Abstract

A mathematical model that describes the behavior of low-resolution Fresnel lenses encoded in any low-resolution device (e.g., a spatial light modulator) is developed. The effects of low-resolution codification, such the appearance of new secondary lenses, are studied for a general case. General expressions for the phase of these lenses are developed, showing that each lens behaves as if it were encoded through all pixels of the low-resolution device. Simple expressions for the light distribution in the focal plane and its dependence on the encoded focal length are developed and commented on in detail. For a given codification device an optimum focal length is found for best lens performance. An optimization method for codification of a single lens with a short focal length is proposed.

© 1994 Optical Society of America

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References

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1992 (6)

1990 (3)

1989 (2)

1985 (1)

1974 (1)

Andrées, P.

April, G. V.

Arsenault, H. H.

Bergeron, A.

Bryngdahl, O.

Connely, S. W.

Cottrell, D. M.

Davis, J. A.

Davis, J. E.

Drayton, S. H.

Feldman, M. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

J. W. Goodman, Introduction a I'Optique de Fourier et a l'Holographie (Masson, Paris, 1972).

Gregory, D. A.

J. C. Kirsch, D. A. Gregory, M. W. Thie, B. K. Jones, “Modulation characteristics of the Epson liquid-crystal television,” Opt. Eng. 31, 963–970 (1992).
[CrossRef]

F. T. S. Yu, X. Yang, D. A. Gregory, “Polychromatic neural networks,” Opt. Commun. 88, 81–86 (1992).
[CrossRef]

Hedman, T. R.

Horner, J. L.

Jaroszewicz, Z.

Jones, B. K.

J. C. Kirsch, D. A. Gregory, M. W. Thie, B. K. Jones, “Modulation characteristics of the Epson liquid-crystal television,” Opt. Eng. 31, 963–970 (1992).
[CrossRef]

Kirsch, J. C.

J. C. Kirsch, D. A. Gregory, M. W. Thie, B. K. Jones, “Modulation characteristics of the Epson liquid-crystal television,” Opt. Eng. 31, 963–970 (1992).
[CrossRef]

Kolodziejczyk, A.

Leger, J. R.

Lilly, R. A.

Martínez-Corral, M.

Ojeda-Castañeda, J.

Schley-Seebold, H. M.

Sypek, M.

Tam, E. C.

Thie, M. W.

J. C. Kirsch, D. A. Gregory, M. W. Thie, B. K. Jones, “Modulation characteristics of the Epson liquid-crystal television,” Opt. Eng. 31, 963–970 (1992).
[CrossRef]

Viñas, S. B.

Yang, X.

F. T. S. Yu, X. Yang, D. A. Gregory, “Polychromatic neural networks,” Opt. Commun. 88, 81–86 (1992).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, X. Yang, D. A. Gregory, “Polychromatic neural networks,” Opt. Commun. 88, 81–86 (1992).
[CrossRef]

Zhou, S.

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

F. T. S. Yu, X. Yang, D. A. Gregory, “Polychromatic neural networks,” Opt. Commun. 88, 81–86 (1992).
[CrossRef]

Opt. Eng. (1)

J. C. Kirsch, D. A. Gregory, M. W. Thie, B. K. Jones, “Modulation characteristics of the Epson liquid-crystal television,” Opt. Eng. 31, 963–970 (1992).
[CrossRef]

Opt. Lett. (2)

Other (2)

J. W. Goodman, Introduction a I'Optique de Fourier et a l'Holographie (Masson, Paris, 1972).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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Figures (10)

Fig. 1
Fig. 1

Binary LRFEL containing 5 × 5 lenses with N = M = 480: (a) Wx = Wy = 96, (b) Wx = Wy = 96.5.

Fig. 2
Fig. 2

Three-dimensional representation of light-intensity distributions in the focal plane for Wx = Wy = 5, cx = cy = 1, and Δx = Δy: (a) order (0, 0), (b) order (1, 0), (c) order (2, 0), (d) order (2, 2), (e) order (3, 0).

Fig. 3
Fig. 3

Illustration of the asymptotic behavior of light-intensity distributions in the focal plane with cx = cy = 1 and Δx = Δy: (a) Wx = Wy = 0.53, (b) Wx = Wy = 13.3.

Fig. 4
Fig. 4

Light intensity at the main focus versus the focal length encoded (f). The focal length is normalized to f opt. The specific conditions are cx = cy = 1, Δx = Δy, and N = M: (a) evolution in the principal focus (xi , = yi = 0), (b) evolution for yi = 0 on the x axis for increasing f, (c) as in (b) for decreasing f.

