Abstract

Advantages of the lensless Fourier holography setup for the reconstruction of digitally recorded holograms in holographic interferometry are presented. This very simple setup helps to achieve a maximum lateral resolution of the object under investigation. Also, the numerical-reconstruction algorithm is very simple and fast to compute. A mathematical model based on Fourier optics is used to describe discretization effects and to determine the lateral resolution. The recording and the reconstruction processes are regarded as an optical imaging system, and the point-spread function is calculated. Results are verified by an experimental setup for a combined shape and deformation measurement.

© 1999 Optical Society of America

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References

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  1. R. J. Pryputniewicz, “Heterodyne holography applications in studies of small components,” Opt. Eng. 24, 849–854 (1985).
    [CrossRef]
  2. M. Kujawinska, R. J. Pryputniewicz, “Micromeasurements: a challenge for photomechanics,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 15–24 (1996).
    [CrossRef]
  3. S. Seebacher, W. Osten, W. Jüptner, “3-D deformation analysis of micro-components using digital holography,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 382–391 (1997).
    [CrossRef]
  4. W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
    [CrossRef]
  5. S. Seebacher, W. Osten, W. Jüptner, “Measuring shape and deformation of small objects using digital holography,” in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, W. O. Jüptner, eds., Proc. SPIE3479, 104–115 (1998).
    [CrossRef]
  6. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011–2015 (1994).
    [CrossRef]
  7. T. Demetrakopoulos, R. Mittra, “Digital and optical reconstruction of images from suboptical diffraction patterns,” Appl. Opt. 13, 665–670 (1974).
    [CrossRef] [PubMed]
  8. M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.
  9. P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), pp. 19–21.
  10. W. Osten, W. Jüptner, “Measurement of displacement vector fields of extended objects,” Opt. Lasers Eng. 24, 261–285 (1996).
    [CrossRef]
  11. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 4–31.
  12. C. Wagner, “Untersuchungen zum optischen 2-Wellenlängen-Contouring an Mikroelementen mittels Digitaler Holografie,” M.S. thesis (University of Karlsruhe, Karlsruhe, Germany, 1998).
  13. T. Kreis, W. Jüptner, “Principles of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, Vol. 3 of Verlag Series in Optical Metrology, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 353–360.
  14. Note that, to avoid a superposition of the two symmetrical virtual images in lensless Fourier holography, the square area must be reduced to 50% of its surface. A rectangle of the size (NΔx)(NΔx/2) can be used, but other area designs are also possible.

1996 (1)

W. Osten, W. Jüptner, “Measurement of displacement vector fields of extended objects,” Opt. Lasers Eng. 24, 261–285 (1996).
[CrossRef]

1994 (1)

1985 (1)

R. J. Pryputniewicz, “Heterodyne holography applications in studies of small components,” Opt. Eng. 24, 849–854 (1985).
[CrossRef]

1974 (1)

Demetrakopoulos, T.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 4–31.

Hariharan, P.

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), pp. 19–21.

Hirayama, T.

M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.

Jüptner, W.

W. Osten, W. Jüptner, “Measurement of displacement vector fields of extended objects,” Opt. Lasers Eng. 24, 261–285 (1996).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “Measuring shape and deformation of small objects using digital holography,” in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, W. O. Jüptner, eds., Proc. SPIE3479, 104–115 (1998).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “3-D deformation analysis of micro-components using digital holography,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 382–391 (1997).
[CrossRef]

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

T. Kreis, W. Jüptner, “Principles of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, Vol. 3 of Verlag Series in Optical Metrology, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 353–360.

Kohgo, H.

M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.

Kreis, T.

T. Kreis, W. Jüptner, “Principles of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, Vol. 3 of Verlag Series in Optical Metrology, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 353–360.

Kujawinska, M.

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

M. Kujawinska, R. J. Pryputniewicz, “Micromeasurements: a challenge for photomechanics,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 15–24 (1996).
[CrossRef]

Mittra, R.

Osten, W.

W. Osten, W. Jüptner, “Measurement of displacement vector fields of extended objects,” Opt. Lasers Eng. 24, 261–285 (1996).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “Measuring shape and deformation of small objects using digital holography,” in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, W. O. Jüptner, eds., Proc. SPIE3479, 104–115 (1998).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “3-D deformation analysis of micro-components using digital holography,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 382–391 (1997).
[CrossRef]

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

Pryputniewicz, R. J.

R. J. Pryputniewicz, “Heterodyne holography applications in studies of small components,” Opt. Eng. 24, 849–854 (1985).
[CrossRef]

M. Kujawinska, R. J. Pryputniewicz, “Micromeasurements: a challenge for photomechanics,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 15–24 (1996).
[CrossRef]

Salbut, L.

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

Schnars, U.

Seebacher, S.

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “Measuring shape and deformation of small objects using digital holography,” in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, W. O. Jüptner, eds., Proc. SPIE3479, 104–115 (1998).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “3-D deformation analysis of micro-components using digital holography,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 382–391 (1997).
[CrossRef]

Takeda, M.

M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.

Taniguchi, K.

M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.

Wagner, C.

C. Wagner, “Untersuchungen zum optischen 2-Wellenlängen-Contouring an Mikroelementen mittels Digitaler Holografie,” M.S. thesis (University of Karlsruhe, Karlsruhe, Germany, 1998).

