Abstract

We discuss both numerically and analytically how the space–bandwidth product and the information density of lenses scale as functions of their diameter and f-number over many orders of magnitude. This information may be useful for the design of optical computing and interconnection systems. For diffractive lenses, cost is defined as the number of resolution elements the lithographic production system must have; the relationship of this quantity to the space–bandwidth product and information density is also given.

© 1994 Optical Society of America

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References

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  1. A. W. Lohmann, “Scaling laws for lens systems,” Appl. Opt. 28, 4996–4998 (1989).
    [CrossRef] [PubMed]
  2. H. Nishihara, T. Suhara, “Micro Fresnel lenses,” Prog. Opt. 24, 1–37 (1987).
    [CrossRef]
  3. J. T. Winthrop, “Propagation of structural information in optical wave fields,” J. Opt. Soc. Am. 61, 15–30 (1971).
    [CrossRef]
  4. H. M. Ozaktas, D. Mendlovic, “Multistage optical interconnection architectures with least possible growth of system size,” Opt. Lett. 18, 296–298 (1993).
    [CrossRef] [PubMed]
  5. D. Mendlovic, H. M. Ozaktas, “Optical coordinate transformation methods and optical interconnection architectures,” Appl. Opt. 32, 5119–5124 (1993).
    [CrossRef] [PubMed]
  6. R. K. Kostuk, J. W. Goodman, L. Hesselink, “Design considerations for holographic optical interconnects,” Appl. Opt. 26, 3947–3953 (1987).
    [CrossRef] [PubMed]

1993 (2)

1989 (1)

1987 (2)

1971 (1)

Goodman, J. W.

Hesselink, L.

Kostuk, R. K.

Lohmann, A. W.

Mendlovic, D.

Nishihara, H.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” Prog. Opt. 24, 1–37 (1987).
[CrossRef]

Ozaktas, H. M.

Suhara, T.

H. Nishihara, T. Suhara, “Micro Fresnel lenses,” Prog. Opt. 24, 1–37 (1987).
[CrossRef]

Winthrop, J. T.

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Figures (10)

Fig. 1
Fig. 1

Geometrical spot size for a refractive lens with incidence angle β as a parameter: (a) tangential; (b) sagittal. The refractive index has been taken as 1.5.

Fig. 2
Fig. 2

Geometrical spot size for a diffractive lens with incidence angle β as a parameter: (a) tangential; (b) sagittal.

Fig. 3
Fig. 3

Definition of linear field extent Xmax and field angle βmax.

Fig. 4
Fig. 4

Contour plot of N as a function of D and f#. The contour levels give log N.

Fig. 5
Fig. 5

Value of βmax maximizing N as a function f# with D as a parameter.

Fig. 6
Fig. 6

Contour plot of I as a function of D and f#. The contour levels give log I.

Fig. 7
Fig. 7

Value of βmax maximizing I as a function f# with D as a parameter.

Fig. 8
Fig. 8

Contour plot of Nlitho as a function of D and f#. The contour levels give log Nlitho. The region around and below the lowermost line is meaningless since here Nlitho approaches or falls below unity.

Fig. 9
Fig. 9

N versus Nlitho. The curves with positive inclination are curves of equal f-number (for the solid, dotted, dashed–dotted, and dashed curves, f# equals 1, 10, 100, and 1000, respectively). The curves with negative inclination are curves of equal diameter (for the same curve types, D/λ equals 10, 103, 105, and 107, respectively).

Fig. 10
Fig. 10

Calculation of lateral aberration. All dimensions are normalized by D.

Equations (23)

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4 λ 2 π [ ( f # ) 2 + 1 / 4 ] ,
σ d 1 σ ^ d 1 λ = λ f # cos 3 β .
σ d 2 σ ^ d 2 λ = λ f # cos β .
σ 1 = σ ^ g 1 ( f # , β ) D + σ ^ d 1 ( f # , β ) λ ,
σ 2 = σ ^ g 2 ( f # , β ) D + σ ^ d 2 ( f # , β ) λ .
max β { max [ σ 1 ( β ) , σ 2 ( β ) ] } ,
σ = max [ σ 1 ( β max ) , σ 2 ( β max ) ] .
N = ( X max σ ) 2 ,
X max 2 = f tan β max = f # D tan β max .
I = N max ( X max 2 , D 2 ) .
sin α = sin β - x f = sin β - ξ f # ,
u ^ ( ξ ) = u ( x = ξ D ) D = ξ + f # ( sin β - ξ / f # ) [ 1 - ( sin β - ξ / f # ) 2 ] 1 / 2 - f # tan β .
u ^ ( ξ ) ( 1 - sin 2 β 2 cos 3 β - 1 cos β ) A - ξ 2 ( sin β 2 f # cos 3 β ) B - ξ 3 ( 1 2 ( f # ) 2 cos 3 β ) C + .
σ ^ g 2 = [ u ^ ( ξ ) - u ^ ¯ ] 2 d ξ ,
A 2 ( 3 ) ( 4 ) - 2 A B ( 4 ) ( 8 ) + B 2 - 2 A C ( 5 ) ( 16 ) + 2 B C ( 6 ) ( 32 ) + C 2 ( 7 ) ( 64 ) ,
σ ^ g = C 8 7 = 1 16 7 ( f # ) 2 .
σ ^ g = A 12 = 1 12 | 1 - sin 2 β 2 cos 3 β - 1 cos β | ,
N 1 / 2 = 2 f # D tan β max A / 12 D + f # λ / cos 3 β max ,
N 1 / 2 ( 2 sin β max cos 2 β max ) D λ .
N = 16 27 ( D λ ) 2 .
N = ( X max / σ ^ g D ) 2 ( f # ) 4 ,
I 1 / 2 = N 1 / 2 max ( X max , D ) = N 1 / 2 X max = 1 σ ^ d λ ,
I 1 / 2 = N 1 / 2 max ( X max , D ) = N 1 / 2 D ( f # ) 2 D ,

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