Abstract

An analytical method to design holographic optical elements for focusing laser scanners, especially disk scanners, with minimum aberrations and optimum scan line definition is reported. The results reveal that the focused spot constraint to a straight line is always astigmatic. However, by accepting small deviations from the straight line, the astigmatism can be eliminated. The second-order analytical solutions are examined with the help of geometrical ray tracing and compared with experimental results. By extending the method to higher-order approximations, it was found that the correction of the aberrations is essentially limited to the direction perpendicular to the scan line.

© 1988 Optical Society of America

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References

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  1. L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
    [CrossRef]
  2. H. Ikeda, M. Ando, T. Inagaki, “Aberration Corrections for a POS Hologram Scanner,” Appl. Opt. 18, 2166 (1979).
    [CrossRef] [PubMed]
  3. H. Funato, “Holographic Scanner for Laser Printer,” Proc. Soc. Photo.-Opt. Instrum. Eng. 390, 174 (1983).
  4. L. Beiser, “Imaging with Laser Scanners,” Opt. News (Nov.1986), pp. 10–16.
    [CrossRef]
  5. Y. Ono, N. Nishida, “Holographic Laser Scanners for Multi-directional Scanning,” Appl. Opt. 22, 2128 (1983).
    [CrossRef] [PubMed]
  6. H. Iwaoka, T. Shiozawa, “Aberration-Free Linear Holographic Scanner and Its Application to a Diode-Laser Printer,” Appl. Opt. 25, 123 (1986).
    [CrossRef] [PubMed]
  7. Y. Ono, N. Nishida, “Holographic Optical Elements with Optimized Phase-Transfer Functions,” J. Opt. Soc. Am. A 3, 139 (1986).
    [CrossRef]
  8. K. A. Winick, J. R. Fienup, “Optimum Holographic Elements Recorded with Nonspherical Wave Fronts,” J. Opt. Soc. Am. 73, 208 (1983).
    [CrossRef]
  9. J. Kedmi, A. A. Friesem, “Optimized Holographic Optical Elements,” J. Opt. Soc. Am. A 3, 2011 (1986).
    [CrossRef]
  10. H. P. Herzig, R. Dandliker, “Holographic Optical Scanning Elements: Analytical Method for Determining the Phase Function,” J. Opt. Soc. Am. A 4, 1063 (1987).
    [CrossRef]
  11. H. P. Herzig, R. Dandliker, “Design Rules for Holographic Optical Scanning Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 812, 86 (1987).
  12. H. P. Herzig, “Holographic Optical Scanning Elements,” Ph.D. Thesis U. Neuchâtel, Switzerland (1987).

1987 (2)

H. P. Herzig, R. Dandliker, “Holographic Optical Scanning Elements: Analytical Method for Determining the Phase Function,” J. Opt. Soc. Am. A 4, 1063 (1987).
[CrossRef]

H. P. Herzig, R. Dandliker, “Design Rules for Holographic Optical Scanning Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 812, 86 (1987).

1986 (4)

1983 (3)

1982 (1)

L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
[CrossRef]

1979 (1)

Ando, M.

Beiser, L.

L. Beiser, “Imaging with Laser Scanners,” Opt. News (Nov.1986), pp. 10–16.
[CrossRef]

Dandliker, R.

H. P. Herzig, R. Dandliker, “Holographic Optical Scanning Elements: Analytical Method for Determining the Phase Function,” J. Opt. Soc. Am. A 4, 1063 (1987).
[CrossRef]

H. P. Herzig, R. Dandliker, “Design Rules for Holographic Optical Scanning Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 812, 86 (1987).

Dickson, L. D.

L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
[CrossRef]

Fienup, J. R.

Friesem, A. A.

Funato, H.

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

Herzig, H. P.

H. P. Herzig, R. Dandliker, “Holographic Optical Scanning Elements: Analytical Method for Determining the Phase Function,” J. Opt. Soc. Am. A 4, 1063 (1987).
[CrossRef]

H. P. Herzig, R. Dandliker, “Design Rules for Holographic Optical Scanning Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 812, 86 (1987).

H. P. Herzig, “Holographic Optical Scanning Elements,” Ph.D. Thesis U. Neuchâtel, Switzerland (1987).

Ikeda, H.

Inagaki, T.

Iwaoka, H.

Kedmi, J.

Nishida, N.

Ono, Y.

Shiozawa, T.

Sincerbox, G. T.

