Abstract

An optical implementation of iterative fractional Fourier transform algorithm is proposed and demonstrated. In the proposed implementation, the phase-shifting digital holography technique and the phase-type spatial light modulator are adopted for the measurement and the modulation of complex optical fields, respectively. With the devised iterative fractional Fourier transform system, we demonstrate two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes.

© 2006 Optical Society of America

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References

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  1. J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).
  2. B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).
  3. H. Kim, K. Choi, and B. Lee, “Diffractive optic synthesis and analysis of light fields and recent applications,” Jpn. J. Appl. Phys. 45, 6555–6575 (2006).
    [Crossref]
  4. V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).
  5. T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A 19, 369–377 (2002).
    [Crossref]
  6. J. Hahn, H. Kim, K. Choi, and B. Lee, “Real-time digital holographic beam-shaping system with a genetic feedback tuning loop,” Appl. Opt. 45, 915–924 (2006).
    [Crossref] [PubMed]
  7. H. Kim and B. Lee, “Optimal nonmonotonic convergence of the iterative Fourier-transform algorithm,” Opt. Lett. 30, 296–298 (2005).
    [Crossref] [PubMed]
  8. H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 12, 2353–2365 (2004).
    [Crossref]
  9. K. Choi, H. Kim, and B. Lee, “Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA,” Opt. Express 12, 2454–2462 (2004).
    [Crossref] [PubMed]
  10. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [Crossref]
  11. A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
    [Crossref]
  12. A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [Crossref]
  13. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).
  14. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
    [Crossref] [PubMed]
  15. T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000).
    [Crossref]
  16. P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
    [Crossref]
  17. E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
    [Crossref]
  18. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [Crossref] [PubMed]

2006 (2)

H. Kim, K. Choi, and B. Lee, “Diffractive optic synthesis and analysis of light fields and recent applications,” Jpn. J. Appl. Phys. 45, 6555–6575 (2006).
[Crossref]

J. Hahn, H. Kim, K. Choi, and B. Lee, “Real-time digital holographic beam-shaping system with a genetic feedback tuning loop,” Appl. Opt. 45, 915–924 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (2)

K. Choi, H. Kim, and B. Lee, “Synthetic phase holograms for auto-stereoscopic image display using a modified IFTA,” Opt. Express 12, 2454–2462 (2004).
[Crossref] [PubMed]

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 12, 2353–2365 (2004).
[Crossref]

2002 (1)

2000 (2)

T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000).
[Crossref]

E. Tajahuerce, G. Saavedra, W. D. Furlan, E. E. Sicre, and P. Andrés, “White-light optical implementation of the fractional Fourier transform with adjustable order control,” Appl. Opt. 39, 238–245 (2000).
[Crossref]

1998 (1)

1997 (2)

1995 (3)

Andrés, P.

Barnes, T. H.

Bitran, Y.

Choi, K.

Dorsch, R. G.

Doskolovich, L.

V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).

Ferreira, C.

Furlan, W. D.

Garcia, J.

Hahn, J.

Kim, H.

Kim, T.

T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000).
[Crossref]

Kotlyar, V. V.

V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).

Kress, B.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

Lee, B.

Lohmann, A. W.

Mendlovic, D.

Meyrueis, P.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).

Ozaktas, H. M.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[Crossref]

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[Crossref]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[Crossref]

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

Ozaktaz, H. M.

Poon, T-C.

T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000).
[Crossref]

Saavedra, G.

Sahin, A.

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[Crossref]

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[Crossref]

Shirai, T.

Sicre, E. E.

Soifer, V. A.

V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).

Tajahuerce, E.

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).

Yamaguchi, I.

Yang, B.

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 12, 2353–2365 (2004).
[Crossref]

Zalevsky, Z.

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

Zhang, T.

Appl. Opt. (4)

J. Opt. Soc. Am. A (5)

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am. A 12, 2353–2365 (2004).
[Crossref]

T. Shirai and T. H. Barnes, “Adaptive restoration of a partially coherent blurred image using an all-optical feedback interferometer with a liquid-crystal device,” J. Opt. Soc. Am. A 19, 369–377 (2002).
[Crossref]

T. Kim and T-C. Poon, “Three-dimensional matching by use of phase-only holographic information and the Wigner distribution,” J. Opt. Soc. Am. A 12, 2520–2528 (2000).
[Crossref]

P. Andrés, W. D. Furlan, G. Saavedra, and A. W. Lohmann, “Variable fractional Fourier processor: a simple implementation,” J. Opt. Soc. Am. A 14, 853–858 (1997).
[Crossref]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. Kim, K. Choi, and B. Lee, “Diffractive optic synthesis and analysis of light fields and recent applications,” Jpn. J. Appl. Phys. 45, 6555–6575 (2006).
[Crossref]

Opt. Commun. (1)

A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Other (4)

V. A. Soifer, V. V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Tayor & Francis Ltd., London, 1997).

J. Turunen and F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (John Wiley and Sons Ltd., New York, 1997).

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffraction Optics and Related Technology (John Wiley and Sons Ltd., New York, 2000).

