Abstract

Babinet’s principle states that two complementary objects that do not differ in structure but that differ by an exchange of black and white will produce essentially the same diffraction pattern. We study the case in which the black–white exchange is performed in some parts of the objects but not in other parts. Although the study was triggered in the realm of computer holography by specific experiments, it is predominantly fundamental in nature. Our aim is to generalize Babinet’s original principle in various ways. Ties to computer holography are explained but not exploited. Conceivable implications for spatial light modulators, for optical logic, and for array illuminators are mentioned.

© 1992 Optical Society of America

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References

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  1. J. Tanida, Y. Ichioka, “OPALS: optical parallel array logic system,” Appl. Opt. 25, 1565–1570 (1986).
    [Crossref] [PubMed]
  2. R. L. Morrison, S. L. Walker, “Binary phase gratings generating even-numbered spot arrays,” in Optical Society of America 1989 Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

1986 (1)

Ichioka, Y.

Morrison, R. L.

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even-numbered spot arrays,” in Optical Society of America 1989 Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

Tanida, J.

Walker, S. L.

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even-numbered spot arrays,” in Optical Society of America 1989 Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

Appl. Opt. (1)

Other (1)

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even-numbered spot arrays,” in Optical Society of America 1989 Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

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Figures (10)

Fig. 1
Fig. 1

Babinet flipping (black–white exchanging) of a binary object. In theory the extent of the domain is infinite.

Fig. 2
Fig. 2

Two-f setup for Fraunhofer diffraction.

Fig. 3
Fig. 3

Case in which the object u(x) and its Babinet counterpart have different mean values.

Fig. 4
Fig. 4

Object riding on a slowly varying bias.

Fig. 5
Fig. 5

Babinet flipping, but only within a finite frame.

Fig. 6
Fig. 6

Partial, or local, Babinet flipping.

Fig. 7
Fig. 7

Local Babinet flipping with one of three cells of a computer hologram.

Fig. 8
Fig. 8

Functional plot in support of Fig. 7.

Fig. 9
Fig. 9

Actual computer hologram with approximately half of the cells having been flipped.

Fig. 10
Fig. 10

Graphic code for logic yes and no in the OPALS processor.

Equations (51)

