Abstract

The detour phase method for the design of computer-generated holograms can be modified to achieve multichannel reconstruction along various diffraction orders. It is shown how a single hologram can be used to display two patterns of different intensities along two diffraction orders. This is achieved by the release of any requirement on the phase distributions of these patterns, thus leaving them as free parameters. Various algorithms are suggested to make possible nonidentical reconstructions along two different off-axis diffraction orders. The two reconstruction orders can be chosen arbitrarily. The case of four-channel reconstructions for generating four different images is discussed as well. Computer simulations and optical experiments were carried out to demonstrate the capabilities of the proposed approaches.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  7. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttgart) 35, 237–246 (1972).
  8. D. Mendlovic, I. Kiryuschev, “Two channel computer generated hologram and its application for optical correlation,” Opt. Commun. 116, 322–325 (1995).
    [CrossRef]
  9. J. Garcia, D. Mas, R. G. Dorsch, “Fractional Fourier transform calculation through the fast Fourier transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
    [CrossRef]
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1996 (1)

1995 (1)

D. Mendlovic, I. Kiryuschev, “Two channel computer generated hologram and its application for optical correlation,” Opt. Commun. 116, 322–325 (1995).
[CrossRef]

1994 (1)

1988 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttgart) 35, 237–246 (1972).

1970 (2)

1967 (1)

1966 (1)

G. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 6, 967–969 (1966).
[CrossRef]

Brown, G. R.

G. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 6, 967–969 (1966).
[CrossRef]

Bryngdahl, O.

Burkhardt, C. B.

Dorsch, R. G.

Garcia, J.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttgart) 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Kiryuschev, I.

D. Mendlovic, I. Kiryuschev, “Two channel computer generated hologram and its application for optical correlation,” Opt. Commun. 116, 322–325 (1995).
[CrossRef]

Lee, W. H.

Lohmann, A. W.

Mas, D.

Mendlovic, D.

D. Mendlovic, I. Kiryuschev, “Two channel computer generated hologram and its application for optical correlation,” Opt. Commun. 116, 322–325 (1995).
[CrossRef]

Paris, D. P.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttgart) 35, 237–246 (1972).

Sinzinger, S.

Wyrowsky, F.

Appl. Opt. (6)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

D. Mendlovic, I. Kiryuschev, “Two channel computer generated hologram and its application for optical correlation,” Opt. Commun. 116, 322–325 (1995).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane figures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (18)

Fig. 1
Fig. 1

Schematic diagram of the proposed setup for two-channel image reconstruction.

Fig. 2
Fig. 2

Block diagram of the proposed algorithm for achieving different reconstructions along two symmetrical diffraction orders. BFSP, back-FSP.

Fig. 3
Fig. 3

Block diagram of the algorithm for reconstructing distributions along nonsymmetric diffraction orders. IFT, inverse Fourier transform.

Fig. 4
Fig. 4

Two objects used for computer simulations and experimental tests.

Fig. 5
Fig. 5

Reconstruction achieved by use of the basic approach (the algorithm shown in Fig. 2) after 100 iterations (w 1 = w 2 = 0.5).

Fig. 6
Fig. 6

Center-line horizontal intensity profile of Fig. 5.

Fig. 7
Fig. 7

MIE factor as a function of the number of iterations for the images reconstructed in (a) Fig. 5(a) and (b) Fig. 5(b).

Fig. 8
Fig. 8

Reconstruction achieved by use of the basic approach after 100 iterations with the weights w 1 = 0.3 and w 2 = 0.7.

Fig. 9
Fig. 9

MIE factor as a function of the iteration number for the images reconstructed in (a) Fig. 8(a) and (b) Fig. 8(b).

Fig. 10
Fig. 10

Reconstruction achieved by use of the simplified approach after 100 iterations (w 1 = w 2= 0.5).

Fig. 11
Fig. 11

Center-line horizontal intensity profiles of (a) Fig. 10(a) and (b) Fig. 10(b).

Fig. 12
Fig. 12

MIE factor as a function of the number of iterations for (a) Fig. 10(a) and (b) Fig. 10(b).

Fig. 13
Fig. 13

Reconstruction achieved by use of the nonsymmetric approach after 150 iterations (w 1 = 0.25 and w 2 = 0.75).

Fig. 14
Fig. 14

MIE factor as a function of the number of iterations for (a) Fig. 13(a) and (b) Fig. 13(b).

Fig. 15
Fig. 15

Optical reconstruction of Fig. 4 achieved by use of the original approach (w 1 = w 2 = 0.5).

Fig. 16
Fig. 16

Optical experimental reconstruction of Fig. 4 achieved by use of the nonsymmetric approach.

Fig. 17
Fig. 17

New set of four objects used for the four-channel experiment.

Fig. 18
Fig. 18

Optical experimental reconstruction of Fig. 17 achieved by use of the four-channel approach.

Equations (17)

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x max Δ x = 1 2 ,
ϕ = 2 π xM Δ x ,
H x ,   y = - -   a x ,   y exp i ϕ x ,   y × exp - i 2 π xx + 2 π yy d x d y ,
- -   a x ,   y exp - i ϕ x ,   y exp - i 2 π xx + 2 π yy d x d y = - -   a x ,   y exp i ϕ x ,   y exp + i 2 π xx + 2 π yy d x d y * = H * - x ,   - y ,
f 1 x ,   y = FSP H x ,   y ,   Z 0 = exp ikZ 0 i λ Z 0 exp i   k 2 Z 0 x 2 + y 2 × - -   H x ,   y exp i   k 2 Z 0 x 2 + y 2 × exp - i   2 π λ Z 0 xx + yy d x d y ,
f 2 x ,   y = FSP H * - x ,   - y ,   Z 0 = exp ikZ 0 i λ Z 0 exp i   k 2 Z 0 x 2 + y 2 × - -   H * - x ,   - y exp i   k 2 Z 0 x 2 + y 2 × exp - i   2 π λ Z 0 xx + yy d x d y .
MIE = m n   | A m , n desired 2 - A m , n obtained 2 | ,
m n A mn desired 2 =     A m , n obtained 2 = 1 .
H 1 new x ,   y = w 1 H 1 + w 2 H 2 ,
f 3 x ,   y = FSP H * - x ,   - y ,   - Z 0 = exp - ikZ 0 - i λ Z 0 exp - i   k 2 Z 0 x 2 + y 2 × - -   H * - x ,   - y exp - i   k 2 Z 0 x 2 + y 2 × exp + i   2 π λ Z 0 xx + yy d x d y = exp + ikZ 0 + i λ Z 0 exp + i   k 2 Z 0 x 2 + y 2 × - -   H - x ,   - y exp + i   k 2 Z 0 x 2 + y 2 × exp - i   2 π λ Z 0 xx + yy d x d y * = exp ikZ 0 i λ Z 0 exp i   k 2 Z 0 x 2 + y 2 × - -   H x ,   y exp + i   k 2 Z 0 x 2 + y 2 × exp - 2 π λ Z 0 x - x + y - y * = f 1 * - x ,   - y .
H = w 1 H 1 + w 2 H 2 ,
f 1 x ,   y = FT A x ,   y exp i ϕ x ,   y M 1 ,
f 2 x ,   y = FT A x ,   y exp i ϕ x ,   y M 2 .
θ = λ / δ x ,
L X = N δ x ,
Z L x / θ = L x 2 / λ N ,
Z L x / θ = L x 2 / λ N .

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