Abstract

Wavefronts incident on a random phase plate are reconstructed via phase retrieval utilizing axially displaced speckle intensity measurements and the wave propagation equation. Retrieved phases and phase subtraction facilitate the investigations of wavefronts from test objects before and after undergoing a small rotation or deformation without sign ambiguity. Angular displacement (Δθ) between incident planar wavefronts is determined from the light source vacuum wavelength (λ) divided by the fringe spacing (Λ). Fourier analysis of the wavefront phase difference yields a peak frequency that is inversely proportional to Λ, and the sign gives the direction of rotation. Numerical simulations confirm the experimental results. In the experiments, the smallest Δθ measured is 0.031°. The technique also permits deformation analysis of a reflecting test object under thermal loading. The technique offers simple, high resolution, noncontact, and whole field evaluation of three-dimensional objects before and after undergoing rotation or deformation.

© 2009 Optical Society of America

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References

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    [CrossRef]
  2. S. Duym and M. Lumori, “Distorted laser interferometric angle measurements of a disk drive pivot,” IEEE/ASME Trans. Mechatron. 3, 265-274 (1998).
    [CrossRef]
  3. W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
    [CrossRef]
  4. B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
    [CrossRef]
  9. L. Kremer, D. Budelsky, D. Platte, and P. von Brentano, “Autocollimator for spectroscopy of broad resonances with pulsed lasers,” Appl. Opt. 34, 4827-4834 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. P. R. Yoder, Jr., E. R. Schlesinger, and J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890-1895 (1975).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. M. H. Chiu and D. C. Su, “Angle measurement using total internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
    [CrossRef]
  15. S. R. Kitchen and C. Dam-Hansen, “Holographic common-path interferometer for angular displacement measurements with spatial phase stepping and extended measurement range,” Appl. Opt. 42, 51-59 (2003).
    [CrossRef] [PubMed]
  16. S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  18. L. Yu, G. Pedrini, W. Osten, and M. K. Kim, “Three-dimensional angle measurement based on propagation vector analysis of digital holography,” Appl. Opt. 46, 3539-3545(2007).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2008 (2)

2007 (2)

2005 (3)

S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

2003 (2)

2002 (2)

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 186-193.

1999 (2)

1998 (2)

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, and R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119-2129 (1998).
[CrossRef]

S. Duym and M. Lumori, “Distorted laser interferometric angle measurements of a disk drive pivot,” IEEE/ASME Trans. Mechatron. 3, 265-274 (1998).
[CrossRef]

1997 (2)

P. Rastogi, Optical Measurement Techniques and Applications (Artech, 1997).

M. H. Chiu and D. C. Su, “Angle measurement using total internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

1996 (1)

L. Zeng, H. Matsumoto, and K. Kawachi, “Divergent-ray projection method for measuring the flapping angle, lag angle, and torsional angle of a bumblebee wing,” Opt. Eng. 35, 3135-3139 (1996).
[CrossRef]

1995 (1)

1993 (1)

1992 (1)

1988 (1)

1984 (1)

G. G. Luther and R. D. Deslattes, “Single-axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

1979 (1)

C. Vest, Holographic Interferometry (Wiley, 1979).

1975 (1)

1970 (1)

Almoro, P.

Anand, A.

Budelsky, D.

Chickvary, J. L.

Chiu, M. H.

M. H. Chiu and D. C. Su, “Angle measurement using total internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

Dam-Hansen, C.

Deng, W.

Deslattes, R. D.

G. G. Luther and R. D. Deslattes, “Single-axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Duym, S.

S. Duym and M. Lumori, “Distorted laser interferometric angle measurements of a disk drive pivot,” IEEE/ASME Trans. Mechatron. 3, 265-274 (1998).
[CrossRef]

Freer, B.

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Gao, W.

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

Ge, Z.

Graf, M.

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Guo, J.

Hansen, R. S.

Hanson, S. G.

Harris, O.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 186-193.

Huang, P.

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

Huang, P. S.

Imam, H.

Kamada, O.

Kawachi, K.

L. Zeng, H. Matsumoto, and K. Kawachi, “Divergent-ray projection method for measuring the flapping angle, lag angle, and torsional angle of a bumblebee wing,” Opt. Eng. 35, 3135-3139 (1996).
[CrossRef]

Kim, M. K.

Kitchen, S. R.

Kiyono, S.

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047-6055 (1992).
[CrossRef] [PubMed]

Kremer, L.

Long, X. W.

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

Lumori, M.

S. Duym and M. Lumori, “Distorted laser interferometric angle measurements of a disk drive pivot,” IEEE/ASME Trans. Mechatron. 3, 265-274 (1998).
[CrossRef]

Luther, G. G.

G. G. Luther and R. D. Deslattes, “Single-axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Malacara, D.

Matsumoto, H.

L. Zeng, H. Matsumoto, and K. Kawachi, “Divergent-ray projection method for measuring the flapping angle, lag angle, and torsional angle of a bumblebee wing,” Opt. Eng. 35, 3135-3139 (1996).
[CrossRef]

Osten, W.

