Abstract

A method is presented for designing optimal holographic optical elements. The method is based on an analytic ray-tracing procedure that uses the minimization of the mean-squared difference of the propagation vector components, between the actual output wave fronts and the desired output wave fronts. The minimization yields integral equations for the grating vector components that can be solved analytically, in some cases without any approximation. This leads to a well-behaved grating function that defines a holographic optical element.

© 1989 Optical Society of America

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References

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  1. D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
    [Crossref]
  2. R. C. Fairchild, R. J. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).
    [Crossref]
  3. K. A. Winick, J. R. Fienup, “Optimum holographic elements recorded with nonspherical wave fronts,”J. Opt. Soc. Am. 73, 208–217 (1983).
    [Crossref]
  4. J. Kedmi, A. A. Friesem, “Optimized holographic optical elements,” J. Opt. Soc. Am. A 3, 2011–2018 (1986).
    [Crossref]
  5. J. N. Cederquist, J. R. Fienup, “Analytic design of optimum holographic optical elements,” J. Opt. Soc. Am. A 4, 699–705 (1987).
    [Crossref]
  6. J. Kedmi, A. A. Friesem, “Optimal holographic Fourier-transform lens,” Appl. Opt. 23, 4015–4019 (1984).
    [Crossref] [PubMed]
  7. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698–2710 (1971).
    [Crossref] [PubMed]
  8. J. N. Latta, “Computer-based analysis of hologram imagery and aberrations. I. Hologram types and their nonchromatic aberrations,” Appl. Opt. 10, 599–608 (1971).
    [Crossref] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1987 (1)

1986 (1)

1984 (1)

1983 (1)

1982 (1)

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

1975 (1)

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
[Crossref]

1971 (2)

Cederquist, J. N.

Close, D. H.

D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408–419 (1975).
[Crossref]

Fairchild, R. C.

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

Fienup, J. R.

Fienup, R. J.

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

Friesem, A. A.

Goodman, J. W.

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

Kedmi, J.

Latta, J. N.

Winick, K. A.

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Figures (11)

Fig. 1
Fig. 1

Readout geometry for an off-axis FTL.

Fig. 2
Fig. 2

Spot size as a function of the input angle for on-axis FTL: (a) spherical quadratic and optimal grating function; (b) magnified of optimal and quadratic grating functions.

Fig. 3
Fig. 3

Spot size as a function of the input angle for an off-axis FTL: spherical, quadratic, and optimal grating functions.

Fig. 4
Fig. 4

Aberrations as a function of the input angle for the on-axis FTL. D, distortion; A, astigmatism and field curvature; C, coma; S, spherical aberration. (a) Spherical grating function; (b) optimal grating function.

Fig. 5
Fig. 5

Spot diagrams for the off-axis FTL. (a) Spherical grating function; (b) quadratic grating function; (c) optimal grating function.

Fig. 6
Fig. 6

Readout geometry for an off-axis HIL.

Fig. 7
Fig. 7

Spot size as a function of the location of the object points (β) for a HIL with a stop aperture of 2 mm (W = 1 mm): spherical (sph), quadratic (q), and optimal (opt) grating functions.

Fig. 8
Fig. 8

Distortions as a function of the location of the object points (β) for a HIL with a stop aperture of 2 mm (W = 1 mm): spherical (sph), quadratic (q), and optimal (opt) grating functions.

Fig. 9
Fig. 9

Spot size as a function of the location of the object points (β), for a HIL without a stop-aperture configuration: spherical (sph), quadratic (q), and optimal (opt) grating functions.

Fig. 10
Fig. 10

Distortions as a function of the location of the object points (β), for a HIL without a stop-aperture configuration; spherical (sph), quadratic (q), and optimal (opt) grating functions.

Fig. 11
Fig. 11

Mean-squared difference E2 as a function of the stop aperture (W), for spherical, quadratic, and optimal HIL’s.

Equations (72)

