Abstract

A method for designing and recording transmission holographic lenses, having low aberrations and high diffraction efficiencies, in the presence of a recording–readout wavelength shift, is presented. The method is based on a recursive design technique, in which the final hologram is recorded with complex wave fronts that are derived from intermediate holograms. The design is illustrated for a lens with an f-number of 2.5 and a large offset angle, recorded at 488 nm and read out at 633 nm. A nearly diffraction-limited spot size and an efficiency of >80% are measured.

© 1990 Optical Society of America

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References

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  1. B. J. Chang, “Dichromated gelatin holograms and their applications,”Opt. Eng. 19, 642–648 (1980).
    [CrossRef]
  2. R. W. Meier, “Magnification and third-order aberrations in holography,” J. Opt. Soc. Am. 55, 987–992 (1965).
  3. E. B. Champagne, “Nonparaxial imaging, magnification, and aberration properties in holography,” J. Opt. Soc. Am. 57, 51–55 (1967).
    [CrossRef]
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,”Bell Syst. Tech. J. 48, 2909–2947 (1969).
  5. L. H. Lin, E. T. Doherty, “Efficient and aberration-free wavefront reconstruction from holograms illuminated at wavelengths differing from the forming wavelength,” Appl. Opt. 10, 1314–1318 (1971).
    [CrossRef] [PubMed]
  6. M. Assenheimer, Y. Amitai, A. A. Friesem, “Recursive design of an efficient holographic optical element with different recording and readout wavelengths,” Appl. Opt. 27, 4747–4752 (1988).
    [CrossRef] [PubMed]
  7. H. Chen, R. R. Hershey, E. Leith, “Design of a holographic lens for the infrared,” Appl. Opt. 26, 1983–1988 (1987).
    [CrossRef] [PubMed]
  8. H. P. Herzig, “Holographic optical elements (HOE) for semiconductor lasers,” Opt. Commun. 58, 144–148 (1986).
    [CrossRef]
  9. M. R. Latta, R. V. Pole, “Design techniques for forming 488-nm holographic lenses with reconstruction at 633 nm,” Appl. Opt. 18, 2418–2421 (1979).
    [CrossRef] [PubMed]
  10. I. A. Mikhailov, “A geometrical analysis of thick holograms,” Opt. Spektrosk. 58, 374–377 (1985).
  11. H. Chen, Q. Shan, “Using holographically generated corrector plates to fabricate low f/no. HOE objectives and collimators,” Appl. Opt. 27, 3542–3550 (1988).
    [CrossRef] [PubMed]
  12. K. Winick, “Designing efficient aberration-free holographic lenses in the presence of a construction–reconstruction wavelength shift,” J. Opt. Soc. Am. 72, 143–148 (1982).
    [CrossRef]
  13. Y. Amitai, A. A. Friesem, “Design of holographic optical elements by using recursive technique,” J. Opt. Soc. Am. A 5, 702–712 (1988).
    [CrossRef]
  14. J. N. Latta, “Computer-based analysis of hologram imagery and aberration,” Appl. Opt. 10, 599–608 (1971).
    [CrossRef] [PubMed]
  15. P. C. Mehta, K. Syam Sunder Rao, R. Hradaynath, “Higher order aberrations in holographic lenses,” Appl. Opt. 21, 4553–4558 (1982).
    [CrossRef] [PubMed]

1988 (3)

1987 (1)

1986 (1)

H. P. Herzig, “Holographic optical elements (HOE) for semiconductor lasers,” Opt. Commun. 58, 144–148 (1986).
[CrossRef]

1985 (1)

I. A. Mikhailov, “A geometrical analysis of thick holograms,” Opt. Spektrosk. 58, 374–377 (1985).

1982 (2)

1980 (1)

B. J. Chang, “Dichromated gelatin holograms and their applications,”Opt. Eng. 19, 642–648 (1980).
[CrossRef]

1979 (1)

1971 (2)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,”Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967 (1)

1965 (1)

Amitai, Y.