Appl. Opt. (1)

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

Opt. Eng. (1)

R. J. Pryputniewicz, “Heterodyne holography applications in studies of small components,” Opt. Eng. 24, 849–854 (1985).
[CrossRef]

Opt. Lasers Eng. (1)

W. Osten, W. Jüptner, “Measurement of displacement vector fields of extended objects,” Opt. Lasers Eng. 24, 261–285 (1996).
[CrossRef]

Other (10)

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 4–31.

C. Wagner, “Untersuchungen zum optischen 2-Wellenlängen-Contouring an Mikroelementen mittels Digitaler Holografie,” M.S. thesis (University of Karlsruhe, Karlsruhe, Germany, 1998).

T. Kreis, W. Jüptner, “Principles of digital holography,” in Fringe ’97: Automatic Processing of Fringe Patterns, Vol. 3 of Verlag Series in Optical Metrology, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 353–360.

Note that, to avoid a superposition of the two symmetrical virtual images in lensless Fourier holography, the square area must be reduced to 50% of its surface. A rectangle of the size (NΔx)(NΔx/2) can be used, but other area designs are also possible.

M. Kujawinska, R. J. Pryputniewicz, “Micromeasurements: a challenge for photomechanics,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 15–24 (1996).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “3-D deformation analysis of micro-components using digital holography,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 382–391 (1997).
[CrossRef]

W. Jüptner, M. Kujawinska, W. Osten, L. Salbut, S. Seebacher, “Combined measurement of silicon microbeams by grating interferometry and digital holography,” in International Conference on Applied Optical Metrology, P. K. Rastogi, F. Gyímesi, eds., Proc. SPIE3407, 348–357 (1998).
[CrossRef]

S. Seebacher, W. Osten, W. Jüptner, “Measuring shape and deformation of small objects using digital holography,” in Laser Interferometry IX: Applications, R. J. Pryputniewicz, G. M. Brown, W. O. Jüptner, eds., Proc. SPIE3479, 104–115 (1998).
[CrossRef]

M. Takeda, K. Taniguchi, T. Hirayama, H. Kohgo, “Single-transform Fourier–Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1996), pp. 67–73.

P. Hariharan, Optical Holography (Cambridge U. Press, Cambridge, 1984), pp. 19–21.

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Figures (18)

Fig. 1
Fig. 1

Schematic of the simple and flexible setup for lenless Fourier holography with a tunable dye laser, a fiber coupler, and a CCD camera.

Fig. 2
Fig. 2

Schematic of the geometry of the Fresnel reconstruction algorithm that is the basis for the lensless Fourier holography algorithm.

Fig. 3
Fig. 3

Schematic of the geometry of deformation analysis by use of lensless Fourier holography.

Fig. 4
Fig. 4

Schematic of the geometry for lensless Fourier holography showing that the angle between the object beam and the reference beam is approximately constant over the whole of the sensor area.

Fig. 5
Fig. 5

Plane reference wave showing the variation of the angle between the reference beam and the object beam over the sensor’s surface.

Fig. 6
Fig. 6

Light-sensitive area γΔξ2 (shaded square) of the CCD chip that is used as the recording medium.

Fig. 7
Fig. 7

Amplitude distortion for γ = 100% resulting from the averaging operation performed by the light-sensitive area. The averaging operation affects the intensity of the reconstructed image primarily at the image corners.

Fig. 8
Fig. 8

Comb function (the infinite sum of Dirac functions that are equally spaced) in which the spacing of the individual Dirac functions corresponds to the pixel spacing. Peaks of the Dirac functions represent sampling points.

Fig. 9
Fig. 9

Contour plot of the original image wave field.

Fig. 10
Fig. 10

Contour plot of the periodic image wave field that results when the original image wave field is convolved with a comb function.

Fig. 11
Fig. 11

Allowed object size: the object is restricted to a square area of size NΔ x. Larger objects must be placed further from the CCD to avoid aliasing.

Fig. 12
Fig. 12

Example of the rect function that is multiplied with the comb function to permit consideration of the CCD chip aperture.

Fig. 13
Fig. 13

Amplitude of the point-spread function of an optical imaging system with a rectangular aperture.

Fig. 14
Fig. 14

Diagram of the 1-mm steel plate used to demonstrate the combined shape and deformation measurement of a small object.

Fig. 15
Fig. 15

Interferogram obtained from the deformation analysis.

Fig. 16
Fig. 16

Phase data obtained from shape analysis.

Fig. 17
Fig. 17

Combined results of the shape and the deformation analyses. Lower plot: the surface of the unloaded steel plate. Upper plot: the shape of the deformed object calculated from the shape and the deformation data.

Fig. 18
Fig. 18

Out-of-plane component w of the deformation field of the object.

Tables (1)

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Table 1 Measurements Needed for Determining the Deformation and the Shape of an Object

Equations (11)

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bx, y=const exp-i πλdx2+y2×Fλdhξ, ηrξ, ηexp-i πλdξ2+η2,
Ufξ, fη=Fλduξ, η=-+-+ uξ, η× exp-i 2πλdξfξ+ηfηdxdy.
rξ, η=const expi πλdξ2+η2.
bx, y=const exp-i πλdx2+y2Fλdhξ, η.
Δϕ=argb1-argb2=argFλdh1-argFλdh2.
D=λ12 cosΘ/2
Λ=λ1λ2|λ1-λ2|.
h=Λ2 cosΘ/2
Δx=λdNΔξ.
bx, y=n=-m=- bx-NΔxn, y-NΔxm×sincγNΔxx-NΔxnsincγNΔx×y-NΔxm*sinxΔxsinyΔxΦx, y,
Φx, y=exp-i2π 12NxΔx+yΔx,

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