Abstract

In reconstruction of in-line recorded holograms, zero-order and conjugate images appear on the same physical location as the object image. Here we propose a method, new to our knowledge, to separate the object image from the others by using two quadrature phase-shifted holograms. The method uses the Hartley transform and a phase retrieval type of algorithm on the difference hologram.

© 2011 Optical Society of America

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References

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  1. D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
    [CrossRef]
  2. J. W. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
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  8. E. Nitanai, T. Nomura, S. Murata, and T. Numata, “Phase-shifting digital holography with a phase difference between orthogonal polarizations,” Appl. Opt. 45, 4873–4877(2006).
    [CrossRef]
  9. M. Sasada, Y. Awatsuji, and T. Kubota, “Parallel quasi-phase shifting digital holography,” Appl. Phys. Lett. 85, 1069–1071(2004).
    [CrossRef]
  10. T. Kim, T.-C. Poon, and G. Indebetouw, “Twin-image removal by digital filtering and optical scanning holography,” in Proceedings of the Thirtieth Southeastern Symposium on System Theory (IEEE, 1998), pp. 191–195.
  11. L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
  12. T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
    [CrossRef]
  13. T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
    [CrossRef]
  14. J.-P. Liu and T.-C. Poon, “Two-step only quadrature phase-shifting holography,” Opt. Lett. 34, 250–252 (2009).
    [CrossRef]
  15. G.-S. Jhou, J.-P. Liu, T.-C. Poon, and P.-J. Chen, “Comparison of two-, three-, and four-exposure quadrature phase-shifting holography,” Appl. Opt. 50, 2443–2450(2011).
    [CrossRef]
  16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  17. R. N. Bracewell, The Hartley Transform (Oxford University, 1986).
  18. R. N. Bracewell, “Aspects of the Hartley transform,” Proc. IEEE 82, 381–387 (1994).
    [CrossRef]
  19. A. W. Lohmann, R. N. Bracewell, H. Bartelt, and N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
    [CrossRef]
  20. J. D. Villasenor, “Optical Hartley transforms,” Proc. IEEE 82, 391–399 (1994).
    [CrossRef]
  21. H. Hamam, “Hartley holograms,” Appl. Opt. 35, 5286–5292(1996).
    [CrossRef]
  22. V. P. Titar’, T. V. Bogdanova, and E. Ya. Tomchuk, “Hartley holograms,” Opt. Spectrosc. 85, 956–962 (1998).
  23. T. V. Bogdanova and V. P. Titar’, “Complex optical holograms,” J. Opt. Technol. 71, 298–306 (2004).
    [CrossRef]
  24. M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).
  25. D. Sayre, J. Miao, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
    [CrossRef]

2011

2009

J.-P. Liu and T.-C. Poon, “Two-step only quadrature phase-shifting holography,” Opt. Lett. 34, 250–252 (2009).
[CrossRef]

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).

2007

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef]

2006

2004

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

T. V. Bogdanova and V. P. Titar’, “Complex optical holograms,” J. Opt. Technol. 71, 298–306 (2004).
[CrossRef]

2003

T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

2001

2000

1999

1998

1996

1994

R. N. Bracewell, “Aspects of the Hartley transform,” Proc. IEEE 82, 381–387 (1994).
[CrossRef]

J. D. Villasenor, “Optical Hartley transforms,” Proc. IEEE 82, 391–399 (1994).
[CrossRef]

1987

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

1985

1982

1971

J. W. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[CrossRef]

1948

D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
[CrossRef]

Awatsuji, Y.

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

Bartelt, H.

Bayraktar, M.

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).

Blu, T.

T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Bogdanova, T. V.

T. V. Bogdanova and V. P. Titar’, “Complex optical holograms,” J. Opt. Technol. 71, 298–306 (2004).
[CrossRef]

V. P. Titar’, T. V. Bogdanova, and E. Ya. Tomchuk, “Hartley holograms,” Opt. Spectrosc. 85, 956–962 (1998).

Bracewell, R. N.

R. N. Bracewell, “Aspects of the Hartley transform,” Proc. IEEE 82, 381–387 (1994).
[CrossRef]

A. W. Lohmann, R. N. Bracewell, H. Bartelt, and N. Streibl, “Optical synthesis of the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
[CrossRef]

R. N. Bracewell, The Hartley Transform (Oxford University, 1986).

Chapman, H. N.

Chen, P.-J.

Cuche, E.

Depeursinge, C.

Fienup, J. R.

Fink, H.-W.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
[CrossRef]

Goodman, J. W.

J. W. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[CrossRef]

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

Hamam, H.

Indebetouw, G.

T. Kim, T.-C. Poon, and G. Indebetouw, “Twin-image removal by digital filtering and optical scanning holography,” in Proceedings of the Thirtieth Southeastern Symposium on System Theory (IEEE, 1998), pp. 191–195.

Jhou, G.-S.

Jüptner, W.

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Kato, J.

Kawai, H.

Kim, T.

T. Kim, T.-C. Poon, and G. Indebetouw, “Twin-image removal by digital filtering and optical scanning holography,” in Proceedings of the Thirtieth Southeastern Symposium on System Theory (IEEE, 1998), pp. 191–195.

Kubota, T.

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

Latychevskaia, T.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef]

Liebling, M.

T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Liu, J.-P.

Lohmann, A. W.

Marquet, P.

Miao, J.

Mizuno, J.

Murata, S.

Nitanai, E.

Nomura, T.

Numata, T.

Ohta, S.

Ohzu, H.

Onural, L.

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

Özcan, M.

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).

Poon, T.-C.