Assenheimer, M.

Champagne, E. B.

Chang, B. J.

B. J. Chang, “Dichromated gelatin holograms and their applications,”Opt. Eng. 19, 642–648 (1980).
[CrossRef]

Chen, H.

Doherty, E. T.

Friesem, A. A.

Hershey, R. R.

Herzig, H. P.

H. P. Herzig, “Holographic optical elements (HOE) for semiconductor lasers,” Opt. Commun. 58, 144–148 (1986).
[CrossRef]

Hradaynath, R.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,”Bell Syst. Tech. J. 48, 2909–2947 (1969).

Latta, J. N.

Latta, M. R.

Leith, E.

Lin, L. H.

Mehta, P. C.

Meier, R. W.

Mikhailov, I. A.

I. A. Mikhailov, “A geometrical analysis of thick holograms,” Opt. Spektrosk. 58, 374–377 (1985).

Pole, R. V.

Shan, Q.

Syam Sunder Rao, K.

Winick, K.

Appl. Opt. (7)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,”Bell Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am. (3)

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

Opt. Commun. (1)

H. P. Herzig, “Holographic optical elements (HOE) for semiconductor lasers,” Opt. Commun. 58, 144–148 (1986).
[CrossRef]

Opt. Eng. (1)

B. J. Chang, “Dichromated gelatin holograms and their applications,”Opt. Eng. 19, 642–648 (1980).
[CrossRef]

Opt. Spektrosk. (1)

I. A. Mikhailov, “A geometrical analysis of thick holograms,” Opt. Spektrosk. 58, 374–377 (1985).

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Figures (8)

Fig. 1
Fig. 1

Recording and readout geometries.

Fig. 2
Fig. 2

Schematic arrangement for recording and readout of intermediate and final holograms. Open circles represent spherical waves; horizontal bars represent holograms.

Fig. 3
Fig. 3

Predicted spot size for the uncorrected hologram.

Fig. 4
Fig. 4

Predicted spot size for the corrected hologram.

Fig. 5
Fig. 5

Transferring a wave front with a unity-magnification telescope.

Fig. 6
Fig. 6

Transferring a wave front with an intermediate hologram. (a) Recording of the intermediate hologram m. (b) Recording of the next hologram p2.

Fig. 7
Fig. 7

Actual spot size for the noncorrected hologram.

Fig. 8
Fig. 8

Actual spot size for the corrected hologram.

Equations (64)

