Abstract

This paper presents a method based on the use of an image sensor for obtaining the complex amplitudes of beams diffracted from an object at two different wavelengths. The complex amplitude for each wavelength is extracted by the Doppler phase-shifting method. The principle underlying the proposed method is experimentally verified by using the method with two lasers having different wavelengths to measure the surface shape of a concave mirror.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  16. T. Kiire, S. Nakadate, and M. Shibuya, “Simultaneous formation of four fringes by using a polarization quadrature phase-shifting interferometer with wave plates and a diffraction grating,” Appl. Opt. 47, 4787–4792 (2008).
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  17. P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
    [CrossRef]
  18. R. Jang, C. S. Kang, J. A. Kim, J. W. Kim, J. E. Kim, and H. Y. Park, “High-speed measurement of three-dimensional surface profiles up to 10 μm using two-wavelength phase-shifting interferometry utilizing an injection locking technique,” Appl. Opt. 50, 1541–1547 (2011).
    [CrossRef]
  19. A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
    [CrossRef]
  20. J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).
  21. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
    [CrossRef]
  22. Y. Kikuchi, D. Barada, T. Kiire, and T. Yatagai, “Doppler phase-shifting digital holography and its application to surface shape measurement,” Opt. Lett. 35, 1548–1550 (2010).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), pp. 57–61.

2011

2010

2009

P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
[CrossRef]

S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48, 09LB01 (2009).

2008

2004

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[CrossRef]

2003

2000

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[CrossRef]

1997

1995

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

1994

1991

1987

K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26, 2810–2816(1987).
[CrossRef]

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

1985

1984

1974

1972

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with computer,” Soviet Physics Technical Physics 17, 333–334 (1972).

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[CrossRef]

Barada, D.

Bingham, P. R.

Brangaccio, D. J.

Brock, N.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Bruning, J. H.

Cheng, Y. Y.

Creath, K.

Dakoff, A.

Doh, K.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Gallagher, J. E.

Gass, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), pp. 57–61.

Groot, P.

Harada, D.

S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48, 09LB01 (2009).

Hayes, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Herriott, D. R.

Hettwer, A.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[CrossRef]

Ishii, Y.

Jang, R.

Jones, J. D. C.

Kang, C. S.

Kiire, T.

Kikuchi, Y.

Kim, J. A.

Kim, J. E.

Kim, J. W.

Kim, M. K.

Kishner, S.

Kothiyal, M. P.

P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
[CrossRef]

Kranz, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[CrossRef]

Kronrod, M.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with computer,” Soviet Physics Technical Physics 17, 333–334 (1972).

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[CrossRef]

Mann, C. J.

Merzlyakov, N.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with computer,” Soviet Physics Technical Physics 17, 333–334 (1972).

Millerd, J.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Mohan, N. K.

P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
[CrossRef]

Murata, S.

S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48, 09LB01 (2009).

Nakadate, S.

North-Morris, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Novak, M.

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Onodera, R.

Onural, L.

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Paquit, V. C.

Park, H. Y.

Poon, T.-C.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Rosenfeld, D. P.

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[CrossRef]

Schilling, B.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Schnars, U.

Schwider, J.

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[CrossRef]

Scott, P. D.

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

Shibuya, M.

Shinoda, K.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Suzuki, Y.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Tanaka, Y.

S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48, 09LB01 (2009).

Tobin, K. W.

Towers, C. E.

Towers, D. P.

Upputuri, P. K.

P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
[CrossRef]

White, A. D.

Wu, M.

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Wyant, J. C.

Yamaguchi, I.

Yaroslavskii, L.

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with computer,” Soviet Physics Technical Physics 17, 333–334 (1972).

Yatagai, T.

Zhang, T.

Appl. Opt.

Appl. Phys. Lett.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phaseshifting digital holography,” Appl. Phys. Lett. 85, 1069–1071 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48, 09LB01 (2009).

Opt. Eng.

P. K. Upputuri, N. K. Mohan, and M. P. Kothiyal, “Measurement of discontinuous surface using multiple-wavelength interferometry,” Opt. Eng. 48073603 (2009).
[CrossRef]

T.-C. Poon, K. Doh, B. Schilling, M. Wu, K. Shinoda, and Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).

A. Hettwer, J. Kranz, and J. Schwider, “Three channel phaseshifting interferometer using polarization-optics and a diffraction grating,” Opt. Eng. 39, 960–966 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

J. Millerd, N. Brock, J. Hayes, M. North-Morris, M. Novak, and J. C. Wyant, “Pixelated phase-mask dynamic interferometer,” Proc. SPIE 5531, 304–314 (2004).

Soviet Physics Technical Physics

M. Kronrod, N. Merzlyakov, and L. Yaroslavskii, “Reconstruction of a hologram with computer,” Soviet Physics Technical Physics 17, 333–334 (1972).

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005), pp. 57–61.

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of simultaneous two-wavelength phase-shifting digital holography. BS, OL, and SF denote beam splitter, objective lens, and spatial filter, respectively.

Fig. 2.
Fig. 2.

Experimental setup for simultaneous two-wavelength phase-shifting digital holography. LD, BS, OL, and SF denote diode laser, beam splitter, objective lens, and spatial filter, respectively. The optical system is installed on a breadboard.

Fig. 3.
Fig. 3.

Temporal variation of digital hologram. (a) and (b) are images captured on the CMOS image sensor in different timing. (c) and (d) are the temporal variation of the pixel values at two pixels A and B as shown in (a) and (b).

