Abstract

A novel artificial diffuser for Fourier transform hologram recording is proposed, and its imaging and spectral properties are analyzed. This new diffuser gives high SNR and resolution characteristics to the holographically reconstructed images and high reliability and redundancy characteristics to the system where holograms are used as information storing media. These characteristics are confirmed by experiments.

© 1980 Optical Society of America

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References

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  1. A. Iwamoto, Y. Mizobuchi, in Proceedings of the Twelfth SMPTE Television Conference (SMPTE, Los Angeles, 1978).
  2. C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
    [CrossRef] [PubMed]
  3. Y. Tsunoda, Y. Takeda, Appl. Opt. 13, 2046 (1974).
    [CrossRef] [PubMed]
  4. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  5. B. R. Frieden, Prog. Opt. 9, 313 (1971).
  6. D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).
  7. D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).
  8. J. D. Gaskill, Linear Systems Fourier Transforms, and Optics (Wiley, New York, 1978).
  9. H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, Princeton, N.J., 1965).
  10. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  11. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

1974 (1)

1971 (1)

B. R. Frieden, Prog. Opt. 9, 313 (1971).

1970 (1)

1964 (1)

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

1961 (1)

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

Burckhardt, C. B.

C. B. Burckhardt, Appl. Opt. 9, 695 (1970).
[CrossRef] [PubMed]

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Frieden, B. R.

B. R. Frieden, Prog. Opt. 9, 313 (1971).

Gaskill, J. D.

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

Iwamoto, A.

A. Iwamoto, Y. Mizobuchi, in Proceedings of the Twelfth SMPTE Television Conference (SMPTE, Los Angeles, 1978).

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Mizobuchi, Y.

A. Iwamoto, Y. Mizobuchi, in Proceedings of the Twelfth SMPTE Television Conference (SMPTE, Los Angeles, 1978).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pollack, H. O.

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

Rowe, H. E.

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, Princeton, N.J., 1965).

Slepian, D.

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

Takeda, Y.

Tsunoda, Y.

Appl. Opt. (2)

Bell Syst. Tech. J. (2)

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

D. Slepian, Bell Syst. Tech. J. 43, 3009 (1964).

Prog. Opt. (1)

B. R. Frieden, Prog. Opt. 9, 313 (1971).

Other (6)

A. Iwamoto, Y. Mizobuchi, in Proceedings of the Twelfth SMPTE Television Conference (SMPTE, Los Angeles, 1978).

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

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

H. E. Rowe, Signals and Noise in Communication Systems (Van Nostrand, Princeton, N.J., 1965).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (7)

Fig. 1
Fig. 1

Hologram composing and reconstructing optical system. The object and photographic material are symmetrically located, with respect to a FT lens, focal distance f apart, and an artificial diffuser superimposed on the object. The FT lens decomposes the composite object into its spatial frequency components; these interfere with the reference light to form a FT hologram. The distance between the artificial diffuser and the object is negligibly small compared with f. The actual length (x′) in the FT plane is related to the corresponding spatial frequency (ξ) as ξ = x′(λf)−1.

Fig. 2
Fig. 2

Normalized circular prolate spheroidal functions ϕ0,0(r,c) for c = 1, 2, 5, and 10. All functions are normalized to unity at the origin. For c = 10, ϕ0,02(r) is also plotted.

Fig. 3
Fig. 3

Calculated spectrum for the optimized artificial diffuser. The calculated power spectrum for the optimized artificial diffuser has a point symmetry with respect to ground frequency (omnidirectional), whose outer profile is determined by ϕ0,02(r), and whose circular spectral component is generated by omnidirectional sampling and broadened by random sampling. This spectrum is calculated from Eq. (17).

Fig. 4
Fig. 4

Realized optimized pattern for the artificial diffuser. This pattern was generated and plotted by a computer and plotter. To make an omnidirectional spectral distribution, a piecewise rotationally symmetric pattern was adopted. Individual positions for black spots were randomized to give broad spectral components. This pattern was reduced in size and changed to phase variation by conventional photographic chemical processes.

Fig. 5
Fig. 5

Spectral distribution of the fabricated artificial diffuser. The intense circular spectrum component spatial frequency is set to 20 lp/mm. The optical intensity distribution (a) shows close similarity to the calculated spectrum for the optimized diffuser shown in Fig. 3. The optical distribution is converted into an electrical signal by a high-resolution Vidicon camera, and its central part is displayed on an oscilloscope selecting one scanning line (b).

Fig. 6
Fig. 6

One of the images reconstructed from a hologram using the optimized artificial diffuser. Owing to its high SNR value and wide spectral response, reconstructed images from holograms made with the optimized artificial diffuser can reproduce many halftone levels.

