Abstract

A comprehensive two-step approach to design staircase-type multilevel diffractive phase elements (DPEs) that generate arbitrary desired diffraction patterns with the highest possible accuracy is presented. First a preliminary periodic grating with an unconstrained phase delay and an optimized nonuniform amplitude profile is designed by means of a customized iterative Fourier-transform algorithm. Then this preliminary grating is subjected to a phase quantization in which strict periodicity is forgone in favour of the best possible preservation of the shape of the power spectrum yielding a final phase only DPE with only rudimentary periodicity. An arbitrarily high similarity between the diffraction patterns of the final DPE and the preliminary grating can be achieved independently of the number D of discrete phase delay levels as long as D ≥ 3. The signal-to-noise ratio of the final DPE is close to the theoretical upper limit. These properties are confirmed in computer simulations and demonstrated in optical experiments. Pseudoperiodic DPEs may have applications in optical computing, optical communication and networking, optical authentication, or coherent laser coupling.

© 2002 Optical Society of America

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References

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  2. S. Sinzinger, J. Jahns, Microoptics (Wiley VCH, Weinheim, Germany, 1999).
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  6. J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
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  7. U. Krackhardt, “Optimum quantization rules for computer generated holograms,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Seriesm139–142 (1994)
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2001 (2)

1999 (2)

1997 (1)

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generators,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

1995 (2)

1994 (1)

1992 (1)

1990 (1)

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. 5, 1058–1065 (1988).
[CrossRef]

1982 (2)

1972 (1)

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

1970 (1)

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Arrizón, V.

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generators,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Brenner, K.-H.

Bryngdahl, O.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Error-diffusion algorithm in phase synthesis and retrieval techniques,” Opt. Lett. 17, 235–237 (1992).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. 5, 1058–1065 (1988).
[CrossRef]

Cohen, N.

Fienup, J. R.

Gaylord, T. K.

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1996).

Grann, E. B.

Gruber, M.

Jahns, J.

S. Sinzinger, J. Jahns, Microoptics (Wiley VCH, Weinheim, Germany, 1999).

Knoesen, A.

Krackhardt, U.

U. Krackhardt, “Optimum quantization rules for computer generated holograms,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Seriesm139–142 (1994)

Levy, U.

Mait, J. N.

Mendlovic, D.

Moharam, M. G.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 1984).

Pommet, D. A.

Saxton, O. W.

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

Sidick, E.

Silvestri, A. M.

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Singer, W.

Sinzinger, S.

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generators,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

S. Sinzinger, J. Jahns, Microoptics (Wiley VCH, Weinheim, Germany, 1999).

Stern, M. B.

M. B. Stern, “Binary optics fabrication,” pp. 53–85 in Micro-Optics: Elements, Systems and ApplicationsH.-P. Herzig, ed. (Taylor & Francis, London, 1997).

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Testorf, M.

Weissbach, S.

Wyrowski, F.

Appl. Opt. (4)

IBM J. Res. Dev. (1)

J. W. Goodman, A. M. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. Mod. Opt. (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. 5, 1058–1065 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

V. Arrizón, S. Sinzinger, “Modified quantization schemes for Fourier-type array generators,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Opt. Lett. (2)

Optik (1)

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

Other (7)

M. B. Stern, “Binary optics fabrication,” pp. 53–85 in Micro-Optics: Elements, Systems and ApplicationsH.-P. Herzig, ed. (Taylor & Francis, London, 1997).

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1996).

A. Papoulis, Probability, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 1984).

U. Krackhardt, “Optimum quantization rules for computer generated holograms,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Seriesm139–142 (1994)

H.-P. Herzig ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

S. Sinzinger, J. Jahns, Microoptics (Wiley VCH, Weinheim, Germany, 1999).

J. Turunen, F. Wyrowski eds. Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, Germany, 1997).

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Figures (23)

Fig. 1
Fig. 1

Stepwise constant phase-delay profile as assumed for the mathematical model of diffraction.

Fig. 2
Fig. 2

Illustration of a phase quantization according to rule (18) in the complex plane. Phasor E φ is projected onto the nearest of D phasors E d that represent the allowed phase delay values.

Fig. 3
Fig. 3

a) Projection of phasor S pm (φ) onto S pm (ψ1) as a result of a phase quantization according to rule (18); b) N + 1 different possible linear combinations of phasors E 1 and E 2 if the periodicity demand is dropped; c) Virtual increase of the phase resolution and smaller error ΔS as result of a pseudoperiodic phase quantization according to rule (19).

Fig. 4
Fig. 4

Graphical representation of phasors Δs n in the complex plane. The N phasors are isotropically distributed with an angular separation of ρ = 2πq/ N.

Fig. 5
Fig. 5

Gray-level coded phase-delay profiles of finite (4×4 periods) two-dimensional gratings. The phase resolutions D = 4, 8, and 16 were obtained through pseudoperiodic quantization of the grating with D = ∞ according to rule (19). Note the increasing disruption of the strict periodicity with decreasing D.

Fig. 6
Fig. 6

Uniformity error σ J of the array generated by quantized FAI gratings versus N, the number of periods involved in the pseudoperiodic quantization scheme. The four curves represent quantizations onto D = 4, 6, 8, and 16 discrete levels starting from a (conventional) pre-DE with unit amplitude.

