Abstract

We describe a double wavelength shift recording scheme for constructing holographic optical element (HOE) objectives and collimating lenses. The lenses are fabricated at intermediate wavelengths for use in a longer wavelength region. A holographic corrector plate is generated with a third wavelength that is the shortest of the three. Aberration of the HOE is compensated by inserting the corrector plate in one of the beams used in the construction process.

© 1988 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled Wave Theory for Thick Holographic Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).
  2. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
    [Crossref]
  3. R. J. Collier, C. B. Burkhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  4. R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
    [Crossref]
  5. H. Chen, R. R. Hershey, E. N. Leith, “Design of a Holographic Lens for the Infrared,” Appl. Opt. 26, 1983 (1987).
    [Crossref] [PubMed]
  6. J. Upatnieks, A. VanderLugt, E. N. Leith, “Correction of Lens Aberrations by Means of Holograms,” Appl. Opt. 5, 589 (1966).
    [Crossref] [PubMed]
  7. J. N. Latta, “Analysis of Multiple Hologram Optical Elements with Low Dispersion and Low Aberrations,” Appl. Opt. 11, 1686 (1972).
    [Crossref] [PubMed]
  8. C. W. Helstrom, “Image Luminance and Ray Tracing in Holography,” J. Opt. Soc. Am. 56, 433 (1966).
    [Crossref]
  9. E. B. Champagne, “Nonparaxial Imaging in Holography,” J. Opt. Soc. Am. 57, 51 (1967).
    [Crossref]

1987 (1)

1982 (1)

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[Crossref]

1981 (1)

1972 (1)

1969 (1)

H. Kogelnik, “Coupled Wave Theory for Thick Holographic Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

1967 (1)

1966 (2)

Burkhardt, C. B.

R. J. Collier, C. B. Burkhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Champagne, E. B.

Chen, H.

Collier, R. J.

R. J. Collier, C. B. Burkhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[Crossref]

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, “Computer-Originated Aspheric Holographic Optical Elements,” Opt. Eng. 21, 133 (1982).
[Crossref]

Gaylord, T. K.

Helstrom, C. W.

Hershey, R. R.

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Holographic Gratings,” Bell Syst. Tech. J. 48, 2909 (1969).

Latta, J. N.

Leith, E. N.

Lin, L. H.

R. J. Collier, C. B. Burkhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Moharam, M. G.

Upatnieks, J.

VanderLugt, A.

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Figures (9)

Fig. 1
Fig. 1

Recording geometry of the double wavelength shift scheme.

Fig. 2
Fig. 2

Ray tracing scheme of the recording geometry.

Fig. 3
Fig. 3

Phase error terrain of the corrector plate, recorded in ZTOA configuration, with 5145 Å green and read out with 6328 Å red. The plate has a f/No. of 1.25 and a focal length of 18.3 mm, maximum phase discrepancy, 20 read out wavelength at x = 6.7 mm, y = 6.7 mm.

Fig. 4
Fig. 4

Phase error terrain of an f/1.25 HOE objective lens recorded in a ZTOA configuration with 6328 Å red and read out with 8300 Å IR.

Fig. 5
Fig. 5

Dependence of optimum v value on the f/No. of the HOE objective lenses.

Fig. 6
Fig. 6

Comparison of the average rms phase error ZTOA configuration; Here λ1 = 6328 å, λ2 = 8300 Å. 1, ZTOA configuration; 2, computer optimized ZTOA configuration; 3, double wavelength shifting scheme.

Fig. 7
Fig. 7

Phase error terrain of a f/1.25 HOE objective lens recorded in the computer optimized ZTOA configuration with 6328 Å red and read out with 8300 Å IR.

Fig. 8
Fig. 8

Phase error terrain of a f/1.25 HOE objective lens recorded using the wavelength shift scheme. Here λ1 = 5145 Å, λ2 = 6328 Å, and λ3 = 8300 Å, focal length, 12.5 mm; maximum phase discrepancy, 0.3 read out wavelength at near corners.

Fig. 9
Fig. 9

(a) Point spread function of the reconstructed image point of a f/1.48 HOE objective lens recorded using the wavelength shift scheme; (b) point spread function of the corrector plate for the HOE in (a).

