Abstract

A novel technique for designing holographic optical elements that can perform general types of coordinate transformation is presented. The design is based on analytic ray-tracing techniques for finding the grating vector of the element, from which the holographic grating function is obtained as a solution of a Poissonlike equation. The grating function can be formed either as a computer-generated or as a computer-originated hologram. The design and realization procedure are illustrated for a specific holographic element that performs a logarithmic coordinate transformation on two-dimensional patterns.

© 1992 Optical Society of America

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References

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  1. O. Bryngdahl, “Geometrical transformation in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
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    [CrossRef]
  3. J. Cederquist, A. M. Tai, “Computer-generated holograms for geometric transformations,” Appl. Opt. 23, 3099–3104 (1984).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
    [CrossRef]
  6. D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” Prog. Opt. 17, 289–356 (1978).
    [CrossRef]
  7. S. K. Case, P. R. Haugen, O. J. Løkberg, “Multifacet holographic optical elements for wave front transformations,” Appl. Opt. 20, 2670–2675 (1981).
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  13. J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, New York, 1981).
    [CrossRef]
  14. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
    [CrossRef] [PubMed]
  15. J. Kedmi, A. A. Friesem, “Optimized holographic optical elements,” J. Opt. Soc. Am. A 3, 2011–2018 (1986).
    [CrossRef]
  16. W. H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
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1990 (1)

1989 (1)

1987 (2)

1986 (1)

1984 (1)

1983 (1)

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

1982 (1)

G. Hausler, N. Streibl, “Optical compensation of geometrical distortion by a deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

1981 (1)

1978 (1)

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” Prog. Opt. 17, 289–356 (1978).
[CrossRef]

1974 (2)

Becker, M.

M. Becker, The Principles and Applications of Variational Methods (MIT Press, Cambridge, Mass., 1975).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Bryngdahl, O.

Casasent, D.

Case, S. K.

Cederquist, J.

Cederquist, J. N.

Friesem, A. A.

Goodman, J. W.

R. K. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
[CrossRef] [PubMed]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, New York, 1981).
[CrossRef]

Hasman, E.

Haugen, P. R.

Hausler, G.

G. Hausler, N. Streibl, “Optical compensation of geometrical distortion by a deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Hesselink, L.

Kedmi, J.

Komatsu, S.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Kostuk, R. K.

Lee, A. J.

Lee, W. H.

Løkberg, O. J.

Ohzu, H.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Psaltis, D.

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” Prog. Opt. 17, 289–356 (1978).
[CrossRef]

Saito, Y.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Song, J. Z.

Streibl, N.

G. Hausler, N. Streibl, “Optical compensation of geometrical distortion by a deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Stuff, M. A.

Tai, A. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Xia, S. F.

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

Y. Saito, S. Komatsu, H. Ohzu, “Scale and rotation invariant real time optical correlator using computer generated hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

G. Hausler, N. Streibl, “Optical compensation of geometrical distortion by a deformable mirror,” Opt. Commun. 42, 381–385 (1982).
[CrossRef]

Prog. Opt. (1)

D. Casasent, D. Psaltis, “Deformation invariant, space-variant optical pattern recognition,” Prog. Opt. 17, 289–356 (1978).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

M. Becker, The Principles and Applications of Variational Methods (MIT Press, Cambridge, Mass., 1975).

J. W. Goodman, “Linear space-variant optical data processing,” in Optical Information Processing, S. H. Lee, ed. (Springer-Verlag, New York, 1981).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Coordinate transformation arrangement with a HOE.

Fig. 2
Fig. 2

Optical arrangement for separable 2-D coordinate transformations. The first HOE performs the x-coordinate transformation and the second HOE performs the y-coordinate transformation.

Fig. 3
Fig. 3

Geometry of the coordinate transformation with high inclination angles.

Fig. 4
Fig. 4

Lateral errors for logarithmic coordinate transformation for a HOE with an optimal grating function (continuous curve) and for a HOE with a paraxially approximated grating function (dashed curve).

