Abstract

We propose a new technique for achromatic 3-D field correlation that makes use of the characteristics of both axial and lateral magnifications of imaging through a common-path Sagnac shearing interferometer. With this technique, we experimentally demonstrate, for the first time to our knowledge, 3-D image reconstruction of coherence holography with generic thermal light. By virtue of the achromatic axial shearing implemented by the difference in axial magnifications in imaging, the technique enables coherence holography to reconstruct a 3-D object with an axial depth beyond the short coherence length of the thermal light.

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References

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  1. M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express13(23), 9629–9635 (2005).
    [CrossRef] [PubMed]
  2. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express17(13), 10633–10641 (2009).
    [CrossRef] [PubMed]
  3. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett.35(10), 1728–1730 (2010).
    [CrossRef] [PubMed]
  4. D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express18(13), 13782–13787 (2010).
    [CrossRef] [PubMed]
  5. W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt.41(10), 1962–1971 (2002).
    [CrossRef] [PubMed]
  6. Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express14(25), 12109–12121 (2006).
    [CrossRef] [PubMed]
  7. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt.39(23), 4107–4111 (2000).
    [CrossRef] [PubMed]
  8. P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt.48(34), H40–H47 (2009).
    [CrossRef] [PubMed]
  9. W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006).
    [CrossRef] [PubMed]
  10. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]
  11. M. V. R. K. Murty, “A compact radial shearing interferometer based on the law of refraction,” Appl. Opt.3(7), 853–857 (1964).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Chap. 10.
  13. J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5.
  14. P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000).
    [CrossRef]

2010 (2)

2009 (2)

2006 (2)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006).
[CrossRef] [PubMed]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express14(25), 12109–12121 (2006).
[CrossRef] [PubMed]

2005 (1)

2002 (1)

2000 (2)

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt.39(23), 4107–4111 (2000).
[CrossRef] [PubMed]

P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000).
[CrossRef]

1982 (1)

1964 (1)

Duan, Z.

Ezawa, T.

Halouzka, M.

Handel, P.

P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000).
[CrossRef]

Hanson, S. G.

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006).
[CrossRef] [PubMed]

Ina, H.

Kobayashi, S.

Kozaki, H.

Miyamoto, Y.

Murty, M. V. R. K.

Naik, D. N.

Pavlicek, P.

Rosen, J.

Takeda, M.

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Real-time coherence holography,” Opt. Express18(13), 13782–13787 (2010).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “Phase-shift coherence holography,” Opt. Lett.35(10), 1728–1730 (2010).
[CrossRef] [PubMed]

P. Pavliček, M. Halouzka, Z. Duan, and M. Takeda, “Spatial coherence profilometry on tilted surfaces,” Appl. Opt.48(34), H40–H47 (2009).
[CrossRef] [PubMed]

D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express17(13), 10633–10641 (2009).
[CrossRef] [PubMed]

Z. Duan, Y. Miyamoto, and M. Takeda, “Dispersion-free optical coherence depth sensing with a spatial frequency comb generated by an angular spectrum modulator,” Opt. Express14(25), 12109–12121 (2006).
[CrossRef] [PubMed]

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006).
[CrossRef] [PubMed]

M. Takeda, W. Wang, Z. Duan, and Y. Miyamoto, “Coherence holography,” Opt. Express13(23), 9629–9635 (2005).
[CrossRef] [PubMed]

W. Wang, H. Kozaki, J. Rosen, and M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt.41(10), 1962–1971 (2002).
[CrossRef] [PubMed]

J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt.39(23), 4107–4111 (2000).
[CrossRef] [PubMed]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
[CrossRef]

Wang, W.

Appl. Opt. (4)

IEEE Trans. Instrum. Meas. (1)

P. Handel, “Properties of the IEEE-STD-1057 four-parameter sine wave fit algorithm,” IEEE Trans. Instrum. Meas.49(6), 1189–1193 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

W. Wang, Z. Duan, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental study of coherence vortices: local properties of phase singularities in a spatial coherence function,” Phys. Rev. Lett.96(7), 073902 (2006).
[CrossRef] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Chap. 10.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5.