Parrill, T.

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Pedrini, G.

Platte, D.

Polner, D.

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Prakash, S.

Rana, S.

Rastogi, P.

P. Rastogi, Optical Measurement Techniques and Applications (Artech, 1997).

Reece, R.

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Rose, B.

Schlesinger, E. R.

Shi, P.

Singh, S.

Stijns, E.

Su, D. C.

M. H. Chiu and D. C. Su, “Angle measurement using total internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

Takeda, M.

Vest, C.

C. Vest, Holographic Interferometry (Wiley, 1979).

von Brentano, P.

Yamada, T.

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

Yang, K. Y.

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

Yoder, P. R.

Yu, L.

Yuan, J.

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

Yura, H. T.

Zeng, L.

L. Zeng, H. Matsumoto, and K. Kawachi, “Divergent-ray projection method for measuring the flapping angle, lag angle, and torsional angle of a bumblebee wing,” Opt. Eng. 35, 3135-3139 (1996).
[CrossRef]

Zhu, Z.

Appl. Opt. (14)

D. Malacara and O. Harris, “Interferometric measurement of angles,” Appl. Opt. 9, 1630-1633 (1970).
[CrossRef] [PubMed]

P. R. Yoder, Jr., E. R. Schlesinger, and J. L. Chickvary, “Active annular-beam laser autocollimator system,” Appl. Opt. 14, 1890-1895 (1975).
[CrossRef] [PubMed]

P. Shi and E. Stijns, “New optical method for measuring small-angle rotations,” Appl. Opt. 27, 4342-4344 (1988).
[CrossRef] [PubMed]

P. Shi and E. Stijns, “Improving the linearity of the Michelson interferometric angular measurement by a parameter compensation method,” Appl. Opt. 32, 44-51 (1993).
[CrossRef] [PubMed]

P. S. Huang, “Use of thin films for high-sensitivity angle measurement,” Appl. Opt. 38, 4831-4836 (1999).
[CrossRef]

J. Guo, Z. Zhu, and W. Deng, “Small-angle measurement based on surface-plasmon resonance and the use of magneto-optical modulation,” Appl. Opt. 38, 6550-6555 (1999).
[CrossRef]

L. Kremer, D. Budelsky, D. Platte, and P. von Brentano, “Autocollimator for spectroscopy of broad resonances with pulsed lasers,” Appl. Opt. 34, 4827-4834 (1995).
[CrossRef] [PubMed]

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, and R. S. Hansen, “Laser-speckle angular-displacement sensor: theoretical and experimental study,” Appl. Opt. 37, 2119-2129 (1998).
[CrossRef]

S. R. Kitchen and C. Dam-Hansen, “Holographic common-path interferometer for angular displacement measurements with spatial phase stepping and extended measurement range,” Appl. Opt. 42, 51-59 (2003).
[CrossRef] [PubMed]

P. S. Huang, S. Kiyono, and O. Kamada, “Angle measurement based on the internal-reflection effect: a new method,” Appl. Opt. 31, 6047-6055 (1992).
[CrossRef] [PubMed]

Z. Ge and M. Takeda, “High-resolution two-dimensional angle measurement technique based on fringe analysis,” Appl. Opt. 42, 6859-6868 (2003).
[CrossRef] [PubMed]

S. Prakash, S. Singh, and S. Rana, “Automated small tilt-angle measurement using Lau interferometry,” Appl. Opt. 44, 5905-5909 (2005).
[CrossRef] [PubMed]

L. Yu, G. Pedrini, W. Osten, and M. K. Kim, “Three-dimensional angle measurement based on propagation vector analysis of digital holography,” Appl. Opt. 46, 3539-3545(2007).
[CrossRef] [PubMed]

P. Almoro and S. G. Hanson, “Random phase plate for wavefront sensing via phase retrieval and a volume speckle field,” Appl. Opt. 47, 2979-2987 (2008).
[CrossRef] [PubMed]

IEEE/ASME Trans. Mechatron. (1)

S. Duym and M. Lumori, “Distorted laser interferometric angle measurements of a disk drive pivot,” IEEE/ASME Trans. Mechatron. 3, 265-274 (1998).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. B (1)

B. Freer, R. Reece, M. Graf, T. Parrill, and D. Polner, “In situ beam angle measurement in a multi-wafer high current ion implanter,” Nucl. Instrum. Methods Phys. Res. B 237, 378-383 (2005).
[CrossRef]

Opt. Eng. (2)

M. H. Chiu and D. C. Su, “Angle measurement using total internal-reflection heterodyne interferometry,” Opt. Eng. 36, 1750-1753 (1997).
[CrossRef]

L. Zeng, H. Matsumoto, and K. Kawachi, “Divergent-ray projection method for measuring the flapping angle, lag angle, and torsional angle of a bumblebee wing,” Opt. Eng. 35, 3135-3139 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Precis. Eng. (1)

W. Gao, P. Huang, T. Yamada, and S. Kiyono, “A compact and sensitive two-dimensional angle probe for flatness measurement of large silicon wafers,” Precis. Eng. 26, 396-404 (2002).
[CrossRef]

Rev. Sci. Instrum. (2)

J. Yuan, X. W. Long, and K. Y. Yang, “Temperature-controlled autocollimator with ultrahigh angular measuring precision,” Rev. Sci. Instrum. 76, 125106 (2005).
[CrossRef]

G. G. Luther and R. D. Deslattes, “Single-axis photoelectronic autocollimator,” Rev. Sci. Instrum. 55, 747-750 (1984).
[CrossRef]

Other (3)

C. Vest, Holographic Interferometry (Wiley, 1979).

P. Rastogi, Optical Measurement Techniques and Applications (Artech, 1997).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 186-193.