Fig. 5
Fig. 5

Experimental results of the 2-D logarithmic geometric transformation for three different inputs. The inputs are shown on the left and the corresponding output planes are shown on the right. The actual size for all is 20 mm by 20 mm.

Equations (34)

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K i = λ 2 π ϕ i , K o = λ 2 π ϕ o ,
K h = λ 2 π ϕ h = K h x x ˆ + K h y y ˆ = λ Λ x x ˆ + λ Λ y y ˆ ,
K o x = K i x + K h x , K o y = K i y + K h y , K o z = ( 1 K o x 2 K o y 2 ) 1 / 2
K o x ( x , y ) = u ( x , y ) x r , K o y ( x , y ) = υ ( x , y ) y r ,
r 2 = [ u ( x , y ) x ] 2 + [ υ ( x , y ) y ] 2 + z 2 .
K h x ( x , y ) = u ( x , y ) x r sin ( α i ) , K h y ( x , y ) = υ ( x , y ) y r .
T ( u , υ , z ) = + + d x d y { g ( x , y ) exp [ i k h ( x , y ) ] } ,
g ( x , y ) = z i λ r 2 t ( x , y ) , h ( x , y ) = ϕ ˆ h ( x , y ) + r ,
h x = h y = 0 .
K h x ( x , y ) = ϕ ˆ h x = r x = u ( x , y ) x r , K h y ( x , y ) = ϕ ˆ h y = r y = υ ( x , y ) y r .
T ( u , υ , z ) t ( x s , y s ) exp { i [ k h ( x s , y s ) + π 4 ] } × ( h x x h y y h x y 2 ) 1 / 2 z r ( x s , y s ) 2 ,
ϕ h ( x , y ) = 2 π λ K h ( x , y ) .
ϕ h ( x ) = 2 π λ K h ( x ) d x ,
K h x ( x , y ) y = K h y ( x , y ) x .
ϕ h ( x , y ) = 2 π λ K h x ( x ) d x + 2 π λ K h y ( y ) d y .
K h = K h x ( x ) x ˆ + K h y ( y ) y ˆ ,
L [ ϕ h ( x , y ) ] = A d x d y [ ( ϕ h x 2 π λ K h x ) 2 + ( ϕ h y 2 π λ K h y ) 2 ] ,
δ L [ ϕ h ( x , y ) ] δ ϕ h ( x , y ) = 0 ,
L = d x d y [ F ( x , y , ϕ h , ϕ h x , ϕ h y ) ] .
δ F δ ϕ h δ δ x δ F δ ( ϕ h / x ) δ δ y δ F δ ( ϕ h / y ) = 0 .
2 ϕ = 2 π λ div ( K h ) ,
2 ϕ = 2 π λ r 3 { ( u x 1 ) [ z 2 + ( υ y ) 2 ] + ( υ y 1 ) [ z 2 + ( u x ) 2 ] ( u x ) ( υ y ) ( u y + υ x ) } ·
2 ϕ = 2 π λ z ( u x + υ y 2 ) .
SBP c t D p D 2 λ z = D λ F # = SBP imag ,
ξ = λ r p 1 cos 2 α ,
z opt = 2 Δ .
SBP c t ( z = z opt ) ( D 2 1 . 41 λ Δ max ) 1 / 2 = ( D 1 . 41 λη ) 1 / 2 ,
α max = arcsin ( λ / Λ min ) ,
tan ( α max ) s Δ max z = s η D z ,
F # min 2 η Λ min λ .
max ( SPB c t ) ( D 2 η Λ min ) 1 / 2 = ( SBP plot 2 η ) 1 / 2 ,
u ( x ) = a * ln ( x ) + b , υ ( y ) = a * ln ( y ) + b ,
ϕ paraxial ( x , y ) = 2 π λ z [ a ( x ln x x ) + b x 0 . 5 x 2 + a ( y ln y y ) + b y 0 . 5 y 2 ] .
2 ϕ h = 2 π λ r 3 ( ( a x 1 ) { z 2 + [ υ ( y ) y ] 2 } + ( a y 1 ) { z 2 + [ u ( x ) x ] 2 } ) .

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