Equations (66)

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x h = C P h PC · x = v x ,
y h = v y .
Φ = 2 π λ 1 ( QP - SP ) + 2 π λ 2 ( PC + C P h - D P h ) - 2 π λ 3 P h I = 2 π λ 3 { λ 3 λ 1 [ λ 2 λ 1 ( ( x - x 0 ) 2 + y 2 + z q 2 - ( x - x 0 ) 2 + y 2 + z s 2 ) + ( x - x 0 ) 2 + y 2 + z c 2 ] - v [ λ 3 λ 2 ( x - x 0 ) 2 + y 2 + ( z d / v ) 2 - λ 3 λ 2 ( x - x 0 ) 2 + y 2 + z c 2 + ( x - x 0 ) 2 + y 2 + ( z i / v ) 2 ] } .
λ 3 λ 2 = λ 3 λ 1 = v .
α c = λ 2 λ 1 ( α q - α s ) ,
β c = λ 2 λ 1 ( β q - β s ) ,
α s = x - x 0 ( x - x 0 ) 2 + y 2 + z s 2 ,
β s = y ( x - x 0 ) 2 + y 2 + z s 2 ,
α q = x - x 0 ( x - x 0 ) 2 + y 2 + z q 2 ,
β q = y ( x - x 0 ) 2 + y 2 + z q 2 .
γ c = - 1 - α c 2 - β c 2 .
x = L α c γ c - x + x 0 + x 0 ,
y = L β c γ c - y .
ψ = 2 π λ 1 ( QP - SP ) + 2 π λ 2 ( PP - DP ) - 2 π λ 3 P I = 2 π λ 3 [ λ 3 λ 1 ( QP - SP ) - λ 3 λ 2 ( L γ c + DP ) - P I ] ,
QP = ( x - x 0 ) 2 + y 2 + z q 2 , SP = ( x - x 0 ) 2 + y 2 + z s 2 , γ c = - 1 - ( λ 2 λ 1 ) 2 [ ( α q - α s ) 2 + ( β q - β s ) 2 ] , DP = ( x - x 0 ) 2 + y 2 + z d 2 , = ( L α c / γ c - x + x 0 ) 2 + ( L β c γ c - y ) 2 + z d 2 , P I = ( L α c / γ c - x + x 0 ) 2 + ( L β c γ c - y ) 2 + z d 2
( Δ ψ ) rms = { [ ψ ( x , y ) - ψ ¯ ] 2 d x d y } 1 / 2 d x d y i = j N [ ψ ( x i , y i ) - 1 N 2 i = j N ψ ( x i , Y j ) ] 2 N 2 .
x i = x 0 - i x 0 / N ,             y j = y 0 - j y 0 / N .
v = CO / OC ,
( Δ ψ ) rms V = 0.
DP = ( x - x 0 ) 2 + y 2 + z d 2 = D P h 2 + ( x - x h ) 2 + ( y - y h ) 2 + 2 ( x + x h ) ( x h - x 0 ) + 2 y h ( y - y h ) = D P h [ 1 + 1 2 ( x + x h ) 2 D P h 2 + 1 2 ( y - y h ) 2 D P h 2 + ( x + x h ) ( x h - x 0 ) D P h 2 + y h ( y - y h ) D P h 2 - 1 2 ( x - x h ) 2 ( x h - x 0 ) 2 D P h 4 - 1 2 y h 2 ( y - y h ) 2 D P h 4 - y h ( y - y h ) ( x - x h ) ( x h - x 0 ) 2 D P h 4 ] .
D P h = v K 1 R c ,
K 1 = 1 U 2 - ( x 0 R c ) 2 + ( x - x 0 R c ) 2 + ( y R c ) 2 .
x - x h = ( 1 + v ) K 3 R c .
y - y h = ( 1 + v ) K 4 R c ,
x h - x 0 = v ( x - x 0 ) ,
y h = v y .
DP = R c { v K 1 + ( 1 + v ) 2 2 v K 1 ( K 2 2 + K 4 2 ) + 1 + v K 1 ( K 3 x - x 0 R c + K 4 y R c ) - ( 1 - v ) 2 2 v K 1 3 [ K 3 ( x - x 0 R c ) + K 4 ( y R c ) ] 2 } .
A 4 V 4 + A 3 V 3 + A 1 V + A 0 = 0 ,
A 0 = - i = j N [ G 1 ( x i , y i ) - 1 N 2 i j N G 1 ( x i , y j ) ] 2 ,
A 1 = - i = j N [ G 2 ( x i , y i ) - 1 N 2 i j N G 2 ( x i , y j ) ] × [ G 1 ( x i , y i ) - 1 N 2 i j N G 1 ( x i , y j ) ]
A 3 = - i = j N [ G 2 ( x i , y i ) - 1 N 2 i j N G 2 ( x i , y j ) ] × [ G 3 ( x i , y i ) - 1 N 2 i j N G 3 ( x i , y j ) ] ,
