Abstract

The design of a holographic optical element for focusing a collimated off-axis beam to an on-axis point is described. It is designed according to a novel recursive technique in which the recording is done at a wavelength which differs from the readout. In this recursive technique the final holographic optical element is recorded by using other holograms to provide the aspheric recording wavefronts necessary for reducing the aberrations and maximizing the diffraction efficiency. The design is illustrated with an example where an f/3.0 focusing element is recorded at 514.5 nm and read out at 1064 nm. A spot size of 15 μm and a diffraction efficiency of ~60% were measured.

© 1988 Optical Society of America

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References

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  1. D. H. Close, “Holographic Optical Elements,” Opt. Eng. 14, 408 (1975).
  2. H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).
  3. H. Ikeda, M. Ando, T. Inagaki, “Aberration Corrections for a POS Hologram Scanner,” Appl. Opt. 18, 2166 (1979).
    [CrossRef] [PubMed]
  4. J. N. Latta, “Computer-Based Analysis of Hologram Imagery and Aberrations. 2: Aberrations Induced by a Wavelength Shift,” Appl. Opt. 10, 609 (1971).
    [CrossRef] [PubMed]
  5. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
  6. M. R. Latta, R. V. Pole, “Design Techniques for Forming 488-nm Holographic Lenses with Reconstruction at 633 nm,” Appl. Opt. 18, 2418 (1979).
    [CrossRef] [PubMed]
  7. L. H. Lin, E. T. Doherty, “Efficient and Aberration-Free Wavefront Reconstruction from Holograms Illuminated at Wavelengths Differing from the Forming Wavelength,” Appl. Opt. 10, 1314 (1971).
    [CrossRef] [PubMed]
  8. K. A. Winick, “Designing Efficient Aberration-Free Holographic Lenses in the Presence of a Construction-Reconstruction Wavelength Shift,” J. Opt. Soc. Am. 72, 143 (1982).
    [CrossRef]
  9. H. Chen, R. R. Hershey, E. N. Leith, “Design of a Holographic Lens for the Infrared,” Appl. Opt. 26, 1983 (1987).
    [CrossRef] [PubMed]
  10. H. P. Herzig, “Holographic Optical Elements (HOE) for Semiconductor Lasers,” Opt. Commun. 58, 144 (1986).
    [CrossRef]
  11. Y. Amitai, A. A. Friesem, “Recursive Design Techniques for Fourier Transform Holographic Lenses,” Opt. Eng. 26, 1133 (1987).
    [CrossRef]
  12. Y. Amitai, A. A. Friesem, “Design of Holographic Optical Elements by Using Recursive Techniques,” J. Opt. Soc. Am. A 5, 702 (1988).
    [CrossRef]
  13. J. N. Latta, “Computer-Based Analysis of Hologram Imagery and Aberrations. 1: Hologram Types and Their Nonchromatic Aberrations,” Appl. Opt. 10, 599 (1971).
    [CrossRef] [PubMed]
  14. E. B. Champagne, “Nonparaxial Imaging, Magnification, and Aberration Properties in Holography,” J. Opt. Soc. A. 57, 51 (1967).
    [CrossRef]
  15. I. A. Mikhailov, “A Geometrical Analysis of Thick Holograms,” Opt. Spektrosk. 58, 612 (1985).
  16. P. C. Mehta, K. S. S. Rao, R. Hradaynath, “Higher Order Aberrations in Holographic Lenses,” Appl. Opt. 21, 4553 (1982).
    [CrossRef] [PubMed]

1988 (1)

1987 (2)

H. Chen, R. R. Hershey, E. N. Leith, “Design of a Holographic Lens for the Infrared,” Appl. Opt. 26, 1983 (1987).
[CrossRef] [PubMed]

Y. Amitai, A. A. Friesem, “Recursive Design Techniques for Fourier Transform Holographic Lenses,” Opt. Eng. 26, 1133 (1987).
[CrossRef]

1986 (1)

H. P. Herzig, “Holographic Optical Elements (HOE) for Semiconductor Lasers,” Opt. Commun. 58, 144 (1986).
[CrossRef]

1985 (1)

I. A. Mikhailov, “A Geometrical Analysis of Thick Holograms,” Opt. Spektrosk. 58, 612 (1985).

1983 (1)

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

1982 (2)

1979 (2)

1975 (1)

D. H. Close, “Holographic Optical Elements,” Opt. Eng. 14, 408 (1975).

1971 (3)

1969 (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

1967 (1)

E. B. Champagne, “Nonparaxial Imaging, Magnification, and Aberration Properties in Holography,” J. Opt. Soc. A. 57, 51 (1967).
[CrossRef]

Amitai, Y.

