Abstract

The cross talk between wavelength-multiplexed holograms is analyzed. The signal-to-noise ratio (SNR) is calculated for a recording schedule that places the center of each image at the null (in wavelength space) of the adjacent hologram. An asymptotic closed-form expression for the minimum SNR is derived in a general reflection geometry. The reflection geometry with counterpropagating signal and reference beams is shown to have the best SNR.

© 1993 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  2. J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).
  3. F. H. Mok, M. C. Tackitt, H. M. Stoll, Opt. Lett. 16, 605 (1991).
    [CrossRef] [PubMed]
  4. F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
    [CrossRef]
  5. G. A. Rakuljic, V. Leyva, A. Yariv, Opt. Lett. 17, 1471 (1992).
    [CrossRef] [PubMed]
  6. C. Gu, J. Hong, I. McMichael, R. Saxena, F. H. Mok, J. Opt. Soc. Am. A 9, 1978 (1992).
    [CrossRef]
  7. C. Gu, J. Hong, in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WT2, p. 111.
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 57.
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 427.
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 39.
  11. M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 807.

1992 (2)

1991 (2)

F. H. Mok, M. C. Tackitt, H. M. Stoll, Opt. Lett. 16, 605 (1991).
[CrossRef] [PubMed]

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

1972 (1)

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Abramowitz, M.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 807.

Amodei, J. J.

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 57.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 39.

Gu, C.

C. Gu, J. Hong, I. McMichael, R. Saxena, F. H. Mok, J. Opt. Soc. Am. A 9, 1978 (1992).
[CrossRef]

C. Gu, J. Hong, in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WT2, p. 111.

Hong, J.

C. Gu, J. Hong, I. McMichael, R. Saxena, F. H. Mok, J. Opt. Soc. Am. A 9, 1978 (1992).
[CrossRef]

C. Gu, J. Hong, in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WT2, p. 111.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 427.

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Leyva, V.

Mayers, A. W.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

McMichael, I.

Mok, F. H.

Rajan, S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

Rakuljic, G. A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 39.

Saxena, R.

Staebler, D. L.

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

Stegun, I. E.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 807.

Stoll, H. M.

Tackitt, M. C.

Wu, S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

Yariv, A.

Yu, F. T. S.

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

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

Opt. Commun. (1)

F. T. S. Yu, S. Wu, A. W. Mayers, S. Rajan, Opt. Commun. 81, 343 (1991).
[CrossRef]

Opt. Lett. (2)

RCA Rev. (1)

J. J. Amodei, D. L. Staebler, RCA Rev. 33, 71 (1972).

Other (5)

C. Gu, J. Hong, in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper WT2, p. 111.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 57.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 427.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 39.

M. Abramowitz, I. E. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, Washington, D.C., 1972), p. 807.

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Figures (5)

Fig. 1
Fig. 1

Recording and readout geometry for wavelength multiplexing.

Fig. 2
Fig. 2

NSR at the output plane for the j = 0 hologram with θ = 0.0°.

Fig. 3
Fig. 3

NSR at the output plane for the j = 0 hologram with θ = 15.0°.

Fig. 4
Fig. 4

SNR versus hologram number (j) for N = 10,001 holograms.

Fig. 5
Fig. 5

SNR versus the total number of holograms.

Equations (12)

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Δ m = - M M R m * S m + c . c . ,
R m = exp ( i k m r ) ,
S m ( r ) exp ( i 2 π λ m z ) d x 0 d y 0 f m ( x 0 , y 0 ) exp [ - i 2 π λ m F ( x x 0 + y y 0 ) ] exp [ - i π z λ m F 2 ( x 0 2 + y 0 2 ) ] ,
E ( r ) k 0 2 4 π exp ( i k r ) r d r exp ( - i K r ) Δ ( r ) ,
k d , j = [ 2 π x 2 λ j F , 2 π y 2 λ j F , 2 π λ j ( 1 - x 2 2 2 F 2 - y 2 2 2 F 2 ) ] .
E ( x 2 , y 2 ) m = - M M f ( - λ m λ j x 2 - F λ m 2 π Δ K m j x , - λ m λ j y 2 - F λ m 2 π Δ K m j y ) sinc { t 2 π [ Δ K m j z + ( 2 π λ j - 2 π λ m ) + π ( x 2 2 + y 2 2 ) F 2 ( λ m λ j 2 - 1 λ j ) + λ m λ j F ( Δ K m j x x 2 + Δ K m y j y 2 ) + λ m 4 π ( Δ K m j x 2 + Δ K m j y 2 ) ] } ,
NSR = m j sinc 2 [ - t ( 1 λ m - 1 λ j ) cos θ - t ( 1 λ m - 1 λ j ) + t ( x 2 2 + y 2 2 ) 2 F 2 ( λ m λ j 2 - 1 λ j ) - t λ m y 2 F λ j ( 1 λ m - 1 λ j ) sin θ + t λ m 2 ( 1 λ m - 1 λ j ) 2 sin 2 θ ] .
NSR = m j sinc 2 [ - ( m - j ) - ( x 2 2 + y 2 2 ) ( j Δ + ν 0 ) 2 F 2 ( 1 + cos θ ) ( m Δ + ν 0 ) ( m - j ) - ( j Δ + v 0 m Δ + ν 0 ) y 2 ( m - j ) F ( 1 + cos θ ) sin θ + c ( m - j ) 2 2 t ( m Δ + ν 0 ) ( 1 + cos θ ) 2 sin 2 θ ] .
a = ( x 2 2 + y 2 2 ) 2 F 2 ( 1 + cos θ ) + y 2 F ( 1 + cos θ ) sin θ .
NSR max = 1 π 2 ( 1 + a ) 2 [ m = 1 M 1 m 2 - m = 1 M cos ( 2 π a m ) m 2 ] .
NSR max a ( x 2 2 + y 2 2 ) 2 F 2 ( 1 + cos θ ) + y 2 F ( 1 + cos θ ) sin θ ,
SNR min 4 F 2 ( x 2 2 + y 2 2 ) .

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