Fig. 5
Fig. 5

FWHM of light-intensity distributions in the focal plane versus the focal length encoded. The focal length is normalized to f opt and the width is normalized to the size of the pixel: (a) cx = cy = 1 and N = M. (b) cx = 0.69, = 0.67, N = 320, and M = 220 (data corresponding to an Epson SLM13). Curve 1 corresponds to the width in the x coordinate, and curve 2 corresponds to the y coordinate.

Fig. 6
Fig. 6

Binary LRFEL corresponding to f opt: (a) N = M = 256, cx = cy = 1, and Δx = Δy; (b) data of an Epson SLM.

Fig. 7
Fig. 7

Binary LRFEL optimized for: (a) cx = cy = 1, Δx = Δy, and Wx = Wy = 5; (b) the data of an Epson SLM, but N = M = 480. The lines dividing blocks are indicated by arrows.

Fig. 8
Fig. 8

Comparison of light intensities at xi = yi = 0 for a LRFEL (lower curve) and for an optimized LRFEL (upper curve) (cx = cy = 1, N = M, and Δx = Δy). The latter always gives more energy.

Fig. 9
Fig. 9

Comparison of FWHM of light distributions for (a) the optimized and (b) the nonoptimized lens. The dashed line corresponds to the Fraunhofer diffraction of the pupil.

Fig. 10
Fig. 10

Evolution of light-intensity distributions on the x axis for increasing focal length for the optimized lens.

Tables (2)

Tables Icon

Table 1 Values of the First Integral of Eq. (35) for Several n

Tables Icon

Table 2 Intensity in the Focus for the Optimized and Nonoptimized Lenses a

Equations (38)

