Abstract

The creation of paraxial arbitrary focal lines by a Fourier computer-generated hologram is demonstrated. The desired focal line is represented by a series of connected straight line segments, each of which is implemented by a radial harmonic function located on a different radial portion of the entire hologram. Each subhologram is multiplied by appropriate linear and quadratic phase functions and is shifted by some distance from the center. The two phase factors determine the location of each line segment, while the in-plane shift determines the tilt angle of the segment.

© 1995 Optical Society of America

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References

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  1. J. Rosen, B. Salik, A. Yariv, J. Opt. Soc. Am. A 12, 2446 (1995).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  3. J. Ojeda-Castaneda, L. R. Berriel Valdos, Opt. Lett. 13, 183 (1988).
    [CrossRef] [PubMed]
  4. J. Rosen, A. Yariv, Opt. Lett. 19, 843 (1994); R. Piestun, J. Shamir, Opt. Lett. 19, 771 (1994).
    [CrossRef] [PubMed]
  5. L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
    [CrossRef]
  6. C. Frère, O. Bryngdahl, Opt. Commun. 60, 369 (1986).
    [CrossRef]

1995 (1)

1994 (1)

1988 (1)

1987 (1)

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

1986 (1)

C. Frère, O. Bryngdahl, Opt. Commun. 60, 369 (1986).
[CrossRef]

1969 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
[CrossRef]

Berriel Valdos, L. R.

Bryngdahl, O.

C. Frère, O. Bryngdahl, Opt. Commun. 60, 369 (1986).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Frère, C.

C. Frère, O. Bryngdahl, Opt. Commun. 60, 369 (1986).
[CrossRef]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
[CrossRef]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Ojeda-Castaneda, J.

Rosen, J.

Salik, B.

Yariv, A.

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969); T. J. Suleski, D. C. O’Shea, “Gray-scale masks for diffractive-optics fabrication. I. Commercial slide imagers,” Appl. Opt. (to be published).
[CrossRef]

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

Opt. Commun. (1)

C. Frère, O. Bryngdahl, Opt. Commun. 60, 369 (1986).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Schematic of the system used to obtain arbitrary paraxial focal lines. An example of a snake beam is shown by the thick line.

Fig. 2
Fig. 2

(a) Approximation of straight lines of the focal curve shown in Fig. 1. All the distances are in arbitrary units. (b) The real part of the Fourier hologram (128 × 128 pixels) that generates the beam shown in (a). (c) Output intensity distribution in the xz plane obtained by computer simulation of the system shown in Fig. 1. The small white cross indicates the front focus location.

Fig. 3
Fig. 3

(a) Real part of the Fourier hologram (128 × 128 pixels) that generates the treelike beam. (c) Output intensity distribution of the treelike beam in the xz plane obtained by computer simulation.

Equations (9)

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u ( x , y , z ) = exp [ j k ( z + 2 f ) ] j λ f - - g ( x i , y i ) × exp [ j 2 π ( x x i + y y i ) λ f - j π z ( x i 2 + y i 2 ) λ f 2 ] d x i d y i .
FST { g ( x i , y i ) exp [ j 2 π ( ξ x i + η y i ) ] } = u ( x - λ f ξ , y - λ f η , z ) .
FST { g ( x i , y i ) exp [ j 2 π γ ( x i 2 + y i 2 ) ] } 2 = u ( x , y , z - 2 λ f 2 γ ) 2 .
FST { g ( x i - α , y i - β ) } 2 = | u ( x - z α f , y - z β f , z ) | 2 = u ( x ¯ sec θ , y ¯ sec φ , z ¯ cos θ cos φ ) 2 ,
g ( r i ) = exp [ j 2 π ( r i / b ) 4 ] ,             R 1 r i R 2 , r i = x i 2 + y i 2
u ( z ) 2 = FST { g ( r i ) } x = y = 0 2 { C z 1 z z 2 C otherwise ,
z 2 - z 1 Δ z 8 λ f 2 R 0 Δ R b 4 ,
g ( x i , y i ) = n = 1 N rect { [ ( x i - α n ) 2 + ( y i - β n ) 2 ] 1 / 2 - R n Δ R n } × exp ( j 2 π { [ ( x i - α n ) 2 + ( y i - β n ) 2 ] 2 b 4 + ξ n x i + η n y i + γ n r i 2 } ) ,
rect ( x 2 Δ ) { 1 - Δ x Δ 0 otherwise , R n = b 2 2 f λ ( z 0 + L n 2 + i = 1 n - 1 L i ) 1 / 2 + i = 1 n max { α n , β n } , Δ R n = L n b 4 8 λ f 2 R n ,             z 0 = 2 λ f 2 b 2 , γ n = d n - ( z 0 + i = 1 n L i ) 2 λ f 2 ,             ξ n = h x , n λ f , η n = h y , n λ f ,             α n = f tan θ x , n ,

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