Y. Amitai, A. A. Friesem, “Design of Holographic Optical Elements by Using Recursive Techniques,” J. Opt. Soc. Am. A 5, 702 (1988).
[CrossRef]

Y. Amitai, A. A. Friesem, “Recursive Design Techniques for Fourier Transform Holographic Lenses,” Opt. Eng. 26, 1133 (1987).
[CrossRef]

Ando, M.

Champagne, E. B.

E. B. Champagne, “Nonparaxial Imaging, Magnification, and Aberration Properties in Holography,” J. Opt. Soc. A. 57, 51 (1967).
[CrossRef]

Chen, H.

Close, D. H.

D. H. Close, “Holographic Optical Elements,” Opt. Eng. 14, 408 (1975).

Doherty, E. T.

Friesem, A. A.

Y. Amitai, A. A. Friesem, “Design of Holographic Optical Elements by Using Recursive Techniques,” J. Opt. Soc. Am. A 5, 702 (1988).
[CrossRef]

Y. Amitai, A. A. Friesem, “Recursive Design Techniques for Fourier Transform Holographic Lenses,” Opt. Eng. 26, 1133 (1987).
[CrossRef]

Funato, H.

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

Hershey, R. R.

Herzig, H. P.

H. P. Herzig, “Holographic Optical Elements (HOE) for Semiconductor Lasers,” Opt. Commun. 58, 144 (1986).
[CrossRef]

Hradaynath, R.

Ikeda, H.

Inagaki, T.

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Latta, J. N.

Latta, M. R.

Leith, E. N.

Lin, L. H.

Mehta, P. C.

Mikhailov, I. A.

I. A. Mikhailov, “A Geometrical Analysis of Thick Holograms,” Opt. Spektrosk. 58, 612 (1985).

Pole, R. V.

Rao, K. S. S.

Winick, K. A.

Appl. Opt. (7)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

J. Opt. Soc. A. (1)

E. B. Champagne, “Nonparaxial Imaging, Magnification, and Aberration Properties in Holography,” J. Opt. Soc. A. 57, 51 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

H. P. Herzig, “Holographic Optical Elements (HOE) for Semiconductor Lasers,” Opt. Commun. 58, 144 (1986).
[CrossRef]

Opt. Eng. (2)

Y. Amitai, A. A. Friesem, “Recursive Design Techniques for Fourier Transform Holographic Lenses,” Opt. Eng. 26, 1133 (1987).
[CrossRef]

D. H. Close, “Holographic Optical Elements,” Opt. Eng. 14, 408 (1975).

Opt. Spektrosk. (1)

I. A. Mikhailov, “A Geometrical Analysis of Thick Holograms,” Opt. Spektrosk. 58, 612 (1985).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo-Opt. Instrum. Eng. 390, 174 (1983).

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Figures (6)

Fig. 1
Fig. 1

Geometry for recording and readout. An arbitrary point source Q at x q ,y q ,z q situated in the xyz space in front of a hologram in the xy plane.

Fig. 2
Fig. 2

Ratio of the object to reference wavefronts radii fulfilling the Bragg condition and the S = 0 condition as a function of the wavelength shift μ. The solid line represents R o B / R o S , whereas the dot–dash line represents R r B / R r S .

Fig. 3
Fig. 3

(a) Recording and (b) readout arrangements for the first-step holograms.

