Abstract

A measurement system based on digital holography for the simultaneous measurement of out-of-plane and radial in-plane displacement fields for the assessment of residual stress is presented. Two holograms are recorded at the same time with a single image taken by a digital camera, allowing the separate evaluation of in-plane and out-of-plane movement. An axis-symmetrical diffractive optical element is used for the illumination of the object, which causes radial sensitivity vectors. By the addition and, respectively, the subtraction, of the four phase maps calculated from two camera frames, the in-plane and out-of-plane deformation of an object can be calculated separately. The device presented is suitable for high-speed, high-resolution measurement of residual stress. In addition to the setup, first measurement results and a short comparison to a mature digital speckle pattern interferometry setup are shown.

© 2010 Optical Society of America

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References

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  1. J. W. Dally and W. F. Riley, Experimental Stress Analysis, 3rd ed. (McGraw-Hill, 1991).
  2. N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residual stress,” Exp. Mech. 6, 577–586(1966).
    [CrossRef]
  3. H. Underwood, “Residual-stress measurement using surface displacements around an indentation,” Exp. Mech. 13, 373–380 (1973).
    [CrossRef]
  4. A. Makino and D. V. Nelson, “Residual stress determination by single axis holographic interferometry and hole drilling, part I: theory,” Exp. Mech. 34, 66–78 (1994).
    [CrossRef]
  5. D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
    [CrossRef]
  6. J. Zhang and T. C. Chong, “Fiber electronic speckle pattern interferometry and its applications in residual stress measurements,” Appl. Opt. 37, 6707–6715 (1998).
    [CrossRef]
  7. M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
    [CrossRef] [PubMed]
  8. S. T. Lin, “Blind-hole stress determination using optical interferometry,” Exp. Mech. 40, 60–67 (2000).
    [CrossRef]
  9. D. R. Schmitt and R. W. Hunt, “Inversion of speckle interferometer fringes for hole-drilling residual stress determination,” Exp. Mech. 40, 129–137 (2000).
    [CrossRef]
  10. C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
    [CrossRef]
  11. F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
    [CrossRef]
  12. V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
    [CrossRef]
  13. A. J. Moore and J. R. Tyrer, “An electronic speckle pattern interferometer for complete in-plane measurement,” Meas. Sci. Technol. 1, 1024–1030 (1990).
    [CrossRef]
  14. A. J. Moore and J. R. Tyrer, “Two-dimensional strain measurement with ESPI,” Opt. Lasers Eng. 24, 381–402 (1996).
    [CrossRef]
  15. A. Albertazzi, Jr., M. R. Borges, and C. Kanda, “A radial in-plane interferometer for residual stresses measurement using ESPI,” in Proceedings of the Society of Experimental Mechanics IX International Congress on Experimental Mechanics (Society for Experimental Mechanics, 2000), pp. 108–111.
  16. A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
    [CrossRef]
  17. M. Viotti, W. Kapp, and A. Albertazzi, Jr., “Achromatic digital speckle pattern interferometer with constant radial in-plane sensitivity by using a diffractive optical element,” Appl. Opt. 48, 2275–2281 (2009).
    [CrossRef] [PubMed]
  18. G. Pedrini, Y.-L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional carrier,” Appl. Opt. 36, 786–792(1997).
    [CrossRef] [PubMed]
  19. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231–7242(2007).
    [CrossRef] [PubMed]
  20. T. Kreis, Holographic Interferometry (Wiley-VCH, 2005).
  21. J. A. Gilbert and J. W. Herrick, “Dual-beam holographic deflection measurement,” Exp. Mech. 21, 349–354 (1981).
    [CrossRef]
  22. M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
    [CrossRef]

2009 (1)

2007 (1)

2004 (1)

V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
[CrossRef]

2001 (2)

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
[CrossRef]

2000 (3)

S. T. Lin, “Blind-hole stress determination using optical interferometry,” Exp. Mech. 40, 60–67 (2000).
[CrossRef]

D. R. Schmitt and R. W. Hunt, “Inversion of speckle interferometer fringes for hole-drilling residual stress determination,” Exp. Mech. 40, 129–137 (2000).
[CrossRef]

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

1999 (1)

M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
[CrossRef] [PubMed]

1998 (1)

1997 (2)

G. Pedrini, Y.-L. Zou, and H. J. Tiziani, “Simultaneous quantitative evaluation of in-plane and out-of-plane deformations by use of a multidirectional carrier,” Appl. Opt. 36, 786–792(1997).
[CrossRef] [PubMed]

