shear interferometer for infrared and visible light,” Appl. Opt. 28, 2372–2379 (1989).
    [CrossRef]
  4. D. W. Templeton, Y. Y. Hung, “Shearographic fringe carrier method for data reduction computerization,” Opt. Eng. 28, 30–34 (1989).
  5. A. R. Ganeshan, D. K. Sharma, M. P. Kothiyal, “Universal digital speckle shearing interferometer,” Appl. Opt. 27, 4371–4374 (1988).
  6. G. Hausler, J. Jutfless, M. Maul, H. Weissmann, “Range sensing based on shearing interferometry,” Appl. Opt. 27, 4638–4644 (1988).
    [CrossRef] [PubMed]
  7. K. Patorski, “Shearing interferometry and the moiré method for shear strain determination,” Appl. Opt. 27, 3567–3572 (1988).
    [CrossRef] [PubMed]
  8. K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986).
    [CrossRef] [PubMed]
  9. J. Takezaki, Y. Y. Hung, “Direct measurement of strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
    [CrossRef]
  10. Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plates,” in Proceedings of the Society of Experimental Stress Analysis, B. E. Rossi, ed. Society of Experimental Stress Analysis, Bethel, Conn., 1982), p. 38.
  11. K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
    [CrossRef] [PubMed]
  12. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613–2524 (1972).
    [CrossRef] [PubMed]
  13. M. V. R. K. Murthy, “The use of a single plane parallel plate as a lateral shearing interferometer with a visible laser source,” Appl. Opt. 3, 531–534 (1964).
    [CrossRef]
  14. H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
    [CrossRef]
  15. H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).
  16. S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
    [CrossRef]
  17. H. V. Tippur, A. J. Rosakis, “Quasi-static and dynamic crack growth along bimaterial interfaces: a note on crack tip field measurements,” Rep. SM90-18 (California Institute of Technology, Pasadena, Calif., 1991); Exp. Mech. 31, 243–251 (1991).
  18. K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, New York, 1985), Chap. 3, p. 65.
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), Chap. 5, p. 77.
  20. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 3, p. 40.
  21. M. L. Williams, “On the stress distribution at the base of a stationary crack,” J. Appl. Mech. 24, 109–114 (1959).
  22. D. P. Rooke, D. J. Cartwright, Compendium of Stress Intensity Factors (Her Majesty’s Stationary Office, London, 1975).
  23. A. J. Rosakis Chandar, K. Ravi, “On the crack tip stress state: an experimental evaluation of three dimensional effects,” Int. J. Solids Struct. 22, 121–134 (1986).
    [CrossRef]

1992 (1)

S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
[CrossRef]

1991 (3)

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).

1989 (3)

J. Lewandowski, “Lateral shear interferometer for infrared and visible light,” Appl. Opt. 28, 2372–2379 (1989).
[CrossRef]

D. W. Templeton, Y. Y. Hung, “Shearographic fringe carrier method for data reduction computerization,” Opt. Eng. 28, 30–34 (1989).

Q. Ru, N. Ohyama, T. Honda, J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28, 3350–3353 (1989).
[CrossRef] [PubMed]

1988 (3)

1986 (3)

K. Patorski, “Grating shearing interferometer with variable shear and fringe orientation,” Appl. Opt. 25, 4192–4198 (1986).
[CrossRef] [PubMed]

A. J. Rosakis Chandar, K. Ravi, “On the crack tip stress state: an experimental evaluation of three dimensional effects,” Int. J. Solids Struct. 22, 121–134 (1986).
[CrossRef]

J. Takezaki, Y. Y. Hung, “Direct measurement of strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

1976 (1)

1972 (1)

1964 (1)

1959 (1)

M. L. Williams, “On the stress distribution at the base of a stationary crack,” J. Appl. Mech. 24, 109–114 (1959).

Cartwright, D. J.

D. P. Rooke, D. J. Cartwright, Compendium of Stress Intensity Factors (Her Majesty’s Stationary Office, London, 1975).

Chao, Y. J.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plates,” in Proceedings of the Society of Experimental Stress Analysis, B. E. Rossi, ed. Society of Experimental Stress Analysis, Bethel, Conn., 1982), p. 38.

Chau, F. S.

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

Ganeshan, A. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 3, p. 40.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), Chap. 5, p. 77.

