Abstract

Speckle pattern decorrelation reduces the accuracy of interferometric shape and deformation measurements. We introduce a technique for the reduction of speckle noise in digital holography. The method is not based on classical filtering techniques such as median filters. Instead it utilizes the shift theorem of the Fourier transform. For this method several holograms of the same object under test are recorded. The reconstruction leads to a set of object wave fields with different speckle patterns. A proper averaging procedure, taking into account the properties of the wrapped phases, leads to an improvement of the accuracy in the resulting phase difference. The theory of the applied method is described and our first results for technical components with an improvement of accuracy up to 1/57 of the wavelength are presented.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. V. Kebbel, J. Müller, and W. Jüptner, "Characterization of aspherical microoptics using digital holography: improvement of accuracy," in Interferometry XI: Applications, W.Osten, ed., Proc. SPIE 4778,188-197 (2002).
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    [CrossRef]
  16. T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, 2005).
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    [CrossRef]
  18. I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).
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2006 (1)

2005 (1)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

2002 (2)

1999 (1)

1998 (1)

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

1997 (1)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

1994 (1)

1989 (1)

1980 (1)

1974 (1)

Baumbach, T.

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).

Brozeit, A.

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Bryngdahl, O.

Burke, J.

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Demetrakopoulos, T. H.

Fjigaki, M.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Gahr, M.

J. Kauffmann, M. Gahr, and H. J. Tiziani, "Noise reduction in speckle pattern interferometry," in Speckle Metrology 2003, K. Gastinger, O.J. Lokberg, and S. Winther, eds., Proc. SPIE 4933,9-14 (2003).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Helmers, H.

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Herriau, J. P.

Hirayama, T.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, "Single-transform Fourier/Hartley fringe analysis for holographic interferometry," in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W.Osten, eds. (Akademie Verlag, 1996), Vol. 2, pp. 67-73.

Huignard, J. P.

Javidi, B.

Jüptner, W.

T. Baumbach, W. Osten, C. v. Kopylow, and W. Jüptner, "Remote metrology by comparative digital holography," Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

V. Kebbel, J. Müller, and W. Jüptner, "Characterization of aspherical microoptics using digital holography: improvement of accuracy," in Interferometry XI: Applications, W.Osten, ed., Proc. SPIE 4778,188-197 (2002).

T. Kreis and W. Jüptner, "Principles of digital holography," in 3rd International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W.Osten, eds. (Akademie Verlag, 1997), Vol. 3, pp. 353-363.

Kauffmann, J.

J. Kauffmann, M. Gahr, and H. J. Tiziani, "Noise reduction in speckle pattern interferometry," in Speckle Metrology 2003, K. Gastinger, O.J. Lokberg, and S. Winther, eds., Proc. SPIE 4933,9-14 (2003).

Kebbel, V.

V. Kebbel, J. Müller, and W. Jüptner, "Characterization of aspherical microoptics using digital holography: improvement of accuracy," in Interferometry XI: Applications, W.Osten, ed., Proc. SPIE 4778,188-197 (2002).

Kohgo, H.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, "Single-transform Fourier/Hartley fringe analysis for holographic interferometry," in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W.Osten, eds. (Akademie Verlag, 1996), Vol. 2, pp. 67-73.

Kopylow, C. v.

Kreis, T.

T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

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

T. Kreis and H. Kreitlow, "Quantitative evaluation of holographic interference patterns under image processing aspects," in 2nd European Congress on Optics Applied to Metrology, P.Meyrueis and M.Grosmann, eds., Proc. SPIE 210,196-202 (1979).

T. Kreis and W. Jüptner, "Principles of digital holography," in 3rd International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W.Osten, eds. (Akademie Verlag, 1997), Vol. 3, pp. 353-363.

Kreitlow, H.

T. Kreis and H. Kreitlow, "Quantitative evaluation of holographic interference patterns under image processing aspects," in 2nd European Congress on Optics Applied to Metrology, P.Meyrueis and M.Grosmann, eds., Proc. SPIE 210,196-202 (1979).

Marrakchi, A.

Mittra, R.

Morimoto, Y.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Mühlig, H.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).

Müller, J.

V. Kebbel, J. Müller, and W. Jüptner, "Characterization of aspherical microoptics using digital holography: improvement of accuracy," in Interferometry XI: Applications, W.Osten, ed., Proc. SPIE 4778,188-197 (2002).

Musiol, G.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).

Nomura, T.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Osten, W.

Osten, W. J. Wolfgang

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Pichon, L.

Sagehorn, H.

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Schnars, U.

Schuh, R.

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Seebacher, S.

