Abstract

The methods of generating the cross derivatives of in-plane displacements by lateral shear interferometry and their subsequent addition are presented. Two approaches are introduced: (1) spatial filtering of composed diffraction structures and (2) moire superimposition of conjugate-type lateral shear interferograms. Experimental corroboration of the principle is presented.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, Oxford, 1969).
  2. J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
    [CrossRef]
  3. K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
    [CrossRef]
  4. K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
    [CrossRef]
  5. K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
    [CrossRef]
  6. D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
    [CrossRef]
  7. D. Post, “Moire Interferometry,” in SEM Handbook of Experimental Mechanics, A. S. Kobayashi, Ed. (Prentice-Hall, Englewood Cliffs, NJ, 1986).
  8. K. Patorski, “Shearing Interferometry Approach for Producing Shear Strain Maps,” Opt. Appl. 17, 299 (1987).
  9. K. Patorski, “Conjugate Lateral Shear Interferometry and Its Implementation,” J. Opt. Soc. Am. A 3, 1862 (1986).
    [CrossRef]
  10. M. Pluta, “On Double-Refracting Microinterferometers Which Suffer from a Variable Interfringe Spacing Across the Image Plane,” J. Microsc. 146, 41 (1987).
    [CrossRef]
  11. N. M. Spornik, V. I. Yanichkin, “Grating Interferometer with Variable Shearing of Wavefronts and Arbitrary Fringe Detection,” Sov. J. Opt. Technol. 38, 487 (1971).
  12. J. C. Wyant, “Lateral-Shear Interferometer Having Variable Shear and Tilt,” J. Opt. Soc. Am. 63, 1312 (1973).
  13. P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
    [CrossRef]
  14. M. P. Rimmer, J. C. Wyant, “Evaluation of Large Aberrations Using a Lateral-Shear Interferometer Having Variable Shear,” Appl. Opt. 14, 142 (1975).
    [PubMed]
  15. K. Patorski, A. Ulinowicz, “Conjugate Lateral Shear Interferometry: Further Considerations,” Appl. Opt. 26, 4506 (1987).
    [CrossRef] [PubMed]

1987 (3)

K. Patorski, “Shearing Interferometry Approach for Producing Shear Strain Maps,” Opt. Appl. 17, 299 (1987).

M. Pluta, “On Double-Refracting Microinterferometers Which Suffer from a Variable Interfringe Spacing Across the Image Plane,” J. Microsc. 146, 41 (1987).
[CrossRef]

K. Patorski, A. Ulinowicz, “Conjugate Lateral Shear Interferometry: Further Considerations,” Appl. Opt. 26, 4506 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
[CrossRef]

1984 (1)

K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
[CrossRef]

1979 (1)

K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
[CrossRef]

1978 (1)

K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
[CrossRef]

1975 (1)

1974 (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
[CrossRef]

1973 (1)

J. C. Wyant, “Lateral-Shear Interferometer Having Variable Shear and Tilt,” J. Opt. Soc. Am. 63, 1312 (1973).

1971 (2)

N. M. Spornik, V. I. Yanichkin, “Grating Interferometer with Variable Shearing of Wavefronts and Arbitrary Fringe Detection,” Sov. J. Opt. Technol. 38, 487 (1971).

J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
[CrossRef]

Clark, J. A.

J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
[CrossRef]

Czarnek, R.

D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
[CrossRef]

Durelli, A. J.

J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
[CrossRef]

Hariharan, P.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
[CrossRef]

Ishida, A.

K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
[CrossRef]

Joh, D.

D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
[CrossRef]

Kato, K.

K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
[CrossRef]

Kumamoto, Y.

K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
[CrossRef]

Murota, T.

K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
[CrossRef]

Parks, V. J.

J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
[CrossRef]

Patorski, K.

Pluta, M.

M. Pluta, “On Double-Refracting Microinterferometers Which Suffer from a Variable Interfringe Spacing Across the Image Plane,” J. Microsc. 146, 41 (1987).
[CrossRef]

Post, D.

D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
[CrossRef]

D. Post, “Moire Interferometry,” in SEM Handbook of Experimental Mechanics, A. S. Kobayashi, Ed. (Prentice-Hall, Englewood Cliffs, NJ, 1986).

Rimmer, M. P.

Spornik, N. M.

N. M. Spornik, V. I. Yanichkin, “Grating Interferometer with Variable Shearing of Wavefronts and Arbitrary Fringe Detection,” Sov. J. Opt. Technol. 38, 487 (1971).

Steel, W. H.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
[CrossRef]

Theocaris, P. S.

See, for example, P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, Oxford, 1969).

Ulinowicz, A.

Wyant, J. C.

M. P. Rimmer, J. C. Wyant, “Evaluation of Large Aberrations Using a Lateral-Shear Interferometer Having Variable Shear,” Appl. Opt. 14, 142 (1975).
[PubMed]

P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
[CrossRef]

J. C. Wyant, “Lateral-Shear Interferometer Having Variable Shear and Tilt,” J. Opt. Soc. Am. 63, 1312 (1973).

Yanichkin, V. I.

N. M. Spornik, V. I. Yanichkin, “Grating Interferometer with Variable Shearing of Wavefronts and Arbitrary Fringe Detection,” Sov. J. Opt. Technol. 38, 487 (1971).

