Abstract

We present a new, to our knowledge, analysis of the performances of common encoding schemes that are used to design computer-generated diffractive optical elements. A statistical analysis of the fringe width and the positional errors, which are introduced by the encoding process, is used to calculate their effects on the diffraction efficiency and the noise. The analysis is applied to interferogram-type encoding methods that are most suited to analytically designed elements, in particular, the polygon-fringe-tracing and the bitmap, or mosaic, methods. Predictions are verified by use of both numerical simulation and existing empirical results. The analysis is used to compare the relative merits of the different encoding schemes quantitatively in terms of optical performance and computational efficiency criteria.

© 1999 Optical Society of America

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References

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  1. M. S. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
    [CrossRef]
  2. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  3. S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
    [CrossRef]
  4. R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 11–23 (1990).
    [CrossRef]
  5. H. Farhoosh, M. R. Feldman, S. H. Lee, C. C. Guest, Y. Fainman, R. Eschbach, “Comparison of binary encoding schemes for electron-beam fabrication of computer generated holograms,” Appl. Opt. 26, 4361–4372 (1987).
    [CrossRef] [PubMed]
  6. M. A. McCord, M. J. Rooks, “Electron beam lithography,” in Handbook of Microlithography, Micromachining and Microfabrication, P. Rai-Choudhury, ed. (SPIE Press, Bellingham, Wash., 1997), pp. 139–249.
  7. W.-H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North Holland, Amsterdam, 1978), pp. 120–232.
  8. W.-H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef] [PubMed]
  9. N. C. Gallagher, J. A. Bucklew, “Nondetour phase digital holograms: an analysis: errata,” Appl. Opt. 19, 4266–4272 (1980).
    [CrossRef] [PubMed]
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  11. A. W. Lohmann, D. P. Paris, “Variable Fresnel zone plate,” Appl. Opt. 6, 1567–1570 (1967).
    [CrossRef] [PubMed]
  12. M. Kajanto, E. Byckling, J. Fagerholm, J. Heikonen, J. Turunen, A. Vasara, A. Salin, “Photolithographic fabrication method of computer-generated holographic interferograms,” Appl. Opt. 28, 778–784 (1989).
    [CrossRef] [PubMed]
  13. E. Barnard, “Optimal error diffusion for computer-generated holograms,” J. Opt. Soc. Am. A 5, 1803–1817 (1988).
    [CrossRef]
  14. C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
    [CrossRef]
  15. C. Paterson, “Computer-generated diffractive optics with spiral phase dislocations,” Ph.D. dissertation (University of London, London, 1996).

1996

M. S. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

1994

C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

1989

1988

1987

1985

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

1980

1974

1967

1965

Arnold, S. M.

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Barnard, E.

Bryngdahl, O.

Bucklew, J. A.

Byckling, E.

Clark, M. S.

M. S. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

Eschbach, R.

Fagerholm, J.

Fainman, Y.

Farhoosh, H.

Feldman, M. R.

Gallagher, N. C.

N. C. Gallagher, J. A. Bucklew, “Nondetour phase digital holograms: an analysis: errata,” Appl. Opt. 19, 4266–4272 (1980).
[CrossRef] [PubMed]

R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 11–23 (1990).
[CrossRef]

Guest, C. C.

Hawley, R. W.

R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 11–23 (1990).
[CrossRef]

Heikonen, J.

Kajanto, M.

Kelly, D. L.

Kozma, A.

Lee, S. H.

Lee, W.-H.

W.-H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
[CrossRef] [PubMed]

W.-H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North Holland, Amsterdam, 1978), pp. 120–232.

Lohmann, A. W.

McCord, M. A.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in Handbook of Microlithography, Micromachining and Microfabrication, P. Rai-Choudhury, ed. (SPIE Press, Bellingham, Wash., 1997), pp. 139–249.

Paris, D. P.

Paterson, C.

C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

C. Paterson, “Computer-generated diffractive optics with spiral phase dislocations,” Ph.D. dissertation (University of London, London, 1996).

Rooks, M. J.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in Handbook of Microlithography, Micromachining and Microfabrication, P. Rai-Choudhury, ed. (SPIE Press, Bellingham, Wash., 1997), pp. 139–249.

Salin, A.

Smith, R.

M. S. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

Turunen, J.

Vasara, A.

Wyrowski, F.

Appl. Opt.

J. Mod. Opt.

C. Paterson, “Diffractive optical elements with spiral phase dislocations,” J. Mod. Opt. 41, 757–765 (1994).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. S. Clark, R. Smith, “A direct-search method for the computer design of holograms,” Opt. Commun. 124, 150–164 (1996).
[CrossRef]

Opt. Eng.