Hausler, G.

Honda, T.

Hung, Y. Y.

D. W. Templeton, Y. Y. Hung, “Shearographic fringe carrier method for data reduction computerization,” Opt. Eng. 28, 30–34 (1989).

J. Takezaki, Y. Y. Hung, “Direct measurement of strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

Iizuka, K.

K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, New York, 1985), Chap. 3, p. 65.

Jutfless, J.

Kothiyal, M. P.

Krishnaswamy, S.

S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).

Lewandowski, J.

J. Lewandowski, “Lateral shear interferometer for infrared and visible light,” Appl. Opt. 28, 2372–2379 (1989).
[CrossRef]

Maul, M.

Murthy, M. V. R. K.

Ohyama, N.

Patorski, K.

Ravi, K.

A. J. Rosakis Chandar, K. Ravi, “On the crack tip stress state: an experimental evaluation of three dimensional effects,” Int. J. Solids Struct. 22, 121–134 (1986).
[CrossRef]

Rooke, D. P.

D. P. Rooke, D. J. Cartwright, Compendium of Stress Intensity Factors (Her Majesty’s Stationary Office, London, 1975).

Rosakis, A. J.

S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
[CrossRef]

H. V. Tippur, A. J. Rosakis, “Quasi-static and dynamic crack growth along bimaterial interfaces: a note on crack tip field measurements,” Rep. SM90-18 (California Institute of Technology, Pasadena, Calif., 1991); Exp. Mech. 31, 243–251 (1991).

Rosakis Chandar, A. J.

A. J. Rosakis Chandar, K. Ravi, “On the crack tip stress state: an experimental evaluation of three dimensional effects,” Int. J. Solids Struct. 22, 121–134 (1986).
[CrossRef]

Ru, Q.

Shang, H. M.

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

Sharma, D. K.

Shim, V. W.

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

Silva, D. E.

Sutton, M. A.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plates,” in Proceedings of the Society of Experimental Stress Analysis, B. E. Rossi, ed. Society of Experimental Stress Analysis, Bethel, Conn., 1982), p. 38.

Suzuki, T.

Takezaki, J.

J. Takezaki, Y. Y. Hung, “Direct measurement of strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

Tay, C. J.

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

Taylor, C. E.

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plates,” in Proceedings of the Society of Experimental Stress Analysis, B. E. Rossi, ed. Society of Experimental Stress Analysis, Bethel, Conn., 1982), p. 38.

Templeton, D. W.

D. W. Templeton, Y. Y. Hung, “Shearographic fringe carrier method for data reduction computerization,” Opt. Eng. 28, 30–34 (1989).

Tippur, H. V.

S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
[CrossRef]

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
[CrossRef]

H. V. Tippur, A. J. Rosakis, “Quasi-static and dynamic crack growth along bimaterial interfaces: a note on crack tip field measurements,” Rep. SM90-18 (California Institute of Technology, Pasadena, Calif., 1991); Exp. Mech. 31, 243–251 (1991).

Toh, S. L.

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

Tsujiuchi, J.

Weissmann, H.

Williams, M. L.

M. L. Williams, “On the stress distribution at the base of a stationary crack,” J. Appl. Mech. 24, 109–114 (1959).

Yokozeki, S.

Appl. Opt. (9)

Int. J. Frac. (1)

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental measurements,” Int. J. Frac. 48, 193–204 (1991).
[CrossRef]

Int. J. Fract. (1)

H. V. Tippur, S. Krishnaswamy, A. J. Rosakis, “Optical mapping of crack tip deformations using the methods of transmission and reflection coherent gradient sensing: a study of crack tip KI dominance,” Int. J. Fract. 52, 91–117 (1991).

