Abstract

It is shown that it should be possible to produce single exposure holograms as well as multiple exposure holograms with the same diffraction efficiency if appropriate exposure conditions are used. The amplitude ratios between object wave and reference wave which must be employed in forming the hologram are specified for the two storage techniques. The object waves are provided to originate from a periodic array of coherent radiating light sources.

© 1971 Optical Society of America

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References

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  1. J. T. LaMacchia, C. J. Vincelette, Appl. Opt. 7, 1857 (1968).
    [CrossRef] [PubMed]
  2. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

1968

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

LaMacchia, J. T.

Vincelette, C. J.

Wolf, E.

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Appl. Opt.

Other

See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

An array {Gν} of light sources in front of the hologram plane x3 = 0.

Fig. 2
Fig. 2

Symbols used in the scalar product k0·r for plane waves. The hologram plane is x3 = 0.

Fig. 3
Fig. 3

The interference structure sinN½ξ/sinξ for N = 25, and ξ = (kd/2L)x.

Fig. 4
Fig. 4

The interference structure sinN½ξ/sinξ for N = 36, and ξ = (kd/2L)x.

Fig. 5
Fig. 5

The interference structure sin2N½ξ/sin2ξfor N = 25, and ξ = (kd/2L)x.

Fig. 6
Fig. 6

A section of the interference term f(N) for N = 25.

Fig. 7
Fig. 7

Amplitude ratio as a function of the fringe visibility for multiple exposure holograms.

Equations (45)