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u ( x ) v ( x ) = 1 - u ( x )             u = 1             or             0.
u ( x ) u ˜ ( ν ) = u ( x ) exp ( - 2 π i ν x ) d x .
u ( x ) = 1 / 2 + [ u ( x ) - 1 / 2 ] .
u ( x ) = 1 / 2 - [ u ( x ) - 1 / 2 ] = 1 - u ( x ) .
u ( x ) = u ¯ + Δ u ( x ) ,
u ( x ) = 1 / 2 + [ u ¯ - 1 / 2 + Δ u ( x ) ] .
u ( x ) = 1 / 2 - [ u ¯ - 1 / 2 + Δ u ( x ) ] = 1 - u ¯ - Δ u ( x ) .
u ˜ ( ν ) = u ¯ δ ( ν ) + Δ u ˜ ( ν ) ,
v ˜ ( ν ) = ( 1 - u ¯ ) δ ( ν ) - Δ u ˜ ( ν ) .
u ¯ = 1 - u ¯ ,             u ¯ = 1 / 2.
u ( x ) = B + [ u ( x ) - B ] ,
v ( x ) = B - [ u ( x ) - B ] = 2 B - u ( x ) .
u ( x ) = u ¯ + Δ u ( x ) = B + [ u ¯ - B + Δ u ( x ) ] ,
v ( x ) = B - [ u ¯ - B + Δ u ( x ) ] = 2 B - u ¯ - Δ u ( x ) .
u ˜ ( ν ) = u ¯ δ ( ν ) + Δ u ˜ ( ν ) ,
v ˜ ( ν ) = ( 2 B - u ¯ ) δ ( ν ) - Δ u ˜ ( ν ) .
B = u ¯ .
u ¯ 2 = 2 B - u ¯ 2             or             B 2 - Re ( B * u ¯ ) = 0.
i B * ( B - u ¯ ) = Re .
u ( x ) = B ( x ) + Δ u ( x ) ,
v ( x ) = B ( x ) - Δ u ( x ) .
F ( x , y ) = rect ( x / a ) rect ( y / b ) .
U ( x ) = u ( x )             within F ; zero outside .
V ( x ) = v ( x )             within F ; zero outside .
U ( x ) = u ( x ) F ( x ) = { 1 / 2 + [ u ( x ) - 1 / 2 ] } F ( x ) ,
V ( x ) = [ 1 / 2 - ( u - 1 / 2 ) ] F ( x ) = F ( x ) - u ( x ) F ( x ) ,
U ˜ ( ν ) = u ˜ ( ν ) ,
V ˜ ( ν ) = F ˜ ( ν ) - U ˜ ( ν ) ,
V ˜ ( ν ) 2 = F ˜ ( ν ) 2 + U ˜ ( ν ) 2 - Re ( F ˜ U ˜ * ) .
U ˜ ( ν ) = u ¯ F ˜ ( ν ) + Δ u ˜ ( ν ) ,
V ˜ ( ν ) = ( 1 - u ¯ ) F ˜ ( ν ) - Δ u ˜ ( ν ) .
F ( x ) = n = 1 N F n ( x ) .
F m ( x ) F n ( x ) = F m ( x ) δ m n .
U ( x ) = n = 1 N F n ( x ) u n ( x ) .
V ( x ) = n = 1 M F n ( x ) v n ( x ) + M + 1 N F n ( x ) u n ( x ) .
v n ( x ) = B n - [ u n ( x ) - B n ] = 2 B n - u n ( x ) .
u n ( x ) = u ¯ n F n ( x ) + Δ u n ( x ) = B n F n ( x ) + [ ( u ¯ n - B n ) F n ( x ) + Δ u n ( x ) ] ,
v n ( x ) = B n F n ( x ) - [ ( u ¯ n - B n ) F n ( x ) + Δ u n ( x ) ] = ( 2 B n - u ¯ n ) F n ( x ) - Δ u n ( x ) .
B n = 1 / 2.
v n ( x ) = ( 1 - u ¯ n ) F n ( x ) - Δ u n ( x ) .
n = 1 M ( n ) = I ,             M + 1 M ( n ) = I I ,             I + I I = ,
I F n ( x ) = F I ( x ) ,             I I F n ( x ) = F I I ( x ) , F I ( x ) + F I I ( x ) = F ( x ) .
u ¯ n = 1 / 2             for all elements in F I ( x ) .
I u ¯ n F n ( x ) = 1 / 2 F I ( x ) = I ( 1 - u ¯ n ) F n ( x ) .
u ¯ n = 1 / 2             in F I ( x ) and in F I I ( x ) ,
I u ¯ n F n ( x ) + I I u ¯ n F n ( x ) = 1 / 2 F ( x ) = I ( 1 - u ¯ n ) F n ( x ) + I I u ¯ n F n ( x ) .
U ( x ) = 1 / 2 F ( x ) + Δ U I ( x ) + Δ U I I ( x ) ,
V ( x ) = 1 / 2 F ( x ) + Δ V I ( x ) + Δ U I I ( x ) = 1 / 2 F ( x ) - Δ U I ( x ) + Δ U I I ( x ) .
U ˜ ( ν ) = 1 / 2 F ˜ ( ν ) + Δ U ˜ I ( ν ) + Δ U ˜ I I ( ν ) ,
V ˜ ( ν ) = 1 / 2 F ˜ ( ν ) - Δ U ˜ I ( ν ) + Δ U ˜ I I ( ν ) .
I ( 1 - u ¯ n ) F n ( x ) .

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