L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
[CrossRef]

Winick, K. A.

Wolfheimer, A. D.

L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
[CrossRef]

Appl. Opt. (3)

IBM J. Res. Dev. (1)

L. D. Dickson, G. T. Sincerbox, A. D. Wolfheimer, “Holography in the IBM 3687 Supermarket Scanner,” IBM J. Res. Dev. 26, 228 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. News (1)

L. Beiser, “Imaging with Laser Scanners,” Opt. News (Nov.1986), pp. 10–16.
[CrossRef]

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

H. P. Herzig, R. Dandliker, “Design Rules for Holographic Optical Scanning Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 812, 86 (1987).

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

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

Other (1)

H. P. Herzig, “Holographic Optical Scanning Elements,” Ph.D. Thesis U. Neuchâtel, Switzerland (1987).

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Figures (10)

Fig. 1
Fig. 1

General scan configuration. The hologram moves along a line x(s) in the hologram plane and the image point describes another line y(t) in space.

Fig. 2
Fig. 2

Geometrical relation between the hologram plane (u,v) and the wave vector k of a spherical wave. k′ is the projection of k onto the plane (v,w).

Fig. 3
Fig. 3

Circular motion x(s) of the hologram (rotating disk) to generate a straight line y(t) in space.

Fig. 4
Fig. 4

Recording setup to realize a holographic scanning element with the aid of a computer-generated hologram (CGH).

Fig. 5
Fig. 5

Experimental setup for measuring the performances of the holographic scanners. The holographic disk rotates around its axis and generates a line perpendicular to the drawing plane. The observation system is translated along the generated line to determine the image point for different scan positions.

Fig. 6
Fig. 6

Holographic optical element corresponding to one segment of a disk scanner.

Fig. 7
Fig. 7

Calculated and measured deviation Δy of the scan from a straight line as a function of the scan length L for the straight-line scanner. The maximum scan length was L = ±105 mm. Calculated (solid line) is based on the center of gravity of the spots. Experimental (+).

Fig. 8
Fig. 8

Spot quality for the straight-line scanner with D = 5 mm (diffraction-limited spot size DS = 38 μm at half-intensity). Experimental results and geometrical ray tracing for nine scan positions corresponding to scan lengths of L = 0, ±30, ±60, ±90, and ±105 mm. The scan line y(t) is parallel to the u′ axis and the configuration is symmetrical with respect to the v′ axis.

Fig. 9
Fig. 9

Calculated and measured deviation Δy of the scan from a straight line as a function of the scan length L for the astigmatism-free scanner. The maximum scan length was L = ±105 mm. Calculated (solid line) is based on the center of gravity of the spots. Experimental (+).

Fig. 10
Fig. 10

Spot quality for the astigmatism-free scanner with D = 5 mm (diffraction-limited spot size DS = 38 μm at half-intensity). Experimental results and geometrical ray tracing for nine scan positions corresponding to scan lengths of L = 0, ±30, ±60, ±90, and ±105 mm. The scan line y(t) is parallel to the u′ axis and the configuration is symmetrical with respect to the v′ axis.

Equations (42)