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ϕ i = ϕ c ± ( ϕ o ϕ r ) ,
sin ( β i ) = sin ( β c ) ± μ [ sin ( β o ) sin ( β r ) ]
1 R i = 1 R c ± μ ( 1 R o 1 R r ) ,
Δ ξ = ϕ i ϕ d = 2 π λ c ( 1 8 S ξ 4 + C ξ 3 1 2 A ξ 2 ) ,
S = 1 R c 3 1 R i 3 ± μ ( 1 R o 3 1 R r 3 ) ,
C = sin ( β c ) R c 2 sin ( β i ) R i 2 ± μ [ sin ( β o ) R o 2 sin ( β r ) R r 2 ] ,
A = sin 2 ( β c ) R c sin 2 ( β i ) R i ± μ [ sin 2 ( β o ) R o sin 2 ( β r ) R r ] .
D 3 l = sin l β c R c 3 l sin l β i R i 3 l ± μ ( sin l β o R o 3 l sin l β r R r 3 l ) ,
1 R o = a 1 R i + b 1 R c , sin β o = a sin β i + b sin β c ,
1 R r = a 1 R c + b 1 R i , sin β r = a sin β c + b sin β i ,
a μ + 1 2 μ , b μ 1 2 μ .
ϕ p ϕ i p = ϕ c p ± ( ϕ o p ϕ r p ) , p = o , r .
1 R p 1 R i p = 1 R c p ± μ p ( 1 R o p 1 R r p ) ,
sin β p sin β i p = sin β c p ± μ p ( sin β o p sin β r p ) , p = o , r .
ϕ r o = ϕ r r = 0 , sin β r o = sin β r r 1 R r o = 1 R r r = 0 .
1 R o o = a 1 R i , sin β o o = a sin β i , 1 R c o = b 1 R c , sin β c o = b sin β c , 1 R o r = a 1 R c , sin β o r = a sin β c , 1 R c r = b 1 R i , sin β c r = b sin β i .
ϕ i = ϕ c + ( ϕ c o + ϕ o o ) ( ϕ c r + ϕ o r ) .
D 3 l = sin l β c R c 3 l sin l β i R i 3 l + μ [ sin l β c o ( R c o ) 3 l + sin l β o o ( R o o ) 3 l sin l β c r ( R c r ) 3 l sin l β o r ( R o r ) 3 l ] , l = 0 , 1 , 2 .
D 3 l = sin l β c R c 3 l sin l β i R i 3 l + μ ( a 3 sin l β i R i 3 l + b 3 sin l β c R c 3 l b 3 sin l β c R i 3 l a 3 sin l β c R c 3 l ) = ( sin l β c R c 3 l sin l β i R i 3 l ) [ 1 μ ( a 3 b 3 ) ] = D 3 l c D 3 l i ( l = 0 , 1 , 2 ) ,
D 3 l q sin l β q R q 3 l [ 1 μ ( a 3 b 3 ) ] .
ϕ r p ϕ i p r = ϕ c p r ± ( ϕ o p r ϕ r p r ) , p = o , r .
1 R r p 1 R i p r = 1 R c p r ± μ p r ( 1 R o p r 1 R r p r ) = 0 , sin β r p sin β i p r = sin β c p r ± μ p r ( sin β o p r sin β r p r ) = 0 , p = o , r .
ϕ o o r = ϕ o r r = 0 , sin β o o r = sin β o r r = 1 R o o r = 1 R o r r = 0 .
μ o r = μ r r ν 1 .
1 R c p r = ν 1 R r p r , sin β c p r = ν sin β r p r , p = o , r .
ϕ i = ϕ c + [ ϕ c o + ϕ o o ( ϕ c o r ϕ r o r ) ] [ ϕ c r + ϕ o r ( ϕ c r r ϕ r r r ) ] ,
D 3 l = D 3 l c D 3 l i μ ( D 3 l o r D 3 l r r ) ,
D 3 l p r = sin l β c p r ( R c p r ) 3 l ν sin l β r p r ( R r p r ) 3 l , p = o , r .
D 3 l p r = ( ν 3 ν ) sin l β r p r ( R r p r ) 3 l , p = o , r .
μ D 3 l o r = D 3 l i , μ D 3 l r r = D 3 l c .