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ϕ o = ϕ i ± n ϕ h ,
K ¯ ^ o = λ read 2 π ¯ ϕ o ,             K ¯ ^ i = λ read 2 π ¯ ϕ i ,
K ¯ h = λ rec 2 π ¯ ϕ h = λ rec Λ x x ^ + λ rec Λ y y ^ ,
K ^ x o = K ^ x i - μ K x h ,
K ^ y o = K ^ y i - μ K y h ,
K ^ z o = ± ( 1 - K ^ x o 2 - K ^ y o 2 ) 1 / 2 ,
E 2 = W μ ( μ ) W ( a ) P ( x , a , μ ) [ K ^ x d ( x , a , μ ) - K ^ x o ( x , a , μ ) ] 2 d μ d a d x ,
E 2 = W μ ( μ ) W ( a ) P ( x , a , μ ) [ K ^ x d ( x , a , μ ) - K ^ x i ( x , a , μ ) + μ K x h ( x ) ] 2 d μ d a d x .
e 2 ( x o ) = W μ ( μ ) W ( a ) P ( x o , a , μ ) [ K ^ x d ( x o , a , μ ) - K ^ x i ( x o , a , μ ) + μ K x h ( x o ) ] 2 d μ d a ,
W μ ( μ ) W ( a ) P ( x o , a , μ ) [ K ^ x d ( x o , a , μ ) - K ^ x i ( x o , a , μ ) + μ K x h ( x o ) ] μ d μ d a = 0.
K x h ( x ) = - { W μ ( μ ) W ( a ) P ( x , a , μ ) μ [ K ^ x d ( x , a , μ ) - K ^ x i ( x , a , μ ) ] d a d μ } [ W μ ( μ ) W ( a ) P ( x , a , μ ) μ 2 d a d μ ] .
ϕ h ( x ) = 2 π λ rec K x h ( x ) d x .
K x h ( x ) = - W ( a ) P ( x , a ) [ K ^ x d ( x , a ) - K ^ x i ( x , a ) ] d a μ o W ( a ) P ( x , a ) d a .
E 2 = W ( a ) W ( b ) P ( x , y , a , b ) [ K ^ d ( x , y , a , b ) - K ^ o ( x , y , a , b ) ] 2 d a d b d x d y ,
E 2 = W ( a ) W ( b ) P ( x , y , a , b ) [ K ^ d ( x , y , a , b ) - K ^ i ( x , y , a , b ) + μ o K h ( x , y ) ] 2 d a d b d x d y .
K h ( x , y ) = - W ( a ) W ( b ) P ( x , y , a , b ) [ K ^ d ( x , y , a , b ) - K ^ i ( x , y , a , b ) ] d a d b μ o W ( a ) W ( b ) P ( x , y , a , b ) d a d b .
ϕ h ( x , y ) - ϕ h ( 0 , 0 ) = 2 π λ rec c K ¯ h · d r ¯ = 2 π λ rec 0 ( x , y ) K x h ( x , y ) d x + K y h ( x , y ) d y ,
K x h ( x , y ) y = K y h ( x , y ) x .
ϕ h ( x , y ) = [ ϕ h ( r ) ] on - axis + 2 π λ rec ( sin θ r ) x .
[ ϕ h ( r ) ] on - axis = 2 π λ rec K r h ( r ) d r ,
ϕ h ( x o , y o ) = [ ϕ h ( x o ) ] off - axis + [ ϕ h ( y o ) ] on - axis = 2 π λ rec [ x = 0 x o K x h ( x , y = 0 ) d x + y = 0 y o K y h ( x = 0 , y ) d y ] ,
ψ ( x ) = ϕ d ( x ) - ϕ o ( x ) .
ϒ ( x , a ) = λ read 2 π ψ x = K ^ x d ( x , a ) - K ^ x o ( x , a ) D + A x + C x 2 + S x 3 + ,
K ^ x d = K ^ x d + i = 1 m j = 0 l c i j a i x j = K ^ x d + i = 1 m c i 0 a i + i = 1 m c i 1 a i x + .
d E 2 d c i 0 = 0.
a = α = sin θ i .
K ^ x i ( x , a ) = K ^ x i ( α ) = α .
K ^ x d ( x , a ) = K ^ x d ( x , α ) = - [ x - ( α f - α r f ) ] { [ x - ( α f - α r f ) ] 2 + d i 2 } 1 / 2 .
K ^ x d = K ^ x d + i = 1 m c i 0 α i .
K ^ x d - [ x - ( α f - α r f ) ] { [ x - ( α f - α r f ) ] 2 + d i 2 } 1 / 2 + c 10 α .
K x h ( x , c 10 ) = - 1 [ α 2 ( x ) - α 1 ( x ) ] × α 1 ( x ) α 2 ( x ) ( - [ x - ( α f - α r f ) ] { [ x - ( α f - α r f ) ] 2 + d i 2 } 1 / 2 + c 10 α - α ) d α ,
K x h ( x , c 10 ) = ( 1 - c 10 ) [ α 1 ( x ) + α 2 ( x ) ] 2 - 1 [ α 2 ( x ) - α 1 ( x ) ] f × ( { [ x + α r f - α 2 ( x ) f ] 2 + d i 2 } 1 / 2 - { [ x + α r f - α 1 ( x ) f ] 2 + d i 2 } 1 / 2 ) .
