Abstract

Intensities recorded with CCD or CMOS sensors represent spatially averaged values of the intensity across the area of a pixel. In this work we investigate the influence of spatial averaging in interferograms on the evaluated phase from an object wave with well resolved, fully developed speckles. Based on an analytical description of the averaging process, a procedure is developed to create a quality map for the evaluated phase, in order to give an estimation of the expected error at each point. The proposed method uses only the local intensity distribution of the object wave for the qualification of the phase values. The theoretical results are tested and verified by means of numerically generated objective and subjective speckle fields.

© 2007 Optical Society of America

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References

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  1. T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, 2004).
    [CrossRef]
  2. N. Shvartsman and I. Freund, "Speckle spots ride phase saddles sidesaddle," Opt. Commun. 117, 228-234 (1995).
    [CrossRef]
  3. T. Yoshimura, M. Zhou, K. Yamahai, and Z. Liyan, "Optimum determination of speckle size to be used in electronic speckle pattern interferometry," Appl. Opt. 34, 87-91 (1995).
    [CrossRef] [PubMed]
  4. M. Lehmann, "Phase-shifting speckle interferometry with unresolved speckles: A theoretical investigation," Opt. Commun. 128, 325-340 (1996).
    [CrossRef]
  5. M. Lehmann, "Measurement optimization in speckle interferometry: the influence of the imaging lens aperture," Opt. Eng. 36, 1162-1168 (1997).
    [CrossRef]
  6. T. Maack, R. Kowarschik, and G. Notni, "Optimum lens aperture in phase-shifting speckle interferometric setups for maximum accuracy of phase measurement," Appl. Opt. 36, 6217-6224 (1997).
    [CrossRef]
  7. V. Eichhorn and H. Helmers, "Phase-shifting electronic speckle pattern interferometry (ESPI) with unresolved speckle," in DGaO-Proceedings, DGaO (DGaO, 2005).
  8. E. Kolenović, W. Osten, and W. Jüptner, "Influence of unresolved speckles in interferometric phase measurements," Proc. SPIE 4101, 104-112 (2000).
  9. E. Kolenović, W. Osten, and W. Jüptner, "Improvement of interferometric phase measurements by consideration of the speckle field topology," Proc. SPIE 4933, 206-211 (2003).
    [CrossRef]
  10. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  11. Y. Zhu, Z. Luan, Q. Yang, W. Lu, and L. Liu, "Novel method to construct a quality map for phase unwrapping based on modulation and the phase gradient," Opt. Eng. 45(10), 1056.011-1056.015 (2006).
    [CrossRef]
  12. E. Kolenović, "Correlation between intensity and phase in monochromatic light," J. Opt. Soc. Am. A 22, 899-906 (2005).
    [CrossRef]
  13. G. Weigelt and B. Stoffregen, "The longitudinal correlation of a three-dimensional speckle intensity distribution," Optik (Jena) 48, 399-407 (1977).
  14. L. Leushacke and M. Kirchner, "Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures," J. Opt. Soc. Am. A 7, 827-832 (1990).
    [CrossRef]
  15. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer Verlag, 1975).
    [CrossRef]
  16. E. Kolenović, W. Osten, and W. Jüptner, "Non-linear speckle phase changes in the image plane caused by out of plane displacement," Opt. Commun. 171, 333-344 (1999).
    [CrossRef]
  17. K. Fliegel, "Modeling and measurement of image sensor characteristics," Telecommun. Radio Eng. 13, 27-34 (2004).
  18. O. Hadar and G. D. Boreman, "Oversampling requirements for pixelated-imager systems," Opt. Eng. 38, 782-785 (1999).
    [CrossRef]
  19. J. W. Goodman, Introduction to Fourier optics, 2nd ed. (McGraw-Hill, 1996).

2006 (1)

Y. Zhu, Z. Luan, Q. Yang, W. Lu, and L. Liu, "Novel method to construct a quality map for phase unwrapping based on modulation and the phase gradient," Opt. Eng. 45(10), 1056.011-1056.015 (2006).
[CrossRef]

2005 (1)

2004 (1)

K. Fliegel, "Modeling and measurement of image sensor characteristics," Telecommun. Radio Eng. 13, 27-34 (2004).

2003 (1)

E. Kolenović, W. Osten, and W. Jüptner, "Improvement of interferometric phase measurements by consideration of the speckle field topology," Proc. SPIE 4933, 206-211 (2003).
[CrossRef]

2000 (1)

E. Kolenović, W. Osten, and W. Jüptner, "Influence of unresolved speckles in interferometric phase measurements," Proc. SPIE 4101, 104-112 (2000).