A 4 = - i = j N [ G 3 ( x i , y i ) - 1 N 2 i j N G 3 ( x i , y j ) ] 2 .
R s = SO , R q = QO , R c = - OC , R c = CO , R d = DO , R i = - O I .
1 R c = λ 2 λ 1 ( 1 R q - 1 R s ) ,
1 R 1 = λ 3 λ 2 ( 1 R d - 1 R c ) ,
R s R q = 1 + 2 ( λ 2 λ 1 ) 2 - 12 λ 2 2 λ 1 2 - 3 2 ( λ 2 2 λ 1 2 - 1 ) = U ,
R c R d = 1 + 2 ( λ 3 λ 2 ) 2 - 12 λ 3 2 λ 2 2 - 3 2 ( λ 3 2 λ 2 2 - 1 ) = U .
R c = v R c .
R i = v λ 3 λ 2 ( 1 - U ) R c ;
R s = λ 2 λ 1 ( U - 1 ) R c ;
R q = U - 1 U ( λ 2 λ 1 ) R c ;
R d = v U R c ;
L = R c 2 - X 0 2 · ( 1 + v ) .
ψ = 2 π λ 3 λ 3 λ 1 ( QP - SP ) + λ 3 λ 2 ( - L λ c - DP ) - P I ]
QP = ( x - x 0 ) 2 + y 2 + z q 2 , with z q = R q 2 - x 0 2 ;
SP = ( x - x 0 ) 2 + y 2 + z s 2 , with z s = R s 2 - x 0 2 .
DP = ( x - x 0 ) 2 + y 2 + z d 2 , with z d = R d 2 - x 0 2 ;
P I = ( x - x 0 ) 2 + y 2 + z i 2 , with z i = R i 2 - x 0 2 ;
x h = v x ,             y h = v y .
DP = R c { v K 1 + ( 1 + v ) 2 2 v K 1 ( K 3 2 + K 4 2 ) + 1 + v K 1 ( K 3 x - x 0 R c + K 4 y R c ) - ( 1 + v ) 2 2 v K 1 3 [ K 3 ( x - x 0 R c ) + K 4 ( y R c ) ] 2 } ,
K 1 = 1 U 2 - ( x 0 R c ) 2 + ( x - x 0 R c ) 2 + ( y R c ) 2 ,
K 2 = 1 γ c 1 - ( x 0 R c ) 2 ,
K 3 = K 2 α c + x 0 - x R c ,
K 4 = K 2 β c - y R c .
P I = R c { v K 5 + ( 1 + v ) 2 2 v K 5 ( K 3 2 + K 4 2 ) + 1 + v K 5 ( K 3 x - x 0 R c + K 4 y R c ) - ( 1 - v ) 2 2 v K 5 3 [ K 3 ( x - x 0 R c ) + K 4 ( y R c ) ] 2 } ,
K 5 = λ 2 2 λ 3 2 ( 1 - U ) 2 - ( x 0 R c ) 2 + ( x - x 0 R c ) 2 + ( y R c ) 2 .
L γ c = R c K 2 ( 1 + v ) .
ψ = - 2 π λ 3 R c ( G 1 V + G 2 + G 3 V ) .
G 1 ( x , y ) = 1 2 ( K 3 2 + K 4 2 ) ( λ 3 λ 2 1 K 1 + 1 K 5 ) - 1 2 ( K 3 x - x 0 R c + K 4 y R c ) 2 ( λ 3 λ 2 1 K 1 3 + 1 K 5 3 ) ,
G 2 ( x , y ) = λ 3 λ 1 ( SP - QP R c ) + ( λ 3 λ 2 K 1 + 1 K 5 ) ( K 3 2 + K 4 2 + K 3 x - x 0 R c + K 4 v R c ) + λ 3 λ 2 K 2 - ( λ 3 λ 2 K 1 3 + 1 K 5 3 ) ( K 3 x - x 0 R c + K 4 y R c ) 2 ,
G 3 ( x , y ) = ( λ 3 λ 2 1 K 1 + 1 K 5 ) [ 1 2 ( K 3 2 + K 4 2 ) + K 3 x - x 0 R c + K 4 y R c ] + K 5 + λ 3 λ 2 ( K 1 + K 2 ) - 1 2 ( λ 3 λ 2 K 1 3 + 1 K 5 3 ) × ( K 3 x - x 0 R c + K 4 y R c ) 2 .
A 4 V 4 + A 3 V 3 + A 1 V + A 0 = 0.
A 0 = - i = j N [ G 1 ( x i , y i ) - 1 N 2 i j N G 1 ( x i , y j ) ] 2 ,
A 1 = - i = j N [ G 2 ( x i , y i ) - 1 N 2 i j N G 2 ( x i , y j ) ] × [ G 1 ( x i , y i ) - 1 N 2 i j N G 1 ( x i , y j ) ]
A 3 = - i = j N [ G 2 ( x i , y i ) - 1 N 2 i j N G 2 ( x i , y j ) ] × [ G 3 ( x i , y i ) - 1 N 2 i j N G 3 ( x i , y j ) ] ,
A 4 = - i = j N [ G 3 ( x i , y i ) - 1 N 2 i j N G 3 ( x i , y j ) ] 2 .

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