Abstract

A theoretical analysis of the power spectra of diffusers with a series of pseudorandom phase sequences is presented. General equations describing three typical cases of power spectra are derived by making use of the constraints assumed. Profiles of the power spectra showed good agreement with the results of numerical computation using the fast-Fourier-transform algorithm. The images of the diffusers through a double-diffraction optical system were also evaluated by computer simulation. The advantage of pseudorandom phase sequences over random phase sequences is discussed from the viewpoint of signal-to-noise ratio.

© 1979 Optical Society of America

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References

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  1. C. B. Burkhardt, “Use of a Random Phase Mask for the Recording of Fourier Transform Holograms of Data Masks,” Appl. Opt. 9, 695–700 (1970).
    [Crossref]
  2. Y. Takeda, “Random Phase Sifters for Fourier Transformed Holograms,” Appl. Opt. 11, 818–822 (1972).
    [Crossref] [PubMed]
  3. W. J. Dallas, “Deterministic Diffusers for Holography,” Appl. Opt. 12, 1179–1187 (1973).
    [Crossref] [PubMed]
  4. S. Yonezawa, “A Deterministic Phase Shifter for Holographic Memory Devices,” Opt. Commun. 19, 370–373 (1976).
    [Crossref]
  5. Y. Torii, “Synthesis of Deterministic Phase Codes for Phase Shifter in Holography,” Opt. Commun. 24, 157–180 (1978).
    [Crossref]
  6. M. Kato, Y. Nakayama, and T. Suzuki, “Speckle Reduction in Holography with a Spatially Incoherent Source,” Appl. Opt. 14, 1093–1099 (1975).
    [Crossref] [PubMed]
  7. I. Sato and M. Kato, “Speckle-Noise Simulation of Fourier-Transform Holography with Random Phase Sequence,” J. Opt. Soc. Am. 65, 856–857 (1975).
    [Crossref]
  8. M. Kato, I. Sato, and Y. Nakayama, “Speckle Reduction Processing in Holography,” Proceedings of the Tenth Congress of the International Commission for Optics, Palacký University, Olomouc Czech Technical University, Prague, August25–29, 1975.
  9. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968), p. 13.

1978 (1)

Y. Torii, “Synthesis of Deterministic Phase Codes for Phase Shifter in Holography,” Opt. Commun. 24, 157–180 (1978).
[Crossref]

1976 (1)

S. Yonezawa, “A Deterministic Phase Shifter for Holographic Memory Devices,” Opt. Commun. 19, 370–373 (1976).
[Crossref]

1975 (2)

1973 (1)

1972 (1)

1970 (1)

Burkhardt, C. B.

Dallas, W. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968), p. 13.

Kato, M.

I. Sato and M. Kato, “Speckle-Noise Simulation of Fourier-Transform Holography with Random Phase Sequence,” J. Opt. Soc. Am. 65, 856–857 (1975).
[Crossref]

M. Kato, Y. Nakayama, and T. Suzuki, “Speckle Reduction in Holography with a Spatially Incoherent Source,” Appl. Opt. 14, 1093–1099 (1975).
[Crossref] [PubMed]

M. Kato, I. Sato, and Y. Nakayama, “Speckle Reduction Processing in Holography,” Proceedings of the Tenth Congress of the International Commission for Optics, Palacký University, Olomouc Czech Technical University, Prague, August25–29, 1975.

Nakayama, Y.

M. Kato, Y. Nakayama, and T. Suzuki, “Speckle Reduction in Holography with a Spatially Incoherent Source,” Appl. Opt. 14, 1093–1099 (1975).
[Crossref] [PubMed]

M. Kato, I. Sato, and Y. Nakayama, “Speckle Reduction Processing in Holography,” Proceedings of the Tenth Congress of the International Commission for Optics, Palacký University, Olomouc Czech Technical University, Prague, August25–29, 1975.

Sato, I.

I. Sato and M. Kato, “Speckle-Noise Simulation of Fourier-Transform Holography with Random Phase Sequence,” J. Opt. Soc. Am. 65, 856–857 (1975).
[Crossref]

M. Kato, I. Sato, and Y. Nakayama, “Speckle Reduction Processing in Holography,” Proceedings of the Tenth Congress of the International Commission for Optics, Palacký University, Olomouc Czech Technical University, Prague, August25–29, 1975.

Suzuki, T.

Takeda, Y.

Torii, Y.