Supplementary Material (1)

» Media 1: MOV (2684 KB)     

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Figures (12)

Fig. 1
Fig. 1

Plane wave undergoing an angular displacement about the y axis is incident on the diffuser of the wavefront sensor.

Fig. 2
Fig. 2

Deformation analysis using retrieved phase difference. The reconstructed wavefronts for the initial and deformed states of the object surface are subtracted and yield an interference pattern due to optical path difference (OPD). For each surface point, the projection of the displacement vector w along the sensitivity vector K contributes to the OPD. K lies along the bisector of the illumination and observation directions.

Fig. 3
Fig. 3

Setup for measuring angular displacement of test wavefronts using speckle intensity measurements and phase retrieval. Angular displacement of a plane wavefront is introduced by a prism mounted on a rotation stage. The test wavefront illuminating the DPP diffuser generates a volume speckle field. The speckle field is sequentially sampled and input into a computer algorithm to reconstruct the phase of the test wavefront.

Fig. 4
Fig. 4

Simulated phases and phase differences for increasing angular displacements. The test planar wavefronts have varying tilts in the x y plane. Subtraction of the test wavefronts results in an interference pattern. The fringe spacing decreases with increasing angular displacements, Δ θ > Δ θ > Δ θ .

Fig. 5
Fig. 5

Experimental phases and phase differences for increasing angular displacements. The test planar wavefront is rotated about the y axis. Subtraction of the retrieved wavefronts results in an interference pattern. The fringe spacing decreases with increasing angular displacements, Δ θ > Δ θ > Δ θ .

Fig. 6
Fig. 6

Phases and phase differences to determine the magnitude and direction of the angular displacement. (a), (b)  Phases of the wavefronts corresponding to two angular positions. (c) Resulting interferogram with fringe spacing Λ obtained upon subtraction of the rotated wavefronts. The magnitude of rotation is calculated from the light source wavelength divided by Λ. (d) Interferogram obtained when the order of subtraction is reversed. The orientation of the dark and bright bands of the fringe pattern depends on the direction of rotation.

Fig. 7
Fig. 7

Amplitude spectra showing a dominant frequency peak f x for phase differences (a)  Δ ϕ = ϕ 2 ϕ 1 and (b)  Δ ϕ = ϕ 1 ϕ 2 . The sign of Δ f x gives the direction of the angular displacement.

Fig. 8
Fig. 8

Interferograms obtained for increasing PZT translation using the same reference initial wavefront: Δ ϕ a :     0 μm ; Δ ϕ b :     1 μm ; ; Δ ϕ i :     8 μm . Fringe spacing decreases as PZT translation and angular displacement increases.

Fig. 9
Fig. 9

Amplitude spectrum for the various interferograms for increasing PZT translation. The frequency shift from the origin increases (for decreasing fringe spacing) as PZT translation and angular displacement increases.

Fig. 10
Fig. 10

Angular displacement increases linearly with PZT translation. Media 1 shows numerical simulations and experimental results on angular measurements using phase retrieval.

Fig. 11
Fig. 11

Setup for the measurement of surface deformation due to heating using the wavefront sensor. Wavefront corresponding to the initial undeformed state (without exposure of object surface to heating rod) is recorded. To induce thermal deformation, the heating rod at 450 ° C is positioned close to the Plexiglas surface for 1 minute. Deformed wavefront (upon exposure of surface to heating rod) is recorded 10 minutes upon removal of the heating rod to allow the system to achieve thermal equilibrium.

Fig. 12
Fig. 12

(a) Wavefront phases and (b) phase differences for different deformation states. Subtraction of the reconstructed wavefronts results in an interferogram depicting the contour and extent of deformation.

Equations (11)

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α 1 λ = 2 π θ 1 ,
a R exp [ i k θ 1 x ] + a R exp [ i k θ 2 x ] .
1 + cos [ k · ( θ 1 - θ 2 ) x ] .
2 π Λ · x = k · ( θ 1 - θ 2 ) x ,
Λ = λ ( θ 1 - θ 2 ) .
I { exp ( i Δ ϕ ) } ,
Λ = L Δ f x .
Δ θ = λ L / Δ f x .
Δ ϕ = ( 2 π / λ ) · 2 w cos η cos ψ .
w K = w cos η = Δ ϕ · λ / 4 π cos ψ .
Δ θ = 0.633 μm 2800 · 5.2 μm / 88 = 0.004 rad × 57.3 ° rad = 0.219 ° .

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