Abstract

In order to record holographic optical elements, recursive design techniques were developed for obtaining complex aspheric wave fronts from a relatively simple hologram. We present the mathematical formalism of this technique with a specific illustration of a holographic Fourier transform, designed with complex wave fronts that were derived recursively from two previous holograms. Both the calculated and the experimental results demonstrate a holographic lens that can handle input angles of up to ±12° with essentially diffraction-limited performance.

© 1988 Optical Society of America

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References

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  1. R. W. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  2. E. B. Champagne, “Nonparaxial imaging, magnification and aberration properties in holography,” J. Opt. Soc. Am. 57, 51–55 (1967).
    [CrossRef]
  3. A. Offner, “Ray tracing through a holographic system,” J. Opt. Soc. Am. 56, 1509–1512 (1966).
    [CrossRef]
  4. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698–2710 (1971).
    [CrossRef] [PubMed]
  5. J. N. Latta, “Computer-based analysis of hologram imagery and aberration,” Appl. Opt. 10, 599–618 (1971).
    [CrossRef] [PubMed]
  6. O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
    [CrossRef]
  7. W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
    [CrossRef]
  8. R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical element,” Opt. Eng. 21, 133–140 (1982).
    [CrossRef]
  9. K. A. Winick, J. R. Fienup, “Optimum holographic elements with nonspherical wave front,” J. Opt. Soc. Am. 73, 208–217 (1983).
    [CrossRef]
  10. J. Kedmi, A. A. Friesem, “Optimal holographic Fourier-transform lens,” Appl. Opt. 23, 4015–4019 (1984).
    [CrossRef] [PubMed]
  11. J. Kedmi, A. A. Friesem, “Optimized holographic optical elements,” J. Opt. Soc. Am. A 3, 2011–2018 (1986).
    [CrossRef]
  12. H. Chen, R. R. Hershey, E. N. Leith, “Design of a holographic lens for the infrared,” Appl. Opt. 26, 1983–1988 (1987).
    [CrossRef] [PubMed]
  13. J. N. Cederquist, J. R. Fienup, “Analytic design of optimum holographic optical elements,” J. Opt. Soc. Am. A 4, 699–705 (1987).
    [CrossRef]
  14. L. H. Lin, E. T. Doherty, “Efficient and aberration-free wavefront reconstruction from hologram illuminated at wavelength differing from the forming wavelength,” Appl. Opt. 10, 1314–1318 (1971).
    [CrossRef] [PubMed]
  15. Y. Amitai, A. A. Friesem, “Recursive design techniques for Fourier transform holographic lenses,” Opt. Eng. 26, 1133–1139 (1987).
    [CrossRef]
  16. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
    [CrossRef]

1987 (3)

1986 (1)

1984 (1)

1983 (1)

1982 (1)

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical element,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

1978 (1)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

1975 (1)

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

1971 (3)

1967 (1)

1966 (1)

1965 (1)

1964 (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Amitai, Y.

Y. Amitai, A. A. Friesem, “Recursive design techniques for Fourier transform holographic lenses,” Opt. Eng. 26, 1133–1139 (1987).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

Cederquist, J. N.

Champagne, E. B.

Chen, H.

Doherty, E. T.

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical element,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Fienup, J. R.

Friesem, A. A.

Hershey, R. R.

Kedmi, J.

Latta, J. N.

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

Leith, E. N.

Lin, L. H.

Meier, R. W.

Offner, A.

Rayces, J. L.

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Winick, K. A.

Appl. Opt. (5)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964).
[CrossRef]

Opt. Eng. (3)

O. Bryngdahl, “Computer-generated holograms as generalized optical components,” Opt. Eng. 14, 426–435 (1975).
[CrossRef]

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical element,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Y. Amitai, A. A. Friesem, “Recursive design techniques for Fourier transform holographic lenses,” Opt. Eng. 26, 1133–1139 (1987).
[CrossRef]

Prog. Opt. (1)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Geometry for a spherical wave.

Fig. 2
Fig. 2

Arrangement for transferring the wave front.

Fig. 3
Fig. 3

Transferring a wave front with an intermediate hologram. (a) Recording the intermediate hologram, (b) Reconstructing the intermediate hologram.

Fig. 4
Fig. 4

Geometries for (a) recording and (b) readout of a conventional HL.

Fig. 5
Fig. 5

The function G ( μ ).

Fig. 6
Fig. 6

The (a) recording and (b) reconstructing geometries for ro.

Fig. 7
Fig. 7

Aberrations for the uncorrected HL 0. FC, Field curvature; SA, spherical aberration.

Fig. 8
Fig. 8

Aberrations for the singly corrected HL 1. Abbreviations are as in Fig. 7.

Fig. 9
Fig. 9

Aberrations for the doubly corrected HL 2. Abbreviations are as in Fig. 7.