Int. J. Solids Struct. (1)

A. J. Rosakis Chandar, K. Ravi, “On the crack tip stress state: an experimental evaluation of three dimensional effects,” Int. J. Solids Struct. 22, 121–134 (1986).
[CrossRef]

J. Appl. Mech. (2)

M. L. Williams, “On the stress distribution at the base of a stationary crack,” J. Appl. Mech. 24, 109–114 (1959).

J. Takezaki, Y. Y. Hung, “Direct measurement of strains in plates by shearography,” J. Appl. Mech. 53, 125–129 (1986).
[CrossRef]

J. Eng. Mater. Technol. (1)

H. M. Shang, S. L. Toh, F. S. Chau, V. W. Shim, C. J. Tay, “Locating and sizing debonds in glassfibre-reinforced plastic plates using shearography,” J. Eng. Mater. Technol. 113, 99–103 (1991).
[CrossRef]

J. Mech. Phys. Solids (1)

S. Krishnaswamy, H. V. Tippur, A. J. Rosakis, “Measurement of transient crack tip deformation fields using the method of coherent gradient sensing,” J. Mech. Phys. Solids 40, 339–372 (1992).
[CrossRef]

Opt. Eng. (1)

D. W. Templeton, Y. Y. Hung, “Shearographic fringe carrier method for data reduction computerization,” Opt. Eng. 28, 30–34 (1989).

Other (6)

D. P. Rooke, D. J. Cartwright, Compendium of Stress Intensity Factors (Her Majesty’s Stationary Office, London, 1975).

Y. J. Chao, M. A. Sutton, C. E. Taylor, “Interferometric methods for measurement of curvature and twist in thin plates,” in Proceedings of the Society of Experimental Stress Analysis, B. E. Rossi, ed. Society of Experimental Stress Analysis, Bethel, Conn., 1982), p. 38.

H. V. Tippur, A. J. Rosakis, “Quasi-static and dynamic crack growth along bimaterial interfaces: a note on crack tip field measurements,” Rep. SM90-18 (California Institute of Technology, Pasadena, Calif., 1991); Exp. Mech. 31, 243–251 (1991).

K. Iizuka, Engineering Optics, 2nd ed. (Springer-Verlag, New York, 1985), Chap. 3, p. 65.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), Chap. 5, p. 77.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 3, p. 40.

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Figures (11)

Fig. 1
Fig. 1

Schematic for the experimental setup for transmission CGS.

Fig. 2
Fig. 2

Schematic for the experimental setup for reflection CGS.

Fig. 3
Fig. 3

Schematic describing the working principle of CGS.

Fig. 4
Fig. 4

Schematic describing the nomenclature used in the Fourier analysis.

Fig. 5
Fig. 5

Undeformed and deformed object wave fronts.

Fig. 6
Fig. 6

Interference patterns obtained using transmission CGS: contours representing (a) ∂(σxx + σyy)/∂xo and (b) ∂(σxx + σyy)/∂yo. Inset is specimen geometry.

Fig. 7
Fig. 7

Comparison between experimental data and analytical predictions for fringe patterns shown in Fig. 6(a).

Fig. 8
Fig. 8

Interference patterns obtained using reflection CGS: contours representing (a) ∂w/∂xo and (b) ∂w/∂yo.

Fig. 9
Fig. 9

Interference fringes obtained using transmission CGS with a modified fracture specimen; mixed-mode fringes representing contours of constant ∂(σxx + σyy)/∂xo.

Fig. 10
Fig. 10

Time sequence of ∂(σxx + σyy)/∂xo, fringes near a dynamically growing crack in PMMA (transmission CGS)

Fig. 11
Fig. 11

Time sequence of ∂w/∂xo near a dynamically growing crack in PMMA (reflection CGS); interframe time = 7 μs.

Equations (33)