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).

Shin, S.-H.

Takahashi, I.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Takeda, M.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, "Single-transform Fourier/Hartley fringe analysis for holographic interferometry," in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W.Osten, eds. (Akademie Verlag, 1996), Vol. 2, pp. 67-73.

Taniguchi, K.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, "Single-transform Fourier/Hartley fringe analysis for holographic interferometry," in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W.Osten, eds. (Akademie Verlag, 1996), Vol. 2, pp. 67-73.

Tiziani, H. J.

J. Kauffmann, M. Gahr, and H. J. Tiziani, "Noise reduction in speckle pattern interferometry," in Speckle Metrology 2003, K. Gastinger, O.J. Lokberg, and S. Winther, eds., Proc. SPIE 4933,9-14 (2003).

Wagner, C.

Wyrowski, F.

Yoneyama, S.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Appl. Opt. (4)

Exp. Mech. (1)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, "Deformation measurement by phase shifting digital holography," Exp. Mech. 45, 65-70 (2005).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting interferometry: a noise reduction filter for phase-unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

T. Kreis, "Frequency analysis of digital holography," Opt. Eng. 41, 771-778 (2002).
[CrossRef]

Opt. Laser Technol. (1)

A. Brozeit, J. Burke, H. Helmers, H. Sagehorn, and R. Schuh, "Noise reduction in ESPI fringes by merging orthogonally polarised speckle fields," Opt. Laser Technol. 30, 325-329 (1998).
[CrossRef]

Opt. Lett. (1)

Other (8)

T. Kreis and W. Jüptner, "Principles of digital holography," in 3rd International Workshop on Automatic Processing of Fringe Patterns, W. Jüptner and W.Osten, eds. (Akademie Verlag, 1997), Vol. 3, pp. 353-363.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, "Single-transform Fourier/Hartley fringe analysis for holographic interferometry," in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W.Osten, eds. (Akademie Verlag, 1996), Vol. 2, pp. 67-73.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, 5th ed. (Verlag Harri Deutsch, 2001).

T. Kreis and H. Kreitlow, "Quantitative evaluation of holographic interference patterns under image processing aspects," in 2nd European Congress on Optics Applied to Metrology, P.Meyrueis and M.Grosmann, eds., Proc. SPIE 210,196-202 (1979).

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

V. Kebbel, J. Müller, and W. Jüptner, "Characterization of aspherical microoptics using digital holography: improvement of accuracy," in Interferometry XI: Applications, W.Osten, ed., Proc. SPIE 4778,188-197 (2002).

J. Kauffmann, M. Gahr, and H. J. Tiziani, "Noise reduction in speckle pattern interferometry," in Speckle Metrology 2003, K. Gastinger, O.J. Lokberg, and S. Winther, eds., Proc. SPIE 4933,9-14 (2003).

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Figures (15)

Fig. 1
Fig. 1

Recording geometry for lensless Fourier holography.

Fig. 2
Fig. 2

Experimental setup for digital holography with a positioning stage for shifting the camera position.

Fig. 3
Fig. 3

Cross correlation between the intensity reconstructions of two hologram positions.

Fig. 4
Fig. 4

Left: Result of the phase subtraction. The phase map derived from holograms with positions at ξ = 0   mm and ξ = 0.3   mm . Right: Calculated phase values due to the known hologram shift and the setup geometry.

Fig. 5
Fig. 5

Standard deviation of the phase difference between two hologram positions over the hologram shift.

Fig. 6
Fig. 6

Phase difference calculated from holograms of (a) position ξ = 0   mm , η = 0   mm and from holograms of (b) position ξ = 20   mm , η = 20   mm . (c) The difference between (a) and (b).

Fig. 7
Fig. 7

Scheme of the averaging process.

Fig. 8
Fig. 8

Enlarged cutout of the phase-difference results of 1, 10, and 25 camera positions and the corresponding cuts along the horizontal axis.

Fig. 9
Fig. 9

Result of averaging with 25 pictures. Left: resulting phase difference. Right: cut along the horizontal axis.

Fig. 10
Fig. 10

Standard deviation of the phase difference over the number of pictures involved in the averaging process.

Fig. 11
Fig. 11

Image of the relay.

Fig. 12
Fig. 12

Results of the two-wavelength contouring measurements. Left: result of a single measurement. Right: result after averaging with 25 pictures.

Fig. 13
Fig. 13

Enlarged cutout of Fig. 12 processed with different filters: (a) Enlarged cutout of Fig. 12 (left side). (b) Result of (a) processed with a median filter (window size of three; six cycles). (c) Same cutout as marked in Fig. 12 (left side) but from Fig. 12 (right side). (d) Result of (c) processed with a median filter (window size of three; one cycle).