Fig. 10
Fig. 10

Spot diagram for 1.

Fig. 11
Fig. 11

Spot diagram for 2.

Fig. 12
Fig. 12

Experimental spot distribution in the focal plane of 1.

Fig. 13
Fig. 13

Experimental spot distribution in the focal plane of 2.

Equations (81)

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Φ i = Φ c ± ( Φ o Φ r ) ,
Φ q p = Φ i p q = Φ c p q ± ( Φ o p q Φ r p q ) .
Φ q p ( ξ , η ) = 2 π λ q p ( R q p r q p ) ,
r = [ R 2 ( 1 cos 2 α sin 2 β sin 2 α ) + ( η R sin α ) 2 + ( ξ R cos α sin β ) 2 ] 1 / 2 = ( R 2 + η 2 + ξ 2 2 R η sin α 2 R ξ cos α sin β ) 1 / 2 = R [ 1 + ( η R ) 2 + ( ξ R ) 2 2 sin α R η 2 cos α sin β R ξ ] 1 / 2 .
r = R [ 1 + 1 2 ( η R ) 2 + 1 2 ( ξ R ) 2 sin α R η cos α sin β R ξ + 1 8 ( ν 2 + ξ 2 ) 2 R 4 1 2 sin α R 3 ( η 3 + η ξ 2 ) 1 2 cos α sin β R 3 ( ξ 3 + ξ η 2 ) + 1 2 sin 2 α R 2 η 2 + 1 2 cos 2 α sin 2 β R 2 ξ 2 + sin α cos α sin β R 2 ξ η + ] .
Φ ( ξ , η ) = 2 π λ q [ 1 2 ( η 2 + ξ 2 ) R η sin α ξ cos α sin β + 1 8 ( η 2 + ξ 2 ) 2 R 3 1 2 sin α R 2 ( η 3 + η ξ 2 ) 1 2 cos α sin β R 2 ( ξ 3 + ξ η 2 ) + 1 2 sin 2 α R η 2 + 1 2 cos 2 α sin 2 β R ξ 2 + sin α cos α sin β R ξ η + ] .
Δ ( ξ , η ) = Φ i ( ξ , η ) Φ d ( ξ , η ) .
( ψ q p ) = ( ψ c p q ) ± μ p q [ ( ψ o p q ) ( ψ r p q ) ] .
( ψ q p ) = ψ q p ¯ ,
sin α i p = ( A c p ) ± μ p [ ( A o p ) ( A r p ) ] ,
cos α i p sin β i p = ( B c p ) ± μ p [ ( B o p ) ( B r p ) ] ,
1 R i p = ( c p ) ± μ p [ ( o p ) ( r p ) ] .
( A q p ) = sin α q p ,
( q p ) = cos α q p sin β q p ,
( q p ) = 1 R q p .
sin α d = sin α i ,
cos α d sin β d = cos α i sin β i .
Δ = Δ F + Δ S + Δ C + Δ A ,
Δ F = 2 π λ c 1 2 F ( η 2 + ξ 2 ) ;
Δ S = 2 π λ c 1 8 S ( η 2 + ξ 2 ) 2 ;
Δ C = 2 π λ c 1 2 [ Cx ( ξ 3 + ξ η 2 ) + Cy ( η 3 + η ξ 2 ) ;
Δ A = 2 π λ c ( 1 2 Ax ξ 2 + 1 2 Ay η 2 + Axy η ξ ) .
Q = ( Q c ) ± μ [ ( Q o ) ( Q r ) ] ( Q d ) ,
( F q p ) = 1 R q p ,
( S q p ) = ( 1 R q p ) 3 ,
( Cx q p ) = cos α q p sin β q p ( R q p ) 2 ,
( Cy q p ) = sin α q p ( R q p ) 2 ,
( Ax q p ) = cos 2 α q p sin 2 β q p R q p ,
( Ay q p ) = sin 2 α q p R q p ,
( Axy q p ) = cos α q p sin α q p sin β q p R q p ·
l i = ( l c ) ± μ [ ( l o ) ( l r ) ] ,
m i = ( m c ) ± μ [ ( m o ) ( m r ) ] ,
n i = ± ( 1 l i 2 m i 2 ) 1 / 2 .
[ l q p ( ξ , η ) ] = R q p cos α q p sin β q p ξ r q p ,
[ m q p ( ξ , η ) ] = R q p sin α q p η r q p ,
sin β o ( x ) x R o 1 2 x 3 R o 3 ,
R o ( x ) = R o cos β o ( x ) R o 1 1 2 sin 2 β o ( x ) R o 1 1 2 ( x R o ) 2 + 1 2 ( x R o ) 4 ,