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h 1 ( x 1 , y 1 ) = h o ( x 1 , y 1 ) g d 1 ( x 1 , y 1 ) = h o ( x 1 , y 1 ) 1 i λ d 1 exp [ i k ( d 1 + x 1 2 + y 1 2 2 d 1 ) ] ,
h 1 ( x 1 , y 1 ) = { h o ( x 1 , y 1 ) 1 i λ d 1 exp [ i k ( d 1 + x 1 2 + y 1 2 2 d 1 ) ] } × A m exp ( i k m y 1 θ ) .
h 2 ( x 2 , y 2 ) = h 1 ( x 2 , y 2 ) 1 i λ Δ exp [ i k ( Δ + x 2 2 + y 2 2 2 Δ ) ] .
h 2 ( x 2 , y 2 ) = h 2 ( x 2 , y 2 ) × A n exp ( i k n y 2 θ ) .
h 2 ( x 2 , y 2 ) = [ ( { h o ( x 2 , y 2 ) 1 i λ d 1 exp [ i k ( d 1 + x 2 2 + y 2 2 2 d 1 ) ] } × A m exp ( i k m y 2 θ ) ) 1 i λ Δ exp [ i k ( Δ + x 2 2 + y 2 2 2 Δ ) ] ] × A n exp ( i k n y 2 θ ) .
h f ( x f , y f ) = 1 i λ f exp [ i k ( f + d 2 ) ] exp [ i k ( x f 2 + y f 2 2 f ) ( 1 - d 2 f ) ] × H 2 ( w x , w y ) .
F [ h 2 ( x 2 , y 2 ) ] = F [ ( { h o ( x 2 , y 2 ) 1 i λ d 1 × exp [ i k ( d 1 + x 2 2 + y 2 2 2 d 1 ) ] } × A m exp ( i k m y 2 θ ) ) 1 i λ Δ exp [ i k ( Δ + x 2 2 + y 2 2 2 Δ ) ] B n δ ( w x , w y - n θ λ ) ,
F [ h 2 ( x 2 , y 2 ) ] = ( { [ H o ( w x , w y ) × exp ( i k d 1 - i π λ d 1 × { w x 2 + w y 2 } ) ] B m δ ( w x , w y - m θ λ ) } × exp ( i k Δ - i π λ Δ { w x 2 + w y 2 } ) ) B n δ ( w x , w y - n θ λ ) ,
= exp [ i k ( d 1 + Δ ) ] [ ( B m H o ( w x , w y - m θ λ ) × exp { - i π λ d 1 [ w x 2 + ( w y - m θ λ ) 2 ] } ) × exp ( - i π λ Δ { w x 2 + w y 2 } ) ] B n δ ( w x , w y - n θ λ ) .
h o ( x o , y o , z o = 0 ) = C exp [ i k d · ( x o e ^ x + y o e ^ y ) ] = C exp [ i k ( α x o + β y o ) ] ,
d = α e ^ x + β e ^ y + γ e ^ z .
H o ( w x , w y - m θ λ ) = δ ( w x - α λ , w y - m θ + β λ ) .
F [ h 2 ( x 2 , y 2 ) ] = exp [ i k ( d 1 + Δ ) ] × ( B m δ ( w x - α λ , w y - m θ + β λ ) × exp { - i π λ d 1 [ ( α λ ) 2 + ( β λ ) 2 ] } × exp { - i π λ Δ [ ( α λ ) 2 + ( m θ + β λ ) 2 ] } ) B n δ ( w x , w y - n θ λ ) .
H 2 ( w x , w y ) = B o B 1 exp [ i k ( d 1 + Δ ) ] × exp { - i π λ d 1 [ ( α λ ) 2 + ( β λ ) 2 ] } × ( exp { - i π λ Δ [ ( α λ ) 2 + ( β - θ λ ) 2 ] } + exp { - i π λ Δ [ ( α λ ) 2 + ( β λ ) 2 ] } ) × δ ( w x - α λ , w y - β - θ λ ) .
h f ( x f , y f ) = B o B 1 i λ f exp [ i k ( d 1 + Δ + d 2 + f ) ] × exp { i π λ f ( 1 - d 2 f ) ( w x 2 + w y 2 ) } × exp { - i π λ ( d 1 + Δ ) [ ( α λ ) 2 + ( β λ ) 2 ] } × [ 1 + exp { - i π Δ θ ( - 2 β + θ λ ) } ] × δ ( w x - α λ , w y - β - θ λ ) = B o B 1 i λ f exp [ i k ( d 1 + Δ + d 2 + f ) ] × exp { i π λ f ( 1 - d 2 f ) [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × exp { - i π λ ( d 1 + Δ ) [ ( α λ ) 2 + ( β λ ) 2 ] } × [ 1 + exp { - i π Δ θ ( - 2 β + θ λ ) } ] × δ ( w x - α λ , w y - β - θ λ ) .
