Abstract

We derive an upper limit for the efficiency of spatially quantized Fourier array illuminators. This complements a previously reported formula for the efficiency limit of diffractive elements constrained only in phase.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman and A. M. Silvestri, IBM J. Res. Devel. 14, 478 (1970).
    [CrossRef]
  2. F. Wyrowski, Opt. Lett. 16, 1915 (1991).
    [CrossRef] [PubMed]
  3. F. Wyrowski, Opt. Eng. 31, 251 (1992).
    [CrossRef]
  4. U. Krackhardt, J. N. Mait, and N. Streibel, Appl. Opt. 31, 27 (1992).
    [CrossRef] [PubMed]
  5. J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–29.

1992 (2)

1991 (1)

1970 (1)

J. W. Goodman and A. M. Silvestri, IBM J. Res. Devel. 14, 478 (1970).
[CrossRef]

Goodman, J. W.

J. W. Goodman and A. M. Silvestri, IBM J. Res. Devel. 14, 478 (1970).
[CrossRef]

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–29.

Krackhardt, U.

Mait, J. N.

Silvestri, A. M.

J. W. Goodman and A. M. Silvestri, IBM J. Res. Devel. 14, 478 (1970).
[CrossRef]

Streibel, N.

Wyrowski, F.

Appl. Opt. (1)

IBM J. Res. Devel. (1)

J. W. Goodman and A. M. Silvestri, IBM J. Res. Devel. 14, 478 (1970).
[CrossRef]

Opt. Eng. (1)

F. Wyrowski, Opt. Eng. 31, 251 (1992).
[CrossRef]

Opt. Lett. (1)

Other (1)

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 4–29.

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Figures (3)

Fig. 1
Fig. 1

Schematic of a discrete periodic phase element for array generation.

Fig. 2
Fig. 2

Efficiency versus M derived from Eq.  (17) for a symmetrical array of uniform spots. Solid curve, N=9 (one-dimensional array); dashed curve, N=9×9 (two-dimensional array).

Fig. 3
Fig. 3

Example of a one-dimensional array generator optimized to generate nine spots. (a) Phase distribution of the element. (b) The envelope sinc2 modulates the square modulus of (c) the weighting function to form (d) the final output intensity distribution.

Equations (17)

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ηZ=sinc2(1/Z),
ηr=|g(x)|2[sinc(δxx)]2F.
e(x,y)=rect(x/a)rect(y/a),
s(x,y)=rect(x/P)rect(y/P).
t(x,y)=[c(x,y)comb(x,y)]s(x,y),
c(x,y)=e(x,y)n,m=1Mexp(iϕnm)δ(x-na)δ(y-ma).
T(u,v)=a2q,l=-E(q,l)W(q,l)S(u-q,v-l),
E(u,v)=sinc(u/M)sinc(v/M),
W(u,v)=n,m=1Mexp(iϕnm)exp[i2π(nu+mv)/M],
S(u,v)=P2sinc(Pu)sinc(Pv).
ql=a4E2(q,l)|W(q,l)|2,
ηs=(q,l)Ωsql=a4(q,l)ΩsE2(q,l)|W(q,l)|2.
ΩW={(q,l)q,l:-M/2+1,M/2}.
(q,l)Ωsql=1.
(q,l)Ωs|W(q,l)|2=M4.
ηs=(q,l)Ωsqlsinc-2(q/M)sinc-2(l/M)-1.
ηs=N(q,l)Ωssinc-2(q/M)sinc-2(l/M)-1.

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