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F ( u , v , t ) = Φ P ( u , v , t ) - Φ S ( u , v , t ) .
Ψ ( u , v , t ) = Φ P ( u , v , t ) - Φ r ( u , v ) = Φ S ( u , v , t ) + F ( u , v , t ) - Φ r ( u , v ) ,
Φ ( u , v , s ) = Φ [ x ( s ) ] + Φ u i | s u i + 1 2 2 Φ u i u j | s u i u j + , i , j = 1 , 2 ,
Ψ ( u , v , t ) = Ψ ( 0 , 0 , t ) + Ψ u i | t u i + 1 2 2 Ψ u i u j | t u i u j + .
Φ [ x ( s ) ] = Ψ ( 0 , 0 , t ) ,
Φ u i | s = Ψ u i | t h i ( t ) ,             i , j = 1 , 2 ,
2 Φ u i u j | s = 2 Ψ u i u j | t h i j ( t ) ,
Φ x i | s = f i ( Φ u k | s ) ,
2 Φ x i x j | s = f i j ( Φ u k | s , 2 Φ u 1 u m | s ) . . . .
Φ x i | s = f i [ h k ( t ) ] g i ( t ) ,
2 Φ x i x j | s = f i j [ h k ( t ) , h l m ( t ) ] g i j ( t ) . . . .
Φ S ( u , v , t ) = k [ ( u + sin β / a ) 2 + ( v + sin α cos β / a ) 2 + ( cos α cos β / a ) 2 ] 1 / 2 .
Φ r ( u , v , t ) = k [ ( u ) 2 + ( v + ρ sin γ ) 2 + ( ρ cos γ ) 2 ] 1 / 2 ,
h 1 = sin β + F 1 ,
h 2 = sin α cos β - sin γ + F 2 ,
h 11 = a cos 2 β - 1 / ρ + F 11 ,
h 12 = - a sin α sin β cos β + F 12 ,
h 22 = a ( 1 - sin 2 α cos 2 β ) - cos 2 γ / ρ + F 22 ,
h 111 = - 3 a 2 sin β cos 2 β + F 111 ,
h 112 = a 2 sin α cos β ( 3 sin 2 β - 1 ) + sin γ / ρ 2 + F 112 ,
h 122 = a 2 sin β ( 3 sin 2 α cos 2 β - 1 ) + F 122 ,
h 222 = 3 a 2 sin α cos β ( sin 2 α cos 2 β - 1 ) + 3 sin γ cos 2 γ / ρ 2 + F 222 ,
h 1111 = 3 a 3 ( - 1 + 6 sin 2 β - 5 sin 4 β ) + 3 / ρ 3 + F 1111 ,
h 1112 = 3 a 3 ( sin α sin β cos β ( 3 - 5 sin 2 β ) + F 1112 ,
h 1122 = a 3 ( - 1 + 3 sin 2 α cos 2 β + 3 sin 2 β - 15 sin 2 α sin 2 β cos 2 β ) + ( 1 - 3 sin 2 γ ) / ρ 3 + F 1122 ,
h 1222 = 3 a 3 sin α sin β cos β ( 3 - 5 sin 2 α cos 2 β ) + F 1222 ,
h 2222 = 3 a 3 ( - 1 + 6 sin 2 α cos 2 β - 5 sin 4 α cos 4 β ) + 3 ( 1 - 6 sin 2 γ + 5 sin 4 γ ) / ρ 3 + F 2222 ,
Φ ( r , ϕ ) = k [ a 0 + a 1 ( r - R ) + 1 2 a 2 ( r - R ) 2 + 1 6 a 3 ( r - R ) 3 + 1 24 a 4 ( r - R ) 4 ] ,
F 1 = a 0 / R - sin β ,
F 2 = a 1 - sin α cos β + sin γ ,
F 11 = a 0 / R 2 - ( a cos 2 β - 1 / ρ ) + ( sin α cos β - sin γ ) / R + F 2 / R ,
F 12 = a 1 / R + a sin α sin β cos β - sin β / R - F 1 / R ,
F 22 = a 2 - a ( 1 - sin 2 α cos 2 β ) + cos 2 γ / ρ ,
F 111 = a 0 / R 3 + 3 a 2 sin β cos 2 β - 3 a sin α sin β cos β / R + sin β / R 2 + 3 F 12 / R + F 1 / R 2 ,
F 112 = a 1 / R 2 - 2 ( a cos 2 β - 1 / ρ ) / R + [ a ( 1 - sin 2 α cos 2 β ) - cos 2 γ / ρ ] / R + ( sin α cos β - sin γ ) / R - [ a 2 sin α cos β ( 3 sin 2 β - 1 ) + sin γ / ρ 2 ] + ( F 22 - 2 F 11 ) / R + F 2 / R 2 ,
F 122 = a 2 / R + 2 a sin α sin β cos β / R - a 2 sin β ( 3 sin 2 α cos 2 β - 1 ) - 2 F 12 / R ,
F 222 = a 3 - 3 a 2 sin α cos β ( sin 2 α cos 2 β - 1 ) - 3 sin γ cos 2 γ / ρ 2 ,
F 1111 = a 0 / R 4 + , . . .
F 2222 = a 4 + 3 a 3 ( 1 - 6 sin 2 α cos 2 β + 5 sin 4 α cos 4 β ) - 3 ( 1 - 6 sin 2 γ + 5 sin 4 γ / ρ 3 ) ,
d β / d ϕ - R a ( β ) cos β + R / ( ρ cos β ) + sin α - sin γ / cos β = 0 ,
d α / d ϕ = - tan β [ ( tan α / cos β ) ( R / ρ - sin γ ) - cos α ] .
Φ = Φ O - Φ R .

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