Abstract

It is possible to manipulate the structure of computer holograms. Because these holograms are produced synthetically, we are able to influence some parameters directly. In order to demonstrate this flexibility, some situations were considered: A specific modulation technique (two-carrier procedure) was chosen together with some specific types of manipulation (carrier profile and partial inversion). The effects are described analytically and illustrated by experiments.

© 1985 Optical Society of America

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References

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  1. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18, 3661–3669 (1979).
    [CrossRef] [PubMed]
  2. F. S. Perry, “A holographic puzzle,”J. Opt. Soc. Am. 57, viii (1967).

1979 (1)

1967 (1)

F. S. Perry, “A holographic puzzle,”J. Opt. Soc. Am. 57, viii (1967).

Lee, W.-H.

Perry, F. S.

F. S. Perry, “A holographic puzzle,”J. Opt. Soc. Am. 57, viii (1967).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

F. S. Perry, “A holographic puzzle,”J. Opt. Soc. Am. 57, viii (1967).

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Figures (5)

Fig. 1
Fig. 1

Illustration to show how the threshold operation can be used to generate binary structures. A slowly varying signal is indicated together with a triangular-profile carrier in a and a sawtooth-profile carrier in d. The corresponding binary representations are shown in b and e. A periodic inversion of portions of these structures leads to the modified binary ones in c and f.

Fig. 2
Fig. 2

The holograms using a, a triangular-profile carrier and b, a sawtooth-profile carrier in the vertical direction are shown together with c and d, their optical reconstructions. A bias of 0.5 was used.

Fig. 3
Fig. 3

Holograms with a triangular-profile carrier in the vertical direction and a bias of a, 0.75 and b, 0.83 result in reconstructions c and d with certain suppressed diffraction orders.

Fig. 4
Fig. 4

A periodic inversion of portions of the hologram structure of Figs. 2a and 2b are shown in a and b. The even diffraction orders (including the zeroth) are suppressed in the corresponding reconstructions c and d.

Fig. 5
Fig. 5

Holograms with a triangular-profile carrier in the vertical direction and a bias of a, 0.66 and b, 0.75 in which portions of the structure are periodically inverted. Higher diffraction orders in addition to the zeroth are suppressed in c and d, the corresponding reconstructions.

Equations (26)

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u ˜ ( ν , μ ) = F x y { u ( x , y ) } = u ˜ ( ν , μ ) exp [ i ϕ ( ν , μ ) ] .
u ˜ H ( ν , μ ) = u ˜ ( ν , μ ) cos [ 2 π ν x 0 + ϕ ( ν , μ ) ] ;
u ( x , y ) u ( x - x 0 , y ) .
v ˜ ( ν , μ ) = A u ˜ H ( ν , μ ) + B .
h ˜ ( ν , μ ) = step [ v ˜ ( ν , μ ) - c ˜ ( ν , μ ) ] ,
step ( α ) = { 0 if α 0 1 otherwise .
v ˜ ( ν ) = A cos ( 2 π ν x 0 ) + B .
h ˜ ( ν , μ ) = ( 1 / μ 0 ) comb ( μ / μ 0 ) * rect { μ / [ μ 0 v ˜ ( ν ) ] } ,
rect ( α ) = { 1 if α 1 / 2 0 otherwise ,
h ( x , y ) = comb ( μ 0 y ) F ν μ { rect [ μ μ 0 v ˜ ( ν ) ] } .
h ( x , y ) = comb ( μ 0 y ) F ν { μ 0 v ˜ ( ν ) sinc [ μ 0 v ˜ ( ν ) y ] } ,
h ( x , 0 ) = [ B δ ( x ) + A δ ( x - x 0 ) + A δ ( x + x 0 ) ] .
h ( x , y ) = 1 π y comb ( μ 0 y ) [ a 0 2 δ ( x ) + k = - k 0 + a k ( y ) δ ( x - k x 0 ) ] ,
a k ( y ) = Im { 1 π - π π exp [ i μ 0 π y v ˜ ( ν ) ] cos ( k ν ) d ν } ,
a k ( y ) = Im { exp [ i μ 0 π y B ] ( i ) k J k ( A π μ 0 y ) } ,
a k ( m / μ 0 ) = a k m = Im { exp [ i π m B ] ( i ) k J k ( A m π ) } .
a 1 m = cos ( π m B ) J 1 ( π m A ) .
a 1 m ( x , y ) = F ν μ { J 1 [ m π a ˜ ( ν , μ ) ] exp [ i ϕ ( ν , μ ) ] } cos ( m π B ) .
h ( x , y ) = B δ ( x , y ) + a ( x - x 0 , y ) + a * [ - ( x + x 0 ) , y ] ,
h ˜ 1 ( ν , μ ) = h ˜ ( ν , μ ) [ 1 μ 0 comb ( μ μ 0 ) * rect ( μ μ 0 / 2 ) ] .
h ˜ i n 2 ( ν , μ ) = [ 1 - 1 μ 0 comb ( μ μ 0 ) * rect ( μ μ 0 / 2 ) ] h ˜ ( ν , μ ) e i π + [ 1 - 1 μ 0 comb ( μ μ 0 ) * rect ( μ μ 0 / 2 ) ] .
g ˜ ( ν , μ ) = h ˜ 1 ( ν , μ ) + h ˜ i n 2 ( ν , μ ) = - h ˜ ( ν , μ ) + 2 [ 1 μ 0 comb ( μ μ 0 ) * rect ( μ μ 0 / 2 ) ] h ˜ ( ν , μ ) + Δ ˜ ( ν , μ ) ,
Δ ˜ ( ν , μ ) = 1 - [ 1 μ 0 comb ( μ μ 0 ) * rect ( μ μ 0 / 2 ) ]
g ( x , y ) = F ν μ [ g ˜ ( ν , μ ) ] = - h ( x , y ) + 2 { comb ( μ 0 y ) ( μ 0 / 2 ) sinc [ ( μ 0 / 2 ) y ] } * h ( x , y ) + Δ ( x , y ) ,
g ( x , y ) = - h ( x , y ) + n = - + { δ [ y - ( n / μ 0 ) ] sinc [ ( μ 0 / 2 ) y ] } * h ( x , y ) + Δ ( x , y ) = - h ( x , y ) + h ( x , y ) + n = - n 0 + sinc ( n / 2 ) h [ x , y - ( n / μ 0 ) ] + Δ ( x , y ) = n = - n 0 + sinc ( n / 2 ) h [ x , y - ( n / μ 0 ) ] + Δ ( x , y ) .
a 1 m ( x , y ) = sin ( π m B ) F ν μ { J 1 ( π m a ˜ ( ν , μ ) ) exp [ i ϕ ( ν , μ ) ] } .

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