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Figures (8)

Fig. 1
Fig. 1

Geometry of Fourier transform coherence holography using a commercial projector for display of hologram.

Fig. 2
Fig. 2

Generation of a Fourier transform coherence hologram.

Fig. 3
Fig. 3

Radial and axial shearing Sagnac common path interferometer.

Fig. 4
Fig. 4

Telescopic system for 3-D optical field correlation.

Fig. 5
Fig. 5

Longitudinal shear with time delay compensation.

Fig. 6
Fig. 6

One of the phase shifted coherence holograms used in the experiment.

Fig. 7
Fig. 7

Experimental set up for generic coherence holography.

Fig. 8
Fig. 8

(a), (b) and (c) are the interference fringes at depths z ˜ =20mm , z ˜ =0 and z ˜ =+20mm , respectively; (d), (e) and (f) shows the corresponding fringe visibilities; (g), (h) and (i) shows the corresponding fringe phases.

Equations (10)

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G( x ^ , y ^ ) { g( x,y,z )exp[ i 2π λf ( x x ^ +y y ^ ) ] dxdy } exp[ i k z ( x ^ , y ^ )z ]dz,
G( x ^ , y ^ ;m )=G( x ^ , y ^ )exp( i2πm/N ),
H( x ^ , y ^ ;m )| G( x ^ , y ^ ) |+ 1 2 G( x ^ , y ^ )exp( i 2mπ N )+ 1 2 G * ( x ^ , y ^ )exp( i 2mπ N ) | G( x ^ , y ^ ) |{ 1+cos[ Φ( x ^ , y ^ )+ 2mπ N ] }
u( x,y,z;m )= H( x ^ , y ^ ;m ) exp[ i Φ R ( x ^ , y ^ ) ]exp[ i k z ( x ^ , y ^ )z ]exp[ i 2π λf ( x x ^ +y y ^ ) ]d x ^ d y ^ ,
I(Δx,Δy,Δz;m)= | u( x 1 , y 1 , z 1 ;m)+u( x 2 , y 2 , z 2 ;m) | 2 =2Γ(0,0,0;m)+2Re[ Γ(Δx,Δy,Δz;m) ],
Γ(0,0,0;m)= u * ( x 1 , y 1 , z 1 ;m)u( x 1 , y 1 , z 1 ;m) = u * ( x 2 , y 2 , z 2 ;m)u( x 2 , y 2 , z 2 ;m) = H( x ^ , y ^ ;m) d x ^ d y ^
Γ( Δx,Δy,Δz;m )= u * ( x 1 , y 1 , z 1 ;m )u( x 2 , y 2 , z 2 ;m ) = H( x ^ , y ^ ;m ) exp[ i k z ( x ^ , y ^ )Δz ]exp[ i 2π λf ( x ^ Δx+ y ^ Δy ) ]d x ^ d y ^
Γ( Δx,Δy,Δz;m )= g ˜ ( Δx,Δy,Δz )+ 1 2 g( Δx,Δy,Δz )exp( i 2mπ N ) + 1 2 g * ( Δx,Δy,Δz )exp( i 2mπ N )
g ˜ ( Δx,Δy,Δz )= | G( x ^ , y ^ ) | exp[ i k z ( x ^ , y ^ )Δz ]exp[ i 2π λf ( x ^ Δx+ y ^ Δy ) ]d x ^ d y ^ .
I( Δx,Δy,Δz;m )=Re{ 2 g ˜ ( 0,0,0 )+2 g ˜ ( Δx,Δy,Δz ) +g( 0,0,0 )exp( i 2mπ N )+g( Δx,Δy,Δz )exp( i 2mπ N ) + g * ( 0,0,0 )exp( i 2mπ N )+ g * ( Δx,Δy,Δz )exp( i 2mπ N ) }

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