μ ( ν 3 ν ) sin l β r o r ( R r o r ) 3 l = sin l β i R i 3 l [ 1 μ ( a 3 b 3 ) ]
sin l β r o r ( R r o r ) 3 l = sin l β i R r 3 l [ 1 μ ( a 3 b 3 ) v v 3 ] .
1 R r o r = γ 1 R i , sin β r o r = γ sin β i ,
γ = [ 1 μ ( a 3 b 3 ) ν ν 3 ] 1 / 3 .
1 R c o r = ν γ 1 R i , sin β c o r = ν γ sin β i .
1 R r r r = γ 1 R c , sin β r r r = γ sin β c ,
1 R c r r = ν γ 1 R c , sin β c r r = ν γ sin β c .
D 5 l = sin l β c R c 5 l sin l β i R i 5 l ± μ ( sin l β o R o 5 l sin l β r R r 5 l )
D 5 l = sin l β c R c 5 l sin l β i R i 5 l + μ { sin l β c o ( R c o ) 5 l + sin l β o o ( R o o ) 5 l [ sin l β c o r ( R c o r ) 5 l ν sin l β r o r ( R r o r ) 5 l ] sin l β c r ( R c r ) 5 l sin l β o r ( R o r ) 5 l + [ sin l β c r r ( R c r r ) 5 l ν sin l β r r r ( R r r r ) 5 l ] } .
D 5 l = ( sin l β c R c 5 l sin l β i R i 5 l ) [ 1 μ ( a 5 b 5 ) μ γ 5 ( ν ν 5 ) ] .
1 μ ( a 5 b 5 ) μ ( ν ν 5 ) = γ 5 .
[ 1 μ ( a 5 b 5 ) ν ν 5 ] 3 = [ 1 μ ( a 3 b 3 ) ν ν 3 ] 5
( ν ν 3 ) 5 ( ν ν 5 ) 3 = [ 1 μ ( a 3 b 3 ) ] 5 [ 1 μ ( a 5 b 5 ) ] 3 .
ν 1 ± { 2 [ 1 μ ( a 3 b 3 ) ] 5 [ 1 μ ( a 5 b 5 ) ] 3 } 1 / 2 .
R i = 100 mm , β i = 20 ° , R c = 100 mm , β c = 20 ° , λ c = 633 nm , λ o = 488 nm μ = 1 .29 , d h = 20 mm ,
R o = 129.7 mm , β o = 15.29 ° , R r = 129.7 mm , β r = 15.29 °
ν = 1 ± 0.0224 .
( λ o p r ) 1 = 477.1 nm , ( λ o p r ) 2 = 498.9 nm ( p = o , r ) .
ν = 1.028 .
R o o = 112.9 mm , β o o = 17.636 ° , R c o = 874.0 mm , β c o = 2.24 ° , R o r = 112.9 mm , β o r = 17.636 ° R c r = 874.0 mm , β c r = 2.24 ° , R r o r = 87.5 mm , β r o r = 23.01 ° , R c o r = 90.0 mm , β c o r = 22.34 ° , R r r r = 87.5 mm , β r r r = 23.01 ° , R c r r = 90.0 mm , β c r r = 22.34 ° , .
( ν 3 ν 5 ) ( ν 5 ν ) 3 = χ ,
χ [ 1 μ ( a 3 b 3 ) ] [ 1 μ ( a 5 b 5 ) ] 3 5
lim ν 1 x ( ν ) = 0 .
χ = ( ν 3 ν ) 5 ( ν 5 ν ) 3 = ν 5 ( ν 2 1 ) 5 ν 3 ( ν 4 1 ) 3 = ν 2 ( ν 2 1 ) 2 ( ν 2 + 1 ) 3 .
χ ( ν ) ψ = χ ln χ ν = χ ( 2 ν + 4 ν ν 2 1 6 ν ν 2 + 1 ) .
lim ν χ ( ν ) = 1 ,
ν 1 ( χ ) = χ 1 ( ν ) for 0 < χ < 1 , 1 < ν .
ν 2 ( χ ) = χ 1 ( ν ) for 0 < χ < 0.25 , 0.5 < ν < 1 .
ν 1 ( χ ) = 1.0286 , ν 2 ( χ ) = 0.9714 .
ν 1 ( χ ) = 1 + 1 , ν 2 ( χ ) = 1 2 ,
χ = [ ( 1 + ) 3 ( 1 + ) ] 5 [ ( 1 + ) 5 ( 1 + ) ] 3 .
χ = ( 1 + 3 + 3 2 + 3 1 ) 5 ( 1 + 5 + 10 2 + 10 3 + 5 4 + 5 1 ) 3 ( 2 + ) 5 ( 4 + ) 3 32 5 64 3 = 2 2 .
± 2 χ
ν 1 , 2 ( χ ) 1 ± 2 χ ,

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