Abstract

Diffractive elements that modulate both the amplitude and the phase of an optical field are designed with use of an iterative Fourier-transform algorithm with gradual amplitude clipping at the plane of the element. Unlike phase-only diffractive elements, these complex-amplitude modulating elements do not generate noise outside the signal window W, and their diffraction efficiencies are comparable to the efficiencies of phase-only elements with a noiseless frame around W. Encoding techniques of the complex amplitude into variations of the diffraction efficiency and phase of the zeroth order of a carrier grating are introduced: the shape and the depth of a substructure within each rectangular pixel are modulated. Approximate and rigorous diffraction calculations are performed to compare different substructuring schemes, which all deflect the noise far away from W but differ, for example, in the manner in which this noise is distributed in the signal plane.

© 1997 Optical Society of America

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References

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    [CrossRef]

1996

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

E. Noponen, J. Turunen, “Complex-amplitude modulation by high-carrier-frequency diffractive elements,” J. Opt. Soc. Am. A 13, 1422–1428 (1996).
[CrossRef]

1994

1992

1991

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1973

1971

1970

D. Kermisch, “Image reconstruction from phase information only,” J. Opt. Soc. Am. A 60, 15–17 (1970).
[CrossRef]

1967

Bryngdahl, O.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Chu, D. C.

Fienup, J. R.

Gale, M. T.

Goodman, J. W.

Honkanen, M.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Jones, A. L.

Kermisch, D.

D. Kermisch, “Image reconstruction from phase information only,” J. Opt. Soc. Am. A 60, 15–17 (1970).
[CrossRef]

Kirk, J. P.

Krackhardt, U.

Lang, G. K.

Lohmann, A. W.

Lüpken, H.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Mait, J. N.

Noponen, E.

Paris, D. B.

Peter, T.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Prongue, D.

Raynor, J. M.

Salminen, O.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Schütz, H.

Streibl, N.

Turunen, J.

E. Noponen, J. Turunen, “Complex-amplitude modulation by high-carrier-frequency diffractive elements,” J. Opt. Soc. Am. A 13, 1422–1428 (1996).
[CrossRef]

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics: Research, Development and Applications, A. Consortini, ed. (Academic, London, 1996), pp. 111–123.

Vahimaa, P.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Wyrowski, F.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics: Research, Development and Applications, A. Consortini, ed. (Academic, London, 1996), pp. 111–123.

Appl. Opt.

J. Mod. Opt.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Opt. Lett.

Rep. Prog. Phys.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other

J. Turunen, F. Wyrowski, “Diffractive optics: from promise to fruition,” in Trends in Optics: Research, Development and Applications, A. Consortini, ed. (Academic, London, 1996), pp. 111–123.

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Figures (8)

Fig. 1
Fig. 1

(a) Illustration of amplitude clipping in complex-amplitude IFTA, (b) evolution of the diffraction efficiency η as the design proceeds and the clipping factor is increased.

Fig. 2
Fig. 2

Efficiencies of framed phase-only diffractive array illuminators with (a) N=3 and (b) N=4 as a function of the number of diffraction orders in the frame.

Fig. 3
Fig. 3

Geometries of two-dimensional complex-amplitude coding schemes: (a) stripe geometry, (b) subpixel encoding method with 4×4 subpixels in each cell.

Fig. 4
Fig. 4

Phase-amplitude combinations available with 4×4 pixel substructure and 16 depth levels according to scalar amplitude-transmittance approach.

Fig. 5
Fig. 5

Phase-amplitude combinations available with 1D or stripe-geometry encoding, according to rigorous electromagnetic approach, for dc=4λ.

Fig. 6
Fig. 6

Phase-amplitude combinations available with subpixel encoding, according to rigorous electromagnetic approach, for dc=4λ.

Fig. 7
Fig. 7

Structure of a 3×3 beam array illuminator with complex-amplitude modulation: The gray levels indicate profile depth.

Fig. 8
Fig. 8

Scalar analysis of the uniformity error inside W and noise in a frame of nine-order-wide frame around W for a 3×3 beam signal. Solid curves, no depth quantization; dashed curves, 64 quantization levels; dotted curves, 32 levels; dashed–dotted curves, 16 levels.

Tables (3)

Tables Icon

Table 1 Efficiencies ηp of Phase-Only Elements with N Equal Orders and of Complex-Amplitude Modulating Elements Designed by Analytic Optimization (η1) by Finding the Upper Bound (η2), by Use of Phase-Only Designs (η3), and by the Amplitude-Clipping Technique (η4)

Tables Icon

Table 2 Phases of Diffraction Orders of Some Complex-Amplitude Modulating N-Beam Array Illuminators

Tables Icon

Table 3 Efficiencies of Separable and Nonseparable Complex-Amplitude Modulating Elements with N×N Equal Orders

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

U(x, y, 0)=m,n=-Tmn exp[i(umx+vny)],
Tmn=1dxdy0dx0dyU(x, y, 0)exp[-i(umx+vny)]dxdy
Tˆmn=ηˆmn1/2 exp(iϕmn).
η=(m,n)W |Tmn|2.
(m,n)W ηˆmn=1.
Us(x, y, 0)=(m,n)W ηˆmn1/2 exp[i(umx+vny+ϕmn)].
1dxdy0dx0dyIs(x, y, 0)dxdy=1,
ηb=1dxdy0dx0dy|Us(x, y, 0)|dxdy2.
η=Imax-1,
Imax=max Is(x, y, 0),
T=1-f+f exp[-ik(n-1)D],
T=A exp(iΦ),
f=1-1-A22(1-A cos Φ),
exp[ik(n-1)D]=Afexp(iΦ)+1-1f.

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