D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
[CrossRef]

1996 (1)

A. J. Moore and J. R. Tyrer, “Two-dimensional strain measurement with ESPI,” Opt. Lasers Eng. 24, 381–402 (1996).
[CrossRef]

1994 (1)

A. Makino and D. V. Nelson, “Residual stress determination by single axis holographic interferometry and hole drilling, part I: theory,” Exp. Mech. 34, 66–78 (1994).
[CrossRef]

1990 (1)

A. J. Moore and J. R. Tyrer, “An electronic speckle pattern interferometer for complete in-plane measurement,” Meas. Sci. Technol. 1, 1024–1030 (1990).
[CrossRef]

1982 (1)

M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
[CrossRef]

1981 (1)

J. A. Gilbert and J. W. Herrick, “Dual-beam holographic deflection measurement,” Exp. Mech. 21, 349–354 (1981).
[CrossRef]

1973 (1)

H. Underwood, “Residual-stress measurement using surface displacements around an indentation,” Exp. Mech. 13, 373–380 (1973).
[CrossRef]

1966 (1)

N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residual stress,” Exp. Mech. 6, 577–586(1966).
[CrossRef]

Albertazzi, A.

M. Viotti, W. Kapp, and A. Albertazzi, Jr., “Achromatic digital speckle pattern interferometer with constant radial in-plane sensitivity by using a diffractive optical element,” Appl. Opt. 48, 2275–2281 (2009).
[CrossRef] [PubMed]

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

A. Albertazzi, Jr., M. R. Borges, and C. Kanda, “A radial in-plane interferometer for residual stresses measurement using ESPI,” in Proceedings of the Society of Experimental Mechanics IX International Congress on Experimental Mechanics (Society for Experimental Mechanics, 2000), pp. 108–111.

Balalov, V. V.

V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
[CrossRef]

Bondarenko, M. M.

V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
[CrossRef]

Borges, M. R.

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

A. Albertazzi, Jr., M. R. Borges, and C. Kanda, “A radial in-plane interferometer for residual stresses measurement using ESPI,” in Proceedings of the Society of Experimental Mechanics IX International Congress on Experimental Mechanics (Society for Experimental Mechanics, 2000), pp. 108–111.

Charrière, F.

Chong, T. C.

Colomb, T.

Cuche, E.

Dally, J. W.

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 3rd ed. (McGraw-Hill, 1991).

Depeursinge, C.

Díaz, F. V.

F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
[CrossRef]

Duqennoy, M.

M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
[CrossRef] [PubMed]

Emery, Y.

Fuchs, E. A.

D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
[CrossRef]

Gilbert, J. A.

J. A. Gilbert and J. W. Herrick, “Dual-beam holographic deflection measurement,” Exp. Mech. 21, 349–354 (1981).
[CrossRef]

Herrick, J. W.

J. A. Gilbert and J. W. Herrick, “Dual-beam holographic deflection measurement,” Exp. Mech. 21, 349–354 (1981).
[CrossRef]

Hideki, I.

M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
[CrossRef]

Hrebabetzky, F.

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

Hunt, R. W.

D. R. Schmitt and R. W. Hunt, “Inversion of speckle interferometer fringes for hole-drilling residual stress determination,” Exp. Mech. 40, 129–137 (2000).
[CrossRef]

Kanda, C.

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

A. Albertazzi, Jr., M. R. Borges, and C. Kanda, “A radial in-plane interferometer for residual stresses measurement using ESPI,” in Proceedings of the Society of Experimental Mechanics IX International Congress on Experimental Mechanics (Society for Experimental Mechanics, 2000), pp. 108–111.

Kapp, W.

Kaufmann, G. H.

F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
[CrossRef]

Kobayashi, S.

M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
[CrossRef]

Kreis, T.

T. Kreis, Holographic Interferometry (Wiley-VCH, 2005).

Kühn, J.

Lin, S. T.

S. T. Lin, “Blind-hole stress determination using optical interferometry,” Exp. Mech. 40, 60–67 (2000).
[CrossRef]

Makino, A.

D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
[CrossRef]

A. Makino and D. V. Nelson, “Residual stress determination by single axis holographic interferometry and hole drilling, part I: theory,” Exp. Mech. 34, 66–78 (1994).
[CrossRef]

Marquet, P.