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A ( x , y ) = L [ K 2 f ( x 2 + y 2 ) + φ 0 ] L ( φ + φ 0 ) ,
x n = [ n + ½ P ( N ) ] Δ x , n = 0 , 1 , 1 , 2 , 2 , y m = [ m + ½ P ( M ) ] Δ y , m = 0 , 1 , 1 , 2 , 2 ,
x k = k X , X = λ f Δ x , k = 0 , ± 1 , ± 2 , y l = l Y , Y = λ f Δ y , l = 0 , ± 1 , ± 2 .
R x = X Δ x , R y = Y Δ y ,
A ( n , m ) = L ( π { [ n + ½ P ( N ) ] 2 R x + [ m + ½ P ( M ) ] 2 R y } + φ 0 ) .
n = k R x + s D ( k R x ) , m = l R y + t D ( l R y ) , x s = ( s p k ) Δ x , y t = ( t q l ) Δ y , p k = D ( k R x ) ½ P ( N ) , q l = D ( l R y ) ½ P ( M ) ,
A ( s , t ) = L ( φ 0 + φ 1 + φ 2 + φ 3 ) , φ 1 = π ( k 2 R x + l 2 R y ) 2 π { k [ ½ P ( N ) D ( k R x ) ] + l [ ½ P ( M ) D ( l R y ) ] } φ 2 = 2 π ( k s + l t ) , φ 3 = π [ ( s p k ) 2 R x + ( t q l ) 2 R y ] .
Z ( x , y ) n , m δ { x [ n + 1 2 P ( N ) ] Δ x , y [ m + 1 2 P ( M ) ] Δ y } = exp ( i φ 1 ) Z ( x k X , y l Y ) s , t δ [ ( x k X ) ( s p k ) Δ x , ( y l Y ) ( t q l ) Δ y ] .
A ( x , y ) = ( L ( x , y ) n , m δ { x [ n + 1 2 P ( N ) ] Δ x , y [ m + 1 2 P ( M ) ] Δ y } rect ( x L x , y L y ) ) * rect ( x Δ x , y Δ y ) ,
U ( x i , y i ) = ( Z ( x , y ) n , m δ { x [ n + 1 2 P ( N ) ] Δ x + , y [ m + 1 2 P ( M ) ] Δ y } rect ( x L x , y L y ) ) * rect ( x Δ x , y Δ y ) * ( 1 i λ f ) Z 1 ( x , y ) ,
U ( x i , y i ) = ( Z ( x , y ) n , m δ { x [ n + 1 2 P ( N ) ] Δ x , y [ m + 1 2 P ( M ) ] Δ y } rect ( x L x , y L y ) ) * rect ( 1 i λ f ) Z 1 ( x , y ) * rect ( x Δ x , y Δ y ) .
U 0 , 0 ( x i , y i ) = [ Z 1 ( x , y ) i λ f λ f ( n , m δ { x [ n + 1 2 P ( N ) ] Δ x , y [ m + 1 2 P ( M ) ] Δ y } rect ( x L x , y L y ) ) ] * rect ( x Δ x , y Δ y ) ,
U 0 , 0 ( x i , y i ) = ( L x L y Δ x Δ y ) { [ Z 1 ( x , y ) i λ f ] × n , m exp { i π [ n P ( N ) + m P ( M ) ] } × sinc [ L x ( x n X ) λ f , L y ( y m Y ) λ f ] } * rect ( x Δ x , y Δ y ) ,
U 0 , 0 ( x i , y i ) = 1 i λ f ( L x L y Δ x Δ y ) { exp [ i K 2 f ( x 2 + y 2 ) ] × sinc ( L x x λ f , L y y λ f ) } * rect ( x Δ x , y Δ y ) ,
K 2 f [ ( x i + Δ x 2 ) 2 + ( y i + Δ y 2 ) 2 ] K 2 f [ ( x i Δ x 2 ) 2 + ( y i Δ y 2 ) 2 ] 2 π .
1 2 ( Δ x Δ x x i X + Δ y Δ y y i Y ) 1 .
1 2 ( Δ x Δ x 1 N + Δ y Δ y 1 M ) 1 , 1 2 [ ( Δ x Δ x ) 2 1 R x + ( Δ y Δ y ) 2 1 R y ] 1 .
U 0 , 0 ( x i , y i ) = 1 i λ f ( L x L y Δ x Δ y ) sinc ( L x x λ f , L y y λ f ) * rect ( x Δ x , y Δ y ) .
U k , l ( x i , y i ) = { Z ( x , y ) s , t δ [ x ( s p k ) Δ x , y ( t q l ) Δ y ] rect ( x + k X L x , y + l Y L y ) } * ( 1 i λ f ) Z 1 ( x , y ) * rect ( x Δ x , y Δ y ) ,
U k , l ) ( x i , y i ) = ( 1 i λ f L x L y Δ x Δ y ) × { exp [ i 2 π λ f ( x k X + y l Y ) ] sinc ( L x x λ f , L y y λ f ) } * rect ( x Δ x , y Δ y ) .
W x = L x X , W y = L y Y , c x = Δ x Δ x , c y = Δ y Δ y ,
u = W x ( x Δ x ) , υ = W y ( y Δ y ) , u i = W x ( x i Δ x ) , υ i = W y ( y i Δ y ) .
U k , l ( u i , υ i ) = ( λ f i Δ x Δ y ) a + u i a + u i b + υ i b + υ i × exp [ i 2 π ( k W x u + l W y υ ) ] sinc ( u , υ ) d u d υ , a = W x c x 2 , b = W y c y 2 .
U k , l ( 0 , 0 ) W x W y 1 [ sinc ( u , υ ) rect ( u W x c x , υ W y c y ) ] ,
U k , l ( 0 , 0 ) rect ( k W x , l W y ) * sinc ( k c x , l c y ) .
U k , l ( 0 , 0 ) rect ( k W x , l W y ) .
U k , l ( 0 , 0 ) sinc ( k c x , l c y )
U 0 , 0 ( x i , y i ) c x c y L x L y i λ f sinc ( W x x i Δ x , W y y i Δ y ) .
U k , l ( x i , y i ) 1 i λ f Δ x Δ y rect ( x Δ x , y Δ y ) .
U ( x i , y i ) = 1 i λ f A ( x , y ) exp [ i ( K 2 s ) ( x 2 + y 2 ) ] × exp { i ( K 2 s ) [ ( x i x ) 2 + ( y i y ) 2 ] } d x d y ,
U ( x i , y i ) = 1 i λ f exp [ i K 2 f Γ ( 1 1 Γ ) ( x i 2 + y i 2 ) ] × A ( x , y ) exp [ i ( K 2 f ) ( x i Γ x ) 2 + ( y i Γ y ) 2 ] d x d y , Γ = s f .
U 0 , 0 ( 0 , 0 ) = f ( W x c x ) / 2 ( W x c x ) / 2 ( W y c y ) / 2 ( W y c y ) / 2 sinc ( u , υ ) d u d υ .
W x c x 2 = 0.685 .
f opt = 0.73 Δ x Δ x N λ .
U 0 , 0 n , m ( 0 , 0 ) = 1 / 2 1 / 2 cos [ 2 π ( n + 1 / 2 ) u ] sinc ( u ) d u × 1 / 2 1 / 2 cos [ 2 π ( m + 1 / 2 ) υ ] sinc ( υ ) d υ
sign = ( 1 ) n + m .
λ f ( X / c x ) ( Y / c y ) ( A ( x n , y m ) exp { i k 2 f [ ( x i x n ) 2 + ( y i y m ) 2 ] } ) × sinc ( x i x n ( X / c x ) , y i y m ( Y / c y ) ) ,
λ f ( X / c x ) ( Y / c y ) { A ( x n , y m ) exp [ i k 2 f ( x n 2 + y m 2 ) ] } × sinc ( x n ( X / c x ) , y m ( Y / c y ) ) .

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