Fig. 4
Fig. 4

Recording and readout arrangements for the intermediate hologram. (a) The emerging image wavefronts ϕ i ( o ) and ϕ i ( r ) from the first-step holograms (solid and dotted lines) interfere with a reference plane wave ϕ r ( int ) for recording the intermediate hologram. (b) Readout of the intermediate hologram with a plane wave ϕ c ( int ) = - ϕ r ( int ) yields the reference and object wavefronts for recording the final element.

Fig. 5
Fig. 5

Calculated spot diagrams: (a) recursive off-axis element; (b) noncorrected off-axis spherical element.

Fig. 6
Fig. 6

Photographs of the intensity distribution at the focal plane: (a) recursive off-axis element; (b) noncorrected off-axis spherical element.

Equations (35)

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ϕ i = ϕ c ± ( ϕ o - ϕ r ) .
ϕ i - ϕ s 2 π λ c ( Δ S + Δ C + Δ A ) ,
Δ S = - ( x 2 + y 2 ) 2 S ,
Δ C = ½ ( x 2 + y 2 ) ( x C x + y C y ) ,
Δ A = - ½ ( x 2 A x + y 2 A y + x y A x y ) ,
S = 1 R c 3 - 1 R i 3 ± μ ( 1 R o 3 - 1 R r 3 ) ,
C C x = sin α c R c 2 - sin α i R i 2 ± μ ( sin α o R o 2 - sin α r R r 2 ) ,
A A x = sin 2 α c R c - sin 2 a i R i ± μ ( sin 2 α o R o - sin 2 α r R r ) ,
1 R i = 1 R c ± μ ( 1 R o - 1 R r ) ,
sin α i = sin α c ± μ ( sin α o - sin α r ) .
1 R o S 1 R o = 1 2 μ ( 1 R i - 1 R c ) ± 1 2 Z ,
1 R r S 1 R r = 1 2 μ ( 1 R c - 1 R i ) ± 1 2 Z ,
Z = 4 3 ( 1 R i 2 + 1 R i R c + 1 R c 2 ) - 1 3 μ 2 ( 1 R i - 1 R c ) 2 .
k i ( x , y ) = k c ( x , y ) ± [ k o ( x , y ) - k r ( x , y ) ] ,
k q ( x , y ) ϕ q ( x , y ) .
k q z = ( 2 π λ q ) 2 - k q x 2 - k q y 2 ,
1 R o B 1 R o = 1 2 μ ( μ + 1 R i + μ - 1 R c ) ,
1 R r B 1 R r = 1 2 μ ( μ - 1 R i + μ + 1 R c ) .
sin α o B sin α o = 1 2 μ [ ( μ + 1 ) sin α i + ( μ - 1 ) sin α c ] ,
sin α r B sin α r = 1 2 μ [ ( μ - 1 ) sin α i + ( μ + 1 ) sin α c ] .
1 R o S = 1 R i [ 1 2 μ ± 1 2 4 μ 2 - 1 3 μ 2 ] ,
1 R r S = 1 R i [ - 1 2 μ + 1 2 4 μ 2 - 1 3 μ 2 ] ,
1 R o B = μ + 1 2 μ 1 R i ,
1 R r B = μ - 1 2 μ 1 R i .
ϕ q = ϕ i ( q ) = ϕ c ( q ) ± [ ϕ o ( q ) - ϕ r ( q ) ] ,
ϕ i = ϕ c ± { ϕ c ( o ) ± [ ϕ o ( o ) - ϕ r ( o ) ] - ϕ c ( r ) [ ϕ o ( r ) - ϕ r ( o ) ] } .
Q i = Q n c + λ c n c λ c ( o ) Q ( o ) + λ c n c λ c ( r ) Q ( r ) ,
S n c = 0 ,
C n c = ± μ ( sin α o R o 2 - sin α r R r 2 ) ,
A n c = ± μ ( sin 2 α o R - sin 2 α r R r ) .
S ( q ) = - 1 R i ( q ) 3 ± μ ( q ) 1 R o ( q ) 3 ,
C ( q ) = - sin α i ( q ) R i ( q ) 2 ,
A ( q ) = - sin 2 α i ( q ) R i ( q ) ,
R q s = R i ( q ) ,
α q B = α i ( q ) ,

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