Möller, O.

F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
[CrossRef]

Montfort, F.

Moore, A. J.

A. J. Moore and J. R. Tyrer, “Two-dimensional strain measurement with ESPI,” Opt. Lasers Eng. 24, 381–402 (1996).
[CrossRef]

A. J. Moore and J. R. Tyrer, “An electronic speckle pattern interferometer for complete in-plane measurement,” Meas. Sci. Technol. 1, 1024–1030 (1990).
[CrossRef]

Nelson, D. V.

D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
[CrossRef]

A. Makino and D. V. Nelson, “Residual stress determination by single axis holographic interferometry and hole drilling, part I: theory,” Exp. Mech. 34, 66–78 (1994).
[CrossRef]

Pedrini, G.

Pisarev, V. S.

V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
[CrossRef]

Quathouh, M.

M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
[CrossRef] [PubMed]

Qurak, M.

M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
[CrossRef] [PubMed]

Rendler, N. J.

N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residual stress,” Exp. Mech. 6, 577–586(1966).
[CrossRef]

Riley, W. F.

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 3rd ed. (McGraw-Hill, 1991).

Schmitt, D. R.

D. R. Schmitt and R. W. Hunt, “Inversion of speckle interferometer fringes for hole-drilling residual stress determination,” Exp. Mech. 40, 129–137 (2000).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

Sciammarella, F. M.

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

Singh, B.

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

Takeda, M.

M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
[CrossRef]

Tiziani, H. J.

Trentadue, B.

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

Tyrer, J. R.

A. J. Moore and J. R. Tyrer, “Two-dimensional strain measurement with ESPI,” Opt. Lasers Eng. 24, 381–402 (1996).
[CrossRef]

A. J. Moore and J. R. Tyrer, “An electronic speckle pattern interferometer for complete in-plane measurement,” Meas. Sci. Technol. 1, 1024–1030 (1990).
[CrossRef]

Underwood, H.

H. Underwood, “Residual-stress measurement using surface displacements around an indentation,” Exp. Mech. 13, 373–380 (1973).
[CrossRef]

Vigness, I.

N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residual stress,” Exp. Mech. 6, 577–586(1966).
[CrossRef]

Viotti, M.

Zhang, J.

Zou, Y.-L.

Appl. Opt. (3)

Exp. Mech. (8)

J. A. Gilbert and J. W. Herrick, “Dual-beam holographic deflection measurement,” Exp. Mech. 21, 349–354 (1981).
[CrossRef]

N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residual stress,” Exp. Mech. 6, 577–586(1966).
[CrossRef]

H. Underwood, “Residual-stress measurement using surface displacements around an indentation,” Exp. Mech. 13, 373–380 (1973).
[CrossRef]

A. Makino and D. V. Nelson, “Residual stress determination by single axis holographic interferometry and hole drilling, part I: theory,” Exp. Mech. 34, 66–78 (1994).
[CrossRef]

S. T. Lin, “Blind-hole stress determination using optical interferometry,” Exp. Mech. 40, 60–67 (2000).
[CrossRef]

D. R. Schmitt and R. W. Hunt, “Inversion of speckle interferometer fringes for hole-drilling residual stress determination,” Exp. Mech. 40, 129–137 (2000).
[CrossRef]

C. A. Sciammarella, B. Singh, B. Trentadue, and F. M. Sciammarella, “Stress analysis of weldments by holographic moiré and the finite element method,” Exp. Mech. 40, 15–21(2000).
[CrossRef]

F. V. Díaz, G. H. Kaufmann, and O. Möller, “Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing,” Exp. Mech. 41, 319–323 (2001).
[CrossRef]

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

M. Takeda, I. Hideki, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160(1982).
[CrossRef]

Meas. Sci. Technol. (1)

A. J. Moore and J. R. Tyrer, “An electronic speckle pattern interferometer for complete in-plane measurement,” Meas. Sci. Technol. 1, 1024–1030 (1990).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (3)

D. V. Nelson, A. Makino, and E. A. Fuchs, “The holographic-hole drilling Method for residual stress determination,” Opt. Lasers Eng. 27, 3–23 (1997).
[CrossRef]

A. J. Moore and J. R. Tyrer, “Two-dimensional strain measurement with ESPI,” Opt. Lasers Eng. 24, 381–402 (1996).
[CrossRef]