E 2 = D 1 D 2 α 1 ( x ) α 2 ( x ) [ K ^ x d + c 10 α - K ^ x i + K x h ( x , c 10 ) ] 2 d α d x ,
c 10 = - D 1 D 2 α 1 ( x ) α 2 ( x ) [ K ^ x d - K ^ x i + K x h ( c 10 = 0 ) ] [ α - α 1 ( x ) + α 2 ( x ) 2 ] d α d x D 1 D 2 α 1 ( x ) α 2 ( x ) [ α - α 1 ( x ) + α 2 ( x ) 2 ] 2 d α d x .
[ K x h ( x ) ] simplified x f + α r = [ K x h ( x ) ] q .
[ K x h ( x ) ] sph = x ( x 2 + f 2 ) 1 / 2 + α r .
ϒ = K ^ x d - K ^ x i + μ K x h .
K ^ x d = - ( x - α f ) [ ( x - α f ) 2 + f 2 ] 1 / 2 α - 1 2 α 3 + ( 3 2 α 2 - 1 ) x f - 3 2 α ( x f ) 2 + 1 2 ( x f ) 3 ,
( K x h ) opt x f ,
( K x h ) sph = x ( x 2 + f 2 ) 1 / 2 x f - 1 2 ( x f ) 3 .
x o ( θ i ) = f tan θ i = f α ( 1 - α 2 ) 1 / 2 .
ϒ opt D = - 1 2 α 3 + ( 3 2 α 2 + μ - 1 ) α ( 1 - α 2 ) 1 / 2 - 3 2 α 3 ( 1 - α 2 ) + 1 2 α 3 ( 1 - α 2 ) 3 / 2 ,
ϒ opt A = [ ( 3 2 α 2 + μ - 1 ) - 3 α 2 ( 1 - α 2 ) 1 / 2 + 3 2 α 2 ( 1 - α 2 ) ] ( x - x o ) f ,
ϒ opt C = [ 3 2 α ( 1 - α 2 ) 1 / 2 - 3 2 α ] ( x - x o f ) 2 ,
ϒ opt S = 1 2 ( x - x o f ) 3 .
ϒ sph D = - 1 2 α 3 + ( 3 2 α 2 + μ - 1 ) α ( 1 - α 2 ) 1 / 2 - 3 2 α 3 ( 1 - α 2 ) + 1 2 ( 1 - μ ) α 3 ( 1 - α 2 ) 3 / 2 ,
ϒ sph A = [ ( 3 2 α 2 + μ - 1 ) - 3 α 2 ( 1 - α 2 ) 1 / 2 + 3 2 ( 1 - μ ) α 2 ( 1 - α 2 ) ] ( x - x o ) f ,
ϒ sph C = [ - 3 2 α + 3 2 ( 1 - μ ) α ( 1 - α 2 ) 1 / 2 ] ( x - x o f ) 2 ,
ϒ sph S = [ 1 2 ( 1 - μ ) ] ( x - x o f ) 3 .
x i = - ( β + η ) M ,
K ^ x i ( x , a ) = K ^ x i ( x , β ) = ( x - β ) [ ( x - β ) 2 + d o 2 ] 1 / 2 ,
K ^ x d ( x , a ) = K ^ x d ( x , β ) = - [ x + M ( β + η ) ] { [ x + M ( β + η ) ] 2 + d i 2 } 1 / 2 .
K x h ( x ) = - 1 [ β 2 ( x ) - β 1 ( x ) ] β 1 ( x ) β 2 ( x ) ( - [ x + M ( β + η ) ] { [ x + M ( β + η ) ] 2 + d i 2 } 1 / 2 - ( x - β ) [ ( x - β ) 2 + d o 2 ] 1 / 2 ) d β ,
K x h ( x ) = 1 [ β 2 ( x ) - β 1 ( x ) ] ( 1 M { [ x + M ( β 2 ( x ) + η ) ] 2 + d i 2 } 1 / 2 - 1 M { [ x + M ( β 1 ( x ) + η ) ] 2 + d i 2 } 1 / 2 + { [ x - β 1 ( x ) ] 2 + d o 2 } 1 / 2 - { [ x - β 2 ( x ) ] 2 + d o 2 } 1 / 2 ) .
[ K x h ( x ) ] simplified x f - [ A c d o + - M ( A c + η ) d i ] ,
1 f = 1 d o + 1 d i .
[ K x h ( x ) ] sph = ( x - A c ) [ ( x - A c ) 2 + d o 2 ] 1 / 2 + [ x + M ( A c + η ) ] { [ x + M ( A c + η ) ] 2 + d i 2 } 1 / 2 .
[ K x h ( x ) ] q = x f + α r o + α r i ,
α r o = sin θ r o = - A c ( A c 2 + d o 2 ) 1 / 2
α r i = sin θ r = M ( A c + η ) { [ M ( A c + η ) ] 2 + d i 2 } 1 / 2 .
α 1 ( x ) = ( x - W 2 ) [ ( x - W 2 ) 2 + ( Δ y ) 2 + d o 2 ] 1 / 2
α 1 ( x ) > α min ;
α 1 ( x ) = α min .
α 2 ( x ) = ( x - W 1 ) [ ( x - W 1 ) 2 + ( Δ y ) 2 + d o 2 ] 1 / 2
α 2 ( x ) < α max ;
α 2 ( x ) = α max .
β 1 ( x ) = d o 1 d o 2 [ - W + A c d o 2 ( d o 1 + d o 2 ) - x ] - W + A c d o 2 ( d o 1 + d o 2 )
β 1 ( x ) A 1 ;
β 1 ( x ) = A 1 .
β 2 ( x ) = d o 1 d o 2 [ W + A c d o 2 ( d o 1 + d o 2 ) - x ] + W + A c d o 2 ( d o 1 + d o 2 )
β 2 ( x ) A 2 ;
β 2 ( x ) = A 2 .

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