Fig. 14
Fig. 14

Results of a deformation measurement. Left: result of a single measurement. Right: result after averaging with 25 pictures.

Fig. 15
Fig. 15

(a) Intensity reconstruction of the relay, (b) cutouts of the marked area from a single measurement, (c) after averaging with 25 pictures.

Equations (21)

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b ( x , y ) = i λ d exp ( i 2 π d λ ) exp [ i π λ d ( x 2 + y 2 ) ] × 1 { h ( ξ , η ) r ( ξ , η ) × exp [ i π λ d ( ξ 2 + η 2 ) ] } ( x λ d , y λ d ) .
r ( ξ , η ) exp [ i π λ d ( ξ 2 + η 2 ) ] .
b ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] 1 [ h ( ξ , η ) ] ( x λ d , y λ d ) .
φ i ( x , y ) = arg [ b i ( x , y ) ] = π λ i d ( x 2 + y 2 ) + arg { 1 [ h i ( ξ , η ) ] } .
Δϕ ( x , y ) = φ 2 ( x , y ) φ 1 ( x , y ) = π Λ d ( x 2 + y 2 ) + arg { 1 [ h 2 ( ξ , η ) ] } arg { 1 [ h 1 ( ξ , η ) ] } ,
Λ = λ 1 λ 2 λ 2 λ 1 .
1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( x λ d , y λ d ) = exp [ i 2 π λ d ( x Δ ξ + y Δ η ) ] × 1 [ h ( ξ , η ) ] ( x λ d , y λ d ) .
b ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] exp [ i 2 π λ d ( x Δ ξ + y Δ η ) ] × 1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( x λ d , y λ d ) .
φ i ( x , y ) = arg [ b i ( x , y ) ] = π λ i d ( x 2 + y 2 ) + π λ i d ( 2 x Δ ξ + 2 y Δ η ) + arg { 1 [ h i ( ξ + Δ ξ , η + Δ η ) ] } .
Δϕ ( x , y ) = arg { 1 [ h 1 ( ξ + Δ ξ , η + Δ η ) ] } arg { 1 [ h 2 ( ξ + Δ ξ , η + Δη ) ] } .
Δϕ ( x , y ) = π Λ d ( x 2 + y 2 ) π Λ d ( 2 x Δ ξ + 2 y Δ η ) + arg { 1 [ h 1 ( ξ + Δ ξ , η + Δ η ) ] } arg { 1 [ h 2 ( ξ + Δ ξ , η + Δ η ) ] } .
ϕ offset ( x i , y i ) = π Λ d ( 2 x i Δ ξ + 2 y i Δ η ) .
x = x λ d , y = y λ d
1 [ h ( ξ + Δ ξ , η + Δ η ) ω ( ξ , η ) ] ( x , y ) = 1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( x , y ) 1 [ ω ( ξ , η ) ] ( x , y ) ,
1 [ ω ( ξ , η ) ] ( x , y ) = sin ( π x ) π x sin ( π y ) π y = sinc ( x ) sinc ( y )
b ˜ ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] × ( { exp [ i 2 π ( x Δ ξ + y Δ η ) ] × 1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( x , y ) }  sinc ( x ) sinc ( y ) ) ,
b ˜ ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] × + 1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( u , v ) × exp [ i 2 π ( u Δ ξ + v Δ η ) ] × sinc ( x u ) sinc ( y v ) d u d v .
b ˜ ( x , y ) exp [ i π λ d ( x 2 + y 2 ) ] × exp [ i 2 π ( x Δ ξ + y Δ η ) ] × + 1 [ h ( ξ + Δ ξ , η + Δ η ) ] ( u , v ) × exp { i 2 π [ ( x u ) Δ ξ + ( y v ) Δ η ] } × sinc ( x u ) sinc ( y v ) d u d v ,
t ( x , y ) = exp [ i 2 π ( x Δ ξ + y Δ η ) ] sinc ( x ) sinc ( y ) ,
ϕ mean ( x , y ) = arctan i = 1 N sin [ ϕ i ( x , y ) ] i = 1 N cos [ ϕ i ( x , y ) ] ,
γ ( u , v ) = x , y [ I 0 ( x , y ) I ¯ 0 ] [ I Δ ξ ( u x , v y ) I ¯ Δ ξ ] x , y [ I 0 ( x , y ) I ¯ 0 ] 2 x , y [ I Δ ξ ( x , y ) I ¯ Δ ξ ] 2 .

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