Abstract

This paper presents a method for designing optimal holographic optical elements. The method is based on the minimization of the mean-squared difference between the desired and the actual output waves. The minimization yields an integral equation for the grating function of the designed optical element. The integral equation is converted into a set of linear equations that can be easily solved. The resulting coefficients form the final solution as a sum of polynomials. This procedure yields a well-behaved grating function that defines a holographic optical element that can be realized with the help of computer-generated holograms. The method is illustrated with a design of an imaging lens. The performance of this lens is then compared, both theoretically and experimentally, with that of a spherical holographic lens. The results show that the newly designed lens is clearly superior to the spherical lens.

© 1986 Optical Society of America

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References

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  1. D. H. Close, “Holographic optical elements,” Opt. Eng. 14, 408 (1975).
    [Crossref]
  2. A. Offner, “Ray tracing through a holographic system,” J. Opt. Soc. Am. 56, 1509–1512 (1966).
    [Crossref]
  3. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2698–2710 (1971).
    [Crossref] [PubMed]
  4. R. C. Fairchild, R. J. Fienup, “Computer-originated hologram lenses,” Opt. Eng. 21, 133 (1982).
  5. K. A. Winick, J. R. Fienup, “Optimum holographic elements recorded with nonspherical wave fronts,” J. Opt. Soc. Am. 73, 208–217 (1983).
    [Crossref]
  6. W-H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119 (1978).
    [Crossref]
  7. P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973), p. 111.
  8. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 464.
  9. W.-H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [Crossref] [PubMed]

1983 (1)

1982 (1)

R. C. Fairchild, R. J. Fienup, “Computer-originated hologram lenses,” Opt. Eng. 21, 133 (1982).

1978 (1)

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

1975 (1)

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

1974 (1)

1971 (1)

1966 (1)

Beckmann, P.

P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973), p. 111.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 464.

Close, D. H.

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

Fairchild, R. C.

R. C. Fairchild, R. J. Fienup, “Computer-originated hologram lenses,” Opt. Eng. 21, 133 (1982).

Fienup, J. R.

Fienup, R. J.

R. C. Fairchild, R. J. Fienup, “Computer-originated hologram lenses,” Opt. Eng. 21, 133 (1982).

Latta, J. N.

Lee, W.-H.

Lee, W-H.

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

Offner, A.

Winick, K. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 464.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

Opt. Eng. (2)

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

R. C. Fairchild, R. J. Fienup, “Computer-originated hologram lenses,” Opt. Eng. 21, 133 (1982).

Prog. Opt. (1)

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

Other (2)

P. Beckmann, Orthogonal Polynomials for Engineers and Physicists (Golem, Boulder, Colo., 1973), p. 111.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), p. 464.

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Figures (8)

Fig. 1
Fig. 1

The optical system for the imaging lens design. The object is imaged with a unity magnification.

Fig. 2
Fig. 2

A plot of the calculated spot size of the image points in the open-aperture configuration for a spherical holographic lens (dashed line) and the corrected lens (dotted line). The spot size was calculated as the standard deviation of the distributions at the image plane.

Fig. 3
Fig. 3

A plot of the calculated spot size of the image points in the 0.5-aperture configuration for a spherical holographic lens (dashed line) and the corrected lens (dotted line). The spot size was calculated as the standard deviation of the distributions at the image plane.

Fig. 4
Fig. 4

A plot of the calculated spot size of the image points in the 0.2-aperture configuration for a spherical holographic lens (dashed line) and the corrected lens (dotted line). The spot size was calculated as the standard deviation of the distributions at the image plane.

Fig. 5
Fig. 5

A plot of the calculated distortions in the image for the 0.5-aperture configuration for a spherical holographic lens (dashed line) and the corrected lens (dotted line). The distortions were calculated as the difference between the average location of the spots at the image plane and their ideal location.

Fig. 6
Fig. 6

The CGH for the imaging lens. The figure is stretched in the y (vertical) direction.

Fig. 7
Fig. 7

Schematic diagram of the recording setup. The modulated plane wave is spatially filtered to yield the desired recording wave that interferes with a spherical wave to form the final holographic lens.

Fig. 8
Fig. 8

Photograph taken at the image plane of the imaging lenses: (a) the imagery produced by the spherical holographic lens, sharpened at α = 0 (the bottom of the imagery); (b) the imagery produced by the spherical holographic lens, sharpened at a = 0.4 (the center of the imagery); (c) the imagery produced by the corrected holographic lens, sharpened at a = 0.4. Note the improvement in resolution and uniformity.

