Abstract

Quantization of the phase-delay profile of a diffractive optical element often leads to unwanted deviations of considerable extent in the diffraction pattern. A method for avoiding this flaw for periodic phase gratings is described. The idea is to forgo strict periodicity and use the new degrees of freedom thus obtained to compensate for quantization-related deviations. The method is demonstrated for Fourier-array illuminator gratings designed with an iterative Fourier-transform algorithm.

© 2001 Optical Society of America

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References

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  1. H. P. Herzig, ed., Micro-Optics (Taylor & Francis, London, 1997).
  2. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH, Weinheim, Germany, 1999).
  3. J. Turunen and F. Wyrowski, Diffractive Optics (Akademie, Berlin, 1997).
  4. J. N. Mait, J. Opt. Soc. Am. A 12, 2145 (1995).
    [CrossRef]
  5. U. Krackhardt, in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 139–142.
  6. V. Arrizón and S. Sinzinger, Opt. Commun. 140, 309 (1997).
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  8. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Auckland, New Zealand, 1987).

1997 (1)

V. Arrizón and S. Sinzinger, Opt. Commun. 140, 309 (1997).

1995 (1)

Arrizón, V.

V. Arrizón and S. Sinzinger, Opt. Commun. 140, 309 (1997).

Goodman, J. W.

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

Jahns, J.

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

Krackhardt, U.

U. Krackhardt, in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 139–142.

Mait, J. N.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Auckland, New Zealand, 1987).

Sinzinger, S.

V. Arrizón and S. Sinzinger, Opt. Commun. 140, 309 (1997).

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

Turunen, J.

J. Turunen and F. Wyrowski, Diffractive Optics (Akademie, Berlin, 1997).

Wyrowski, F.

J. Turunen and F. Wyrowski, Diffractive Optics (Akademie, Berlin, 1997).

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

V. Arrizón and S. Sinzinger, Opt. Commun. 140, 309 (1997).

Other (6)

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

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, Auckland, New Zealand, 1987).

U. Krackhardt, in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 139–142.

H. P. Herzig, ed., Micro-Optics (Taylor & Francis, London, 1997).

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

J. Turunen and F. Wyrowski, Diffractive Optics (Akademie, Berlin, 1997).

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Figures (5)

Fig. 1
Fig. 1

Stepwise constant phase-delay profile of a pixelated DPE.

Fig. 2
Fig. 2

Graphic representation of phase quantization in a complex plane.

Fig. 3
Fig. 3

Phase-delay profile and (discrete) power spectrum of one period of a 1×9 FAI generated through an iterative Fourier-transform algorithm.

Fig. 4
Fig. 4

Diffraction patterns of finite 1×9 FAI gratings. Gu2 is normalized with respect to an idealized FAI (perfect uniformity and η=100% diffraction efficiency).

Fig. 5
Fig. 5

Closeups of the regions of interest depicted as small rectangles in Fig.  4.

Equations (12)

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gx=txsx=pxwxsx,
px=rectNMx,
wx=n=0N-1m=1Mexpiϕnmδx-nM+mNM.
Gu=PuWuSu,
Pu=1NMsincuNM,
Wu=n=0N-1m=1Mexpiϕnmexpi2πunM+mNM.
Wk=m=1Mexpi2πkmNMSkm,
Skm=n=0N-1expiϕnmexpi2πknN.
Skm=Nexpiϕ1mk/N an integer0otherwise.
Sκm=n=0N-1expiϕnm,κ=N,2N,,MN.
ϕ-ψqmin.
Sκmϕ-Sκmψmin.

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