Abstract

A fast calculation method for computer generation of cylindrical holograms is proposed. The calculation method is based on wave propagation in spectral domain and in cylindrical co-ordinates, which is otherwise similar to the angular spectrum of plane waves in cartesian co-ordinates. The calculation requires only two FFT operations and hence is much faster. The theoretical background of the calculation method, sampling conditions and simulation results are presented. The generated cylindrical hologram has been tested for reconstruction in different view angles and also in plane surfaces.

© 2010 Optical Society of America

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References

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  1. T. H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am. 57, 31396–31398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396.
    [CrossRef]
  2. O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194.
    [CrossRef] [PubMed]
  3. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739.
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  5. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299.
    [CrossRef]
  6. Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and Applications, Tung H. Jeong and Hans I. Bjelkhagen, eds., Proc.SPIE 5742, 267–274 (2005).
  7. A. Kashiwagi and Y. Sakamoto, “A Fast calculation method of cylindrical computer-generated holograms which perform image reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7.
    [PubMed]
  8. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
    [CrossRef] [PubMed]
  9. T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Computer-generated cylindrical rainbow hologram,” in Practical Holography XXII: Materials and Applications, Hans I. Bjelkhagen and Raymond K. Kostuk, eds., Proc.SPIE 6912, 69121C (2009).
  10. Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.
  11. G. E. Williams, Fourier Acoustics, Sound Radiation, and Near-field Acoustical Holography (Academic Press, 1999).
  12. N. N. Lebedev, Special Functions and Their Applications (Prentice Hall, 1965), pp. 98–160.
  13. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 2001), pp. 702–705.

2008 (1)

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
[CrossRef] [PubMed]

2005 (1)

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

1993 (1)

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299.
[CrossRef]

1982 (1)

O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194.
[CrossRef] [PubMed]

1967 (2)

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739.
[CrossRef]

T. H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am. 57, 31396–31398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396.
[CrossRef]

Bianco, B.

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299.
[CrossRef]

Fernandes, J. C. A.

O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194.
[CrossRef] [PubMed]

Fujii, T.

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
[CrossRef] [PubMed]

Itoh, M.

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

Jeong, T. H.

T. H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am. 57, 31396–31398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396.
[CrossRef]

Lohmann, A. W.

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739.
[CrossRef]

Paris, D. P.

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739.
[CrossRef]

Sando, Y.

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

Soares, O. D. D.

O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194.
[CrossRef] [PubMed]

Tommasi, T.

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299.
[CrossRef]

Yamaguchi, T.

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
[CrossRef] [PubMed]

Yatagai, T.

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

Yoshikawa, H.

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
[CrossRef] [PubMed]

Appl. Opt. (4)

O. D. D. Soares and J. C. A. Fernandes, “Cylindrical hologram of 360o field of view,” Appl. Opt. 21, 3194–3196 (1982), http://www.opticsinfobase.org/abstract.cfm?URI=ao-21-17-3194.
[CrossRef] [PubMed]

A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 739–1748 (1967), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-6-10-1739.
[CrossRef]

Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Appl. Opt. 13, 1418–1423 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-5-1418.

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Fast calculation method for computer-generated cylindrical holograms,” Appl. Opt. 47, D63–D70 (2008), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-47-19-D63.
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

T. H. Jeong, “Cylindrical holography and some proposed applications,” J. Opt. Soc. Am. 57, 31396–31398 (1967), http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-57-11-1396.
[CrossRef]

J. Opt. Soc. Am. A (1)

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993), http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-2-299.
[CrossRef]

Other (7)

Y. Sakamoto and M. Tobise, “Computer generated cylindrical hologram,” in Practical Holography XIX: Materials and Applications, Tung H. Jeong and Hans I. Bjelkhagen, eds., Proc.SPIE 5742, 267–274 (2005).

A. Kashiwagi and Y. Sakamoto, “A Fast calculation method of cylindrical computer-generated holograms which perform image reconstruction of volume data,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DWB7, http://www.opticsinfobase.org/abstract.cfm?URI=DH-2007-DWB7.
[PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

G. E. Williams, Fourier Acoustics, Sound Radiation, and Near-field Acoustical Holography (Academic Press, 1999).

N. N. Lebedev, Special Functions and Their Applications (Prentice Hall, 1965), pp. 98–160.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, 2001), pp. 702–705.

