Abstract

An upper bound of the diffraction efficiency of diffractive elements that only influence the phase of the illumination wave is derived. The derivation only utilizes the specification of the desired diffraction pattern. It is independent of the technique to design and fabricate the diffractive element. The theory is based on the transmittance approach to describe the effect of the element on the illumination wave.

© 1991 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, Proc. IEEE 73, 894 (1985).
    [CrossRef]
  2. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).
  3. R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 68, 809 (1978).
    [CrossRef]
  4. R. Magnusson, T. K. Gaylord, Opt. Commun. 28, 1 (1979).
    [CrossRef]
  5. O. Bryngdahl, R. Wyrowski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 65–76.
    [CrossRef]
  6. F. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1211, 2 (1990).
  7. F. Wyrowski, Opt. Commun. 81, 353 (1991).
    [CrossRef]
  8. R. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. (to be published).
  9. H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.
  10. K. Krackhardt, J. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. (to be published).
  11. R. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1507, 128 (1991).

1991 (2)

F. Wyrowski, Opt. Commun. 81, 353 (1991).
[CrossRef]

R. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1507, 128 (1991).

1990 (1)

F. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1211, 2 (1990).

1985 (1)

T. K. Gaylord, M. G. Moharam, Proc. IEEE 73, 894 (1985).
[CrossRef]

1979 (1)

R. Magnusson, T. K. Gaylord, Opt. Commun. 28, 1 (1979).
[CrossRef]

1978 (1)

Bryngdahl, O.

O. Bryngdahl, R. Wyrowski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 65–76.
[CrossRef]

H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, Proc. IEEE 73, 894 (1985).
[CrossRef]

R. Magnusson, T. K. Gaylord, Opt. Commun. 28, 1 (1979).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 68, 809 (1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

Krackhardt, K.

K. Krackhardt, J. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. (to be published).

Liipken, H.

H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.

Magnusson, R.

R. Magnusson, T. K. Gaylord, Opt. Commun. 28, 1 (1979).
[CrossRef]

R. Magnusson, T. K. Gaylord, J. Opt. Soc. Am. 68, 809 (1978).
[CrossRef]

Mait, J.

K. Krackhardt, J. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. (to be published).

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, Proc. IEEE 73, 894 (1985).
[CrossRef]

Peter, T.

H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.

Streibl, N.

K. Krackhardt, J. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. (to be published).

Wyrowski, F.

F. Wyrowski, Opt. Commun. 81, 353 (1991).
[CrossRef]

F. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1211, 2 (1990).

H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.

Wyrowski, R.

R. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1507, 128 (1991).

R. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. (to be published).

O. Bryngdahl, R. Wyrowski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 65–76.
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

R. Magnusson, T. K. Gaylord, Opt. Commun. 28, 1 (1979).
[CrossRef]

F. Wyrowski, Opt. Commun. 81, 353 (1991).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, Proc. IEEE 73, 894 (1985).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

F. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1211, 2 (1990).

R. Wyrowski, Proc. Soc. Photo-Opt. Instrum. Eng. 1507, 128 (1991).

Other (5)

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1968).

O. Bryngdahl, R. Wyrowski, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 65–76.
[CrossRef]

R. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. (to be published).

H. Liipken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illumination,” submitted to Opt. Commun.

K. Krackhardt, J. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. (to be published).

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Figures (1)

Fig. 1
Fig. 1

Illustration of coding operators as a mapping of (a) F(u) and (b) αF(u) onto the unit circle. A change (a) C(u) and (b) Ĉ(u) causes a modification of the phase of F(u) by ΔΦ(u).

Equations (22)

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g ( x ) = α f ( x ) ,             x F ,
η = g ( x ) 2 F ,
F ( u ) = F f ( x )
F ( u ) 1.
G ( u ) = C F ( u ) = F ( u ) + C ( u )
g ( x ) = f ( x ) + c ( x ) .
c ( x ) = ( α - 1 ) f ( x ) + c ^ ( x ) ,             x F ,
c ^ ( x ) = 0 ,             x F .
g ( x ) = f ( x ) + c ( x ) = α f ( x ) + c ^ ( x ) ,
η = α 2 η 0 ,
η 0 = f ( x ) 2 F .
η 0 = f ( x ) 2 F f ( x ) 2 = F ( u ) 2
g ( x ) 2 = α 2 f ( x ) 2 + c ^ ( x ) 2 .
C ^ ( u ) 2 = 1 - α 2 F ( u ) 2 ,
α 2 = 1 - C ^ ( u ) 2 F ( u ) 2
C ^ ( u ) 2 = 1 + α 2 F ( u ) 2 - 2 α F ( u ) cos [ Δ Φ ( u ) ]
α = F ( u ) cos [ Δ Φ ( u ) ] F ( u ) 2 .
α l = F ( u ) F ( u ) 2
η l = F ( u ) 2 F ( u ) 2 .
G ( u ) = C p ( u ) F ( u ) = exp { i arg [ F ( u ) ] } .
η < η l
F ( u ) = 1 P max { P 0 * ( u ) / P 0 ( u ) 2 1 / P 0 ( u ) P max P max exp { - i arg [ P 0 ( u ) ] } otherwise .

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