Abstract

Some geometrical transforms cannot be implemented as computer-generated holograms (CGH’s) with Bryngdahl’s technique. The phase functions of these geometrical transforms have phase singularities that cause branch points in the transmission functions of CGH’s. An example of such a geometrical transform is the rotation transform. This transform is implemented by including a superposition of phase singularities in the phase function of the CGH. I present results of this implementation.

© 1993 Optical Society of America

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References

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  1. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [Crossref]
  2. W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
    [Crossref]
  3. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
    [Crossref]
  4. C-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [Crossref] [PubMed]
  5. F. S. Roux, “Implementation of general point transforms like the Hough transform, with diffractive optics,” Appl. Opt. (to be published).
  6. F. S. Roux, “Implementation of general optical transforms in terms of point spread transforms,” Ph.D. dissertation (University of Pretoria, Pretoria, South Africa, 1990).
  7. H. Bartlet, S. K. Case, “Coordinate transformations via multifacet holographic optical elements,” Opt. Eng. 22, 497–500 (1983).

1991 (1)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[Crossref]

1987 (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[Crossref]

1983 (2)

H. Bartlet, S. K. Case, “Coordinate transformations via multifacet holographic optical elements,” Opt. Eng. 22, 497–500 (1983).

C-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[Crossref] [PubMed]

1974 (1)

Bartlet, H.

H. Bartlet, S. K. Case, “Coordinate transformations via multifacet holographic optical elements,” Opt. Eng. 22, 497–500 (1983).

Bryngdahl, O.

Case, S. K.

H. Bartlet, S. K. Case, “Coordinate transformations via multifacet holographic optical elements,” Opt. Eng. 22, 497–500 (1983).

Dahdouh, A.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[Crossref]

Darling, A. M.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[Crossref]

Han, C-Y.

Hossack, W. J.

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[Crossref]

Ishii, Y.

Murata, K.

Roux, F. S.

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[Crossref]

F. S. Roux, “Implementation of general point transforms like the Hough transform, with diffractive optics,” Appl. Opt. (to be published).

F. S. Roux, “Implementation of general optical transforms in terms of point spread transforms,” Ph.D. dissertation (University of Pretoria, Pretoria, South Africa, 1990).

Appl. Opt. (1)

J. Mod. Opt. (1)

W. J. Hossack, A. M. Darling, A. Dahdouh, “Coordinate transforms with multiple computer-generated optical elements,” J. Mod. Opt. 34, 1235–1250 (1987).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

H. Bartlet, S. K. Case, “Coordinate transformations via multifacet holographic optical elements,” Opt. Eng. 22, 497–500 (1983).

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[Crossref]

Other (2)

F. S. Roux, “Implementation of general point transforms like the Hough transform, with diffractive optics,” Appl. Opt. (to be published).

F. S. Roux, “Implementation of general optical transforms in terms of point spread transforms,” Ph.D. dissertation (University of Pretoria, Pretoria, South Africa, 1990).

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Figures (2)

Fig. 1
Fig. 1

Part of the transmission function of a 60° rotation transform.

Fig. 2
Fig. 2

Simulated output plane diffraction pattern of the 60° rotation transform CGH.

Equations (25)

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2 Ψ x y = 2 Ψ y x ,
c Ψ · d s = 0 ,
Ψ = 2 π F .
u ( x , y ) y = υ ( x , y ) y ,
c Ψ · d s = n 2 π ,
arg ( x a , y b ) = arctan ( y b x a ) for x > 0 , y > 0 , arg ( x a , y b ) = arctan ( y b x a ) + π for x < 0 , arg ( x a , y b ) = arctan ( y b x a ) + 2 π for x > 0 , y < 0 ,
arg ( x , y ) = ϕ ,
Ψ ( x , y ) = Ψ B ( x , y ) + n σ n arg ( y y n , x x n ) ,
u ( x , y ) = x ( cos α ) y ( sin α ) , υ ( x , y ) = x ( sin α ) + y ( cos α ) ,
u ( x , y ) y = sin α, υ ( x , y ) x = sin α .
x = ρ cos ϕ , y = ρ sin ϕ ,
u ( ρ , ϕ ) = ρ cos ( ϕ + α ) , υ ( ρ , ϕ ) = ρ sin ( ϕ + α ) ,
F ( ρ , ϕ ) 1 λ L { [ ρ cos ( ϕ + α ) ρ cos ϕ ] e ˆ x + [ ρ sin ( ϕ + α ) ρ sin ϕ ] e ˆ y } ,
e ˆ x = ( cos ϕ ) e ˆ ρ ( sin ϕ ) e ˆ ϕ , e ˆ y = ( sin ϕ ) e ˆ ρ + ( cos ϕ ) e ˆ ϕ ,
F ( ρ , ϕ ) = ρ λ L { [ ( cos α ) 1 ] e ˆ ρ + ( sin α ) e ˆ ϕ } .
2 π F = Ψ = Ψ ρ e ˆ ρ + 1 ρ Ψ ϕ e ˆ ϕ .
Ψ ρ = ρ k L ( cos α 1 ) ,
Ψ ϕ = ρ 2 k L sin α .
Ψ ρ = ρ 2 k 2 L [ ( cos α ) 1 ] + f ( ϕ ) ,
0 2 π Ψ ϕ d ϕ = n 2 π .
0 2 π Ψ ϕ d ϕ = 2 π ρ 2 k L sin α .
ρ n = ( ( n 0 . 5 ) L k sin α ) 1 / 2 .
ϕ n = ϕ n 1 + ( ρ 1 / ρ n ) .
Ψ ( ρ , ϕ ) = ρ 2 k 2 L ( cos α 1 ) + n B n ( ρ , ϕ ) ,
B n ( ρ , ϕ ) = arg [ ρ ( cos ϕ ) ρ n ( cos ϕ n ) , ρ ( sin ϕ ) ρ n ( sin ϕ n ) ] .

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