V. S. Pisarev, V. V. Balalov, and M. M. Bondarenko, “Classification of holographic fringe patterns inherent in through hole drilling in residual stress field,” Opt. Lasers Eng. 42, 673–702 (2004).
[CrossRef]

Proc. SPIE (1)

A. Albertazzi, Jr., C. Kanda, M. R. Borges, and F. Hrebabetzky, “Portable residual stresses measurement device using ESPI and a radial in-plane interferometer,” Proc. SPIE 4420, 112–122 (2001).
[CrossRef]

Ultrasonics (1)

M. Duqennoy, M. Quathouh, and M. Qurak, “Determination of stresses in aluminium alloy using optical detection of Rayleigh waves,” Ultrasonics 37, 365–372 (1999).
[CrossRef] [PubMed]

Other (3)

J. W. Dally and W. F. Riley, Experimental Stress Analysis, 3rd ed. (McGraw-Hill, 1991).

A. Albertazzi, Jr., M. R. Borges, and C. Kanda, “A radial in-plane interferometer for residual stresses measurement using ESPI,” in Proceedings of the Society of Experimental Mechanics IX International Congress on Experimental Mechanics (Society for Experimental Mechanics, 2000), pp. 108–111.

T. Kreis, Holographic Interferometry (Wiley-VCH, 2005).

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Figures (11)

Fig. 1
Fig. 1

Scheme of the DH measurement setup.

Fig. 2
Fig. 2

Ring apertures RA2 and RA1 used for the illumination of the DOE. In (a), a middle ring is darkened while the inner part and the outer part are transparent; in (b) the outer ring is darkened.

Fig. 3
Fig. 3

Optical path from the first laser diode, LD1, to object Obj, and the resulting sensitivity vector s 1 .

Fig. 4
Fig. 4

Optical path from the second laser diode, LD2, to object Obj, and the resulting sensitivity vector s 2 .

Fig. 5
Fig. 5

Detailed view of the DOE with the two illuminations of lasers LD1 and LD2 and the resulting in-plane and out-of-plane sensitivity vectors s IP and s OP for the addition and subtraction of the phase-difference phase maps.

Fig. 6
Fig. 6

Object used for measurement of in-plane movements. (a) Unloaded state, (b) loaded state, and (c) front view of (b).

Fig. 7
Fig. 7

Metal sheet, loaded with a micrometer screw, used for in-plane and out-of-plane deformation measurements.

Fig. 8
Fig. 8

(a) Magnified part of a recorded hologram showing both spatial carrier fringes and (b) power spectrum of the Fourier transform of (a) with the approximate size and position of the applied filters FF1 and FF2.

Fig. 9
Fig. 9

Measurement of an almost in-plane only rigid body movement (see Fig. 6). Images (a) and (b) show the phase difference for both lasers. Image (c) shows the addition of (a) and (b), i.e., out-of-plane movement, and image (d) shows the difference of (a) and (b), i.e., the in-plane movement.

Fig. 10
Fig. 10

Images (a) and (b) show the phase difference induced by the deformation for both illuminations. Image (c) shows the addition of the phase maps (a) and (b), and (d) shows their difference.

Fig. 11
Fig. 11

Comparison of rigid body movement measured with (a) DH and (b) DSPI.

Equations (7)

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

Δ ϕ = ϕ 1 ϕ 2 = 2 π λ ( k O k i ) d = 2 π λ s d ,
Δ ϕ 1 = ϕ LD 1 , 1 ϕ LD 1 , 2 = 2 π λ s LD 1 d = 2 π λ ( s LD 1 , x d x + s LD 1 , y d y ) = 2 π λ ( s x d x + s y d y ) .
Δ ϕ 2 = ϕ LD 2 , 1 ϕ LD 2 , 2 = 2 π λ s LD 2 d = 2 π λ ( s x d x + s y d y ) .
Δ ϕ IP = Δ ϕ 1 Δ ϕ 2 = 2 π λ 2 s x d x ,
Δ ϕ OP = Δ ϕ 1 + Δ ϕ 2 = 2 π λ 2 s y d y ,
Δ ϕ 1 = ϕ LD 1 , 1 ϕ LD 1 , 2 , Δ ϕ 2 = ϕ LD 2 , 1 ϕ LD 2 , 2 ,
Δ ϕ IP = Δ ϕ 1 Δ ϕ 2 , Δ ϕ OP = Δ ϕ 1 + Δ ϕ 2 ,

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