Equations (36)

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U in ( x , a ) = exp [ i 2 π λ ψ in ( x , a ) ] ,
U d ( x , a ) = exp [ i 2 π λ ψ d ( x , a ) ] .
U out ( x , a ) = exp [ i 2 π λ ψ out ( x , a ) ] ,
ψ out = ψ in ψ h ,
T ( x ) = 1 2 + 1 2 cos [ 2 π λ ψ h ( x ) ] .
E 2 = a 1 a 2 x 1 ( a ) x 2 ( a ) [ ψ d ( x , a ) ψ out ( x , a ) + ϕ ( a ) ] 2 d x d a .
E 2 = a 1 a 2 x 1 ( a ) x 2 ( a ) [ ψ d ( x , a ) ψ in ( x , a ) + ψ h ( x ) + ϕ ( a ) ] 2 d x d a .
E 2 = a 1 a 2 b 1 b 2 x 1 ( a , b ) x 2 ( a , b ) y 1 ( a , b ) y 2 ( a , b ) [ ψ d ( x , y ; a , b ) ψ in ( x , y ; a , b ) + ψ h ( x , y ) + ϕ ( a , b ) ] 2 d x d y d a d b .
e a 2 ( η ) = x 1 ( a ) x 2 ( a ) [ ψ d ( x , a ) ψ in ( x , a ) + ψ ˆ h ( x ) + η ] 2 d x .
x 1 ( a ) x 2 ( a ) [ ψ d ( x , a ) ψ in ( x , a ) + ψ ˆ h ( x ) + η min ] d x = 0.
ϕ ˆ ( a ) = η min = 1 x a ( a ) x 1 ( a ) × x 1 ( a ) x 2 ( a ) [ ψ d ( x , a ) ψ in ( x , a ) + ψ ˆ h ( x ) ] d x .
e x 2 ( ξ ) = a 1 ( x ) a 2 ( x ) [ ψ d ( x , a ) ψ in ( x , a ) + ξ + ϕ ˆ ( a ) ] 2 d a .
a 1 ( x ) a 2 ( x ) [ ψ d ( x , a ) ψ in ( x , a ) + ϕ ˆ ( a ) + ξ min ] d a = 0 .
ψ ˆ h ( x ) = ξ min = 1 a 2 ( x ) a 1 ( x ) × a 1 ( x ) a 2 ( x ) [ ψ d ( x , a ) ψ in ( x , a ) + ϕ ˆ ( a ) ] d a .
F u 1 ( u 2 u 1 ) u 1 u 2 F d u .
ϕ ˆ ( a ) = ψ d ( x , a ) ψ in ( x , a ) x ψ ˆ h ( x ) x ,
ψ ˆ h ( x ) = ψ d ( x , a ) ψ in ( x , a ) a ϕ ˆ ( a ) a .
ψ ˆ h ( x ) = f ( x ) + ψ ˆ h ( x ) x a ,
f ( x ) = ψ d ψ in x a ψ d ψ in a .
f ( x ) = 0 i n a i T i ( x ) ,
( I O ) x = b ,
Mx = b
f ( x , y ) = 0 i n 0 j n a i j T i ( x ) T j ( y ) .
T 0 ( x ) = 1 , T 2 ( x ) = 2 x 2 1 , T 4 ( x ) = 8 x 4 8 x 2 + 1 , T 6 ( x ) = 32 x 6 48 x 4 + 18 x 2 1.
ψ in ( x , a ) = [ ( x a ) 2 + d 0 2 ] 1 / 2 ,
ψ d ( x , a ) = [ ( x + a ) 2 + d i 2 ] 1 / 2 ,
x 1 ( a ) = a 2 w , x 2 ( a ) = a + 2 w , a 1 ( a ) = x 2 w , a 2 ( a ) = x + 2 w .
ψ ˆ h ( x ) = c 2 T 2 ( x ) + c 4 T 4 ( x ) + c 6 T 6 ( x ) .
ψ d ( x , a ) = [ ( x + a δ a 3 ) 2 + d 1 2 ] 1 / 2 ,
T ( x , y ) = 1 2 + 1 2 cos [ 2 π λ ( c 2 T 2 ( x ) + c 4 T 4 ( x ) + c 6 T 6 ( x ) + ρ y ) ] .
U m = exp { i 2 π λ [ x 2 + f 2 c 2 T 2 ( x ) c 4 T 4 ( x ) c 6 T 6 ( x ) ρ y ] } ,
U c = exp ( i 2 π λ x 2 + f 2 ) .
T h ( x , y ) = 1 2 + 1 2 cos { i 2 π λ x 2 + f 2 c 2 T 2 ( x ) c 4 T 4 ( x ) c 6 T 6 ( x ) + c y ] } .
2 π λ [ x 2 + f 2 c 2 T 2 ( x ) c 4 T 4 ( x ) c 6 T 6 ( x ) + c y ] = 2 π n ,
y = 1 c [ λ n x 2 + f 2 + c 2 T 2 ( x ) + c 4 T 4 ( x ) + c 6 T 6 ( x ) ] .
U s = exp [ i 2 π λ x 2 + y 2 + f 2 ] .

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