Fig. 7
Fig. 7

D sides of a polygon (here an equilateral triangle) as the set of locations that can be reached by phasor S pm (ψ) with the pseudoperiodic quantization scheme. ΔS is the residual error when the optimal approximation for S pm (φ) is chosen.

Fig. 8
Fig. 8

Block diagram of the IFTA.

Fig. 9
Fig. 9

Illustration of an adaptation to two different object-domain constraints in the complex plane. Left-hand side: Radial projection of phasor m onto the unit circle according to rule (26). Right-hand side: Radial projection of phasor * m onto the inscribed polygon.

Fig. 10
Fig. 10

Illustration of the object-domain adaptation rules (31) and (32) in the complex plane.

Fig. 11
Fig. 11

Illustration of the directions along which phasors m are projected in the complex plane when adaptation rules (31) and (32) are applied. The corners of the equilateral triangle represent the D = 3 discrete levels onto which all phase values φ nm are mapped in the quantization step.

Fig. 12
Fig. 12

Phase-delay profile 〈φ m 〉 and corresponding amplitude profile 〈a m 〉 of a 1 × 9 FAI obtained by use of IFTA and the object-domain adaptation rule (26).

Fig. 13
Fig. 13

Phase-delay profile (φ m ) and corresponding amplitude profile (a m ) of a 1 × 9 FAI obtained by use of IFTA and the object-domain adaptation rules (31) and (32) for four equidistant phase-delay levels.

Fig. 14
Fig. 14

Difference of the phase-delay profiles of Figs. 12 and 13.

Fig. 15
Fig. 15

Uniformity error σ J of the array generated by quantized FAI gratings, versus N. The five curves represent quantizations onto D = 3, 4, 6, 8, and 16 discrete levels starting from a pre-DE with matched polygonal amplitude distribution. A comparison with Fig. 6 clearly confirms the improved performance.

Fig. 16
Fig. 16

Efficiency η of periodic (open circles) and pseudoperiodic (filled circles) quantized 1 × 9 FAI gratings according to definition (34) versus D. The solid line indicates the function sinc2(1/D).

Fig. 17
Fig. 17

Efficiency η* of periodic (open circles) and pseudoperiodic (filled circles) quantized 1 × 9 FAI gratings according to definition (37) versus D. The solid line indicates the function sinc2(1/D).

Fig. 18
Fig. 18

Signal-to-noise ratio SNR versus N. The four curves represent quantized 1 × 9 FAI gratings with D = 3, 4, 6, and 8 equidistant phase-delay levels.

Fig. 19
Fig. 19

Gray-level image of the optical diffraction pattern of a periodic 1 × 9 FAI grating with four equidistant phase-delay levels derived from the pre-DE of Fig. 12 with unit amplitudes and the corresponding intensity plot (obtained through vertical integration).

Fig. 20
Fig. 20

Gray-level image of the optical diffraction pattern of a pseudoperiodic 1 × 9 FAI grating with four equidistant phase-delay levels derived from the pre-DE of Fig. 12 with unit amplitudes, and the corresponding intensity plot.

Fig. 21
Fig. 21

Gray-level image of the optical diffraction pattern of a pseudoperiodic 1 × 9 FAI grating with four equidistant phase-delay levels derived from the pre-DE of Fig. 13 with matched polygonal amplitude distribution and the corresponding intensity plot.

Fig. 22
Fig. 22

Gray-level image of the optical diffraction pattern of a (conventional) periodic quantized 9 × 9 FAI grating with four equidistant phase-delay levels derived from a pre-DE with unit amplitudes.

Fig. 23
Fig. 23

Gray-level image of the optical diffraction pattern of a pseudoperiodic quantized 9 × 9 FAI grating with four equidistant phase-delay levels derived from a pre-DE with matched polygonal amplitude distribution.

Equations (37)

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gx=txfx=pxwxfx,
px=rectNMx,
wx=n=0N-1m=1Mexpiφnmδx-nM+mNM.
φnm=φ1mn.
Gu=PuWu  Fu,
Pu=1NMsincuNM,
Wu=n=0N-1m=1Mexpiφnmexpi2π unM+mNM.
wk=wx=k/NM, Wk=Wu=k,
kK, K=1, 2, 3,  , NM,
Wk=m=1Mexpi2π kmNMSkm,
Skm=n=0N-1expiφnmexpi2π knN.
pKN, KN=N, 2N, 3N,  , MN,
qK¯N, K¯N=K\KN.
Spm=n=0N-1expiφnm.
Spm=N expiφ1m,
Sqm=0.
φψd d=1,  , D,
|φ-ψd|  min.
|Spmφ-Spmψ|  min
Sqmψ=n1 snψ1+n2 snψ2,
snψd=expiψdexpi2π qnN.
0=n2 snψ1-n2 snψ1
Sqmψ=N sn0ψ10+n2 snψ2-snψ1=defn2 Δsn,
ΔSSpmφ=1- cosπD,
W˜m=αmTm expiϕm+1-αmWm,
wm=expiφ˜m,
|w˜m-wm|  min.
n=cosψ1+ψ22sinψ1+ψ22,
w˜2=n · w˜kn,
w˜1=w˜m-w˜2.
w2=n cosψ2-ψ12,
w1=w˜1|w˜1|  w1maxifw˜1w1max|w˜1||w˜1|  w1max,
w1max=sinψ2-ψ12.
η=J|WmPm|2M|WmPm|2.
Iin=Jθ.
Imin1η*=θ.
η*=Iminθ-JIminIin.

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