H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley and Sons Ltd., New York, 2001).

Supplementary Material (3)

» Media 1: GIF (234 KB)     
» Media 2: GIF (132 KB)     
» Media 3: GIF (884 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Iterative FRFT algorithm.

Fig. 2.
Fig. 2.

Optical implementation of the ath order and its complementary (2-a)th order two-dimensional FRFT. The incident optical field may be diverging, converging or normally incident to the input plane.

Fig. 3.
Fig. 3.

Optical implementation of the iterative FRFT algorithm.

Fig. 4.
Fig. 4.

Flow chart of the iterative FRFT algorithm.

Fig. 5.
Fig. 5.

Soft clipping in the iterative FRFT algorithm.

Fig. 6.
Fig. 6.

The relationship for the two transforms: input with the spherical radius R (dotted lines) and the plane wave input with the corresponding defocus Δd.

Fig. 7.
Fig. 7.

Improvement of diffraction images as the number of iteration increases in the iterative FRFT algorithm for the two conditions of the radius of the spherical phase: (a) R=∞ (0.24 MB movie) and (b) R=62mm (0.13 MB movie).

Fig. 8.
Fig. 8.

Multiplexing four DOE phase profiles (a) the patterned masks and (b) the applied spherical phase profiles.

Fig. 9.
Fig. 9.

(0.88 MB movie) Diffraction images of the multiplexed DOE captured by a CCD as the capturing location is changed.

Equations (16)

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F α ( u , v ) = K α ( u , v , u , v ) G ( u , v ) d u d v ,
for a 2 m , K a ( u , v , u , v ) = [ 1 j cot ( π a 2 ) ] exp ( j π [ cot ( π a 2 ) ( u 2 + v 2 ) 2 csc ( π a 2 ) ( u u + v v ) + cot ( π a 2 ) ( u 2 + v 2 ) ] ) ,
for a = 4 m , K a ( u , v , u , v ) = δ ( u u ) δ ( v v ) ,
for a = 4 m ± 2 , K a ( u , v , u , v ) = δ ( u + u ) δ ( v + v ) ,
F a 1 [ F a 2 ( f ) ] = F a 2 [ F a 1 ( f ) ] = F a 1 + a 2 ( f ) .
F ( x , y ) = h ( x , y , x , y ) G ( x , y ) d x d y ,
h ( x , y , x , y ) = csc ( a π 2 ) s 2 M e j π 2 exp ( j π ( x 2 + y 2 ) λ R ) exp ( j π s 2 [ cot ( π a 2 ) ( x 2 + y 2 ) M 2 2 csc ( π a 2 ) M ( x x + y y ) + cot ( π a 2 ) ( x 2 + y 2 ) ] ) .
P ( x 2 , y 2 ) = j λ f [ exp ( j π ( x 1 2 + y 1 2 ) λ R ) G ( x 1 , y 1 ) ] exp ( j 2 π λ f ( x 2 x 1 + y 2 y 1 ) ) d x 1 d y 1
= j λ f exp ( j π ( x 2 2 + y 2 2 ) λ R ) exp ( j π λ [ ( x 1 2 + y 1 2 ) R 2 f ( x 2 x 1 + y 2 y 1 ) + ( x 2 2 + y 2 2 ) R ] ) G ( x 1 , y 1 ) d x 1 d y 1 ,
a = 2 π arccos ( f R ) ,
s 4 = λ 2 R 2 f 2 R 2 f 2 .
2 a = 2 π arccos ( f R ) ,
s 4 = λ 2 R 2 f 2 R 2 f 2 .
P ( x 2 , y 2 ) = j λ f [ exp ( j π ( x 1 2 + y 1 2 ) λ R ) G ( x 1 , y 1 ) ] exp ( j 2 π λ f ( x 2 x 1 + y 2 y 1 ) ) d x 1 d y 1 ,
F ( x 2 , y 2 ) = j λ f { exp [ j π λ f ( Δ d f ) ( x 1 2 + y 1 2 ) ] G ( x 1 , y 1 ) } exp ( j 2 π λ f ( x 2 x 1 + y 2 y 1 ) ) d x 1 d y 1 .
Δ d = f 2 R .

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