G.-S. Jhou, J.-P. Liu, T.-C. Poon, and P.-J. Chen, “Comparison of two-, three-, and four-exposure quadrature phase-shifting holography,” Appl. Opt. 50, 2443–2450(2011).
[CrossRef]

J.-P. Liu and T.-C. Poon, “Two-step only quadrature phase-shifting holography,” Opt. Lett. 34, 250–252 (2009).
[CrossRef]

T. Kim, T.-C. Poon, and G. Indebetouw, “Twin-image removal by digital filtering and optical scanning holography,” in Proceedings of the Thirtieth Southeastern Symposium on System Theory (IEEE, 1998), pp. 191–195.

Sasada, M.

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

Sayre, D.

Schnars, U.

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Scott, P. D.

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

Streibl, N.

Takaki, Y.

Titar’, V. P.

T. V. Bogdanova and V. P. Titar’, “Complex optical holograms,” J. Opt. Technol. 71, 298–306 (2004).
[CrossRef]

V. P. Titar’, T. V. Bogdanova, and E. Ya. Tomchuk, “Hartley holograms,” Opt. Spectrosc. 85, 956–962 (1998).

Tomchuk, E. Ya.

V. P. Titar’, T. V. Bogdanova, and E. Ya. Tomchuk, “Hartley holograms,” Opt. Spectrosc. 85, 956–962 (1998).

Unser, M. A.

T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

Villasenor, J. D.

J. D. Villasenor, “Optical Hartley transforms,” Proc. IEEE 82, 391–399 (1994).
[CrossRef]

Yamaguchi, I.

Appl. Opt.

Appl. Phys. Lett.

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

J. Opt. Soc. Am. A

J. Opt. Technol.

Nature

D. Gabor, “A new microscopic principle,” Nature 161, 777–778(1948).
[CrossRef]

Opt. Eng.

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

Opt. Lett.

Opt. Spectrosc.

V. P. Titar’, T. V. Bogdanova, and E. Ya. Tomchuk, “Hartley holograms,” Opt. Spectrosc. 85, 956–962 (1998).

Phys. Rev. Lett.

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901(2007).
[CrossRef]

Proc. IEEE

R. N. Bracewell, “Aspects of the Hartley transform,” Proc. IEEE 82, 381–387 (1994).
[CrossRef]

J. D. Villasenor, “Optical Hartley transforms,” Proc. IEEE 82, 391–399 (1994).
[CrossRef]

J. W. Goodman, “An introduction to the principles and applications of holography,” Proc. IEEE 59, 1292–1304 (1971).
[CrossRef]

Proc. SPIE

T. Blu, M. Liebling, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE 5207, 553–559 (2003).
[CrossRef]

M. Özcan and M. Bayraktar, “Digital holography image reconstruction methods,” Proc. SPIE 7233, 72330B (2009).

Other

T. Kim, T.-C. Poon, and G. Indebetouw, “Twin-image removal by digital filtering and optical scanning holography,” in Proceedings of the Thirtieth Southeastern Symposium on System Theory (IEEE, 1998), pp. 191–195.

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

U. Schnars and W. Jüptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

R. N. Bracewell, The Hartley Transform (Oxford University, 1986).

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Figures (3)

Fig. 1
Fig. 1

In-line holographic recording setup. A plane reference wave and an object wave are normally incident to the CCD. The reference wave has a phase shifter in its path. Beam splitters BS1, BS2 and mirrors M1, M2 are used to split and recombine the reference and object waves.

Fig. 2
Fig. 2

Flow chart of the proposed algorithm. Initial f k values are random, whereas initial g k values are calculated using the Hartley transform constraint as it is explained in the text. Next step is the application of the Fourier domain constraints. Afterward, if the iteration level is not reached, object-space constraints are applied to obtain the new f k + 1 values, which is the input to the next iteration loop.

Fig. 3
Fig. 3

(Top) Real and imaginary parts of the ( 512 × 512 ) pixel size object. Reconstruction results are shown after 1, 10, and 40 iterations. MSE errors were 0.082, 0.062, and 0.015 for the real parts and 0.77, 0.57, and 0.14 for the imaginary parts at these iteration counts.

Equations (15)

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I = | O + R | 2 = | O | 2 + | R | 2 + O R * + O * R ,
T = R I = | O | 2 R + | R | 2 R + O | R | 2 + O * R 2 .
H ( u , v ) = + + f ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y ,
f ( x , y ) = + + H ( u , v ) cas [ 2 π ( u x + v y ) ] d u d v ,
Re [ O ( u , v ) ] Im [ O ( u , v ) ] = + + f ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y + + g ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y .
H [ Re [ O ( u , v ) ] Im [ O ( u , v ) ] ] = Δ ( x , y ) = f ( x , y ) g ( x , y ) ,
I 1 = | O + R | 2 = | O | 2 + | R | 2 + O R * + O * R ,
I 2 = | O + i R | 2 = | O | 2 + | R | 2 i O R * + i O * R .
I 1 I 2 = Δ I = ( O + O * ) + i ( O O * ) .
Δ I = Re ( O ) Im ( O ) .
f k + 1 ( x , y ) = { f k ( x , y ) ( x , y ) γ , f k ( x , y ) β f k ( x , y ) ( x , y ) γ ,
MSE = m n ( f a ( m , n ) f c ( m , n ) ) 2 m n ( f a ( m , n ) ) 2 ,
O ( u , v ) = i exp ( k d ) λ d exp ( i π λ d ( u 2 + v 2 ) ) F 1 [ exp ( i π λ d ( x 2 + y 2 ) ) o ( x , y ) ] ,
F [ α ( x , y ) ] = O ( u , v ) .
H [ Re ( O ) Im ( O ) ] = f α ( x , y ) g α ( x , y ) .

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