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I = E R + E G 2 = E R 2 + E G 2 + 2 R e E R E G * .
W = I · τ
a = W max - W min 2 W ¯ with 0 a 1.
W ¯ = W max + W min 2
E R ( p ) = e 0 U 0 ( p ) e i ( α 0 - ω t )
E G ( p ) = ν = 1 N e ν U ν ( p ) e i ( α ν - ω t )
E R ( s ) = e 0 U 0 ( s ) e i ( α 0 - ω t )
E ν ( s ) = e ν U ν ( s ) e i ( α ν - ω t ) ,
I ( p ) = | ν = 0 N e ν U ν ( p ) e i ( α ν - ω t ) | 2 = U 0 ( p ) 2 + ν = 1 N U ν ( p ) 2 + ν = 2 ν > ν N ν = 1 N - 1 2 e ν e ν U ν ( p ) U ν ( p ) cos ( α ν - α ν ) + ν = 1 N 2 e 0 e ν U 0 ( p ) U ν ( p ) cos ( α 0 - α ν ) .
I ν = U 0 ( s ) 2 + U ν ( s ) 2 + 2 e 0 e ν U 0 ( s ) U ν ( s ) cos ( α 0 - α ν ) .
W ( p ) = τ ( p ) U 0 ( p ) 2 [ 1 + ( U ( p ) U 0 ( p ) ) 2 f ( N ) + 2 U ( p ) U 0 ( p ) g ( N ) ]
W ( s ) = τ ( s ) U 0 ( s ) 2 [ N { 1 + ( U ( s ) U 0 ( s ) ) 2 } + 2 U ( s ) U 0 ( s ) g ( N ) ] ,
g ( N ) = ν = 1 N cos ( α 0 - α ν )
f ( N ) = ν = 1 N ν = 1 N cos ( α ν - α ν ) = | ν = 1 N e i α ν | 2 = N + 2 ν > ν cos ( α ν - α ν ) .
W ¯ ( p ) = τ ( p ) U 0 ( p ) 2 [ 1 + U ( p ) 2 U 0 ( p ) ( { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } max + { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } min ) ]
W ¯ ( s ) = τ ( s ) U 0 ( s ) 2 [ N ( 1 + U ( s ) 2 U 0 ( s ) 2 ) + U ( s ) U 0 ( s ) ( g max ( N ) + g min ( N ) ) ] .
a ( p ) = U ( p ) U 0 ( p ) ( { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } max - { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } min ) 2 + U ( p ) U 0 ( p ) ( { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } max + { U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } min )
a ( s ) = U ( s ) U 0 ( s ) ( g max ( N ) - g min ( N ) ) N ( 1 + U ( s ) 2 U 0 ( s ) 2 ) + U ( s ) U 0 ( s ) ( g max ( N ) + g min ( N ) ) .
U ( s ) U 0 ( s ) = G ( N ) 2 N a ( s ) ± G ( N ) 2 N a ( s ) { 1 - [ 2 N a ( s ) G ( N ) ] 2 } 1 2
G ( N ) ( g max ( N ) - g min ( N ) ) - a ( s ) ( g max ( N ) + g min ( N ) )
g min ( N ) = - g max ( N ) ,
{ U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } max = U ( p ) U 0 ( p ) f max ( N ) + 2 g max ( N ) ,
{ U ( p ) U 0 ( p ) f ( N ) + 2 g ( N ) } min = U ( p ) U 0 ( p ) f max ( N ) + 2 g min ( N ) .
W ¯ ( p ) = τ ( p ) U 0 ( p ) 2 [ 1 + ( U ( p ) U 0 ( p ) ) 2 f max ( N ) ]
W ¯ ( s ) = τ ( s ) U 0 ( s ) 2 N [ 1 + ( U ( s ) U 0 ( s ) ) 2 ]
a ( p ) = 2 g max ( N ) · U ( p ) U 0 ( p ) 1 + ( U ( p ) U 0 ( p ) ) 2 f max ( N )
a ( s ) = 2 g max ( N ) · U ( s ) U 0 ( s ) N [ 1 + ( U ( s ) U 0 ( s ) ) 2 ] .
U ( p ) U 0 ( p ) = g max ( N ) a ( p ) · f max ( N ) ( 1 ± { 1 - [ a ( p ) g max ( N ) ] 2 f max ( N ) } 1 2 )
U ( s ) U 0 ( s ) = g max ( N ) a ( s ) · N ( 1 ± { 1 - [ a ( s ) · N g max ( N ) ] 2 } 1 2 ) .
r = ( x 1 x 2 0 ) ,             and             R ν = ( ν 1 d ν 2 d - L )
ρ ν = r - R ν = L [ 1 + ( x 1 - ν 1 d ) 2 + ( x 2 - ν 2 d ) 2 L 2 ] 1 2 .
ρ ν L + r 2 2 L - d L ( ν 1 x 1 + ν 2 x 2 ) .
g ( N ) = cos [ k ρ ¯ + ( N 1 2 - 1 ) ( ξ 1 + ξ 2 ) ] · sin N 1 2 ξ 1 sin ξ 1 · sin N 1 2 ξ 2 sin ξ 2 ,
f ( N ) = sin 2 N 1 2 ξ 1 sin 2 ξ 1 · sin 2 N 1 2 ξ 2 sin 2 ξ 2 ,
ξ 1 = k d 2 L x 1 ,             ξ 2 = k d 2 L x 2 ,
ρ ¯ = ρ 0 - ( L + r 2 2 L ) .
g min ( N ) = - N , g max ( N ) = + N f min ( N ) = 0 , and f max ( N ) = N 2 .    
U ( p ) U 0 ( p ) = 1 a N [ 1 ± ( 1 - a 2 ) 1 2 ]
U ( s ) U 0 ( s ) = 1 a [ 1 ± ( 1 - a 2 ) 1 2 ] .
U ( p ) U 0 ( p ) a ( 4 + a 2 ) 8 N , U ( s ) U 0 ( s ) a ( 4 + a 2 ) 8 .
τ ( p ) = τ ( s ) = τ U ( p ) = U ( s ) = U U 0 ( p ) 2 U 2 f max ( N ) U 0 ( s ) 2 U 2 ,
W ¯ ( p ) τ U 0 ( p ) 2 , W ¯ ( s ) τ U 0 ( s ) 2 · N a ( p ) 2 g max ( N ) · U U 0 ( p ) , a ( s ) 2 g max ( N ) · U N U 0 ( s ) .
U 0 ( p ) = N 1 2 U 0 ( s ) ,
a ( p ) = N 1 2 a ( s ) .
U ( s ) U 0 ( s ) = N U ( p ) U 0 ( p ) , and τ ( s ) U 0 ( s ) 2 = 1 N τ ( p ) U 0 ( p ) 2 .

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