Abstract

A novel type of multiplexed computer-generated hologram (CGH) is designed with more than one billion of pixels per period. It consists of elementary cells divided into arbitrary-shaped polygonal apertures, the division being identical in all cells. The cells are further digitized into pixel arrays to exploit the huge space–bandwidth product of electron-beam lithography. The polygonal apertures in the same location inside the cells constitute a subhologram. With the Abbe transform that has never, to our knowledge, been used in other CGH designs, the subhologram images (subimages) are obtained with fast Fourier transforms. It is therefore possible to design a multiplexed CGH that has a size thousands of times larger than the manageable size of a conventional CGH designed with the iterative Fourier transform algorithm (IFTA). A much larger object window than that of the conventional CGH can also be achieved with the multiplexed polygonal-aperture CGH, owing to its extremely large dimensions. The multiplexed polygonal-aperture CGH is designed with the novel iterative subhologram design algorithm, which considers the coherent summation of the subimages and applies constraints on the total image, subimages, and subholograms. As a result, the noise appearing in the preceding multiplexed-CGH designs is avoided. The multiplexed polygonal-aperture CGH has a much higher diffraction efficiency than that resulting from either the preceding multiplexed-CGH designs or the conventional CGH designed by the IFTA.

© 2002 Optical Society of America

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References

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    [CrossRef]
  3. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  4. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  5. J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
    [CrossRef]
  6. J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
    [CrossRef]
  7. W. T. Cathey, “Multiplexed holograms,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), Sec. 5.2, pp. 191–197.
  8. O. K. Ersoy, J. Y. Zhuang, J. Brede, “Iterative interlacing approach for synthesis of computer-generated holograms,” Appl. Opt. 31, 6894–6901 (1992).
    [CrossRef] [PubMed]
  9. R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berück-sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckere, Jena, Germany, 1888).
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    [CrossRef]
  11. M. P. Chang, O. K. Ersoy, “Iterative interlacing error diffusion for synthesis of computer-generated holograms,” Appl. Opt. 32, 3122–3128 (1993).
    [CrossRef] [PubMed]
  12. M. v. Laue, “Bemerkung über Fraunhofersche Beugung,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.89–91 (1936).
  13. M. v. Laue, “Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern,” Ann. Phys. (Leipzig) 26, 55–68 (1936).
  14. R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967).
  15. W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.
  16. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.
  17. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  18. L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

2000 (1)

1999 (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

1993 (1)

1992 (1)

1990 (1)

1988 (1)

1982 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1936 (2)

M. v. Laue, “Bemerkung über Fraunhofersche Beugung,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.89–91 (1936).

M. v. Laue, “Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern,” Ann. Phys. (Leipzig) 26, 55–68 (1936).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.

Brede, J.

Bryngdahl, O.

Cathey, W. T.

W. T. Cathey, “Multiplexed holograms,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), Sec. 5.2, pp. 191–197.

Chang, M. P.

Cirino, G. A.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Ersoy, O. K.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gillet, J.-N.

J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967).

Komrska, J.

Laue, M. v.

M. v. Laue, “Bemerkung über Fraunhofersche Beugung,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.89–91 (1936).

M. v. Laue, “Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern,” Ann. Phys. (Leipzig) 26, 55–68 (1936).

Mansano, R. D.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

Mousinho, A. P.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

Neto, L. G.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

Press, W. H.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sheng, Y.

J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Straubel, R.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berück-sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckere, Jena, Germany, 1888).

Teukolski, S. A.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Verdonck, P.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

Vetterling, W. T.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Wyrowski, F.

Zhuang, J. Y.

Ann. Phys. (Leipzig) (1)

M. v. Laue, “Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern,” Ann. Phys. (Leipzig) 26, 55–68 (1936).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun. (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. (1)

M. v. Laue, “Bemerkung über Fraunhofersche Beugung,” Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.89–91 (1936).

Other (6)

W. T. Cathey, “Multiplexed holograms,” in Handbook of Optical Holography, H. J. Caulfield, ed. (Academic, New York, 1979), Sec. 5.2, pp. 191–197.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berück-sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckere, Jena, Germany, 1888).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967).

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: the Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.

L. G. Neto, G. A. Cirino, A. P. Mousinho, R. D. Mansano, P. Verdonck, “Novel complex-amplitude modulation diffractive optical element,” in Gradient Index, Miniature, and Diffractive Optical Systems II, T. J. Suleski, ed., Proc. SPIE4437 (to be published).