1999 (2)

E. Kolenović, W. Osten, and W. Jüptner, "Non-linear speckle phase changes in the image plane caused by out of plane displacement," Opt. Commun. 171, 333-344 (1999).
[CrossRef]

O. Hadar and G. D. Boreman, "Oversampling requirements for pixelated-imager systems," Opt. Eng. 38, 782-785 (1999).
[CrossRef]

1997 (2)

T. Maack, R. Kowarschik, and G. Notni, "Optimum lens aperture in phase-shifting speckle interferometric setups for maximum accuracy of phase measurement," Appl. Opt. 36, 6217-6224 (1997).
[CrossRef]

M. Lehmann, "Measurement optimization in speckle interferometry: the influence of the imaging lens aperture," Opt. Eng. 36, 1162-1168 (1997).
[CrossRef]

1996 (1)

M. Lehmann, "Phase-shifting speckle interferometry with unresolved speckles: A theoretical investigation," Opt. Commun. 128, 325-340 (1996).
[CrossRef]

1995 (2)

1990 (1)

1977 (1)

G. Weigelt and B. Stoffregen, "The longitudinal correlation of a three-dimensional speckle intensity distribution," Optik (Jena) 48, 399-407 (1977).

Appl. Opt. (2)

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

Opt. Commun. (3)

E. Kolenović, W. Osten, and W. Jüptner, "Non-linear speckle phase changes in the image plane caused by out of plane displacement," Opt. Commun. 171, 333-344 (1999).
[CrossRef]

N. Shvartsman and I. Freund, "Speckle spots ride phase saddles sidesaddle," Opt. Commun. 117, 228-234 (1995).
[CrossRef]

M. Lehmann, "Phase-shifting speckle interferometry with unresolved speckles: A theoretical investigation," Opt. Commun. 128, 325-340 (1996).
[CrossRef]

Opt. Eng. (3)

M. Lehmann, "Measurement optimization in speckle interferometry: the influence of the imaging lens aperture," Opt. Eng. 36, 1162-1168 (1997).
[CrossRef]

Y. Zhu, Z. Luan, Q. Yang, W. Lu, and L. Liu, "Novel method to construct a quality map for phase unwrapping based on modulation and the phase gradient," Opt. Eng. 45(10), 1056.011-1056.015 (2006).
[CrossRef]

O. Hadar and G. D. Boreman, "Oversampling requirements for pixelated-imager systems," Opt. Eng. 38, 782-785 (1999).
[CrossRef]

Optik (1)

G. Weigelt and B. Stoffregen, "The longitudinal correlation of a three-dimensional speckle intensity distribution," Optik (Jena) 48, 399-407 (1977).

Proc. SPIE (2)

E. Kolenović, W. Osten, and W. Jüptner, "Influence of unresolved speckles in interferometric phase measurements," Proc. SPIE 4101, 104-112 (2000).

E. Kolenović, W. Osten, and W. Jüptner, "Improvement of interferometric phase measurements by consideration of the speckle field topology," Proc. SPIE 4933, 206-211 (2003).
[CrossRef]

Telecommun. Radio Eng. (1)

K. Fliegel, "Modeling and measurement of image sensor characteristics," Telecommun. Radio Eng. 13, 27-34 (2004).

Other (5)

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

T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, 2004).
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer Verlag, 1975).
[CrossRef]

V. Eichhorn and H. Helmers, "Phase-shifting electronic speckle pattern interferometry (ESPI) with unresolved speckle," in DGaO-Proceedings, DGaO (DGaO, 2005).

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Figures (6)

Fig. 1
Fig. 1

Setup geometry and coordinate systems for objective speckle fields.

Fig. 2
Fig. 2

Setup geometry and coordinate systems for subjective speckle fields.

Fig. 3
Fig. 3

Numerically generated phase (a) and intensity (b) of an objective speckle field. The images in the second row show | T ϕ ( r ) | obtained from the phase (c) and calculated from the intensity (d), according to Eq. (13). The results in (c) and (d) are scaled to the same maximum values. Brighter values represent higher phase gradients. The field was generated with a wavelength of λ = 632.8   nm for a circular shaped object with a diameter of d o = 4   mm in a distance of d = 300   mm . Average speckle size: 56.9   μm . Depicted area: 4 × 4   mm 2 , sampled with 512 × 512 pixels. Pixel spacing: Δ x , Δ y = 7.8   μm . Average speckle resolution: 7.3 pixels in each dimension.

Fig. 4
Fig. 4

Numerically generated phase (a) and intensity (b) of a subjective speckle field. The images in the second row show | T ϕ ( r ) | obtained from the phase (c) and calculated from the intensity (d), according to Eq. (14). The results in (c) and (d) are scaled to the same maximum values. Brighter values represent higher phase gradients. The field was generated with a wavelength of λ = 632.8   nm for a circular shaped object with a diameter of d o = 4   mm . Focal length of the lens: f = 200   mm . Lens diameter: d l = 6   mm . Distances: d 1 = 300   mm , d 2 = 600   mm . Average speckle size: 75 .9   μm . Depicted area: 8 × 8   mm 2 , sampled with 512 × 512 pixels. Pixel spacing: Δ x , Δ y = 15.6   μm . Average speckle resolution: 4.9 pixels in each dimension.

Fig. 5
Fig. 5

Histograms with the relative number of pixels versus their phase errors Δ ϕ e . The solid curves show the error distribution before, and the dashed curves after masking with the binarized quality map, which was thresholded at Δ ϕ ˜ e = 0.015   rad .

