Abstract

A simple three-aperture speckle shearing arrangement is proposed for simultaneous measurement of slope and curvature of the specimen. The double exposed shearogram of the laser-illuminated specimen yields five distinct diffraction halos when subjected to Fourier filtering. The first-order halos on either side of the central halo give slope fringes of the specimen, together with relatively broad low-visibility curvature fringes. The second-order halos provide pure slope fringes with a sensitivity double that of the slope fringes obtained from the first-order halo.

© 1984 Optical Society of America

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References

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  1. P. M. Boone, Exp. Mech. 15, 295 (1975).
    [CrossRef]
  2. J. A. Leendertz, J. N. Butters, J. Phys. E 6, 1107 (1973).
    [CrossRef]
  3. Y. Y. Hung, C. Y. Liang, Appl. Opt. 18, 1046 (1979).
    [CrossRef] [PubMed]
  4. R. Krishnamurthy, Ph.D. Thesis, Indian Institute of Technology, Madras (1983).
  5. A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
    [CrossRef]
  6. R. Krishamurthy, R. S. Sirohi, M. P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Optical Society of India, Jan. 1983), paper 2.1.
  7. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), pp. 156 and 157.
  8. Y. Y. Hung, R. E. Rowlands, I. M. Daniel, Appl. Opt. 14, 618 (1975).
    [CrossRef] [PubMed]
  9. F. P. Chiang, R. M. Juang, Appl. Opt. 15, 2199 (1976).
    [CrossRef] [PubMed]
  10. S. Nakadate, T. Yatagai, H. Saito, Appl. Opt. 19, 4241 (1980).
    [CrossRef] [PubMed]
  11. Y. Y. Hung, in Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), p. 67.
  12. J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
    [CrossRef]

1980 (1)

1979 (2)

Y. Y. Hung, C. Y. Liang, Appl. Opt. 18, 1046 (1979).
[CrossRef] [PubMed]

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
[CrossRef]

1978 (1)

J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
[CrossRef]

1976 (1)

1975 (2)

1973 (1)

J. A. Leendertz, J. N. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Assa, A.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
[CrossRef]

Betser, A. A.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
[CrossRef]

Boone, P. M.

P. M. Boone, Exp. Mech. 15, 295 (1975).
[CrossRef]

Brdicko, J.

J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
[CrossRef]

Butters, J. N.

J. A. Leendertz, J. N. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Chiang, F. P.

Daniel, I. M.

Hazell, C. R.

J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
[CrossRef]

Hung, Y. Y.

Juang, R. M.

Kothiyal, M. P.

R. Krishamurthy, R. S. Sirohi, M. P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Optical Society of India, Jan. 1983), paper 2.1.

Krishamurthy, R.

R. Krishamurthy, R. S. Sirohi, M. P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Optical Society of India, Jan. 1983), paper 2.1.

Krishnamurthy, R.

R. Krishnamurthy, Ph.D. Thesis, Indian Institute of Technology, Madras (1983).

Leendertz, J. A.

J. A. Leendertz, J. N. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Liang, C. Y.

Nakadate, S.

Olson, M. D.

J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
[CrossRef]

Politch, J.

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
[CrossRef]

Rowlands, R. E.

Saito, H.

Sirohi, R. S.

R. Krishamurthy, R. S. Sirohi, M. P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Optical Society of India, Jan. 1983), paper 2.1.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), pp. 156 and 157.

Yatagai, T.

Appl. Opt. (4)

Exp. Mech. (2)

A. Assa, J. Politch, A. A. Betser, Exp. Mech. 19, 129 (1979).
[CrossRef]

P. M. Boone, Exp. Mech. 15, 295 (1975).
[CrossRef]

J. Phys. E (1)

J. A. Leendertz, J. N. Butters, J. Phys. E 6, 1107 (1973).
[CrossRef]

Opt. Acta (1)

J. Brdicko, M. D. Olson, C. R. Hazell, Opt. Acta 25, 963 (1978).
[CrossRef]

Other (4)

R. Krishnamurthy, Ph.D. Thesis, Indian Institute of Technology, Madras (1983).

Y. Y. Hung, in Speckle Metrology, R. K. Erf, Ed. (Academic, New York, 1978), p. 67.

R. Krishamurthy, R. S. Sirohi, M. P. Kothiyal, presented at Eleventh All India Symposium on Optics and Optoelectronics (Optical Society of India, Jan. 1983), paper 2.1.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), pp. 156 and 157.

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Figures (6)

Fig. 1
Fig. 1

Schematic representation for slope and curvature from finite differences.

Fig. 2
Fig. 2

Schematic of defocusing-type three-aperture speckle shear interferometer.

Fig. 3
Fig. 3

Optical layout of wedge-type three-aperture speckle shear interferometer.

Fig. 4
Fig. 4

Schematic of three-aperture speckle shear interferometer for phase-difference calculations.

Fig. 5
Fig. 5

Slope and curvature fringes of a centrally loaded diaphragm obtained from first-order halo (central deflection = 15 μm).

Fig. 6
Fig. 6

Slope fringes with double sensitivity obtained from second-order halo (central deflection = 15 μm).