Y. Torii, “Synthesis of Deterministic Phase Codes for Phase Shifter in Holography,” Opt. Commun. 24, 157–180 (1978).
[Crossref]

Yonezawa, S.

S. Yonezawa, “A Deterministic Phase Shifter for Holographic Memory Devices,” Opt. Commun. 19, 370–373 (1976).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

S. Yonezawa, “A Deterministic Phase Shifter for Holographic Memory Devices,” Opt. Commun. 19, 370–373 (1976).
[Crossref]

Y. Torii, “Synthesis of Deterministic Phase Codes for Phase Shifter in Holography,” Opt. Commun. 24, 157–180 (1978).
[Crossref]

Other (2)

M. Kato, I. Sato, and Y. Nakayama, “Speckle Reduction Processing in Holography,” Proceedings of the Tenth Congress of the International Commission for Optics, Palacký University, Olomouc Czech Technical University, Prague, August25–29, 1975.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968), p. 13.

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Figures (8)

FIG. 1
FIG. 1

Double-diffraction optical system. An object at the plane P1 is imaged through an aperture of the size DH onto the plane P2.

FIG. 2
FIG. 2

Schematic model of a one-dimensional diffuser with a pseudorandom phase sequence.

FIG. 3
FIG. 3

Normalized forms of the power spectra for the respective diffusers with three typical phase sequences; L = P = 50 μ, λ = 488 nm, and f = 70 mm.

FIG. 4
FIG. 4

Relation between the three-level and six-level diffusers; the imaginary three-level diffuser with sampling areas of size 3L and period 2L (a) is just equivalent to the six-level diffuser with sampling areas of size L and period L (b).

FIG. 5
FIG. 5

Power-spectra obtained by a computer simulation: (a), the four-level diffuser; (b), the three-level diffuser; (c), and the six-level diffuser. Other parameters are the same as in Fig. 3.

FIG. 6
FIG. 6

Original phase levels of the four-level diffuser and their normalized images obtained by a computer simulation. DH indicates the aperture size corresponding to that of the hologram. Other parameters are the same as in Fig. 3.

FIG. 7
FIG. 7

Original phase levels of the six-level diffuser and their normalized images obtained by a computer simulation. DH indicates the aperture size corresponding to that of the hologram. Other parameters are also the same as in Fig. 3.

FIG. 8
FIG. 8

Calculated signal-to-noise ratio of the image versus hologram size (S/N vs DH) with respect to the four typical phase sequences; P0 = 50 μ and other parameters are also the same as in Fig. 3.

Tables (3)

Tables Icon

TABLE I Examples of the PRPS.

Tables Icon

TABLE II Values of the probability Pr(J,M) that ϕnϕn±M in the four-level PRPS takes values πJ/2 (J = 0, 1, 2, 3).

Tables Icon

TABLE III Values of the probability Pr(J,M) that ϕnϕn±M in the three-level PRPS takes value 2πJ/3 (J = 0,1,2). [ ] denotes Gauss’s symbol.

Equations (29)