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Figures (7)

Fig. 1
Fig. 1

Polygonal aperture Ω of pentagonal contour C.

Fig. 2
Fig. 2

Multiplexed CGH divided into M × N = 4 × 4 elementary square cells composed of four triangular apertures: (a) CGH period and (b) layout of the elementary cell (j, r).

Fig. 3
Fig. 3

Multiplexed CGH divided into M × N = 4 × 4 elementary square cells (per period) composed of eight arbitrary-shaped polygonal apertures.

Fig. 4
Fig. 4

ISDA for S multiplexed polygonal-aperture subholograms. (The bold symbols denote M × N complex-valued arrays; “×” and “/” are the element-by-element array product and division, respectively, and “‖ · ‖2” corresponds to a quadratic double summation.)

Fig. 5
Fig. 5

Digitization into P × Q = 16 × 16 pixels of the elementary cell (j, r) in a multiplexed CGH with the polygonal layout of Fig. 2.

Fig. 6
Fig. 6

Desired image of 256 × 256 pixels with 256 gray levels.

Fig. 7
Fig. 7

Diffraction image of 258 × 258 pixels generated by a multiplexed triangular-aperture CGH designed with 258 × 258 elementary cells per period (M = 258, A = 256).

Tables (1)

Tables Icon

Table 1 Multiplexed Triangular-Aperture CGHsa Designed with the Novel ISDA Versus Conventional CGHs Designed with the IFTA

Equations (30)

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Up=Ωexp-i2πpξdS,
Up=-12π2p2 Ω 2ψ dS=-12π2p2 Cψnˆ dl,
Up=i2πp2 C exp-i2πpξpnˆ dl.
Up=i2πp2q=1QpnˆqLqsincptˆqLq×exp-i2πpξMq,
U1m, n, j, r=exp-iπmM+nNΘ1m, n×1MNexp-i2πmM j+nN r×expiφ1j, r,
Θ1m, n=i2πm2+n2mM+nN2exp-i π2×mM-nNsinc12mM-nN-mM-nN2expi π2mM+nN×sinc12mM+nN-nN expiπ nNsincmM.
U2m, n, j, r=exp-iπmM+nNΘ2m, n×1MNexp-i2πmM j+nN r×expiφ2j, r,
Θ2m, n=i2πm2+n2-mM-nN2×exp-i π2mM+nN×sinc12mM+nN-mM+nN2×exp-i π2mM-nNsinc12mM-nN+mM expiπ mMsincnN.
U3m, n, j, r=exp-iπmM+nNΘ3m, n×1MNexp-i2πmM j+nN r×expiφ3j, r,
Θ3m, n=Θ1*m, n
U4m, n, j, r=exp-iπmM+nNΘ4m, n×1MNexp-i2πmM j+nN r×expiφ4j, r,
Θ4m, n=Θ2*m, n.
Tm, n=s=14 Θsm, n1MNj=0M-1r=0N-1expiφsj, r×exp-i2πmM j+nN r,
Fsm, n=1MNj=0M-1r=0N-1expiφsj, r×exp-i2πmM j+nN r
Tm, n=s=14 Θsm, nFsm, n,
Tm, n=s=1S Θsm, nFsm, n,
Fs1m, n=Hm, nexpi2πΓsm, n.
fskj, r=FFT-1Fskm, n.
f˜skj, r=fskj, r/|fskj, r|.
T˜km, n=s=14 Θsm, nF˜skm, n,
Tkm, n=βkHm, nT˜km, n|T˜km, n|if m-A/2, A/2-1 and n-B/2, B/2-10 or T˜km, notherwise,
Gsk+1m, n=Tkm, n-s=1ss4 Θsm, n×F˜skm, n,
Ek=m=-M/2M/2-1n=-N/2N/2-1 |Tkm, n-T˜km, n|2.
Ek=m=-A/2A/2-1n=-B/2B/2-1β Hm, n|T˜km, n|-12|T˜km, n|2=β2-2cβ+η0,
m=-A/2A/2-1n=-B/2B/2-1Hm, n2=1,η=m=-A/2A/2-1n=-B/2B/2-1 |T˜km, n|2
c=m=-A/2A/2-1n=-B/2B/2-1 |T˜km, n|Hm, n
Tkm, n-T˜km, n=Θsm, n×Fsk+1m, n-F˜skm, n.
|Tm, n|2=s=1S Ism, n+s=1Ss>s Cssm, n,
Ism, n=|Θsm, nFsm, n|2
Cssm, n=2 ReΘsm, nFsm, n×Θs*m, nFs*m, n,

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