Fig. 4.
Fig. 4.

Beat frequency spectrum and beat signals. (a) is the beat frequency spectrum obtained by temporal Fourier transform of the temporally varied pixel values as shown in Fig. 3(c) and (d). (b) and (c) are extracted beat signals for the wavelength of 635 nm and 650 nm, respectively. Solid and dashed lines are beat frequency spectrum and extracted beat signal at the pixels A and B, respectively.

Fig. 5.
Fig. 5.

Phase distributions for the wavelengths of (a) 635 nm and (b) 650 nm.

Fig. 6.
Fig. 6.

Surface shape of the concave mirror reconstructed from the phase difference between two phase distributions as shown in Figs.  5(a) and (b). (a) is the three-dimensional shape. Wire frame shows an ideal concave shape. (b) is a profile along the x-direction through the center pixels in y-direction. Solid and dashed lines are the experimental result and the ideal concave shape, respectively.

Fig. 7.
Fig. 7.

Surface shape of the concave mirror reconstructed from the phase distribution in Fig. 5(a). The phase is unwrapped by determining the number of the uncertainty. (a) is the three-dimensional shape. Wire frame shows an ideal concave shape. (b) is a profile along the x-direction through the center pixels in y-direction. Solid and dashed lines are the experimental result and the ideal concave shape, respectively.

Equations (24)

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UO(t,x,y)=i=12UO,i(t,x,y),
UR(t)=i=12UR,i(t),
UO,i(t,x,y)=AO,i(x,y)cos[ωitϕO,i(t,x,y)],
UR,i(t)=AR,icos[ωitϕR,i(t)],
ϕO,i(t,x,y)ϕO,i,0(x,y)+0t4πλiυO(τ)dτ,
ϕR,i(t)ϕR,i,0+0t4πλiυR(τ)dτ,
λi=2πcωi,
H(t,x,y)1ΔttΔtt[UO(τ,x,y)+UR(τ)]2dτ=1ΔttΔtt[UO2(τ,x,y)+UR2(τ)+2UO(τ,x,y)UR(τ)]dτ,
H(t,x,y)=1ΔttΔttij[UO,i(τ,x,y)+UR,i(τ)][UO,j(τ,x,y)+UR,j(τ)]dτ=1ΔttΔttij[UO,i(τ,x,y)UO,j(τ,x,y)+UR,i(τ)UR,j(τ)+UO,i(τ,x,y)UR,j(τ)+UO,j(τ,x,y)UR,i(τ)]dτ.
UO,i(t,x,y)UO,j(t,x,y)=AO,i(x,y)AO,j(x,y)2{cos(Δωijt)cos[ΔϕO,ij(t,x,y)]+sin(Δωijt)sin[ΔϕO,ij(t,x,y)]+cos(ωijt)cos[ϕO,ij(t,x,y)]+sin(ωijt)sin[ϕO,ij(t,x,y)]},
UR,i(t)UR,j(t)=AR,iAR,j2{cos(Δωijt)cos[ΔϕR,ij(t)]+sin(Δωijt)sin[ΔϕR,ij(t)]+cos(ωijt)cos[ϕR,ij(t)]+sin(ωijt)sin[ϕR,ij(t)]},
UO,i(t,x,y)UR,j(t)=AO,i(x,y)AR,j2{cos[ϕO,ij(t,x,y)ϕR,ij(t)2]·{cos(Δωijt)cos[ΔϕO,ij(t,x,y)+ΔϕR,ij(t)2]+sin(Δωijt)sin[ΔϕO,ij(t,x,y)+ΔϕR,ij(t)2]}+cos[ΔϕO,ij(t,x,y)ΔϕR,ij(t)2]·{cos(ωijt)cos[ϕO,ij(t,x,y)+ϕR,ij(t)2]+sin(ωijt)sin[ϕO,ij(t,x,y)+ϕR,ij(t)2]}},
Δωij=ωiωj,
ωij=ωi+ωj,
ΔϕO,ij(t,x,y)=ϕO,i(t,x,y)ϕO,j(t,x,y),
ϕO,ij(t,x,y)=ϕO,i(t,x,y)+ϕO,j(t,x,y),
ΔϕR,ij(t)=ϕR,i(t)ϕR,j(t),
ϕR,ij(t)=ϕR,i(t)+ϕR,j(t).
H(t,x,y)i{AO,i2(x,y)2+AR,i22+AO,i(x,y)AR,icos[ϕO,i(t,x,y)ϕR,i(t)]}.
H(t,x,y)i{AO,i2(x,y)2+AR,i22+AO,i(x,y)AR,icos[4πvRλitϕO,i,0(x,y)+ϕR,i,0]}.
fb,i=2υRλi.
FtH(fb,x,y)=i{[AO,i2(x,y)2+AR,i22]δ(fb)+AO,i(x,y)AR,i2exp[iϕO,i,0(x,y)+iϕR,i,0]δ(fb2υR,iλi)+AO,i(x,y)AR,i2exp[iϕO,i,0(x,y)iϕR,i,0]δ(fb+2υR,iλi)},
FtH(fb,x,y)=i{[AO,i2(x,y)2+AR,i22]δ(fb)+AO,i(x,y)AR,i2exp[iϕO,i,0(x,y)+iϕR,i,0]Bi(fbfb,0)+AO,i(x,y)AR,i2exp[iϕO,i,0(x,y)iϕR,i,0]Bi*(fb+fb,0)},
zi(x,y)=λi4πΦO,i(x,y)+niλi2,

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