Abstract

In the paraxial domain of diffractive optics diffractive elements are, in general, designed by scalar theory. The resulting elements possess transverse features that are large compared with the wavelength. We demonstrate some advantages of using wavelength-scale features and hence rigorous diffraction theory in the paraxial domain of diffractive optics. Our design procedure leads to elements with efficiencies that exceed by a considerable margin the upper bounds of diffraction efficiency predicted by scalar synthesis theory.

© 1995 Optical Society of America

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References

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  1. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  2. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  3. F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef]
  4. F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
    [CrossRef]
  5. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  6. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  7. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef]
  9. H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
    [CrossRef]
  10. L. E. Hargrove, “Optical effects of ultrasonic waves producing phase and amplitude modulation,” J. Acoust. Soc. Am. 34, 1547–1552 (1962).
    [CrossRef]
  11. M. Kuittinen, T. Jaaskelainen, “Synthesis of phase gratings for desired diffraction patterns,” Opt. Commun. 81, 145–149 (1991).
    [CrossRef]
  12. T. Jaaskelainen, M. Kuittinen, “Planar interconnection gratings,” Opt. Comp. Proc. 2, 29–38 (1992).
  13. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [CrossRef] [PubMed]
  14. M. W. Farn, “Binary grating with improved efficiency,” Appl. Opt. 31, 4453–4458 (1992).
    [CrossRef] [PubMed]
  15. D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Phys. 32, 1154–1167 (1993).
  16. M. Collischon, H. Haidner, P. Kipfer, A. Lang, J. T. Sheridan, J. Schwider, N. Streibl, J. Lindolf, “Binary blazed reflection gratings,” Appl. Opt. 33, 3572–3577 (1994).
    [CrossRef] [PubMed]
  17. H. Ichikawa, J. Turunen, M. R. Taghizadeh, “Analysis of hybrid holographic gratings by thin-grating decomposition method,” J. Opt. Soc. Am. A 10, 1176–1183 (1993).
    [CrossRef]
  18. E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
    [CrossRef]
  19. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
    [CrossRef]
  20. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  21. D. Maystre, R. Petit, “Diffraction par un reseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
    [CrossRef]
  22. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  23. J. Turunen, E. Noponen, F. Wyrowski, “Diffractive optics beyond the paraxial domain,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 186–189.

1994 (2)

1993 (4)

H. Ichikawa, J. Turunen, M. R. Taghizadeh, “Analysis of hybrid holographic gratings by thin-grating decomposition method,” J. Opt. Soc. Am. A 10, 1176–1183 (1993).
[CrossRef]

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Phys. 32, 1154–1167 (1993).

F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

1992 (6)

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef]

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

M. W. Farn, “Binary grating with improved efficiency,” Appl. Opt. 31, 4453–4458 (1992).
[CrossRef] [PubMed]

T. Jaaskelainen, M. Kuittinen, “Planar interconnection gratings,” Opt. Comp. Proc. 2, 29–38 (1992).

1991 (4)

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

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

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

M. Kuittinen, T. Jaaskelainen, “Synthesis of phase gratings for desired diffraction patterns,” Opt. Commun. 81, 145–149 (1991).
[CrossRef]

1986 (1)

1978 (1)

1972 (1)

D. Maystre, R. Petit, “Diffraction par un reseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

1962 (1)

L. E. Hargrove, “Optical effects of ultrasonic waves producing phase and amplitude modulation,” J. Acoust. Soc. Am. 34, 1547–1552 (1962).
[CrossRef]

Baird, W. E.

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]

Collischon, M.

Farn, M. W.

Gaylord, T. K.

Haidner, H.

Hargrove, L. E.

L. E. Hargrove, “Optical effects of ultrasonic waves producing phase and amplitude modulation,” J. Acoust. Soc. Am. 34, 1547–1552 (1962).
[CrossRef]

Ichikawa, H.

Jaaskelainen, T.

T. Jaaskelainen, M. Kuittinen, “Planar interconnection gratings,” Opt. Comp. Proc. 2, 29–38 (1992).

M. Kuittinen, T. Jaaskelainen, “Synthesis of phase gratings for desired diffraction patterns,” Opt. Commun. 81, 145–149 (1991).
[CrossRef]

Kipfer, P.

Knop, K.

Krackhardt, U.

Kuittinen, M.

T. Jaaskelainen, M. Kuittinen, “Planar interconnection gratings,” Opt. Comp. Proc. 2, 29–38 (1992).

M. Kuittinen, T. Jaaskelainen, “Synthesis of phase gratings for desired diffraction patterns,” Opt. Commun. 81, 145–149 (1991).
[CrossRef]

Lang, A.