T. Yamaguchi, T. Fujii, and H. Yoshikawa, “Computer-generated cylindrical rainbow hologram,” in Practical Holography XXII: Materials and Applications, Hans I. Bjelkhagen and Raymond K. Kostuk, eds., Proc.SPIE 6912, 69121C (2009).

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Figures (12)

Fig. 1
Fig. 1

Coordinate system.

Fig. 2
Fig. 2

Recording setup: (a) schematic and (b) geometry.

Fig. 3
Fig. 3

Object.

Fig. 4
Fig. 4

Hologram.

Fig. 5
Fig. 5

Reconstruction-Cylindrical surface.

Fig. 6
Fig. 6

Planar reconstruction: (a) schematic and (b) geometry.

Fig. 7
Fig. 7

Planar Reconstruction (z=1).

Fig. 8
Fig. 8

Planar Reconstruction (z=−1).

Fig. 9
Fig. 9

Variable angle reconstruction - Schematic.

Fig. 10
Fig. 10

120° (−60° to 60°) view angle.

Fig. 11
Fig. 11

90° (90° to 180°) view angle.

Fig. 12
Fig. 12

45° (−180° to −135°) view angle.

Tables (1)

Tables Icon

Table 1 Calculation time

Equations (21)

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( 2 + k ) p = 0 ,
p ( r , ϕ , y ) = R ( r ) Φ ( ϕ ) Y ( y ) ,
R ( r ) = R 1 H n ( 1 ) ( k r r ) + R 2 H n ( 2 ) ( k r r ) ,
Φ ( ϕ ) = ϕ 1 e i n ϕ + ϕ 2 e in ϕ ,
Y ( y ) = Y 1 e ik y y + Y 2 e ik y y ,
p ( r , ϕ , y ) = n = e i n ϕ 1 2 π [ A n ( k y ) e i k y y H n ( 1 ) ( k r r ) + B n ( k y ) e ik y y H n ( 2 ) ( k r r ) ] dk y ,
p ( a , ϕ , y ) = n = e in ϕ 1 2 π A n ( k y ) e ik y y H n ( 1 ) ( k r a ) d k y ,
P n ( r , k y ) 1 2 π 0 2 π d ϕ p ( r , ϕ , y ) e i n ϕ e i k y y d y ,
p ( r , ϕ , y ) = n = e i n ϕ 1 2 π p n ( r . k y ) e i k y y d k y ,
P n ( a , k y ) = A n ( k y ) H n ( 1 ) ( k r a ) ,
p ( r , ϕ , y ) = n = e i n ϕ 1 2 π p n ( a , k y ) e i k y y H n ( 1 ) ( k r r ) H n ( 1 ) ( k r a ) d k y ,
p ( x , y , z ) = 1 4 π 2 d k x d k y P ( k x , k y , z 0 ) e i ( k x x + k y y ) e i k z ( z z 0 ) .
P n ( a , k y ) = 1 2 π 0 2 π d ϕ p ( a , ϕ , y ) e in ϕ e i k y y d y ,
T ( a , k a , r , k r ) = H n ( 1 ) ( k r r ) H n ( 1 ) ( k r a ) ,
n | k r 1 ( λ k y ) 2 | n = N / 2 π ,
| k λ 2 r n Δ L 0 2 1 λ 2 n 2 Δ L 0 2 | n = N / 2 π ,
2 n r λ Δ L 0 Δ L 0 2 λ 2 ( N 2 ) 2 ,
Δ L 0 N r λ ( or ) N Δ L 0 2 r λ .
Hologram = | I F F T [ F F T ( Object ) × T F ] + I F F T [ F F T ( Reference ) × T F ] | 2
Reconstruction = | I F F T [ F F T ( Hologram × Conjugate [ Reference ] ) × T F ] | 2
U ( a , ϕ , z ) = U ( r , θ , z ) exp ( ikr ) r d ϕ d z .

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