Appl. Opt. (2)

Bull. JSME (3)

K. Kato, T. Murota, A. Ishida, “On Interferometry for Measurement of Shear Strain and Rotation Using Diffraction Beams,” Bull. JSME 21, 1222 (1978).
[CrossRef]

K. Kato, T. Murota, A. Ishida, “On Compensation of Errors in the Moire Method by Use of Diffraction Beams,” Bull. JSME 22, 319 (1979).
[CrossRef]

K. Kato, T. Murota, Y. Kumamoto, “Improvement of the Method of Measuring Shear Strain Using Interferometry of Diffraction Beams,” Bull. JSME 27, 165 (1984).
[CrossRef]

Exp. Mech. (1)

D. Post, R. Czarnek, D. Joh, “Shear Strain Contours from Moire Interferometry,” Exp. Mech. 25, 282 (1985).
[CrossRef]

J. Microsc. (1)

M. Pluta, “On Double-Refracting Microinterferometers Which Suffer from a Variable Interfringe Spacing Across the Image Plane,” J. Microsc. 146, 41 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. C. Wyant, “Lateral-Shear Interferometer Having Variable Shear and Tilt,” J. Opt. Soc. Am. 63, 1312 (1973).

J. Opt. Soc. Am. A (1)

J. Strain Anal. (1)

J. A. Clark, A. J. Durelli, V. J. Parks, “Shear and Rotation Moire Patterns Obtained by Spatial Filtering of Diffraction Patterns,” J. Strain Anal. 6, 134 (1971).
[CrossRef]

Opt. Appl. (1)

K. Patorski, “Shearing Interferometry Approach for Producing Shear Strain Maps,” Opt. Appl. 17, 299 (1987).

Opt. Commun. (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double Grating Interferometer with Variable Lateral Shear,” Opt. Commun. 11, 317 (1974).
[CrossRef]

Sov. J. Opt. Technol. (1)

N. M. Spornik, V. I. Yanichkin, “Grating Interferometer with Variable Shearing of Wavefronts and Arbitrary Fringe Detection,” Sov. J. Opt. Technol. 38, 487 (1971).

Other (2)

See, for example, P. S. Theocaris, Moire Fringes in Strain Analysis (Pergamon, Oxford, 1969).

D. Post, “Moire Interferometry,” in SEM Handbook of Experimental Mechanics, A. S. Kobayashi, Ed. (Prentice-Hall, Englewood Cliffs, NJ, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Optical system for producing the cross-derivative interferograms ∂u(x,y)∂y and ∂v(x,y)/∂x. The case of generating ∂u(x,y)/∂y is shown: SG, specimen grating; R, shearing grating; L1 and L2, imaging optics; OP, observation plane.

Fig. 2
Fig. 2

Spatial filtering system for producing the interference map of γxy from the cross-derivative gratings: DG, cross-type structure of derivative gratings ∂u(x,y)/∂y and ∂v(x,y)/∂x; SF, spatial filter.

Fig. 3
Fig. 3

Double spatial filtering system for obtaining the map of γxy: DG, cross-type structure of the cross derivatives; MG, reference grating; SF1 and SF2, spatial filters; L1, L2 and L3, L4, imaging optics.

Fig. 4
Fig. 4

Interferogram of cross derivative ∂v(x,y)/∂x of in-plane displacement v(x,y) simulated by the wavefront with spherical aberration.

Fig. 5
Fig. 5

Interferogram of function γxy.

Fig. 6
Fig. 6

Schematic representation of a double grating interferometer with gratings G1 and G2 placed symmetrically with respect to beam focus O. Counter propagate direction of rotation of G1 and G2 about the optical axis results in lateral shearing in the output plane.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

T x S G ( x , y ) = m a m exp { i m 2 π d [ x + u ( x , y ) ] } ,
T R ( x , y ) = c 0 + 2 c 1 cos 2 π d R y ,
E ( x , y ) exp { i [ - k θ y y + 2 π d u ( x , y - Δ y ) ] } + exp { i [ k θ y y - 2 π d u ( x , y + Δ y ) ] } ,
θ y = z f L 2 tan ( λ d R ) ,
Δ y = f L 2 tan ( λ / d R ) .
I 1 ( x , y ) 1 + cos 2 [ k θ y y + 2 π d Δ y u ( x , y ) y ] .
p = ( f L , 2 / z ) d R
I 2 ( x , y ) 1 + cos 2 [ k θ x x + 2 π d Δ x y ( x , y ) x ] .
( a )     ( 1 x , 0 )     and ( 0 x , - 1 y ) , ( b )     ( - 1 x , 0 y ) and ( 0 x , 1 y ) , ( c )     ( 1 x , 1 y )     and ( - 1 x , - 1 y ) , ( d )     ( 1 x , 1 y )     and ( 0 x , 0 y ) ,
E + 1 x , 0 y ( x , y ) + E 0 x , - 1 y ( x , y ) exp { i 2 π p [ x + Δ v ( x , y ) x ] } + exp { - i 2 π p [ y + Δ u ( x , y ) y ] } ,
I ( x , y ) = 1 + 2 cos 2 π p { x + y + Δ [ u ( x , y ) y + u ( x , y ) x ] } .
( a )     ( - 1 x , 0 y ) and ( 0 x , + 1 y ) , ( b )     ( 1 x , 0 y )     and ( 0 x , - 1 y ) .
I ( x , y ) | exp [ - i 2 π p Δ v ( x , y ) y ] + exp [ i 2 π p Δ u ( x , y ) y ] | 2 = 2 { 1 + cos 2 π p Δ [ u ( x , y ) y + v ( x , y ) x ] } ,
ψ ( x , y ) = W 0 ( x 2 + y 2 ) 2 ,
u ( x , y ) = v ( x , y ) = W 0 ( x 2 + y 2 ) 2 ,
u ( x , y ) y = ψ y = 4 W 0 y ( x 2 + y 2 ) ,
u ( x , y ) y = ψ x = 4 W 0 x ( x 2 + y 2 ) .
γ x , y = 4 W 0 ( x + y ) ( x 2 + y 2 ) .

Metrics