sin β r ( x ) = sin β r ,
β c ( x ) = β c + Δ β c 180 ° + β r x cos β r R t ·
sin β c ( x ) = sin ( 180 ° + β r + Δ β c ) sin β r Δ β c cos β r sin β r + x cos 2 β r R t ·
sin β i ( x ) x cos 2 β r R t x R o ·
sin β i ( x ) 0 .
R d ( x ) R o .
S ( x ) = 3 2 x 2 ( R o ) 5 ,
Cx ( x ) = x ( R o ) 3 + 3 2 x 3 ( R o ) 5 ,
Ax ( x ) = x 2 ( R o ) 3 + 3 2 x 4 ( R o ) 5 ,
F ( x ) = 1 2 x 2 ( R o ) 3 1 2 x 4 ( R o ) 5 ·
sin β q r ( x ) = sin β q r ,
R q r ( x ) = R q r = .
1 R q r ( x ) 1 R q r 1 2 x 2 ( R q r ) 3 + 1 2 x 4 ( R q r ) 5 ,
sin β q r ( x ) x R q r 1 2 ( x R q r ) 3 ·
S r ( x ) = [ ( 1 R c r ) 3 μ r ( 1 R r r ) 3 ] ,
Cx r ( x ) = x [ ( 1 R c r ) 3 μ r ( 1 R r r ) 3 ] 3 2 x 3 [ ( 1 R c r ) 5 μ r ( 1 R r r ) 5 ] ,
Ax r ( x ) = x 2 [ ( 1 R c r ) 3 μ r ( 1 R r r ) 3 ] 3 2 x 4 [ ( 1 R c r ) 5 μ r ( 1 R r r ) 5 ] ,
F r ( x ) = 1 2 x 2 [ ( 1 R c r ) 3 μ r ( 1 R r r ) 3 ] + 1 2 x 4 [ ( 1 R c r ) 5 μ r ( 1 R r r ) 5 ] ·
( 1 R o ) 3 = ( 1 R c r ) 3 μ r ( 1 R r r ) 3 ·
1 R c r μ r 1 R r r = 0 ,
sin β r = sin β i r = μ r sin β o r .
β o r = arcsin ( 1 μ r sin β r ) ,
R r r = R o [ ( μ r ) 3 μ r ] 1 / 3 ,
R c r = R o [ ( μ r ) 3 μ r ] 1 / 3 μ r ·
S ( x ) S r ( x ) = ( 1 R o ) 3 ·
Cx ( x ) = 3 2 x 3 R o 5 G ( μ r ) ,
Ax ( x ) = 3 2 x 4 R o 5 G ( μ r ) ,
F ( x ) = 1 2 x 4 R o 5 G ( μ r ) ,
G ( μ ) = μ 5 μ ( μ 3 μ ) 5 / 3 1 .
1 R c r + μ r ( 1 R o r 1 R r r ) = 0 ,
( 1 R c r ) 3 + μ r [ ( 1 R o r ) 3 ( 1 R r r ) 3 ] = ( 1 R o ) 3 ·
1 R o r = 1 R o 1 [ ( 2 μ r ) 3 2 μ r ] 1 / 3 ,
1 R r r = 1 R o 1 [ ( 2 μ r ) 3 2 μ r ] 1 / 3 ,
1 R c r = 2 μ r 1 R o 1 [ ( 2 μ r ) 3 2 μ r ] 1 / 3 ·
S r ( x ) = { ( 1 R c r ) 3 + μ r [ ( 1 R o r ) 3 ( 1 R r r ) 3 ] } ,
Cx r ( x ) = x { ( 1 R c r ) 3 + μ r [ ( 1 R o r ) 3 ( 1 R r r ) 3 ] } 3 2 x 3 { ( 1 R c r ) 5 + μ r [ ( 1 R o r ) 5 ( 1 R r r ) 5 ] } ,
Ax r ( x ) = x 2 { ( 1 R c r ) 3 + μ r [ ( 1 R o r ) 3 ( 1 R r r ) 3 ] } 3 2 x 4 { ( 1 R c r ) 5 + μ r [ ( 1 R o r ) 5 ( 1 R r r ) 5 ] } ,
F r ( x ) = 1 2 x 2 { ( 1 R c r ) 3 + μ r [ ( 1 R o r ) 3 ( 1 R r r ) 3 ] } + 1 2 x 4 { ( 1 R c r ) 5 + μ r [ ( 1 R o r ) 5 ( 1 R r r ) 5 ] } ·
Cx ( x ) = 3 2 x 3 R o 5 G ( 2 μ r ) ,
Ax ( x ) = 3 2 x 4 R o 5 G ( 2 μ r ) ,
F ( x ) = 1 2 x 4 R o 5 G ( 2 μ r ) .
R o r o = R o r , β o r o = 0 , 1 R o r o = 0 , β r r o = β r , λ o r o = λ c r o = λ o r μ r o = 1 , 1 R c r o = 0 , β c r o = 0 .
R i r o = R o r ,
β i r o = β o r = β r .

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