h i ( x i , y i ) = exp { i k x i 2 + y i 2 2 d 3 } × F [ h f ( x f , y f ) × 1 i λ d 3 exp { i k ( d 3 + x f 2 + y f 2 2 d 3 ) } ] = B o B 1 λ 2 f d 3 exp { i k x i 2 + y i 2 2 d 3 } × exp [ i k ( d 1 + Δ + d 2 + f + d 3 ) ] × exp { i π λ f ( 1 - d 2 + d 3 f ) [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × exp { - i π λ ( d 1 + Δ ) [ ( α λ ) 2 + ( β λ ) 2 ] } × [ 1 + exp { - i π Δ θ ( - 2 β + θ λ ) } ] × F [ δ ( w x - α λ , w y - β - θ λ ) ] ,
= C exp [ i k Ψ ] × [ 1 + exp { - i π Δ θ ( - 2 β + θ λ ) } ] × exp { - i k ( x i α + y i β ) } ,
I i ( x i , y i ) = h i h i * = 2 C [ 1 + cos ( π Δ θ 2 β - θ λ ) ] ,
π Δ θ ( 2 β - θ / λ ) = 2 N π ,             N = 0 , ± 1 , ± 2 ,
β = N p Δ + θ 2 .
β = N p Δ ,             N = 0 , ± 1 , ± 2 , .
α = M p Δ ,             M = 0 , ± 1 , ± 2 , .
d ( δ S ) x o e x + ( δ S ) y o e y + e z ,
δ S ( x o , y o ) c h ( σ ^ x x + σ ^ y y ) .
c h ( σ ^ ˙ x x + σ ^ y y ) x o M p Δ ,             M = 0 , ± 1 , ± 2 , ,
c h ( σ ^ x x + σ ^ y y ) y o N p Δ ,             N = 0 , ± 1 , ± 2 , .
d 2 w x o e x + 2 w y o e y + e z ,
w x o M p 2 Δ ,             M = 0 , ± 1 , ± 2 , ,
w y o N p 2 Δ ,             N = 0 , ± 1 , ± 2 , .
c h ( σ ^ x x + σ ^ y y ) x o = c h K I 2 π r - 3 / 2 cos ( 3 ϕ / 2 ) + O ( r - 1 / 2 ) = M P Δ ,
c h ( σ ^ x x + σ ^ y y ) y o = c h K I 2 π r - 3 / 2 sin ( 3 ϕ / 2 ) + O ( r - 1 / 2 ) = N p Δ ,
c h ( σ ^ x x + σ ^ y y ) x o = c h 1 2 π r - 3 / 2 × [ K I cos ( 3 ϕ / 2 ) - K II sin ( 3 ϕ / 2 ) ] + O ( r - 1 / 2 ) = M p Δ ,
F [ h 2 ( x 2 , y 2 ) ] = exp [ i k ( d 1 + Δ ) ] × exp { - i π λ d 1 [ ( α λ ) 2 + ( β λ ) 2 ] } × ( B o exp { - i π λ Δ [ ( α λ ) 2 + ( β λ ) 2 ] } × δ ( w x - α λ , w y - β λ ) + B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β + θ λ ) 2 ] } × δ ( w x - α λ , w y - β + θ λ ) + B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × δ ( w x - α λ , w y - β - θ λ ) + · ) [ B o δ ( w x , w y ) + B 1 δ ( w x , w y - θ λ ) + B 1 δ ( w x , w y + θ λ ) + · ] = exp [ i k ( d 1 + Δ ) ] × exp { - i π λ d 1 [ ( α λ ) 2 + ( β λ ) 2 ] } × ( B o 2 exp { - i π λ Δ [ ( α λ ) 2 + ( β λ ) 2 ] } × δ ( w x - α λ , w y - β λ ) + B o B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β + θ λ ) 2 ] } × δ ( w x - α λ , w y - β + θ λ ) + B o B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × δ ( w x - α λ , w y - β - θ λ ) + B o B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β λ ) 2 ] } × δ ( w x - α λ , w y - β + θ λ ) + B 1 2 exp { - i π λ Δ [ ( α λ ) 2 + ( β + θ λ ) 2 ] } × δ ( w x - α λ , w y - β + 2 θ λ ) + B 1 2 exp { - i π λ Δ [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × δ ( w x - α λ , w y - β λ ) + B o B 1 exp { - i π λ Δ [ ( α λ ) 2 + ( β λ ) 2 ] } × δ ( w x - α λ , w y - β - θ λ ) + B 1 2 exp { - i π λ Δ [ ( α λ ) 2 + ( β + θ λ ) 2 ] } × δ ( w x - α λ , w y - β λ ) + B 1 2 exp { - i π λ Δ [ ( α λ ) 2 + ( β - θ λ ) 2 ] } × δ ( w x - α λ , w y - β - 2 θ λ ) + ) .

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