Fig. 6
Fig. 6

Thresholded quality maps for the objective (left) and the subjective speckle field (right). Thresholding value: Δ ϕ ˜ e = 0.015   rad .

Equations (26)

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I i ( x , y ) = I o ( x , y ) + I r ( x , y ) + 2 I o ( x , y ) I r ( x , y ) × cos ( Δ ϕ ( x , y ) ) ,
Δ ϕ ( x , y ) = ϕ o ( x , y ) ϕ r ( x , y ) ,
I ¯ i = p 1 p I o ( x , y ) + I r ( x , y ) + 2 I o ( x , y ) I r ( x , y ) × cos ( Δ ϕ ( x , y ) ) d p .
  I ¯ m = p 1 p 2 I o ( x , y ) I r ( x , y ) cos ( Δ ϕ ( x , y ) ) d p ,
I ¯ m = 2 I r p p ( I o + I o x x + I o y y ) × cos ( Δ ϕ + ϕ o x x + ϕ o y y ) d p ,
p = Δ x Δ y , d p = d x d y ,
sinc ( x ) = sin ( π x ) π x ,
I ¯ m = 2 I o I r cos ( Δ ϕ ) sinc ( ϕ o x Δ x 2 π ) sinc ( ϕ o y Δ y 2 π ) + 2 I o x I r sin ( Δ ϕ ) sinc ( ϕ o y Δ y 2 π ) × [ cos ( ϕ o x Δ x 2 ) sinc ( ϕ o x Δ x 2 π ) ( ϕ o x ) ] + 2 I o y I r sin ( Δ ϕ ) sinc ( ϕ o x Δ x 2 π ) × [ cos ( ϕ o y Δ y 2 ) sinc ( ϕ o y Δ y 2 π ) ( ϕ o y ) ] .
| ϕ ( r ) | 2 ( ϕ ( r ) x ) 2 + ( ϕ ( r ) y ) 2 + ( ϕ ( r ) z ) 2 = k 2 + 2 I ( r ) I ( r ) ,
2 I ( r ) x 2 2 I ( r ) z 2 , 2 I ( r ) y 2 2 I ( r ) z 2 .
( ϕ ( r ) x ) 2 + ( ϕ ( r ) y ) 2 + ( ϕ ( r ) z ) 2 k 2 + 1 I ( r ) ( 2 I ( r ) x 2 + 2 I ( r ) y 2 ) .
ϕ ( r ) z k π λ d 2 ( x 2 + y 2 ) ,
| T ϕ ( r ) | 2 ( ϕ ( r ) x ) 2 + ( ϕ ( r ) y ) 2 1 I ( r ) ( 2 I ( r ) x 2 + 2 I ( r ) y 2 ) + k 2 [ ( x 2 + y 2 ) d 2 ( x 2 + y 2 ) 2 4 d 4 ] ,
( ϕ ( r ) x ) 2 + ( ϕ ( r ) y ) 2 1 I ( r ) ( 2 I ( r ) x 2 + 2 I ( r ) y 2 ) + k 2 [ ( x 2 + y 2 ) d 2 2 ( x 2 + y 2 ) 2 4 d 2 4 ]
T p ( f x , f y ) = sinc ( Δ x f x ) sinc ( Δ y f y ) ,
Δ ϕ ¯ = I ¯ i 3 I ¯ i 1 I ¯ i 0 I ¯ i 2 ,
Δ ϕ e = | Δ ϕ Δ ϕ ¯ | .
Δ ϕ ˜ e = | I ¯ m 3 I ¯ m 1 I ¯ m 0 I ¯ m 2 | .
U ( r ) = exp ( i k z ) i λ z exp [ i k 2 z ( x 2 + y 2 ) ] × { U ( u , v ) exp [ i k 2 z ( u 2 + v 2 ) ] } × exp [ i k z ( x u + y v ) ] d u d v ,
ϕ ( r ) = k z + k 2 z ( x 2 + y 2 ) + arg [ f ( r ) ] ,
f ( r ) = 1 i λ z { U ( u , v ) exp [ i k 2 z ( u 2 + v 2 ) ] } × exp [ i k z ( x u + y v ) ] d u d v .
ϕ ( r ) z = k k 2 z 2 ( x 2 + y 2 ) + z arg [ f ( r ) ] .
f ( r ) z = 1 i λ z 2 { U ( u , v ) exp [ i k 2 z ( u 2 + v 2 ) ] } × exp [ i k z ( x u + y v ) ] d u d v + k λ z 3 × { [ x u + y v u 2 + v 2 2 ] U ( u , v ) × exp [ i k 2 z ( u 2 + v 2 ) ] } exp [ i k z ( x u + y v ) ] × d u d v .
f ( r ) z 1 i λ z 2 { U ( u , v ) exp [ i k 2 z ( u 2 + v 2 ) ] } × exp [ i k z ( x u + y v ) ] d u d v ,
f ( r ) z 1 z f ( r ) .
ϕ ( r ) z k k 2 z 2 ( x 2 + y 2 ) .

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