Abstract

The theoretical diffraction efficiency upper limit of diffractive phase elements (DPE’s) with finite apertures is investigated. A successful numerical method of evaluating the efficiency upper bound of DPE’s is proposed. The method includes a hybrid optimization procedure that combines a genetic algorithm with the conjugate gradient method. This efficient global optimization technique can also be used to design DPE’s. Simulation computations are detailed for rotationally symmetric beam shaping in which a Gaussian profile laser beam is converted into a uniform beam. Numerical results demonstrate that the estimated diffraction efficiency upper bound is consistent with the design results.

© 2000 Optical Society of America

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References

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  1. F. Wyrowski, Opt. Lett. 16, 1915 (1991).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. H. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995), p. 69.
  6. H. H. Hopkins, Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
    [CrossRef]

1999 (1)

1992 (1)

1991 (1)

1957 (1)

H. H. Hopkins, Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
[CrossRef]

Appl. Opt. (2)

Opt. Lett. (1)

Proc. Phys. Soc. London Sect. B (1)

H. H. Hopkins, Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
[CrossRef]

Other (2)

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991), p. 321.

H. Schwefel, Evolution and Optimum Seeking (Wiley, New York, 1995), p. 69.

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Figures (2)

Fig. 1
Fig. 1

Relationship between the minimum uniformity error of the design and the desired diffraction efficiency.

Fig. 2
Fig. 2

Optimal design result when ηd=0.95, with the input Gaussian beam at z=0 and the output uniform beam at z=zs. Inset, phase profile of the optimal DPE.

Equations (22)

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Uix,y,zs=αUsx,y,zs+Unx,y,zs,
αUsx,y,0+Unx,y,0=Uix,y,0,
AUix,y,02dxdy=α2WUsx,y,zs2dxdy+-Unx,y,zs2dxdy,
-Unx,y,zs2dxdy=-Unx,y,02dxdy.
-Unx,y,02dxdyAUnx,y,02dxdyAUix,y,0-αUsx,y,02dxdy.
α2AUi(x,y,0Usx,y,0dxdyWUsx,y,zs2dxdy+AUsx,y,02dxdy.
η=α2WUsx,y,zs2dxdyAUix,y,02dxdy.
η4ζ1+ζ2×AUix,y,0Usx,y,0dxdy2AUix,y,02dxdyAUsx,y,02dxdy=ηu,
ζ=AUsx,y,02dxdyWUsx,y,zs2dxdy,
Upq=Usxp,yq,0=m=1Mn=1NGpqmn expiϕmn,
Ipq=Upq2, p=1,2,,P, q=1,2,,Q,
Gpqmn=SmnGxp,yq,x,y;zsUsx,y,zsdxdy*.
ηu=maxΦFΦ,
FΦ=4p=1Pq=1QIpqIpqiSpq2m=1Mn=1NImnsSmnp=1Pq=1QIpqSpq+m=1Mn=1NImnsSmn2p=1Pq=1QIpqiSpq,
Fϕmn=j=1Pk=1QFIjkIjkϕmn,
Ijkϕmn=ϕmnUjkUjk*=Ujk*Ujkϕmn+UjkUjk*ϕmn,
Ijk/ϕmn=2 ImGjkmn*Ujk exp-iϕmn.
Fϕmn=2 Imexp-iϕmnj=1Pk=1QGjkmn*UjkFIjk.
cost=CRη+σ,
Rη=0ηηdηd-η2η<ηd,
η=p=1PIpSpm=1MImSm,
σ=1Pp=1PIˆ-Ip2/Iˆ21/2,

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