Equations (27)

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( d w d x ) x i f = w ( x i + 1 ) - w ( x i ) x i + 1 - x i = w i + 1 - w i h i ,
h i = x i + 1 - x i , ( d w d x ) x i b = w ( x i ) - w ( x i - 1 ) x i - x i - 1 = w i - w i - 1 h i - 1 ,
( d w d x ) x i c = w i + 1 - w i - 1 h i + 1 + h i .
( d 2 w d x 2 ) x i c = w i + 1 + w i - 1 - 2 w i h i 2 ,
Δ x = z i ( μ - 1 ) α ,
Δ x = Δ x / M ,
Δ x = z 0 ( μ - 1 ) α .
O 1 ( x 0 , y 0 ) O 1 ( x o + u 1 , y 0 + v 1 , w 1 ) , O 2 ( x 0 + Δ x , y 0 ) O 2 ( x 0 + Δ x + u 2 , y 0 + v 2 , w 2 ) , O 3 ( x 0 - Δ x , y 0 ) O 3 ( x 0 - Δ x + u 3 , y 0 + v 3 , w 3 ) ,
δ 21 = δ 2 - δ 1 = 2 π λ { [ u 2 - u 1 Δ x Δ x sin θ + w 2 - w 1 Δ x × Δ x ( 1 + cos θ ) ] - D z 0 u 1 } ,
δ 13 = δ 1 - δ 3 = 2 π λ { [ u 1 - u 3 Δ x Δ x sin θ + w 1 - w 3 Δ x × Δ x ( 1 + cos θ ) ] - D z 0 u 1 } ,
U T 1 = U 1 ( x i , y i ) + U 2 ( x i , y i ) + U 3 ( x i , y i ) = A exp ( i ϕ i ) + A exp [ i ( ϕ 2 + δ 0 ) ] + A exp [ i ( ϕ 3 - δ 0 ) ] ,
I 1 ( x i , y i ) = U T 1 · U T 1 * = I ( x i , y i ) [ 3 + 2 cos ( ϕ 21 - δ 0 ) + 2 cos ( ϕ 13 - δ 0 ) ] + 2 cos ( ϕ 23 - 2 δ 0 ) ] ,
ϕ 21 = ϕ 2 - ϕ 1 , ϕ 13 = ϕ 1 - ϕ 3 , ϕ 23 = ϕ 2 - ϕ 3 .
U T 2 = A exp [ i ( ϕ 1 + δ 1 ) ] + A exp [ i ( ϕ 2 + δ 0 + δ 2 ) ] + A exp [ i ( ϕ 3 - δ 0 + δ 3 ) ] .
I 2 ( x i , y i ) = I ( x i , y i ) [ 3 + 2 cos ( ϕ 21 - δ 0 + δ 21 ) + 2 cos ( ϕ 13 - δ 0 + δ 13 ) + 2 cos ( ϕ 23 - 2 δ 0 + δ 23 ) ] .
I ( first order ) = C [ 4 + 2 cos ( ϕ 13 - ϕ 21 ) + 2 cos ( ϕ 13 - ϕ 21 + δ 13 ) + 2 cos ( ϕ 21 - ϕ 13 + δ 21 ) + 2 cos ( ϕ 13 - ϕ 21 + δ 13 - δ 21 ) + 2 cos ( δ 21 + δ 13 2 ) cos ( δ 21 - δ 13 2 ) .
δ 21 + δ 13 = 2 π λ { [ u 2 - u 3 2 Δ x 2 Δ x sin θ + w 2 - w 3 2 Δ x 2 Δ x × ( 1 + cos θ ) ] - 2 D z 0 u 1 } .
u 2 - u 3 2 Δ x             and w 2 - w 3 2 Δ x
δ 21 + δ 13 = 4 π λ { [ ( d u d x ) Δ x sin θ + ( d w d x ) Δ x ( 1 + cos θ ) ] - D z 0 u 1 } .
δ 21 - δ 13 = 2 π λ [ u 2 + u 3 - 2 u 1 Δ x 2 Δ x 2 sin θ + w 2 + w 3 - 2 w 1 Δ x 2 Δ x 2 ( 1 + cos θ ) ] .
( u 2 + u 3 - 2 u 1 Δ x 2 ) and ( w 2 + w 3 - 2 w 1 Δ x 2 )
( d 2 u d x 2 ) and ( d 2 w d x 2 ) ,
δ 21 - δ 13 = 2 π λ [ d 2 u d x 2 Δ x 2 sin θ + d 2 w d x 2 Δ x 2 ( 1 + cos θ ) ] .
I ( second order ) = C exp ( i ϕ 23 ) + exp [ i ( ϕ 23 + δ 23 ) ] 2 = C ( 1 + cos δ 23 ) .
δ 23 = 2 π λ { [ ( u 3 - u 2 2 Δ x ) 2 Δ x sin θ + ( w 3 - w 2 2 Δ x ) 2 Δ x × ( 1 + cos θ ) ] - 2 D z 0 u } .
δ 21 + δ 13 = 4 π λ [ 2 ( d w d x ) Δ x ] ,
δ 21 - α 13 = 2 π λ [ 2 ( d 2 w d x 2 ) Δ x 2 ] .

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