Abstract

We describe the design of Fourier-type phase-only array generators. The numerical optimization employs the Fienup algorithm, where the parageometric design of the phase retardation profile, with the form of a lenslet array, is used as the initial guess of the optimization process. This approach provides designs with high performance that can be obtained with comparatively low computing effort. This is particularly true for elements generating large spot arrays. For symmetric reconstruction fields, the optimized phase profile typically has the same symmetry as that for the reconstruction field and can be easily unwrapped.

© 2000 Optical Society of America

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References

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  1. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  2. G.-Y. Yoon, S. Matsuoka, T. Jitsuno, M. Nakatsuka, Y. Kato, “Wave-front design algorithm for shaping a quasi-far-field pattern,” Appl. Opt. 37, 1386–1392 (1998).
    [CrossRef]
  3. N. Streibl, U. Nölscher, J. Jahns, S. Walker, “Array generation with lenslet arrays,” Appl. Opt. 30, 2739–2742 (1991).
    [CrossRef] [PubMed]
  4. V. V. Wong, G. J. Swanson, “Design and fabrication of a Gaussian fan-out optical interconnect,” Appl. Opt. 32, 2502–2511 (1993).
    [CrossRef] [PubMed]
  5. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.
  6. F. Wyrowski, “Modulation schemes of phase gratings,” Opt. Eng. 31, 251–257 (1992).
    [CrossRef]
  7. V. Arrizón, M. Testorf, “Efficiency limit of spatially quantized Fourier array illuminators,” Opt. Lett. 22, 197–199 (1997).
    [CrossRef] [PubMed]
  8. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  9. F. Wyrowski, “Design of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  10. M. Haruna, M. Takahashi, K. Wakahayashi, H. Nishihara, “Laser beam lithographed micro-Fresnel lenses,” Appl. Opt. 29, 5120–5126 (1990).
    [CrossRef] [PubMed]
  11. M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
    [CrossRef]

1998 (1)

1997 (1)

1995 (1)

M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
[CrossRef]

1993 (2)

1992 (1)

F. Wyrowski, “Modulation schemes of phase gratings,” Opt. Eng. 31, 251–257 (1992).
[CrossRef]

1991 (1)

1990 (2)

1980 (1)

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

Arrizón, V.

Ehbets, P.

M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
[CrossRef]

Fienup, J. R.

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

Haruna, M.

Herzig, H. P.

M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
[CrossRef]

Jahns, J.

Jitsuno, T.

Kato, Y.

Kuittinen, M.

M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
[CrossRef]

Matsuoka, S.

Nakatsuka, M.

Nishihara, H.

Nölscher, U.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.

Streibl, N.

Swanson, G. J.

Takahashi, M.

Testorf, M.

Wakahayashi, K.

Walker, S.

Wong, V. V.

Wyrowski, F.

Yoon, G.-Y.

Appl. Opt. (4)

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

Opt. Commun. (1)

M. Kuittinen, H. P. Herzig, P. Ehbets, “Improvements in diffraction efficiency of gratings and microlenses with continuous relief structures,” Opt. Commun. 120, 230–234 (1995).
[CrossRef]

Opt. Eng. (2)

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

F. Wyrowski, “Modulation schemes of phase gratings,” Opt. Eng. 31, 251–257 (1992).
[CrossRef]

Opt. Lett. (1)

Other (1)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.

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Figures (12)

Fig. 1
Fig. 1

Structure of one period of a spatially quantized phase DOE with M×M pixels.

Fig. 2
Fig. 2

One period (with 128 pixels) of (a) the parageometric and (b) the optimized phase reliefs of a Fourier array generator of 1×17 spots.

Fig. 3
Fig. 3

Local efficiencies of the spot arrays respectively generated by (a) the parageometric and (b) the optimized DOE’s in Fig. 2.

Fig. 4
Fig. 4

Gray-scale plots for single periods of (a) the parageometric and (b) the optimized phase reliefs of a (17×17)-spot array generator.

Fig. 5
Fig. 5

(a) Signal uniformity error and (b) reconstruction efficiency versus the number of iterations during the Fienup optimization of the (17×17)-spot array generator. The plots correspond to the parageometric (solid curves), random (dotted curves), and uniform (dashed curves) initial phase reliefs.

Fig. 6
Fig. 6

Number of iterations, versus the number of signal spots, required to obtain a uniformity error of 1% during the Fienup optimization of 1-D FAI’s. The results correspond to the parageometric (solid bars), random (open bars), and uniform (striped bars) initial phase reliefs. The subplots correspond to (a) small and (b) large numbers of signal spots.

Fig. 7
Fig. 7

Reconstruction efficiencies of the 1-D FAI’s considered in Fig. 6. The types of bars [and the subplots (a) and (b)] are defined as in Fig. 6.

Fig. 8
Fig. 8

Unwrapped versions of the phase reliefs in Figs. (a) 2(b) and (b) 4(b).

Fig. 9
Fig. 9

Joint between two adjacent periods for the high-resolution version of the DOE in Fig. 2(b). A complete period of the high-resolution DOE has 16,384 pixels. The joint in the figure shows only 1665 pixels.

Fig. 10
Fig. 10

Nonrectangular array of 3862 spots as a desired reconstruction field of a Fourier-type DOE.

Fig. 11
Fig. 11

One period of the DOE (optimized with the parageometric initial DOE) that generates the spot array in Fig. 10.

Fig. 12
Fig. 12

Flow chart of the Fienup algorithm.

Equations (20)

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t0(x)=exp[iϕ(x)]rect[x/d],
u(x)=12πddxϕ(x),
dudx=12πd2ϕdx2=const.
ϕ(x)=-πλfx2.
Δu=N/d,
|umax|=N/2d.
|umax|=d2λf.
ϕ(x)=-Nπ(x/d)2.
t0(x, y)=exp[iϕ(x, y)]rect(x/d)rect(y/d),
ϕ(x, y)=-Nπ(x2+y2)/d2.
ηql=E2(q, l)|W(q, l)|2,
E(q, l)=sinc(q/M)sinc(l/M),
W(q, l)=1M2n,m=1M×exp(iϕnm)exp[-i2π(nq+ml)/M].
ΩW={(q, l); q, l:-M/2+1,,M/2}.
ηL=Q(q,l)ΩsE-2(q, l)-1.
|WI(q, l)|2=ηLE-2(q, l)/Qif(q, l)Ωs0otherwise.
ϕnm=-Nπ(n2+m2)/M2,
ϕnm=-π(2nMxqc+Nxn2)/Mx2
-π(2mMylc+Nym2)/My2,
W(q, l)=K|WI(q, l)|exp[i arg(W(q, l))]if(q, l)ΩsAW(q, l)if(q, l)Ωs,

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