Lindolf, J.

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.

Maystre, D.

D. Maystre, R. Petit, “Diffraction par un reseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Moharam, M. G.

Morris, G. M.

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Phys. 32, 1154–1167 (1993).

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Noponen, E.

E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, E. Noponen, F. Wyrowski, “Diffractive optics beyond the paraxial domain,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 186–189.

Peter, T.

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

Petit, R.

D. Maystre, R. Petit, “Diffraction par un reseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

Raguin, D. H.

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Phys. 32, 1154–1167 (1993).

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Schwider, J.

Sheridan, J. T.

Streibl, N.

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

H. Ichikawa, J. Turunen, M. R. Taghizadeh, “Analysis of hybrid holographic gratings by thin-grating decomposition method,” J. Opt. Soc. Am. A 10, 1176–1183 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Turunen, J.

E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
[CrossRef]

H. Ichikawa, J. Turunen, M. R. Taghizadeh, “Analysis of hybrid holographic gratings by thin-grating decomposition method,” J. Opt. Soc. Am. A 10, 1176–1183 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, E. Noponen, F. Wyrowski, “Diffractive optics beyond the paraxial domain,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 186–189.

Vasara, A.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Wyrowski, F.

F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
[CrossRef]

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[CrossRef]

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]

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

J. Turunen, E. Noponen, F. Wyrowski, “Diffractive optics beyond the paraxial domain,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 186–189.

Appl. Opt. (4)

Appl. Phys. (1)

D. H. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Phys. 32, 1154–1167 (1993).

J. Acoust. Soc. Am. (1)

L. E. Hargrove, “Optical effects of ultrasonic waves producing phase and amplitude modulation,” J. Acoust. Soc. Am. 34, 1547–1552 (1962).
[CrossRef]

J. Mod. Opt. (1)

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (5)

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[CrossRef]

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

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

M. Kuittinen, T. Jaaskelainen, “Synthesis of phase gratings for desired diffraction patterns,” Opt. Commun. 81, 145–149 (1991).
[CrossRef]

D. Maystre, R. Petit, “Diffraction par un reseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

Opt. Comp. Proc. (1)

T. Jaaskelainen, M. Kuittinen, “Planar interconnection gratings,” Opt. Comp. Proc. 2, 29–38 (1992).

Opt. Lett. (1)

Rep. Prog. Phys. (1)

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

Other (3)

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. Turunen, E. Noponen, F. Wyrowski, “Diffractive optics beyond the paraxial domain,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 186–189.

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Figures (7)

Fig. 1
Fig. 1

Paraxial domain of diffractive optics: geometry and notation.

Fig. 2
Fig. 2

Diffraction of a plane wave by a grating: Tm denotes the complex amplitude associated with the mth-order diffracted plane wave.

Fig. 3
Fig. 3

(a) Amplitude and (b) phase of the signal wave required for generating a five-beam array with 100% efficiency.

Fig. 4
Fig. 4

Efficiency of a two-beam fan-out element with the optimized groove depth as a function of the grating period.

Fig. 5
Fig. 5

(a) Diffraction efficiency and (b) the groove depth of optimized two-beam reflection-type fan-out gratings with several grooves in a period as a function of d/λ.

Fig. 6
Fig. 6

(a) Amplitude and (b) phase of the electric field diffracted by the grating profile shown in (c).

Fig. 7
Fig. 7

(a) Diffraction efficiency and (b) the groove depth of optimized two-beam transmission-type fan-out gratings with several grooves in a period as a function of d/λ.

Tables (2)

Tables Icon

Table 1 Upper Bounds ηl Obtained by Optimization of the Signal Phase and Efficiencies η of Optimized Grating Solutions with N Equal-Efficiency Diffraction Orders

Tables Icon

Table 2 Efficiencies of Orders m = −4, …, 4 Generated by Rigorously Designed Gratings with a Two-Beam Signala and with an Irregular Signalb

Equations (7)

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

U ( x , z ) = m = T m exp [ i ( α m x + t m z ) ] ,
U s ( x ) = m w T m exp [ i n 2 k sin ( θ m ) x ] ,
U ( x , 0 ) = U ( x , h ) exp [ i k h 0 n ( x , z ) d z ] ,
η = m = W η m = m W | T m | 2
η l = | U s ( x ) | 2 | U s ( x ) 2 | ,
f ( x ) = 1 d 0 d f ( x ) d x .
U s ( x ) = 1 2 [ exp ( i 2 π x / d ) + exp ( i 2 π x / d ) ] = 2 cos ( 2 π x / d ) ,

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