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G ( ξ ) = g ( x 1 ) exp ( 2 π i ξ λ f x 1 ) d x 1 .
I H ( ξ ) = G ( ξ ) · G ( ξ ) * ,
t ( ξ ) = rect ( ξ / D H ) ,
I 2 ( x 2 ) = | t ( ξ ) · G ( ξ ) exp ( 2 π i x 2 λ f ξ ) d ξ | 2 .
g ( x 1 ) = n = N N rect ( x 1 n P L ) exp ( i ϕ n ) ,
ϕ n ϕ n 1 = 2 π / S
2 π ( S 1 ) / S ,
A ( x 1 ) = g ( u ) · g * ( u x 1 ) d u .
I H ( ξ ) = A ( x 1 ) exp ( 2 π i ξ λ f x 1 ) d x 1 .
A ( x 1 ) = ( 2 N + 1 ) B ( x 1 ) + M = 1 2 N n = N + M N { B ( x 1 M P ) × exp [ i ( ϕ n ϕ n M ) ] + B ( x 1 + M P ) × exp [ i ( ϕ n ϕ n + M ) ] } ,
B ( x 1 ± M P ) = rect ( u L ) rect ( u x 1 M P L ) d u ,
ϕ n ϕ n 1 = π / 2
( 3 / 2 ) π .
A ( x 1 ) = ( 2 N + 1 ) B ( x 1 ) + 2 N [ B ( x 1 P ) + B ( x 1 + P ) ] × [ P r ( 0 , 1 ) + i P r ( 1 , 1 ) P r ( 2 , 1 ) i P r ( 3 , 1 ) ] + + ( 2 N M + 1 ) [ B ( x 1 M P ) + B ( x 1 + M P ) ] × [ P r ( 0 , M ) + i P r ( 1 , M ) P r ( 2 , M ) i P r ( 3 , M ) ] + + [ B ( x 1 2 N P ) + B ( x 1 + 2 N P ) ] × [ P r ( 0 , 2 N ) + i P r ( 1 , 2 N ) P r ( 2 , 2 N ) i P r ( 3 , 2 N ) ] .
E [ A ( x 1 ) ] = ( 2 N + 1 ) B ( x 1 ) .
E [ I H ( ξ ) ] = ( 2 N + 1 ) L 2 sinc 2 ( L ξ / λ f ) ,
sinc ( L ξ / λ f ) = sin ( π L ξ / λ f ) π L ξ / λ f .
ϕ n ϕ n 1 = 2 π / 3
( 4 / 3 ) π
E [ A ( x 1 ) ] = ( 2 N + 1 ) B ( x 1 ) + 2 N [ B ( x 1 P ) + B ( x 1 + P ) ] × [ P r ( 0 , 1 ) + P r ( 1 , 1 ) exp ( i 2 3 π ) + P r ( 2 , 1 ) exp ( i 4 3 π ) ] + + ( 2 N M + 1 ) [ B ( x 1 M P ) + B ( x 1 + M P ) ] × [ P r ( 0 , M ) + P r ( 1 , M ) exp ( i 2 3 π ) + P r ( 2 , M ) exp ( i 4 3 π ) ] + + [ B ( x 1 2 N P ) + B ( x 1 + 2 N P ) ] × [ P r ( 0 , 2 N ) + P r ( 1 , 2 N ) exp ( i 2 3 π ) + P r ( 2 , 2 N ) exp ( i 4 3 π ) ] .
E [ A ( x 1 ) ] = ( 2 N + 1 ) B ( x 1 ) + M = 1 2 N ( 2 N M + 1 ) [ B ( x 1 M P ) + B ( x 1 + M P ) ] × ( k = 0 [ M / 2 1 ] 2 2 k 1 k = 0 [ ( M 1 ) / 2 ] 2 2 k 1 + k = 1 [ M / 2 ] 2 2 k k = 1 [ ( M 1 ) / 2 ] 2 2 k ) = ( 2 N + 1 ) B ( x 1 ) + M = 1 [ 2 N ] ( 2 N M + 1 ) ( 1 2 ) M × [ B ( x 1 M P ) + B ( x 1 + M P ) ] ,
E [ I H ( ξ ) ] = [ 2 N + 1 + 2 M = 1 2 N ( 2 N M + 1 ) ( 1 2 ) M × cos ( 2 π M P ξ λ f ) ] L 2 sinc 2 ( L ξ λ f ) .
E [ I H ( ξ ) ] = 9 [ N + 1 + 2 M = 1 N ( 2 N M + 1 ) ( 1 2 ) M × cos ( 4 π M L ξ λ f ) ] L 2 sinc 2 ( 3 L ξ λ f ) .
S / N = 20 log 10 [ I 2 ( x 2 ) ] av [ ( { I 2 ( x 2 ) [ I 2 ( x 2 ) ] av } 2 ) av ] 1 / 2 ,
I 2 ( x 2 ) = x 2 P 0 / 2 x 2 + P 0 / 2 I 2 ( x 2 ) d x 2 ,
P r ( J , M ) = ( 1 / 2 ) [ P r ( J 1 , M 1 ) + P r ( J + 1 , M 1 ) ]
P r ( J , 1 ) = { 1 2 , J = 1 or S 1 0 , otherwise ,
E [ A ( x 1 ) ] = ( 2 N + 1 ) B ( x 1 ) + M = 1 2 N ( 2 N M + 1 ) J = 0 S 1 P r ( J , M ) cos ( 2 π S J ) × [ B ( x 1 M P ) + B ( x 1 + M P ) ] .
E [ I H ( ξ ) ] = [ 2 N + 1 + 2 M = 1 2 N ( 2 N M + 1 ) J = 0 S 1 P r ( J , M ) × cos ( 2 π S J ) × cos ( 2 π M P ξ λ f ) ] L 2 sinc 2 ( L ξ λ f ) .