Fig. 7
Fig. 7

Power spectrum of the PPM signal with random modulation. The power spectrum of the PPM signal with random modulation is calculated using Eq. (17) where the pulses, whose shapes correspond to Gaussian profiles of c = 10 in Fig. 2, are distributed according to Gaussian law.

Equations (29)

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H ( ξ , η ) = F ( ξ , η ) S ( ξ , η ) ,
[ S ( x , y ) F S ( ξ , η ) ] , and
[ f ( x , y ) F F ( ξ , η ) ] .
f ( x ) = exp [ j ϕ ( x ) ] .
f d ( x ) f ( x ) exp ( j π x 2 λ d ) .
I d ( x ) 1 - λ d 2 π ϕ ( 2 ) + ( λ d 4 π ) 2 [ ϕ ( 2 ) 2 + ϕ ( 1 ) 4 ] .
sin a x a x ~ 1 - a 2 x 2 6 ~ exp ( - a 2 x 2 6 ) ,
I a ( x ) f ( x ) exp ( - a 2 x 2 / 6 ) 2 ~ 1 - 3 ( ϕ ( 1 ) a ) 2 + ( 3 2 a 2 ) 2 [ ϕ ( 1 ) 4 + ϕ ( 2 ) 2 ] .
f ( x , y ) = w ( x , y ) comb ( x , y ) ,
F ( ξ , η ) = W ( ξ , η ) × comb ( ξ , η ) ,
W ( ξ , η ) F w ( x , y ) , comb ( x , y ) 2 - D comb function , and comb ( ξ , η ) F comb ( x , y ) .
0 r d r J 0 ( 2 π r ξ ) Φ 0 , 0 ( r ) = { ( r 0 / 2 π ξ m ) λ 0 , 0 - 1 / 2 Φ 0 , 0 ( r 0 ξ / ξ m ) ( 0 ξ ξ m ) 0 ( ξ > ξ m ) ,
0 r 0 r d r J 0 ( 2 π r ξ ) Φ 0 , 0 ( r ) = ( r 0 / 2 π ξ m ) λ 0 , 0 1 / 2 Φ 0 , 0 ( r 0 ξ / ξ m ) ,
c = 2 π r 0 ξ m ,
Φ 0 , 0 ( r ) = [ 4 π λ 0 , 0 ξ m ( r 0 ) 1 / 2 ] 1 / 2 exp [ - c 2 ( r r 0 ) 2 ] .
ξ s = 1 2 r 0 .
D comb ( a 1 x + b 1 y , a 2 x + b 2 y ) F comb ( b 2 ξ - a 2 η D , - b 1 ξ + a 1 η D ) ,
P f ( ξ ) = exp [ - Φ ϕ ( 0 ) ] [ δ ( ξ ) + n = 1 1 n ! P ϕ ( ξ ) * * P ϕ ( ξ ) ] .             1 ( n - 1 )
P f ( ξ ) ~ [ 1 - Φ ϕ ( 0 ) ] δ ( ξ ) + P ϕ ( ξ ) .
P x ( ξ ) { [ 1 - R ( ξ ) 2 ] + R ( ξ ) 2 2 r 0 n = - δ ( ξ - n ξ s ) } W ( ξ ) ,
v α ( x ) = v ( x ) exp ( - α x 2 ) ,
v ( x ) = exp [ j ϕ ( x ) ] ,
v ( x ) = - V ( ξ ) exp ( j 2 π ξ x ) d ξ ,
d 2 v / d x 2 = - ( - 4 π 2 ξ 2 ) V ( ξ ) exp ( j 2 π ξ x ) d ξ ,
d 2 exp ( j ϕ ) / d x 2 = { j ϕ ( 2 ) - [ ϕ ( 1 ) ] 2 } exp ( j ϕ ) ,
exp ( - α x 2 ) exp ( - j 2 π ξ ( x ) ) d x = C exp ( - π 2 ξ 2 α ) , C = ( π / α ) 1 / 2 ,
v α ( x ) = V ( ξ ) exp ( - α x 2 ) exp [ j 2 π ξ ( x - X ) ] d X d ξ = C V ( ξ ) exp ( j 2 π ξ x ) exp ( - π 2 ξ 2 α - 1 ) d ξ ~ C V ( ξ ) exp ( j 2 π ξ x ) ( 1 - π 2 α ξ 2 ) d ξ = C V ( ξ ) exp ( j 2 π ξ x ) d ξ - C α π 2 ξ 2 V ( ξ ) exp ( j 2 π ξ x ) d ξ .
ξ 2 V ( ξ ) exp ( j 2 π ξ x ) d ξ = exp ( j ϕ ) 4 π 2 { [ ϕ ( 1 ) ] 2 - j ϕ ( 2 ) } .
v α ( x ) = C exp ( j ϕ ) [ 1 - ϕ ( 1 ) 2 4 α + j ϕ ( 2 ) 4 α ] .

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