S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 (1985).
[CrossRef]

Other

R. W. Hawley, N. C. Gallagher, “An efficient electron beam pattern data format for the production of binary computer generated holograms,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 11–23 (1990).
[CrossRef]

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in Handbook of Microlithography, Micromachining and Microfabrication, P. Rai-Choudhury, ed. (SPIE Press, Bellingham, Wash., 1997), pp. 139–249.

W.-H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North Holland, Amsterdam, 1978), pp. 120–232.

C. Paterson, “Computer-generated diffractive optics with spiral phase dislocations,” Ph.D. dissertation (University of London, London, 1996).

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Figures (2)

Fig. 1
Fig. 1

Drop in the first-order diffraction efficiency η0 - η plotted versus (Δx/ d)2 for bitmap-method binary amplitude gratings. The solid curve represents the theoretical prediction of expression (33); the crosses represent the numerical simulation of periodic gratings (N = 128).

Fig. 2
Fig. 2

Fringe-edge positional error for the bitmap method.

Equations (37)

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τx, y=1if cosϕx, y>cossin-1 γx, y0otherwise,
τx, y=m=-sinmπqx, ymπexpimϕx, y, πqx, y=sin-1 γx, y.
A=a expiϕ0=a0+δaexpiϕ0,
δa=a-a0=sin mπq0+δq-sin mπq0/mπ,
a=a0+δa¯+δa-δa¯,
η=η0a0+δa¯/a02=η01+2δa¯/a0+δa¯/a02=η0sin2 mπq0+δq¯sin2 mπq02,
IδaI01=δa-δa¯2¯a012=δa2¯-δa¯2a012=sin2 mπq0+δq¯-sin mπq0+δq¯2m2π2 sin2 πq0,
η=η0cosπδq¯2η01-π2δq2¯=η01-π2δw/d2¯,
Iδa=I01sin2 mπδq¯-sin mπδq¯2m2π2I01δq2¯-δq¯2=I01δw/d2¯-δw/d¯2,
Iδa=I01cos2 mπδq¯-cos mπδq¯2m2π2I01m2π24δw/d4¯-δw/d2¯2,
A=A0 expiδϕA01+iδϕ-12 δϕ2.
AA01+iδϕ¯-12δϕ2¯+A0iδϕ-δϕ¯-12δϕ2-δϕ2¯.
η=η01-δϕ2¯-δϕ¯2=η01-2πmδr/d2¯+2πmδr/d¯2.
IδϕI01δϕ2¯-δϕ¯2a02a012=2πδr/d2¯-2πδr/d¯2sin2 mπq0,
Iδϕ4π2I01δr/d2¯-δr/d¯2odd m0even m.
ηη01-π2δw/d2¯-4π2δr/d2¯+4π2δr/d¯2.
Iδa+Iδϕ4π2I01δr/d2¯-δr/d¯2odd mI01δw/d2¯-δw/d¯2even m.
δϕ-Δϕ+2Δϕ2x/l2=22x/l2-1Δϕ,
δϕ¯¯ 01 Δϕ2x2-1dx=-¯Δϕ/3,
δϕ2¯01 Δϕ22x2-12dx=7Δϕ2/15.
ηη01-π2δw/d2¯-7/15-¯2/9Δϕ2.
Iδϕ/I0715 Δϕ2-19 ¯2Δϕ2,
δϕ22x/l2-1Δϕ+ξ.
δϕ¯ 01Δϕ2x2-1+ξdx=0  ξ=Δϕ/3.
δϕ2¯01 Δϕ22x2-2/32dx=56Δϕ2/45,
δz¯=0,  δz2¯=δzx2¯+δzy2¯,
δzx2¯=1Δx sin θ-1/2|Δx sin θ|1/2|Δx sin θ|δzx2dδzx=Δx2 sin2 θ12,
δz2¯=112Δx2 sin2 θ+Δy2 cos2 θ.
δw2¯=δz1+δz22¯=2δz2¯=16Δx2 sin2 θ+Δy2 cos2 θ,
δr2¯=12δz1-δz22¯=12δz2¯=124Δx2 sin2 θ+Δy2 cos2 θ.
ηη01-π2δw/d2¯-4π2δr/d2¯η01-π23Δx2 sin2 θ+Δy2 cos2 θd2.
Iδϕδϕ2¯I0=2πd2124Δx2 sin2 θ+Δy2 cos2 θI0.
ηη01-13πΔxd2.
SNR241Δx2d2π20.6dΔx2.
Nbits=1Δx1Δy=|ϕ/x|maxΔϕ|ϕ/y|maxΔϕ,
Nrects=12|ϕ/x|max2π|ϕ/y|maxΔϕ,
Nsegs=ϕ/e2π142ϕ/e|2Δϕ1/2,

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